Integrand size = 27, antiderivative size = 572 \[ \int \frac {(a+b \arccos (c x))^2}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=-\frac {b^2 c^2}{2 d^3 x}+\frac {b^2 c^2}{6 d^3 x \left (1-c^2 x^2\right )}-\frac {b^2 c^4 x}{12 d^3 \left (1-c^2 x^2\right )}+\frac {b c^3 (a+b \arccos (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b c (a+b \arccos (c x))}{3 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {29 b c^3 (a+b \arccos (c x))}{12 d^3 \sqrt {1-c^2 x^2}}-\frac {(a+b \arccos (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 (a+b \arccos (c x))^2}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \arccos (c x))^2}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \arccos (c x))^2}{8 d^3 \left (1-c^2 x^2\right )}-\frac {35 i c^3 (a+b \arccos (c x))^2 \arctan \left (e^{i \arccos (c x)}\right )}{4 d^3}-\frac {38 b c^3 (a+b \arccos (c x)) \text {arctanh}\left (e^{i \arccos (c x)}\right )}{3 d^3}+\frac {17 b^2 c^3 \text {arctanh}(c x)}{6 d^3}+\frac {19 i b^2 c^3 \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{3 d^3}+\frac {35 i b c^3 (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )}{4 d^3}-\frac {35 i b c^3 (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )}{4 d^3}-\frac {19 i b^2 c^3 \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{3 d^3}-\frac {35 b^2 c^3 \operatorname {PolyLog}\left (3,-i e^{i \arccos (c x)}\right )}{4 d^3}+\frac {35 b^2 c^3 \operatorname {PolyLog}\left (3,i e^{i \arccos (c x)}\right )}{4 d^3} \] Output:
-1/2*b^2*c^2/d^3/x+1/6*b^2*c^2/d^3/x/(-c^2*x^2+1)-1/12*b^2*c^4*x/d^3/(-c^2 *x^2+1)+1/6*b*c^3*(a+b*arccos(c*x))/d^3/(-c^2*x^2+1)^(3/2)-1/3*b*c*(a+b*ar ccos(c*x))/d^3/x^2/(-c^2*x^2+1)^(3/2)-29/12*b*c^3*(a+b*arccos(c*x))/d^3/(- c^2*x^2+1)^(1/2)-1/3*(a+b*arccos(c*x))^2/d^3/x^3/(-c^2*x^2+1)^2-7/3*c^2*(a +b*arccos(c*x))^2/d^3/x/(-c^2*x^2+1)^2+35/12*c^4*x*(a+b*arccos(c*x))^2/d^3 /(-c^2*x^2+1)^2+35/8*c^4*x*(a+b*arccos(c*x))^2/d^3/(-c^2*x^2+1)+19/3*I*b^2 *c^3*polylog(2,-c*x-I*(-c^2*x^2+1)^(1/2))/d^3-38/3*b*c^3*(a+b*arccos(c*x)) *arctanh(c*x+I*(-c^2*x^2+1)^(1/2))/d^3+17/6*b^2*c^3*arctanh(c*x)/d^3-35/4* I*c^3*(a+b*arccos(c*x))^2*arctan(c*x+I*(-c^2*x^2+1)^(1/2))/d^3-19/3*I*b^2* c^3*polylog(2,c*x+I*(-c^2*x^2+1)^(1/2))/d^3-35/4*I*b*c^3*(a+b*arccos(c*x)) *polylog(2,I*(c*x+I*(-c^2*x^2+1)^(1/2)))/d^3+35/4*I*b*c^3*(a+b*arccos(c*x) )*polylog(2,-I*(c*x+I*(-c^2*x^2+1)^(1/2)))/d^3-35/4*b^2*c^3*polylog(3,-I*( c*x+I*(-c^2*x^2+1)^(1/2)))/d^3+35/4*b^2*c^3*polylog(3,I*(c*x+I*(-c^2*x^2+1 )^(1/2)))/d^3
Time = 9.24 (sec) , antiderivative size = 1135, normalized size of antiderivative = 1.98 \[ \int \frac {(a+b \arccos (c x))^2}{x^4 \left (d-c^2 d x^2\right )^3} \, dx =\text {Too large to display} \] Input:
Integrate[(a + b*ArcCos[c*x])^2/(x^4*(d - c^2*d*x^2)^3),x]
Output:
-1/3*a^2/(d^3*x^3) - (3*a^2*c^2)/(d^3*x) + (a^2*c^4*x)/(4*d^3*(-1 + c^2*x^ 2)^2) - (11*a^2*c^4*x)/(8*d^3*(-1 + c^2*x^2)) - (35*a^2*c^3*Log[1 - c*x])/ (16*d^3) + (35*a^2*c^3*Log[1 + c*x])/(16*d^3) - (2*a*b*(-1/6*(c*Sqrt[1 - c ^2*x^2])/x^2 + (c^3*((-2 + c*x)*Sqrt[1 - c^2*x^2] - 3*ArcCos[c*x]))/(48*(- 1 + c*x)^2) - (c^3*((2 + c*x)*Sqrt[1 - c^2*x^2] - 3*ArcCos[c*x]))/(48*(1 + c*x)^2) - (11*c^4*(Sqrt[1 - c^2*x^2] - ArcCos[c*x]))/(16*(c + c^2*x)) + A rcCos[c*x]/(3*x^3) - (11*c^4*(Sqrt[1 - c^2*x^2] + ArcCos[c*x]))/(16*(c - c ^2*x)) + (c^3*Log[x])/6 - (c^3*Log[1 + Sqrt[1 - c^2*x^2]])/6 - 3*c^2*(-(Ar cCos[c*x]/x) - c*Log[x] + c*Log[1 + Sqrt[1 - c^2*x^2]]) - (35*c^4*(((-1/2* I)*ArcCos[c*x]^2)/c + (2*ArcCos[c*x]*Log[1 + E^(I*ArcCos[c*x])])/c - ((2*I )*PolyLog[2, -E^(I*ArcCos[c*x])])/c))/16 - ((35*I)/32)*c^3*(ArcCos[c*x]*(A rcCos[c*x] + (4*I)*Log[1 - E^(I*ArcCos[c*x])]) + 4*PolyLog[2, E^(I*ArcCos[ c*x])])))/d^3 - (b^2*c^3*(64 - (16*(-2 + ArcCos[c*x])*ArcCos[c*x])/(-1 + S qrt[1 - c^2*x^2]) + 608*ArcCos[c*x]^2 - 272*ArcCos[c*x]*Cot[ArcCos[c*x]/2] - 2*(2 + 33*ArcCos[c*x]^2)*Csc[ArcCos[c*x]/2]^2 - 2*Sqrt[1 - c^2*x^2]*Arc Cos[c*x]*Csc[ArcCos[c*x]/2]^4 - 3*ArcCos[c*x]^2*Csc[ArcCos[c*x]/2]^4 + 544 *Log[Tan[ArcCos[c*x]/2]] - 1216*(ArcCos[c*x]*(Log[1 - I*E^(I*ArcCos[c*x])] - Log[1 + I*E^(I*ArcCos[c*x])]) + I*(PolyLog[2, (-I)*E^(I*ArcCos[c*x])] - PolyLog[2, I*E^(I*ArcCos[c*x])])) + 840*(ArcCos[c*x]^2*(Log[1 - E^(I*ArcC os[c*x])] - Log[1 + E^(I*ArcCos[c*x])]) + (2*I)*ArcCos[c*x]*(PolyLog[2,...
