Integrand size = 24, antiderivative size = 332 \[ \int \frac {(a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {b^2 x}{12 d^3 \left (1-c^2 x^2\right )}-\frac {b (a+b \arccos (c x))}{6 c d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {3 b (a+b \arccos (c x))}{4 c d^3 \sqrt {1-c^2 x^2}}+\frac {x (a+b \arccos (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {3 x (a+b \arccos (c x))^2}{8 d^3 \left (1-c^2 x^2\right )}-\frac {3 i (a+b \arccos (c x))^2 \arctan \left (e^{i \arccos (c x)}\right )}{4 c d^3}+\frac {5 b^2 \text {arctanh}(c x)}{6 c d^3}+\frac {3 i b (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )}{4 c d^3}-\frac {3 i b (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )}{4 c d^3}-\frac {3 b^2 \operatorname {PolyLog}\left (3,-i e^{i \arccos (c x)}\right )}{4 c d^3}+\frac {3 b^2 \operatorname {PolyLog}\left (3,i e^{i \arccos (c x)}\right )}{4 c d^3} \] Output:
1/12*b^2*x/d^3/(-c^2*x^2+1)-1/6*b*(a+b*arccos(c*x))/c/d^3/(-c^2*x^2+1)^(3/ 2)-3/4*b*(a+b*arccos(c*x))/c/d^3/(-c^2*x^2+1)^(1/2)+1/4*x*(a+b*arccos(c*x) )^2/d^3/(-c^2*x^2+1)^2+3/8*x*(a+b*arccos(c*x))^2/d^3/(-c^2*x^2+1)-3/4*I*(a +b*arccos(c*x))^2*arctan(c*x+I*(-c^2*x^2+1)^(1/2))/c/d^3+5/6*b^2*arctanh(c *x)/c/d^3+3/4*I*b*(a+b*arccos(c*x))*polylog(2,-I*(c*x+I*(-c^2*x^2+1)^(1/2) ))/c/d^3-3/4*I*b*(a+b*arccos(c*x))*polylog(2,I*(c*x+I*(-c^2*x^2+1)^(1/2))) /c/d^3-3/4*b^2*polylog(3,-I*(c*x+I*(-c^2*x^2+1)^(1/2)))/c/d^3+3/4*b^2*poly log(3,I*(c*x+I*(-c^2*x^2+1)^(1/2)))/c/d^3
Time = 7.54 (sec) , antiderivative size = 652, normalized size of antiderivative = 1.96 \[ \int \frac {(a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {a^2 x}{4 d^3 \left (-1+c^2 x^2\right )^2}-\frac {3 a^2 x}{8 d^3 \left (-1+c^2 x^2\right )}-\frac {3 a^2 \log (1-c x)}{16 c d^3}+\frac {3 a^2 \log (1+c x)}{16 c d^3}-\frac {2 a b \left (\frac {(-2+c x) \sqrt {1-c^2 x^2}-3 \arccos (c x)}{48 (-1+c x)^2}-\frac {(2+c x) \sqrt {1-c^2 x^2}-3 \arccos (c x)}{48 (1+c x)^2}-\frac {3 \left (\sqrt {1-c^2 x^2}-\arccos (c x)\right )}{16 (1+c x)}-\frac {3 \left (\sqrt {1-c^2 x^2}+\arccos (c x)\right )}{16 (1-c x)}-\frac {3}{16} \left (-\frac {1}{2} i \arccos (c x)^2+2 \arccos (c x) \log \left (1+e^{i \arccos (c x)}\right )-2 i \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )\right )+\frac {3}{16} \left (2 \arccos (c x) \log \left (1-e^{i \arccos (c x)}\right )-2 i \left (\frac {1}{4} \arccos (c x)^2+\operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )\right )\right )\right )}{c d^3}-\frac {b^2 \left (-80 \arccos (c x) \cot \left (\frac {1}{2} \arccos (c x)\right )-2 \left (2+9 \arccos (c x)^2\right ) \csc ^2\left (\frac {1}{2} \arccos (c x)\right )-2 \sqrt {1-c^2 x^2} \arccos (c x) \csc ^4\left (\frac {1}{2} \arccos (c x)\right )-3 \arccos (c x)^2 \csc ^4\left (\frac {1}{2} \arccos (c x)\right )+160 \log \left (\tan \left (\frac {1}{2} \arccos (c x)\right )\right )+72 \left (\arccos (c x)^2 \left (\log \left (1-e^{i \arccos (c x)}\right )-\log \left (1+e^{i \arccos (c x)}\right )\right )+2 i \arccos (c x) \left (\operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-\operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )\right )+2 \left (-\operatorname {PolyLog}\left (3,-e^{i \arccos (c x)}\right )+\operatorname {PolyLog}\left (3,e^{i \arccos (c x)}\right )\right )\right )+2 \left (2+9 \arccos (c x)^2\right ) \sec ^2\left (\frac {1}{2} \arccos (c x)\right )+3 \arccos (c x)^2 \sec ^4\left (\frac {1}{2} \arccos (c x)\right )-\frac {32 \arccos (c x) \sin ^4\left (\frac {1}{2} \arccos (c x)\right )}{\left (1-c^2 x^2\right )^{3/2}}-80 \arccos (c x) \tan \left (\frac {1}{2} \arccos (c x)\right )\right )}{192 c d^3} \] Input:
Integrate[(a + b*ArcCos[c*x])^2/(d - c^2*d*x^2)^3,x]
Output:
(a^2*x)/(4*d^3*(-1 + c^2*x^2)^2) - (3*a^2*x)/(8*d^3*(-1 + c^2*x^2)) - (3*a ^2*Log[1 - c*x])/(16*c*d^3) + (3*a^2*Log[1 + c*x])/(16*c*d^3) - (2*a*b*((( -2 + c*x)*Sqrt[1 - c^2*x^2] - 3*ArcCos[c*x])/(48*(-1 + c*x)^2) - ((2 + c*x )*Sqrt[1 - c^2*x^2] - 3*ArcCos[c*x])/(48*(1 + c*x)^2) - (3*(Sqrt[1 - c^2*x ^2] - ArcCos[c*x]))/(16*(1 + c*x)) - (3*(Sqrt[1 - c^2*x^2] + ArcCos[c*x])) /(16*(1 - c*x)) - (3*((-1/2*I)*ArcCos[c*x]^2 + 2*ArcCos[c*x]*Log[1 + E^(I* ArcCos[c*x])] - (2*I)*PolyLog[2, -E^(I*ArcCos[c*x])]))/16 + (3*(2*ArcCos[c *x]*Log[1 - E^(I*ArcCos[c*x])] - (2*I)*(ArcCos[c*x]^2/4 + PolyLog[2, E^(I* ArcCos[c*x])])))/16))/(c*d^3) - (b^2*(-80*ArcCos[c*x]*Cot[ArcCos[c*x]/2] - 2*(2 + 9*ArcCos[c*x]^2)*Csc[ArcCos[c*x]/2]^2 - 2*Sqrt[1 - c^2*x^2]*ArcCos [c*x]*Csc[ArcCos[c*x]/2]^4 - 3*ArcCos[c*x]^2*Csc[ArcCos[c*x]/2]^4 + 160*Lo g[Tan[ArcCos[c*x]/2]] + 72*(ArcCos[c*x]^2*(Log[1 - E^(I*ArcCos[c*x])] - Lo g[1 + E^(I*ArcCos[c*x])]) + (2*I)*ArcCos[c*x]*(PolyLog[2, -E^(I*ArcCos[c*x ])] - PolyLog[2, E^(I*ArcCos[c*x])]) + 2*(-PolyLog[3, -E^(I*ArcCos[c*x])] + PolyLog[3, E^(I*ArcCos[c*x])])) + 2*(2 + 9*ArcCos[c*x]^2)*Sec[ArcCos[c*x ]/2]^2 + 3*ArcCos[c*x]^2*Sec[ArcCos[c*x]/2]^4 - (32*ArcCos[c*x]*Sin[ArcCos [c*x]/2]^4)/(1 - c^2*x^2)^(3/2) - 80*ArcCos[c*x]*Tan[ArcCos[c*x]/2]))/(192 *c*d^3)
Time = 1.81 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.91, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5163, 27, 5163, 5165, 3042, 4671, 3011, 2720, 5183, 215, 219, 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.
