Integrand size = 29, antiderivative size = 577 \[ \int \frac {(a+b \arccos (c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {b^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c x (a+b \arccos (c x))}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {(a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {(a+b \arccos (c x))^2}{d^2 \sqrt {d-c^2 d x^2}}+\frac {14 i b \sqrt {1-c^2 x^2} (a+b \arccos (c x)) \arctan \left (e^{i \arccos (c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2 \text {arctanh}\left (e^{i \arccos (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {1-c^2 x^2} (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {7 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {7 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {1-c^2 x^2} (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arccos (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arccos (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}} \] Output:
1/3*b^2/d^2/(-c^2*d*x^2+d)^(1/2)-1/3*b*c*x*(a+b*arccos(c*x))/d^2/(-c^2*x^2 +1)^(1/2)/(-c^2*d*x^2+d)^(1/2)+1/3*(a+b*arccos(c*x))^2/d/(-c^2*d*x^2+d)^(3 /2)+(a+b*arccos(c*x))^2/d^2/(-c^2*d*x^2+d)^(1/2)+14/3*I*b*(-c^2*x^2+1)^(1/ 2)*(a+b*arccos(c*x))*arctan(c*x+I*(-c^2*x^2+1)^(1/2))/d^2/(-c^2*d*x^2+d)^( 1/2)-2*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))^2*arctanh(c*x+I*(-c^2*x^2+1)^( 1/2))/d^2/(-c^2*d*x^2+d)^(1/2)+2*I*b*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))* polylog(2,-c*x-I*(-c^2*x^2+1)^(1/2))/d^2/(-c^2*d*x^2+d)^(1/2)-7/3*I*b^2*(- c^2*x^2+1)^(1/2)*polylog(2,-I*(c*x+I*(-c^2*x^2+1)^(1/2)))/d^2/(-c^2*d*x^2+ d)^(1/2)+7/3*I*b^2*(-c^2*x^2+1)^(1/2)*polylog(2,I*(c*x+I*(-c^2*x^2+1)^(1/2 )))/d^2/(-c^2*d*x^2+d)^(1/2)-2*I*b*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))*po lylog(2,c*x+I*(-c^2*x^2+1)^(1/2))/d^2/(-c^2*d*x^2+d)^(1/2)-2*b^2*(-c^2*x^2 +1)^(1/2)*polylog(3,-c*x-I*(-c^2*x^2+1)^(1/2))/d^2/(-c^2*d*x^2+d)^(1/2)+2* b^2*(-c^2*x^2+1)^(1/2)*polylog(3,c*x+I*(-c^2*x^2+1)^(1/2))/d^2/(-c^2*d*x^2 +d)^(1/2)
Time = 10.19 (sec) , antiderivative size = 782, normalized size of antiderivative = 1.36 \[ \int \frac {(a+b \arccos (c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
Integrate[(a + b*ArcCos[c*x])^2/(x*(d - c^2*d*x^2)^(5/2)),x]
Output:
Sqrt[-(d*(-1 + c^2*x^2))]*(a^2/(3*d^3*(-1 + c^2*x^2)^2) - a^2/(d^3*(-1 + c ^2*x^2))) + (a^2*Log[c*x])/d^(5/2) - (a^2*Log[d + Sqrt[d]*Sqrt[-(d*(-1 + c ^2*x^2))]])/d^(5/2) - (a*b*(1 - c^2*x^2)^(3/2)*(-14*ArcCos[c*x]*Cot[ArcCos [c*x]/2] - Csc[ArcCos[c*x]/2]^2 - (Sqrt[1 - c^2*x^2]*ArcCos[c*x]*Csc[ArcCo s[c*x]/2]^4)/2 + 24*ArcCos[c*x]*Log[1 - I*E^(I*ArcCos[c*x])] - 24*ArcCos[c *x]*Log[1 + I*E^(I*ArcCos[c*x])] - 28*Log[Cos[ArcCos[c*x]/2]] + 28*Log[Sin [ArcCos[c*x]/2]] + (24*I)*PolyLog[2, (-I)*E^(I*ArcCos[c*x])] - (24*I)*Poly Log[2, I*E^(I*ArcCos[c*x])] + Sec[ArcCos[c*x]/2]^2 - (8*ArcCos[c*x]*Sin[Ar cCos[c*x]/2]^4)/(1 - c^2*x^2)^(3/2) - 14*ArcCos[c*x]*Tan[ArcCos[c*x]/2]))/ (12*d*(d*(1 - c^2*x^2))^(3/2)) - (b^2*(1 - c^2*x^2)^(3/2)*(-4*Cot[ArcCos[c *x]/2] - 14*ArcCos[c*x]^2*Cot[ArcCos[c*x]/2] - 2*ArcCos[c*x]*Csc[ArcCos[c* x]/2]^2 - (Sqrt[1 - c^2*x^2]*ArcCos[c*x]^2*Csc[ArcCos[c*x]/2]^4)/2 + 56*Ar cCos[c*x]*Log[1 - E^(I*ArcCos[c*x])] + 24*ArcCos[c*x]^2*Log[1 - I*E^(I*Arc Cos[c*x])] - 24*ArcCos[c*x]^2*Log[1 + I*E^(I*ArcCos[c*x])] - 56*ArcCos[c*x ]*Log[1 + E^(I*ArcCos[c*x])] + (56*I)*PolyLog[2, -E^(I*ArcCos[c*x])] + (48 *I)*ArcCos[c*x]*PolyLog[2, (-I)*E^(I*ArcCos[c*x])] - (48*I)*ArcCos[c*x]*Po lyLog[2, I*E^(I*ArcCos[c*x])] - (56*I)*PolyLog[2, E^(I*ArcCos[c*x])] - 48* PolyLog[3, (-I)*E^(I*ArcCos[c*x])] + 48*PolyLog[3, I*E^(I*ArcCos[c*x])] + 2*ArcCos[c*x]*Sec[ArcCos[c*x]/2]^2 - (8*ArcCos[c*x]^2*Sin[ArcCos[c*x]/2]^4 )/(1 - c^2*x^2)^(3/2) - 4*Tan[ArcCos[c*x]/2] - 14*ArcCos[c*x]^2*Tan[Arc...