Time = 6.35 (sec) , antiderivative size = 773, normalized size of antiderivative = 1.35, number of steps used = 28, number of rules used = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5205, 27, 5205, 253, 264, 219, 5163, 5163, 5165, 3042, 4671, 3011, 2720, 5183, 215, 219, 5209, 215, 219, 5209, 219, 5219, 3042, 4669, 2715, 2838, 7143}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(a+b \arccos (c x))^2}{x^4 \left (d-c^2 d x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 5205 |
| \(\displaystyle \frac {7}{3} c^2 \int \frac {(a+b \arccos (c x))^2}{d^3 x^2 \left (1-c^2 x^2\right )^3}dx-\frac {2 b c \int \frac {a+b \arccos (c x)}{x^3 \left (1-c^2 x^2\right )^{5/2}}dx}{3 d^3}-\frac {(a+b \arccos (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7 c^2 \int \frac {(a+b \arccos (c x))^2}{x^2 \left (1-c^2 x^2\right )^3}dx}{3 d^3}-\frac {2 b c \int \frac {a+b \arccos (c x)}{x^3 \left (1-c^2 x^2\right )^{5/2}}dx}{3 d^3}-\frac {(a+b \arccos (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 5205 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \int \frac {(a+b \arccos (c x))^2}{\left (1-c^2 x^2\right )^3}dx-2 b c \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx-\frac {(a+b \arccos (c x))^2}{x \left (1-c^2 x^2\right )^2}\right )}{3 d^3}-\frac {2 b c \left (\frac {5}{2} c^2 \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx-\frac {1}{2} b c \int \frac {1}{x^2 \left (1-c^2 x^2\right )^2}dx-\frac {a+b \arccos (c x)}{2 x^2 \left (1-c^2 x^2\right )^{3/2}}\right )}{3 d^3}-\frac {(a+b \arccos (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \int \frac {(a+b \arccos (c x))^2}{\left (1-c^2 x^2\right )^3}dx-2 b c \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx-\frac {(a+b \arccos (c x))^2}{x \left (1-c^2 x^2\right )^2}\right )}{3 d^3}-\frac {2 b c \left (\frac {5}{2} c^2 \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx-\frac {1}{2} b c \left (\frac {3}{2} \int \frac {1}{x^2 \left (1-c^2 x^2\right )}dx+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )-\frac {a+b \arccos (c x)}{2 x^2 \left (1-c^2 x^2\right )^{3/2}}\right )}{3 d^3}-\frac {(a+b \arccos (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \int \frac {(a+b \arccos (c x))^2}{\left (1-c^2 x^2\right )^3}dx-2 b c \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx-\frac {(a+b \arccos (c x))^2}{x \left (1-c^2 x^2\right )^2}\right )}{3 d^3}-\frac {2 b c \left (\frac {5}{2} c^2 \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx-\frac {1}{2} b c \left (\frac {3}{2} \left (c^2 \int \frac {1}{1-c^2 x^2}dx-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )-\frac {a+b \arccos (c x)}{2 x^2 \left (1-c^2 x^2\right )^{3/2}}\right )}{3 d^3}-\frac {(a+b \arccos (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {2 b c \left (\frac {5}{2} c^2 \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx-\frac {a+b \arccos (c x)}{2 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {1}{2} b c \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )\right )}{3 d^3}+\frac {7 c^2 \left (5 c^2 \int \frac {(a+b \arccos (c x))^2}{\left (1-c^2 x^2\right )^3}dx-2 b c \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx-\frac {(a+b \arccos (c x))^2}{x \left (1-c^2 x^2\right )^2}\right )}{3 d^3}-\frac {(a+b \arccos (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 5163 |
| \(\displaystyle -\frac {2 b c \left (\frac {5}{2} c^2 \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx-\frac {a+b \arccos (c x)}{2 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {1}{2} b c \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )\right )}{3 d^3}+\frac {7 c^2 \left (5 c^2 \left (\frac {1}{2} b c \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx+\frac {3}{4} \int \frac {(a+b \arccos (c x))^2}{\left (1-c^2 x^2\right )^2}dx+\frac {x (a+b \arccos (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )-2 b c \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx-\frac {(a+b \arccos (c x))^2}{x \left (1-c^2 x^2\right )^2}\right )}{3 d^3}-\frac {(a+b \arccos (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 5163 |
| \(\displaystyle -\frac {2 b c \left (\frac {5}{2} c^2 \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx-\frac {a+b \arccos (c x)}{2 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {1}{2} b c \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )\right )}{3 d^3}+\frac {7 c^2 \left (5 c^2 \left (\frac {1}{2} b c \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx+\frac {3}{4} \left (b c \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {1}{2} \int \frac {(a+b \arccos (c x))^2}{1-c^2 x^2}dx+\frac {x (a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}\right )+\frac {x (a+b \arccos (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )-2 b c \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx-\frac {(a+b \arccos (c x))^2}{x \left (1-c^2 x^2\right )^2}\right )}{3 d^3}-\frac {(a+b \arccos (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 5165 |
| \(\displaystyle -\frac {2 b c \left (\frac {5}{2} c^2 \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx-\frac {a+b \arccos (c x)}{2 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {1}{2} b c \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )\right )}{3 d^3}+\frac {7 c^2 \left (5 c^2 \left (\frac {1}{2} b c \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx+\frac {3}{4} \left (b c \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx-\frac {\int \frac {(a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}}d\arccos (c x)}{2 c}+\frac {x (a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}\right )+\frac {x (a+b \arccos (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )-2 b c \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx-\frac {(a+b \arccos (c x))^2}{x \left (1-c^2 x^2\right )^2}\right )}{3 d^3}-\frac {(a+b \arccos (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 b c \left (\frac {5}{2} c^2 \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx-\frac {a+b \arccos (c x)}{2 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {1}{2} b c \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )\right )}{3 d^3}+\frac {7 c^2 \left (-2 b c \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx+5 c^2 \left (\frac {1}{2} b c \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx+\frac {3}{4} \left (b c \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx-\frac {\int (a+b \arccos (c x))^2 \csc (\arccos (c x))d\arccos (c x)}{2 c}+\frac {x (a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}\right )+\frac {x (a+b \arccos (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )-\frac {(a+b \arccos (c x))^2}{x \left (1-c^2 x^2\right )^2}\right )}{3 d^3}-\frac {(a+b \arccos (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle -\frac {2 b c \left (\frac {5}{2} c^2 \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx-\frac {a+b \arccos (c x)}{2 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {1}{2} b c \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )\right )}{3 