| \(\displaystyle \int \frac {(a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 5163 |
| \(\displaystyle \frac {b c \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx}{2 d^3}+\frac {3 \int \frac {(a+b \arccos (c x))^2}{d^2 \left (1-c^2 x^2\right )^2}dx}{4 d}+\frac {x (a+b \arccos (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx}{2 d^3}+\frac {3 \int \frac {(a+b \arccos (c x))^2}{\left (1-c^2 x^2\right )^2}dx}{4 d^3}+\frac {x (a+b \arccos (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 5163 |
| \(\displaystyle \frac {b c \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx}{2 d^3}+\frac {3 \left (b c \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {1}{2} \int \frac {(a+b \arccos (c x))^2}{1-c^2 x^2}dx+\frac {x (a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}+\frac {x (a+b \arccos (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 5165 |
| \(\displaystyle \frac {b c \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx}{2 d^3}+\frac {3 \left (b c \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx-\frac {\int \frac {(a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}}d\arccos (c x)}{2 c}+\frac {x (a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}+\frac {x (a+b \arccos (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b c \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx}{2 d^3}+\frac {3 \left (b c \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx-\frac {\int (a+b \arccos (c x))^2 \csc (\arccos (c x))d\arccos (c x)}{2 c}+\frac {x (a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}+\frac {x (a+b \arccos (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle \frac {3 \left (-\frac {-2 b \int (a+b \arccos (c x)) \log \left (1-e^{i \arccos (c x)}\right )d\arccos (c x)+2 b \int (a+b \arccos (c x)) \log \left (1+e^{i \arccos (c x)}\right )d\arccos (c x)-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2}{2 c}+b c \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {x (a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}+\frac {b c \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx}{2 d^3}+\frac {x (a+b \arccos (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {3 \left (-\frac {2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-i b \int \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )d\arccos (c x)\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-i b \int \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )d\arccos (c x)\right )-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2}{2 c}+b c \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {x (a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}+\frac {b c \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx}{2 d^3}+\frac {x (a+b \arccos (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {3 \left (-\frac {2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2}{2 c}+b c \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {x (a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}+\frac {b c \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx}{2 d^3}+\frac {x (a+b \arccos (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 5183 |
| \(\displaystyle \frac {3 \left (-\frac {2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2}{2 c}+b c \left (\frac {b \int \frac {1}{1-c^2 x^2}dx}{c}+\frac {a+b \arccos (c x)}{c^2 \sqrt {1-c^2 x^2}}\right )+\frac {x (a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}+\frac {b c \left (\frac {b \int \frac {1}{\left (1-c^2 x^2\right )^2}dx}{3 c}+\frac {a+b \arccos (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}\right )}{2 d^3}+\frac {x (a+b \arccos (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {3 \left (-\frac {2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2}{2 c}+b c \left (\frac {b \int \frac {1}{1-c^2 x^2}dx}{c}+\frac {a+b \arccos (c x)}{c^2 \sqrt {1-c^2 x^2}}\right )+\frac {x (a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}+\frac {b c \left (\frac {b \left (\frac {1}{2} \int \frac {1}{1-c^2 x^2}dx+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c}+\frac {a+b \arccos (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}\right )}{2 d^3}+\frac {x (a+b \arccos (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3 \left (-\frac {2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2}{2 c}+b c \left (\frac {a+b \arccos (c x)}{c^2 \sqrt {1-c^2 x^2}}+\frac {b \text {arctanh}(c x)}{c^2}\right )+\frac {x (a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}+\frac {b c \left (\frac {a+b \arccos (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}+\frac {b \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c}\right )}{2 d^3}+\frac {x (a+b \arccos (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {3 \left (b c \left (\frac {a+b \arccos (c x)}{c^2 \sqrt {1-c^2 x^2}}+\frac {b \text {arctanh}(c x)}{c^2}\right )-\frac {-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2+2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \operatorname {PolyLog}\left (3,-e^{i \arccos (c x)}\right )\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \operatorname {PolyLog}\left (3,e^{i \arccos (c x)}\right )\right )}{2 c}+\frac {x (a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}+\frac {b c \left (\frac {a+b \arccos (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}+\frac {b \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c}\right )}{2 d^3}+\frac {x (a+b \arccos (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
Input:
Int[(a + b*ArcCos[c*x])^2/(d - c^2*d*x^2)^3,x]
Output:
(x*(a + b*ArcCos[c*x])^2)/(4*d^3*(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*d^3) + (3*((x*(a + b*ArcCos[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*ArcCos[c*x])] + 2*b*(I*(a + b*Arc Cos[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*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[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[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_.) + (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[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 = 0.60 (sec) , antiderivative size = 549, normalized size of antiderivative = 1.65
| method | result | size |
| derivativedivides | \(\frac {-\frac {a^{2} \left (-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {3}{16 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{16}+\frac {1}{16 \left (c x +1\right )^{2}}+\frac {3}{16 \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{16}\right )}{d^{3}}-\frac {b^{2} \left (\frac {9 \arccos \left (c x \right )^{2} c^{3} x^{3}+18 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{2} x^{2}-15 \arccos \left (c x \right )^{2} c x +2 c^{3} x^{3}-22 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-2 c x}{24 c^{4} x^{4}-48 c^{2} x^{2}+24}-\frac {5 \,\operatorname {arctanh}\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {3 \arccos \left (c x \right )^{2} \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{8}+\frac {3 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{4}-\frac {3 \operatorname {polylog}\left (3, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{4}+\frac {3 \arccos \left (c x \right )^{2} \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{8}-\frac {3 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{4}+\frac {3 \operatorname {polylog}\left (3, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{4}\right )}{d^{3}}-\frac {2 a b \left (\frac {9 c^{3} x^{3} \arccos \left (c x \right )+9 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}-15 c x \arccos \left (c x \right )-11 \sqrt {-c^{2} x^{2}+1}}{24 c^{4} x^{4}-48 c^{2} x^{2}+24}+\frac {3 \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{8}-\frac {3 \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{8}-\frac {3 i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{8}+\frac {3 i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{8}\right )}{d^{3}}}{c}\) | \(549\) |