d^3}+\frac {7 c^2 \left (5 c^2 \left (\frac {3}{4} \left (-\frac {-2 b \int (a+b \arccos (c x)) \log \left (1-e^{i \arccos (c x)}\right )d\arccos (c x)+2 b \int (a+b \arccos (c x)) \log \left (1+e^{i \arccos (c x)}\right )d\arccos (c x)-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2}{2 c}+b c \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {x (a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}\right )+\frac {1}{2} b c \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx+\frac {x (a+b \arccos (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )-2 b c \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx-\frac {(a+b \arccos (c x))^2}{x \left (1-c^2 x^2\right )^2}\right )}{3 d^3}-\frac {(a+b \arccos (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \left (\frac {3}{4} \left (-\frac {2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-i b \int \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )d\arccos (c x)\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-i b \int \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )d\arccos (c x)\right )-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2}{2 c}+b c \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {x (a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}\right )+\frac {1}{2} b c \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx+\frac {x (a+b \arccos (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )-2 b c \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx-\frac {(a+b \arccos (c x))^2}{x \left (1-c^2 x^2\right )^2}\right )}{3 d^3}-\frac {2 b c \left (\frac {5}{2} c^2 \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx-\frac {a+b \arccos (c x)}{2 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {1}{2} b c \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )\right )}{3 d^3}-\frac {(a+b \arccos (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \left (\frac {3}{4} \left (-\frac {2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2}{2 c}+b c \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {x (a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}\right )+\frac {1}{2} b c \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx+\frac {x (a+b \arccos (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )-2 b c \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx-\frac {(a+b \arccos (c x))^2}{x \left (1-c^2 x^2\right )^2}\right )}{3 d^3}-\frac {2 b c \left (\frac {5}{2} c^2 \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx-\frac {a+b \arccos (c x)}{2 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {1}{2} b c \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )\right )}{3 d^3}-\frac {(a+b \arccos (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 5183 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \left (\frac {3}{4} \left (-\frac {2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2}{2 c}+b c \left (\frac {b \int \frac {1}{1-c^2 x^2}dx}{c}+\frac {a+b \arccos (c x)}{c^2 \sqrt {1-c^2 x^2}}\right )+\frac {x (a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}\right )+\frac {1}{2} b c \left (\frac {b \int \frac {1}{\left (1-c^2 x^2\right )^2}dx}{3 c}+\frac {a+b \arccos (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}\right )+\frac {x (a+b \arccos (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )-2 b c \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx-\frac {(a+b \arccos (c x))^2}{x \left (1-c^2 x^2\right )^2}\right )}{3 d^3}-\frac {2 b c \left (\frac {5}{2} c^2 \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx-\frac {a+b \arccos (c x)}{2 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {1}{2} b c \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )\right )}{3 d^3}-\frac {(a+b \arccos (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \left (\frac {3}{4} \left (-\frac {2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2}{2 c}+b c \left (\frac {b \int \frac {1}{1-c^2 x^2}dx}{c}+\frac {a+b \arccos (c x)}{c^2 \sqrt {1-c^2 x^2}}\right )+\frac {x (a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}\right )+\frac {1}{2} b c \left (\frac {b \left (\frac {1}{2} \int \frac {1}{1-c^2 x^2}dx+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c}+\frac {a+b \arccos (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}\right )+\frac {x (a+b \arccos (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )-2 b c \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx-\frac {(a+b \arccos (c x))^2}{x \left (1-c^2 x^2\right )^2}\right )}{3 d^3}-\frac {2 b c \left (\frac {5}{2} c^2 \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx-\frac {a+b \arccos (c x)}{2 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {1}{2} b c \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )\right )}{3 d^3}-\frac {(a+b \arccos (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \left (\frac {3}{4} \left (-\frac {2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2}{2 c}+b c \left (\frac {a+b \arccos (c x)}{c^2 \sqrt {1-c^2 x^2}}+\frac {b \text {arctanh}(c x)}{c^2}\right )+\frac {x (a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}\right )+\frac {1}{2} b c \left (\frac {a+b \arccos (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}+\frac {b \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c}\right )+\frac {x (a+b \arccos (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )-2 b c \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx-\frac {(a+b \arccos (c x))^2}{x \left (1-c^2 x^2\right )^2}\right )}{3 d^3}-\frac {2 b c \left (\frac {5}{2} c^2 \int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^{5/2}}dx-\frac {a+b \arccos (c x)}{2 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {1}{2} b c \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )\right )}{3 d^3}-\frac {(a+b \arccos (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 5209 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \left (\frac {3}{4} \left (-\frac {2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2}{2 c}+b c \left (\frac {a+b \arccos (c x)}{c^2 \sqrt {1-c^2 x^2}}+\frac {b \text {arctanh}(c x)}{c^2}\right )+\frac {x (a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}\right )+\frac {1}{2} b c \left (\frac {a+b \arccos (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}+\frac {b \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c}\right )+\frac {x (a+b \arccos (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )-2 b c \left (\int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^{3/2}}dx+\frac {1}{3} b c \int \frac {1}{\left (1-c^2 x^2\right )^2}dx+\frac {a+b \arccos (c x)}{3 \left (1-c^2 x^2\right )^{3/2}}\right )-\frac {(a+b \arccos (c x))^2}{x \left (1-c^2 x^2\right )^2}\right )}{3 d^3}-\frac {2 b c \left (\frac {5}{2} c^2 \left (\int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^{3/2}}dx+\frac {1}{3} b c \int \frac {1}{\left (1-c^2 x^2\right )^2}dx+\frac {a+b \arccos (c x)}{3 \left (1-c^2 x^2\right )^{3/2}}\right )-\frac {a+b \arccos (c x)}{2 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {1}{2} b c \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )\right )}{3 d^3}-\frac {(a+b \arccos (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \left (\frac {3}{4} \left (-\frac {2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2}{2 c}+b c \left (\frac {a+b \arccos (c x)}{c^2 \sqrt {1-c^2 x^2}}+\frac {b \text {arctanh}(c x)}{c^2}\right )+\frac {x (a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}\right )+\frac {1}{2} b c \left (\frac {a+b \arccos (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}+\frac {b \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c}\right )+\frac {x (a+b \arccos (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )-2 b c \left (\int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^{3/2}}dx+\frac {1}{3} b c \left (\frac {1}{2} \int \frac {1}{1-c^2 x^2}dx+\frac {x}{2 \left (1-c^2 x^2\right )}\right )+\frac {a+b \arccos (c x)}{3 \left (1-c^2 x^2\right )^{3/2}}\right )-\frac {(a+b \arccos (c x))^2}{x \left (1-c^2 x^2\right )^2}\right )}{3 d^3}-\frac {2 b c \left (\frac {5}{2} c^2 \left (\int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^{3/2}}dx+\frac {1}{3} b c \left (\frac {1}{2} \int \frac {1}{1-c^2 x^2}dx+\frac {x}{2 \left (1-c^2 x^2\right )}\right )+\frac {a+b \arccos (c x)}{3 \left (1-c^2 x^2\right )^{3/2}}\right )-\frac {a+b \arccos (c x)}{2 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {1}{2} b c \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )\right )}{3 d^3}-\frac {(a+b \arccos (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \left (\frac {3}{4} \left (-\frac {2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2}{2 c}+b c \left (\frac {a+b \arccos (c x)}{c^2 \sqrt {1-c^2 x^2}}+\frac {b \text {arctanh}(c x)}{c^2}\right )+\frac {x (a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}\right )+\frac {1}{2} b c \left (\frac {a+b \arccos (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}+\frac {b \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c}\right )+\frac {x (a+b \arccos (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )-2 b c \left (\int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^{3/2}}dx+\frac {a+b \arccos (c x)}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {1}{3} b c \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )\right )-\frac {(a+b \arccos (c x))^2}{x \left (1-c^2 x^2\right )^2}\right )}{3 d^3}-\frac {2 b c \left (\frac {5}{2} c^2 \left (\int \frac {a+b \arccos (c x)}{x \left (1-c^2 x^2\right )^{3/2}}dx+\frac {a+b \arccos (c x)}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {1}{3} b c \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )\right )-\frac {a+b \arccos (c x)}{2 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {1}{2} b c \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )\right )}{3 d^3}-\frac {(a+b \arccos (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 5209 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \left (\frac {3}{4} \left (-\frac {2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2}{2 c}+b c \left (\frac {a+b \arccos (c x)}{c^2 \sqrt {1-c^2 x^2}}+\frac {b \text {arctanh}(c x)}{c^2}\right )+\frac {x (a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}\right )+\frac {1}{2} b c \left (\frac {a+b \arccos (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}+\frac {b \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c}\right )+\frac {x (a+b \arccos (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )-2 b c \left (\int \frac {a+b \arccos (c x)}{x \sqrt {1-c^2 x^2}}dx+b c \int \frac {1}{1-c^2 x^2}dx+\frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}+\frac {a+b \arccos (c x)}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {1}{3} b c \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )\right )-\frac {(a+b \arccos (c x))^2}{x \left (1-c^2 x^2\right )^2}\right )}{3 d^3}-\frac {2 b c \left (\frac {5}{2} c^2 \left (\int \frac {a+b \arccos (c x)}{x \sqrt {1-c^2 x^2}}dx+b c \int \frac {1}{1-c^2 x^2}dx+\frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}+\frac {a+b \arccos (c x)}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {1}{3} b c \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )\right )-\frac {a+b \arccos (c x)}{2 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {1}{2} b c \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )\right )}{3 d^3}-\frac {(a+b \arccos (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \left (\frac {3}{4} \left (-\frac {2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2}{2 c}+b c \left (\frac {a+b \arccos (c x)}{c^2 \sqrt {1-c^2 x^2}}+\frac {b \text {arctanh}(c x)}{c^2}\right )+\frac {x (a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}\right )+\frac {1}{2} b c \left (\frac {a+b \arccos (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}+\frac {b \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c}\right )+\frac {x (a+b \arccos (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )-2 b c \left (\int \frac {a+b \arccos (c x)}{x \sqrt {1-c^2 x^2}}dx+\frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}+\frac {a+b \arccos (c x)}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {1}{3} b c \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )+b \text {arctanh}(c x)\right )-\frac {(a+b \arccos (c x))^2}{x \left (1-c^2 x^2\right )^2}\right )}{3 d^3}-\frac {2 b c \left (\frac {5}{2} c^2 \left (\int \frac {a+b \arccos (c x)}{x \sqrt {1-c^2 x^2}}dx+\frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}+\frac {a+b \arccos (c x)}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {1}{3} b c \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )+b \text {arctanh}(c x)\right )-\frac {a+b \arccos (c x)}{2 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {1}{2} b c \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )\right )}{3 d^3}-\frac {(a+b \arccos (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 5219 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \left (\frac {3}{4} \left (-\frac {2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2}{2 c}+b c \left (\frac {a+b \arccos (c x)}{c^2 \sqrt {1-c^2 x^2}}+\frac {b \text {arctanh}(c x)}{c^2}\right )+\frac {x (a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}\right )+\frac {1}{2} b c \left (\frac {a+b \arccos (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}+\frac {b \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c}\right )+\frac {x (a+b \arccos (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )-2 b c \left (-\int \frac {a+b \arccos (c x)}{c x}d\arccos (c x)+\frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}+\frac {a+b \arccos (c x)}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {1}{3} b c \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )+b \text {arctanh}(c x)\right )-\frac {(a+b \arccos (c x))^2}{x \left (1-c^2 x^2\right )^2}\right )}{3 d^3}-\frac {2 b c \left (\frac {5}{2} c^2 \left (-\int \frac {a+b \arccos (c x)}{c x}d\arccos (c x)+\frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}+\frac {a+b \arccos (c x)}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {1}{3} b c \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )+b \text {arctanh}(c x)\right )-\frac {a+b \arccos (c x)}{2 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {1}{2} b c \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )\right )}{3 d^3}-\frac {(a+b \arccos (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \left (\frac {3}{4} \left (-\frac {2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2}{2 c}+b c \left (\frac {a+b \arccos (c x)}{c^2 \sqrt {1-c^2 x^2}}+\frac {b \text {arctanh}(c x)}{c^2}\right )+\frac {x (a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}\right )+\frac {1}{2} b c \left (\frac {a+b \arccos (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}+\frac {b \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c}\right )+\frac {x (a+b \arccos (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )-2 b c \left (-\int (a+b \arccos (c x)) \csc \left (\arccos (c x)+\frac {\pi }{2}\right )d\arccos (c x)+\frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}+\frac {a+b \arccos (c x)}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {1}{3} b c \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )+b \text {arctanh}(c x)\right )-\frac {(a+b \arccos (c x))^2}{x \left (1-c^2 x^2\right )^2}\right )}{3 d^3}-\frac {2 b c \left (\frac {5}{2} c^2 \left (-\int (a+b \arccos (c x)) \csc \left (\arccos (c x)+\frac {\pi }{2}\right )d\arccos (c x)+\frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}+\frac {a+b \arccos (c x)}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {1}{3} b c \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )+b \text {arctanh}(c x)\right )-\frac {a+b \arccos (c x)}{2 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {1}{2} b c \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )\right )}{3 d^3}-\frac {(a+b \arccos (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle \frac {7 \left (5 \left (\frac {x (a+b \arccos (c x))^2}{4 \left (1-c^2 x^2\right )^2}+\frac {1}{2} b c \left (\frac {a+b \arccos (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}+\frac {b \left (\frac {x}{2 \left (1-c^2 x^2\right )}+\frac {\text {arctanh}(c x)}{2 c}\right )}{3 c}\right )+\frac {3}{4} \left (\frac {x (a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}+b c \left (\frac {a+b \arccos (c x)}{c^2 \sqrt {1-c^2 x^2}}+\frac {b \text {arctanh}(c x)}{c^2}\right )-\frac {-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2+2 b \left (i (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 b \left (i (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )}{2 c}\right )\right ) c^2-2 b \left (2 i \arctan \left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))+\frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}+\frac {a+b \arccos (c x)}{3 \left (1-c^2 x^2\right )^{3/2}}+b \text {arctanh}(c x)+\frac {1}{3} b c \left (\frac {x}{2 \left (1-c^2 x^2\right )}+\frac {\text {arctanh}(c x)}{2 c}\right )+b \int \log \left (1-i e^{i \arccos (c x)}\right )d\arccos (c x)-b \int \log \left (1+i e^{i \arccos (c x)}\right )d\arccos (c x)\right ) c-\frac {(a+b \arccos (c x))^2}{x \left (1-c^2 x^2\right )^2}\right ) c^2}{3 d^3}-\frac {2 b \left (\frac {5}{2} \left (2 i \arctan \left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))+\frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}+\frac {a+b \arccos (c x)}{3 \left (1-c^2 x^2\right )^{3/2}}+b \text {arctanh}(c x)+\frac {1}{3} b c \left (\frac {x}{2 \left (1-c^2 x^2\right )}+\frac {\text {arctanh}(c x)}{2 c}\right )+b \int \log \left (1-i e^{i \arccos (c x)}\right )d\arccos (c x)-b \int \log \left (1+i e^{i \arccos (c x)}\right )d\arccos (c x)\right ) c^2-\frac {1}{2} b \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right ) c-\frac {a+b \arccos (c x)}{2 x^2 \left (1-c^2 x^2\right )^{3/2}}\right ) c}{3 d^3}-\frac {(a+b \arccos (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {7 \left (5 \left (\frac {x (a+b \arccos (c x))^2}{4 \left (1-c^2 x^2\right )^2}+\frac {1}{2} b c \left (\frac {a+b \arccos (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}+\frac {b \left (\frac {x}{2 \left (1-c^2 x^2\right )}+\frac {\text {arctanh}(c x)}{2 c}\right )}{3 c}\right )+\frac {3}{4} \left (\frac {x (a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}+b c \left (\frac {a+b \arccos (c x)}{c^2 \sqrt {1-c^2 x^2}}+\frac {b \text {arctanh}(c x)}{c^2}\right )-\frac {-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2+2 b \left (i (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 b \left (i (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )}{2 c}\right )\right ) c^2-2 b \left (2 i \arctan \left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))+\frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}+\frac {a+b \arccos (c x)}{3 \left (1-c^2 x^2\right )^{3/2}}+b \text {arctanh}(c x)+\frac {1}{3} b c \left (\frac {x}{2 \left (1-c^2 x^2\right )}+\frac {\text {arctanh}(c x)}{2 c}\right )-i b \int e^{-i \arccos (c x)} \log \left (1-i e^{i \arccos (c x)}\right )de^{i \arccos (c x)}+i b \int e^{-i \arccos (c x)} \log \left (1+i e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right ) c-\frac {(a+b \arccos (c x))^2}{x \left (1-c^2 x^2\right )^2}\right ) c^2}{3 d^3}-\frac {2 b \left (\frac {5}{2} \left (2 i \arctan \left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))+\frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}+\frac {a+b \arccos (c x)}{3 \left (1-c^2 x^2\right )^{3/2}}+b \text {arctanh}(c x)+\frac {1}{3} b c \left (\frac {x}{2 \left (1-c^2 x^2\right )}+\frac {\text {arctanh}(c x)}{2 c}\right )-i b \int e^{-i \arccos (c x)} \log \left (1-i e^{i \arccos (c x)}\right )de^{i \arccos (c x)}+i b \int e^{-i \arccos (c x)} \log \left (1+i e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right ) c^2-\frac {1}{2} b \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right ) c-\frac {a+b \arccos (c x)}{2 x^2 \left (1-c^2 x^2\right )^{3/2}}\right ) c}{3 d^3}-\frac {(a+b \arccos (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {7 \left (5 \left (\frac {x (a+b \arccos (c x))^2}{4 \left (1-c^2 x^2\right )^2}+\frac {1}{2} b c \left (\frac {a+b \arccos (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}+\frac {b \left (\frac {x}{2 \left (1-c^2 x^2\right )}+\frac {\text {arctanh}(c x)}{2 c}\right )}{3 c}\right )+\frac {3}{4} \left (\frac {x (a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}+b c \left (\frac {a+b \arccos (c x)}{c^2 \sqrt {1-c^2 x^2}}+\frac {b \text {arctanh}(c x)}{c^2}\right )-\frac {-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2+2 b \left (i (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 b \left (i (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )}{2 c}\right )\right ) c^2-2 b \left (2 i \arctan \left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))+\frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}+\frac {a+b \arccos (c x)}{3 \left (1-c^2 x^2\right )^{3/2}}+b \text {arctanh}(c x)+\frac {1}{3} b c \left (\frac {x}{2 \left (1-c^2 x^2\right )}+\frac {\text {arctanh}(c x)}{2 c}\right )-i b \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )\right ) c-\frac {(a+b \arccos (c x))^2}{x \left (1-c^2 x^2\right )^2}\right ) c^2}{3 d^3}-\frac {2 b \left (\frac {5}{2} \left (2 i \arctan \left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))+\frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}+\frac {a+b \arccos (c x)}{3 \left (1-c^2 x^2\right )^{3/2}}+b \text {arctanh}(c x)+\frac {1}{3} b c \left (\frac {x}{2 \left (1-c^2 x^2\right )}+\frac {\text {arctanh}(c x)}{2 c}\right )-i b \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )\right ) c^2-\frac {1}{2} b \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right ) c-\frac {a+b \arccos (c x)}{2 x^2 \left (1-c^2 x^2\right )^{3/2}}\right ) c}{3 d^3}-\frac {(a+b \arccos (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {7 c^2 \left (-2 b c \left (2 i \arctan \left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))+\frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}+\frac {a+b \arccos (c x)}{3 \left (1-c^2 x^2\right )^{3/2}}-i b \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )+\frac {1}{3} b c \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )+b \text {arctanh}(c x)\right )+5 c^2 \left (\frac {3}{4} \left (b c \left (\frac {a+b \arccos (c x)}{c^2 \sqrt {1-c^2 x^2}}+\frac {b \text {arctanh}(c x)}{c^2}\right )-\frac {-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2+2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \operatorname {PolyLog}\left (3,-e^{i \arccos (c x)}\right )\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \operatorname {PolyLog}\left (3,e^{i \arccos (c x)}\right )\right )}{2 c}+\frac {x (a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}\right )+\frac {1}{2} b c \left (\frac {a+b \arccos (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}+\frac {b \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c}\right )+\frac {x (a+b \arccos (c x))^2}{4 \left (1-c^2 x^2\right )^2}\right )-\frac {(a+b \arccos (c x))^2}{x \left (1-c^2 x^2\right )^2}\right )}{3 d^3}-\frac {2 b c \left (\frac {5}{2} c^2 \left (2 i \arctan \left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))+\frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}+\frac {a+b \arccos (c x)}{3 \left (1-c^2 x^2\right )^{3/2}}-i b \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )+\frac {1}{3} b c \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )+b \text {arctanh}(c x)\right )-\frac {a+b \arccos (c x)}{2 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {1}{2} b c \left (\frac {3}{2} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (1-c^2 x^2\right )}\right )\right )}{3 d^3}-\frac {(a+b \arccos (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
Input:
Int[(a + b*ArcCos[c*x])^2/(x^4*(d - c^2*d*x^2)^3),x]
Output:
-1/3*(a + b*ArcCos[c*x])^2/(d^3*x^3*(1 - c^2*x^2)^2) - (2*b*c*(-1/2*(a + b *ArcCos[c*x])/(x^2*(1 - c^2*x^2)^(3/2)) - (b*c*(1/(2*x*(1 - c^2*x^2)) + (3 *(-x^(-1) + c*ArcTanh[c*x]))/2))/2 + (5*c^2*((a + b*ArcCos[c*x])/(3*(1 - c ^2*x^2)^(3/2)) + (a + b*ArcCos[c*x])/Sqrt[1 - c^2*x^2] + (2*I)*(a + b*ArcC os[c*x])*ArcTan[E^(I*ArcCos[c*x])] + b*ArcTanh[c*x] + (b*c*(x/(2*(1 - c^2* x^2)) + ArcTanh[c*x]/(2*c)))/3 - I*b*PolyLog[2, (-I)*E^(I*ArcCos[c*x])] + I*b*PolyLog[2, I*E^(I*ArcCos[c*x])]))/2))/(3*d^3) + (7*c^2*(-((a + b*ArcCo s[c*x])^2/(x*(1 - c^2*x^2)^2)) - 2*b*c*((a + b*ArcCos[c*x])/(3*(1 - c^2*x^ 2)^(3/2)) + (a + b*ArcCos[c*x])/Sqrt[1 - c^2*x^2] + (2*I)*(a + b*ArcCos[c* x])*ArcTan[E^(I*ArcCos[c*x])] + b*ArcTanh[c*x] + (b*c*(x/(2*(1 - c^2*x^2)) + ArcTanh[c*x]/(2*c)))/3 - I*b*PolyLog[2, (-I)*E^(I*ArcCos[c*x])] + I*b*P olyLog[2, I*E^(I*ArcCos[c*x])]) + 5*c^2*((x*(a + b*ArcCos[c*x])^2)/(4*(1 - c^2*x^2)^2) + (b*c*((a + b*ArcCos[c*x])/(3*c^2*(1 - c^2*x^2)^(3/2)) + (b* (x/(2*(1 - c^2*x^2)) + ArcTanh[c*x]/(2*c)))/(3*c)))/2 + (3*((x*(a + b*ArcC os[c*x])^2)/(2*(1 - c^2*x^2)) + b*c*((a + b*ArcCos[c*x])/(c^2*Sqrt[1 - c^2 *x^2]) + (b*ArcTanh[c*x])/c^2) - (-2*(a + b*ArcCos[c*x])^2*ArcTanh[E^(I*Ar cCos[c*x])] + 2*b*(I*(a + b*ArcCos[c*x])*PolyLog[2, -E^(I*ArcCos[c*x])] - b*PolyLog[3, -E^(I*ArcCos[c*x])]) - 2*b*(I*(a + b*ArcCos[c*x])*PolyLog[2, E^(I*ArcCos[c*x])] - b*PolyLog[3, E^(I*ArcCos[c*x])]))/(2*c)))/4)))/(3*d^3 )
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x )^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m }, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol ] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Si mp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x ))], x], x]) /; FreeQ[{c, d, e, f}, x] && IntegerQ[2*k] && IGtQ[m, 0]
Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x )^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG tQ[m, 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_ Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(2*d*(p + 1 ))), x] + (Simp[(2*p + 3)/(2*d*(p + 1)) Int[(d + e*x^2)^(p + 1)*(a + b*Ar cCos[c*x])^n, x], x] - Simp[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2 *x^2)^p] Int[x*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x ]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbo l] :> Simp[-(c*d)^(-1) Subst[Int[(a + b*x)^n*Csc[x], x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ .), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] I nt[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b *ArcCos[c*x])^n/(d*f*(m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1)) ) Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] + Simp[b* c*(n/(f*(m + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[(f*x)^(m + 1)*( 1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(2*d*f*(p + 1))), x] + (Simp[(m + 2*p + 3)/(2*d*(p + 1)) Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^n, x], x] - Simp[b*c *(n/(2*f*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[(f*x)^(m + 1)* (1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b , c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && !G tQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])
Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)* (x_)^2], x_Symbol] :> Simp[(-(c^(m + 1))^(-1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[ d + e*x^2]] Subst[Int[(a + b*x)^n*Cos[x]^m, x], x, ArcCos[c*x]], x] /; Fr eeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[m]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Time = 0.97 (sec) , antiderivative size = 820, normalized size of antiderivative = 1.43
| method | result | size |