| default | \(\frac {-\frac {a^{2} \left (-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {3}{16 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{16}+\frac {1}{16 \left (c x +1\right )^{2}}+\frac {3}{16 \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{16}\right )}{d^{3}}-\frac {b^{2} \left (\frac {9 \arccos \left (c x \right )^{2} c^{3} x^{3}+18 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{2} x^{2}-15 \arccos \left (c x \right )^{2} c x +2 c^{3} x^{3}-22 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-2 c x}{24 c^{4} x^{4}-48 c^{2} x^{2}+24}-\frac {5 \,\operatorname {arctanh}\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {3 \arccos \left (c x \right )^{2} \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{8}+\frac {3 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{4}-\frac {3 \operatorname {polylog}\left (3, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{4}+\frac {3 \arccos \left (c x \right )^{2} \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{8}-\frac {3 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{4}+\frac {3 \operatorname {polylog}\left (3, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{4}\right )}{d^{3}}-\frac {2 a b \left (\frac {9 c^{3} x^{3} \arccos \left (c x \right )+9 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}-15 c x \arccos \left (c x \right )-11 \sqrt {-c^{2} x^{2}+1}}{24 c^{4} x^{4}-48 c^{2} x^{2}+24}+\frac {3 \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{8}-\frac {3 \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{8}-\frac {3 i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{8}+\frac {3 i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{8}\right )}{d^{3}}}{c}\) | \(549\) |
| parts | \(-\frac {a^{2} \left (-\frac {1}{16 c \left (c x -1\right )^{2}}+\frac {3}{16 c \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{16 c}+\frac {1}{16 c \left (c x +1\right )^{2}}+\frac {3}{16 c \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{16 c}\right )}{d^{3}}-\frac {b^{2} \left (\frac {9 \arccos \left (c x \right )^{2} c^{3} x^{3}+18 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{2} x^{2}-15 \arccos \left (c x \right )^{2} c x +2 c^{3} x^{3}-22 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-2 c x}{24 c^{4} x^{4}-48 c^{2} x^{2}+24}-\frac {5 \,\operatorname {arctanh}\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {3 \arccos \left (c x \right )^{2} \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{8}+\frac {3 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{4}-\frac {3 \operatorname {polylog}\left (3, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{4}+\frac {3 \arccos \left (c x \right )^{2} \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{8}-\frac {3 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{4}+\frac {3 \operatorname {polylog}\left (3, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{4}\right )}{d^{3} c}-\frac {2 a b \left (\frac {9 c^{3} x^{3} \arccos \left (c x \right )+9 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}-15 c x \arccos \left (c x \right )-11 \sqrt {-c^{2} x^{2}+1}}{24 c^{4} x^{4}-48 c^{2} x^{2}+24}+\frac {3 \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{8}-\frac {3 \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{8}-\frac {3 i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{8}+\frac {3 i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{8}\right )}{d^{3} c}\) | \(569\) |
Input:
int((a+b*arccos(c*x))^2/(-c^2*d*x^2+d)^3,x,method=_RETURNVERBOSE)
Output:
1/c*(-a^2/d^3*(-1/16/(c*x-1)^2+3/16/(c*x-1)+3/16*ln(c*x-1)+1/16/(c*x+1)^2+ 3/16/(c*x+1)-3/16*ln(c*x+1))-b^2/d^3*(1/24*(9*arccos(c*x)^2*c^3*x^3+18*(-c ^2*x^2+1)^(1/2)*arccos(c*x)*c^2*x^2-15*arccos(c*x)^2*c*x+2*c^3*x^3-22*arcc os(c*x)*(-c^2*x^2+1)^(1/2)-2*c*x)/(c^4*x^4-2*c^2*x^2+1)-5/3*arctanh(c*x+I* (-c^2*x^2+1)^(1/2))-3/8*arccos(c*x)^2*ln(1+c*x+I*(-c^2*x^2+1)^(1/2))+3/4*I *arccos(c*x)*polylog(2,-c*x-I*(-c^2*x^2+1)^(1/2))-3/4*polylog(3,-c*x-I*(-c ^2*x^2+1)^(1/2))+3/8*arccos(c*x)^2*ln(1-c*x-I*(-c^2*x^2+1)^(1/2))-3/4*I*ar ccos(c*x)*polylog(2,c*x+I*(-c^2*x^2+1)^(1/2))+3/4*polylog(3,c*x+I*(-c^2*x^ 2+1)^(1/2)))-2*a*b/d^3*(1/24*(9*c^3*x^3*arccos(c*x)+9*c^2*x^2*(-c^2*x^2+1) ^(1/2)-15*c*x*arccos(c*x)-11*(-c^2*x^2+1)^(1/2))/(c^4*x^4-2*c^2*x^2+1)+3/8 *arccos(c*x)*ln(1-c*x-I*(-c^2*x^2+1)^(1/2))-3/8*arccos(c*x)*ln(1+c*x+I*(-c ^2*x^2+1)^(1/2))-3/8*I*polylog(2,c*x+I*(-c^2*x^2+1)^(1/2))+3/8*I*polylog(2 ,-c*x-I*(-c^2*x^2+1)^(1/2))))