| derivativedivides | \(c^{3} \left (-\frac {a^{2} \left (-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {11}{16 \left (c x -1\right )}+\frac {35 \ln \left (c x -1\right )}{16}+\frac {1}{3 c^{3} x^{3}}+\frac {3}{c x}+\frac {1}{16 \left (c x +1\right )^{2}}+\frac {11}{16 \left (c x +1\right )}-\frac {35 \ln \left (c x +1\right )}{16}\right )}{d^{3}}-\frac {b^{2} \left (\frac {105 \arccos \left (c x \right )^{2} c^{6} x^{6}+58 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) x^{5} c^{5}-175 \arccos \left (c x \right )^{2} x^{4} c^{4}+10 c^{6} x^{6}-54 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) x^{3} c^{3}+56 \arccos \left (c x \right )^{2} x^{2} c^{2}-18 c^{4} x^{4}-8 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) x c +8 \arccos \left (c x \right )^{2}+8 c^{2} x^{2}}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{3} x^{3}}-\frac {35 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{4}-\frac {35 \arccos \left (c x \right )^{2} \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{8}+\frac {35 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{4}+\frac {35 \arccos \left (c x \right )^{2} \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{8}-\frac {19 i \operatorname {dilog}\left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{3}+\frac {19 i \operatorname {dilog}\left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{3}+\frac {19 \arccos \left (c x \right ) \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{3}-\frac {19 \arccos \left (c x \right ) \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{3}-\frac {17 \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{6}+\frac {17 \ln \left (i \sqrt {-c^{2} x^{2}+1}+c x -1\right )}{6}-\frac {35 \operatorname {polylog}\left (3, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{4}+\frac {35 \operatorname {polylog}\left (3, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{4}\right )}{d^{3}}-\frac {2 a b \left (\frac {105 \arccos \left (c x \right ) c^{6} x^{6}+29 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}-175 c^{4} x^{4} \arccos \left (c x \right )-27 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+56 c^{2} x^{2} \arccos \left (c x \right )-4 c x \sqrt {-c^{2} x^{2}+1}+8 \arccos \left (c x \right )}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{3} x^{3}}+\frac {35 i \operatorname {dilog}\left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{8}+\frac {19 i \arctan \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {35 i \operatorname {dilog}\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )}{8}-\frac {35 \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{8}\right )}{d^{3}}\right )\) | \(820\) |
| default | \(c^{3} \left (-\frac {a^{2} \left (-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {11}{16 \left (c x -1\right )}+\frac {35 \ln \left (c x -1\right )}{16}+\frac {1}{3 c^{3} x^{3}}+\frac {3}{c x}+\frac {1}{16 \left (c x +1\right )^{2}}+\frac {11}{16 \left (c x +1\right )}-\frac {35 \ln \left (c x +1\right )}{16}\right )}{d^{3}}-\frac {b^{2} \left (\frac {105 \arccos \left (c x \right )^{2} c^{6} x^{6}+58 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) x^{5} c^{5}-175 \arccos \left (c x \right )^{2} x^{4} c^{4}+10 c^{6} x^{6}-54 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) x^{3} c^{3}+56 \arccos \left (c x \right )^{2} x^{2} c^{2}-18 c^{4} x^{4}-8 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) x c +8 \arccos \left (c x \right )^{2}+8 c^{2} x^{2}}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{3} x^{3}}-\frac {35 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{4}-\frac {35 \arccos \left (c x \right )^{2} \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{8}+\frac {35 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{4}+\frac {35 \arccos \left (c x \right )^{2} \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{8}-\frac {19 i \operatorname {dilog}\left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{3}+\frac {19 i \operatorname {dilog}\left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{3}+\frac {19 \arccos \left (c x \right ) \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{3}-\frac {19 \arccos \left (c x \right ) \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{3}-\frac {17 \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{6}+\frac {17 \ln \left (i \sqrt {-c^{2} x^{2}+1}+c x -1\right )}{6}-\frac {35 \operatorname {polylog}\left (3, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{4}+\frac {35 \operatorname {polylog}\left (3, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{4}\right )}{d^{3}}-\frac {2 a b \left (\frac {105 \arccos \left (c x \right ) c^{6} x^{6}+29 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}-175 c^{4} x^{4} \arccos \left (c x \right )-27 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+56 c^{2} x^{2} \arccos \left (c x \right )-4 c x \sqrt {-c^{2} x^{2}+1}+8 \arccos \left (c x \right )}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{3} x^{3}}+\frac {35 i \operatorname {dilog}\left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{8}+\frac {19 i \arctan \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {35 i \operatorname {dilog}\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )}{8}-\frac {35 \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{8}\right )}{d^{3}}\right )\) | \(820\) |
| parts | \(-\frac {a^{2} \left (\frac {1}{3 x^{3}}+\frac {3 c^{2}}{x}-\frac {c^{3}}{16 \left (c x -1\right )^{2}}+\frac {11 c^{3}}{16 \left (c x -1\right )}+\frac {35 c^{3} \ln \left (c x -1\right )}{16}+\frac {c^{3}}{16 \left (c x +1\right )^{2}}+\frac {11 c^{3}}{16 \left (c x +1\right )}-\frac {35 c^{3} \ln \left (c x +1\right )}{16}\right )}{d^{3}}-\frac {b^{2} c^{3} \left (\frac {105 \arccos \left (c x \right )^{2} c^{6} x^{6}+58 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) x^{5} c^{5}-175 \arccos \left (c x \right )^{2} x^{4} c^{4}+10 c^{6} x^{6}-54 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) x^{3} c^{3}+56 \arccos \left (c x \right )^{2} x^{2} c^{2}-18 c^{4} x^{4}-8 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) x c +8 \arccos \left (c x \right )^{2}+8 c^{2} x^{2}}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{3} x^{3}}-\frac {35 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{4}-\frac {35 \arccos \left (c x \right )^{2} \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{8}+\frac {35 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{4}+\frac {35 \arccos \left (c x \right )^{2} \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{8}-\frac {19 i \operatorname {dilog}\left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{3}+\frac {19 i \operatorname {dilog}\left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{3}+\frac {19 \arccos \left (c x \right ) \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{3}-\frac {19 \arccos \left (c x \right ) \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{3}-\frac {17 \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{6}+\frac {17 \ln \left (i \sqrt {-c^{2} x^{2}+1}+c x -1\right )}{6}-\frac {35 \operatorname {polylog}\left (3, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{4}+\frac {35 \operatorname {polylog}\left (3, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{4}\right )}{d^{3}}-\frac {2 a b \,c^{3} \left (\frac {105 \arccos \left (c x \right ) c^{6} x^{6}+29 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}-175 c^{4} x^{4} \arccos \left (c x \right )-27 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+56 c^{2} x^{2} \arccos \left (c x \right )-4 c x \sqrt {-c^{2} x^{2}+1}+8 \arccos \left (c x \right )}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{3} x^{3}}+\frac {35 i \operatorname {dilog}\left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{8}+\frac {19 i \arctan \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {35 i \operatorname {dilog}\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )}{8}-\frac {35 \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{8}\right )}{d^{3}}\) | \(837\) |
Input:
int((a+b*arccos(c*x))^2/x^4/(-c^2*d*x^2+d)^3,x,method=_RETURNVERBOSE)
Output:
c^3*(-a^2/d^3*(-1/16/(c*x-1)^2+11/16/(c*x-1)+35/16*ln(c*x-1)+1/3/c^3/x^3+3 /c/x+1/16/(c*x+1)^2+11/16/(c*x+1)-35/16*ln(c*x+1))-b^2/d^3*(1/24*(105*arcc os(c*x)^2*c^6*x^6+58*(-c^2*x^2+1)^(1/2)*arccos(c*x)*x^5*c^5-175*arccos(c*x )^2*x^4*c^4+10*c^6*x^6-54*(-c^2*x^2+1)^(1/2)*arccos(c*x)*x^3*c^3+56*arccos (c*x)^2*x^2*c^2-18*c^4*x^4-8*(-c^2*x^2+1)^(1/2)*arccos(c*x)*x*c+8*arccos(c *x)^2+8*c^2*x^2)/(c^4*x^4-2*c^2*x^2+1)/c^3/x^3-35/4*I*arccos(c*x)*polylog( 2,c*x+I*(-c^2*x^2+1)^(1/2))-35/8*arccos(c*x)^2*ln(1+c*x+I*(-c^2*x^2+1)^(1/ 2))+35/4*I*arccos(c*x)*polylog(2,-c*x-I*(-c^2*x^2+1)^(1/2))+35/8*arccos(c* x)^2*ln(1-c*x-I*(-c^2*x^2+1)^(1/2))-19/3*I*dilog(1+I*(c*x+I*(-c^2*x^2+1)^( 1/2)))+19/3*I*dilog(1-I*(c*x+I*(-c^2*x^2+1)^(1/2)))+19/3*arccos(c*x)*ln(1+ I*(c*x+I*(-c^2*x^2+1)^(1/2)))-19/3*arccos(c*x)*ln(1-I*(c*x+I*(-c^2*x^2+1)^ (1/2)))-17/6*ln(1+c*x+I*(-c^2*x^2+1)^(1/2))+17/6*ln(I*(-c^2*x^2+1)^(1/2)+c *x-1)-35/4*polylog(3,-c*x-I*(-c^2*x^2+1)^(1/2))+35/4*polylog(3,c*x+I*(-c^2 *x^2+1)^(1/2)))-2*a*b/d^3*(1/24*(105*arccos(c*x)*c^6*x^6+29*c^5*x^5*(-c^2* x^2+1)^(1/2)-175*c^4*x^4*arccos(c*x)-27*c^3*x^3*(-c^2*x^2+1)^(1/2)+56*c^2* x^2*arccos(c*x)-4*c*x*(-c^2*x^2+1)^(1/2)+8*arccos(c*x))/(c^4*x^4-2*c^2*x^2 +1)/c^3/x^3+35/8*I*dilog(1+c*x+I*(-c^2*x^2+1)^(1/2))+19/3*I*arctan(c*x+I*( -c^2*x^2+1)^(1/2))+35/8*I*dilog(c*x+I*(-c^2*x^2+1)^(1/2))-35/8*arccos(c*x) *ln(1+c*x+I*(-c^2*x^2+1)^(1/2))))
\[ \int \frac {(a+b \arccos (c x))^2}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{3} x^{4}} \,d x } \] Input:
integrate((a+b*arccos(c*x))^2/x^4/(-c^2*d*x^2+d)^3,x, algorithm="fricas")
Output:
integral(-(b^2*arccos(c*x)^2 + 2*a*b*arccos(c*x) + a^2)/(c^6*d^3*x^10 - 3* c^4*d^3*x^8 + 3*c^2*d^3*x^6 - d^3*x^4), x)
\[ \int \frac {(a+b \arccos (c x))^2}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=- \frac {\int \frac {a^{2}}{c^{6} x^{10} - 3 c^{4} x^{8} + 3 c^{2} x^{6} - x^{4}}\, dx + \int \frac {b^{2} \operatorname {acos}^{2}{\left (c x \right )}}{c^{6} x^{10} - 3 c^{4} x^{8} + 3 c^{2} x^{6} - x^{4}}\, dx + \int \frac {2 a b \operatorname {acos}{\left (c x \right )}}{c^{6} x^{10} - 3 c^{4} x^{8} + 3 c^{2} x^{6} - x^{4}}\, dx}{d^{3}} \] Input:
integrate((a+b*acos(c*x))**2/x**4/(-c**2*d*x**2+d)**3,x)
Output:
-(Integral(a**2/(c**6*x**10 - 3*c**4*x**8 + 3*c**2*x**6 - x**4), x) + Inte gral(b**2*acos(c*x)**2/(c**6*x**10 - 3*c**4*x**8 + 3*c**2*x**6 - x**4), x) + Integral(2*a*b*acos(c*x)/(c**6*x**10 - 3*c**4*x**8 + 3*c**2*x**6 - x**4 ), x))/d**3
\[ \int \frac {(a+b \arccos (c x))^2}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{3} x^{4}} \,d x } \] Input:
integrate((a+b*arccos(c*x))^2/x^4/(-c^2*d*x^2+d)^3,x, algorithm="maxima")
Output:
1/48*a^2*(105*c^3*log(c*x + 1)/d^3 - 105*c^3*log(c*x - 1)/d^3 - 2*(105*c^6 *x^6 - 175*c^4*x^4 + 56*c^2*x^2 + 8)/(c^4*d^3*x^7 - 2*c^2*d^3*x^5 + d^3*x^ 3)) - 1/48*((210*b^2*c^6*x^6 - 350*b^2*c^4*x^4 + 112*b^2*c^2*x^2 + 16*b^2 - 105*(b^2*c^7*x^7 - 2*b^2*c^5*x^5 + b^2*c^3*x^3)*log(c*x + 1) + 105*(b^2* c^7*x^7 - 2*b^2*c^5*x^5 + b^2*c^3*x^3)*log(-c*x + 1))*arctan2(sqrt(c*x + 1 )*sqrt(-c*x + 1), c*x)^2 + 48*(c^4*d^3*x^7 - 2*c^2*d^3*x^5 + d^3*x^3)*inte grate(-1/24*((210*b^2*c^7*x^7 - 350*b^2*c^5*x^5 + 112*b^2*c^3*x^3 + 16*b^2 *c*x - 105*(b^2*c^8*x^8 - 2*b^2*c^6*x^6 + b^2*c^4*x^4)*log(c*x + 1) + 105* (b^2*c^8*x^8 - 2*b^2*c^6*x^6 + b^2*c^4*x^4)*log(-c*x + 1))*sqrt(c*x + 1)*s qrt(-c*x + 1)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) - 48*a*b*arctan2( sqrt(c*x + 1)*sqrt(-c*x + 1), c*x))/(c^6*d^3*x^10 - 3*c^4*d^3*x^8 + 3*c^2* d^3*x^6 - d^3*x^4), x))/(c^4*d^3*x^7 - 2*c^2*d^3*x^5 + d^3*x^3)
Timed out. \[ \int \frac {(a+b \arccos (c x))^2}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate((a+b*arccos(c*x))^2/x^4/(-c^2*d*x^2+d)^3,x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {(a+b \arccos (c x))^2}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=\int \frac {{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2}{x^4\,{\left (d-c^2\,d\,x^2\right )}^3} \,d x \] Input:
int((a + b*acos(c*x))^2/(x^4*(d - c^2*d*x^2)^3),x)
Output:
int((a + b*acos(c*x))^2/(x^4*(d - c^2*d*x^2)^3), x)
\[ \int \frac {(a+b \arccos (c x))^2}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=\frac {-96 \left (\int \frac {\mathit {acos} \left (c x \right )}{c^{6} x^{10}-3 c^{4} x^{8}+3 c^{2} x^{6}-x^{4}}d x \right ) a b \,c^{4} x^{7}+192 \left (\int \frac {\mathit {acos} \left (c x \right )}{c^{6} x^{10}-3 c^{4} x^{8}+3 c^{2} x^{6}-x^{4}}d x \right ) a b \,c^{2} x^{5}-96 \left (\int \frac {\mathit {acos} \left (c x \right )}{c^{6} x^{10}-3 c^{4} x^{8}+3 c^{2} x^{6}-x^{4}}d x \right ) a b \,x^{3}-48 \left (\int \frac {\mathit {acos} \left (c x \right )^{2}}{c^{6} x^{10}-3 c^{4} x^{8}+3 c^{2} x^{6}-x^{4}}d x \right ) b^{2} c^{4} x^{7}+96 \left (\int \frac {\mathit {acos} \left (c x \right )^{2}}{c^{6} x^{10}-3 c^{4} x^{8}+3 c^{2} x^{6}-x^{4}}d x \right ) b^{2} c^{2} x^{5}-48 \left (\int \frac {\mathit {acos} \left (c x \right )^{2}}{c^{6} x^{10}-3 c^{4} x^{8}+3 c^{2} x^{6}-x^{4}}d x \right ) b^{2} x^{3}-105 \,\mathrm {log}\left (c^{2} x -c \right ) a^{2} c^{7} x^{7}+210 \,\mathrm {log}\left (c^{2} x -c \right ) a^{2} c^{5} x^{5}-105 \,\mathrm {log}\left (c^{2} x -c \right ) a^{2} c^{3} x^{3}+105 \,\mathrm {log}\left (c^{2} x +c \right ) a^{2} c^{7} x^{7}-210 \,\mathrm {log}\left (c^{2} x +c \right ) a^{2} c^{5} x^{5}+105 \,\mathrm {log}\left (c^{2} x +c \right ) a^{2} c^{3} x^{3}-210 a^{2} c^{6} x^{6}+350 a^{2} c^{4} x^{4}-112 a^{2} c^{2} x^{2}-16 a^{2}}{48 d^{3} x^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )} \] Input:
int((a+b*acos(c*x))^2/x^4/(-c^2*d*x^2+d)^3,x)
Output:
( - 96*int(acos(c*x)/(c**6*x**10 - 3*c**4*x**8 + 3*c**2*x**6 - x**4),x)*a* b*c**4*x**7 + 192*int(acos(c*x)/(c**6*x**10 - 3*c**4*x**8 + 3*c**2*x**6 - x**4),x)*a*b*c**2*x**5 - 96*int(acos(c*x)/(c**6*x**10 - 3*c**4*x**8 + 3*c* *2*x**6 - x**4),x)*a*b*x**3 - 48*int(acos(c*x)**2/(c**6*x**10 - 3*c**4*x** 8 + 3*c**2*x**6 - x**4),x)*b**2*c**4*x**7 + 96*int(acos(c*x)**2/(c**6*x**1 0 - 3*c**4*x**8 + 3*c**2*x**6 - x**4),x)*b**2*c**2*x**5 - 48*int(acos(c*x) **2/(c**6*x**10 - 3*c**4*x**8 + 3*c**2*x**6 - x**4),x)*b**2*x**3 - 105*log (c**2*x - c)*a**2*c**7*x**7 + 210*log(c**2*x - c)*a**2*c**5*x**5 - 105*log (c**2*x - c)*a**2*c**3*x**3 + 105*log(c**2*x + c)*a**2*c**7*x**7 - 210*log (c**2*x + c)*a**2*c**5*x**5 + 105*log(c**2*x + c)*a**2*c**3*x**3 - 210*a** 2*c**6*x**6 + 350*a**2*c**4*x**4 - 112*a**2*c**2*x**2 - 16*a**2)/(48*d**3* x**3*(c**4*x**4 - 2*c**2*x**2 + 1))