\[ \int \frac {(a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{3}} \,d x } \] Input:
integrate((a+b*arccos(c*x))^2/(-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^6 - 3*c ^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x)
Timed out. \[ \int \frac {(a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate((a+b*acos(c*x))**2/(-c**2*d*x**2+d)**3,x)
Output:
Timed out
\[ \int \frac {(a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{3}} \,d x } \] Input:
integrate((a+b*arccos(c*x))^2/(-c^2*d*x^2+d)^3,x, algorithm="maxima")
Output:
-1/16*a^2*(2*(3*c^2*x^3 - 5*x)/(c^4*d^3*x^4 - 2*c^2*d^3*x^2 + d^3) - 3*log (c*x + 1)/(c*d^3) + 3*log(c*x - 1)/(c*d^3)) - 1/16*((6*b^2*c^3*x^3 - 10*b^ 2*c*x - 3*(b^2*c^4*x^4 - 2*b^2*c^2*x^2 + b^2)*log(c*x + 1) + 3*(b^2*c^4*x^ 4 - 2*b^2*c^2*x^2 + b^2)*log(-c*x + 1))*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^2 + 16*(c^5*d^3*x^4 - 2*c^3*d^3*x^2 + c*d^3)*integrate(-1/8*((6*b ^2*c^3*x^3 - 10*b^2*c*x - 3*(b^2*c^4*x^4 - 2*b^2*c^2*x^2 + b^2)*log(c*x + 1) + 3*(b^2*c^4*x^4 - 2*b^2*c^2*x^2 + b^2)*log(-c*x + 1))*sqrt(c*x + 1)*sq rt(-c*x + 1)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) - 16*a*b*arctan2(s qrt(c*x + 1)*sqrt(-c*x + 1), c*x))/(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^ 3*x^2 - d^3), x))/(c^5*d^3*x^4 - 2*c^3*d^3*x^2 + c*d^3)
Exception generated. \[ \int \frac {(a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((a+b*arccos(c*x))^2/(-c^2*d*x^2+d)^3,x, algorithm="giac")
Output:
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {(a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\int \frac {{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^3} \,d x \] Input:
int((a + b*acos(c*x))^2/(d - c^2*d*x^2)^3,x)
Output:
int((a + b*acos(c*x))^2/(d - c^2*d*x^2)^3, x)
\[ \int \frac {(a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {-32 \left (\int \frac {\mathit {acos} \left (c x \right )}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) a b \,c^{5} x^{4}+64 \left (\int \frac {\mathit {acos} \left (c x \right )}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) a b \,c^{3} x^{2}-32 \left (\int \frac {\mathit {acos} \left (c x \right )}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) a b c -16 \left (\int \frac {\mathit {acos} \left (c x \right )^{2}}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) b^{2} c^{5} x^{4}+32 \left (\int \frac {\mathit {acos} \left (c x \right )^{2}}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) b^{2} c^{3} x^{2}-16 \left (\int \frac {\mathit {acos} \left (c x \right )^{2}}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) b^{2} c -3 \,\mathrm {log}\left (c^{2} x -c \right ) a^{2} c^{4} x^{4}+6 \,\mathrm {log}\left (c^{2} x -c \right ) a^{2} c^{2} x^{2}-3 \,\mathrm {log}\left (c^{2} x -c \right ) a^{2}+3 \,\mathrm {log}\left (c^{2} x +c \right ) a^{2} c^{4} x^{4}-6 \,\mathrm {log}\left (c^{2} x +c \right ) a^{2} c^{2} x^{2}+3 \,\mathrm {log}\left (c^{2} x +c \right ) a^{2}-6 a^{2} c^{3} x^{3}+10 a^{2} c x}{16 c \,d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )} \] Input:
int((a+b*acos(c*x))^2/(-c^2*d*x^2+d)^3,x)
Output:
( - 32*int(acos(c*x)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1),x)*a*b*c* *5*x**4 + 64*int(acos(c*x)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1),x)* a*b*c**3*x**2 - 32*int(acos(c*x)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1),x)*a*b*c - 16*int(acos(c*x)**2/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1),x)*b**2*c**5*x**4 + 32*int(acos(c*x)**2/(c**6*x**6 - 3*c**4*x**4 + 3*c **2*x**2 - 1),x)*b**2*c**3*x**2 - 16*int(acos(c*x)**2/(c**6*x**6 - 3*c**4* x**4 + 3*c**2*x**2 - 1),x)*b**2*c - 3*log(c**2*x - c)*a**2*c**4*x**4 + 6*l og(c**2*x - c)*a**2*c**2*x**2 - 3*log(c**2*x - c)*a**2 + 3*log(c**2*x + c) *a**2*c**4*x**4 - 6*log(c**2*x + c)*a**2*c**2*x**2 + 3*log(c**2*x + c)*a** 2 - 6*a**2*c**3*x**3 + 10*a**2*c*x)/(16*c*d**3*(c**4*x**4 - 2*c**2*x**2 + 1))