Time = 4.24 (sec) , antiderivative size = 473, normalized size of antiderivative = 0.82, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.690, Rules used = {5209, 5163, 241, 5165, 3042, 4671, 2715, 2838, 5209, 5165, 3042, 4671, 2715, 2838, 5219, 3042, 4669, 3011, 2720, 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.
| \(\displaystyle \int \frac {(a+b \arccos (c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 5209 |
| \(\displaystyle \frac {2 b c \sqrt {1-c^2 x^2} \int \frac {a+b \arccos (c x)}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \arccos (c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {(a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5163 |
| \(\displaystyle \frac {2 b c \sqrt {1-c^2 x^2} \left (\frac {1}{2} \int \frac {a+b \arccos (c x)}{1-c^2 x^2}dx+\frac {1}{2} b c \int \frac {x}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {x (a+b \arccos (c x))}{2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \arccos (c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {(a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {2 b c \sqrt {1-c^2 x^2} \left (\frac {1}{2} \int \frac {a+b \arccos (c x)}{1-c^2 x^2}dx+\frac {x (a+b \arccos (c x))}{2 \left (1-c^2 x^2\right )}+\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \arccos (c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {(a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5165 |
| \(\displaystyle \frac {2 b c \sqrt {1-c^2 x^2} \left (-\frac {\int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}d\arccos (c x)}{2 c}+\frac {x (a+b \arccos (c x))}{2 \left (1-c^2 x^2\right )}+\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \arccos (c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {(a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 b c \sqrt {1-c^2 x^2} \left (-\frac {\int (a+b \arccos (c x)) \csc (\arccos (c x))d\arccos (c x)}{2 c}+\frac {x (a+b \arccos (c x))}{2 \left (1-c^2 x^2\right )}+\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \arccos (c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {(a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle \frac {2 b c \sqrt {1-c^2 x^2} \left (-\frac {-b \int \log \left (1-e^{i \arccos (c x)}\right )d\arccos (c x)+b \int \log \left (1+e^{i \arccos (c x)}\right )d\arccos (c x)-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))}{2 c}+\frac {x (a+b \arccos (c x))}{2 \left (1-c^2 x^2\right )}+\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \arccos (c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {(a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {2 b c \sqrt {1-c^2 x^2} \left (-\frac {i b \int e^{-i \arccos (c x)} \log \left (1-e^{i \arccos (c x)}\right )de^{i \arccos (c x)}-i b \int e^{-i \arccos (c x)} \log \left (1+e^{i \arccos (c x)}\right )de^{i \arccos (c x)}-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))}{2 c}+\frac {x (a+b \arccos (c x))}{2 \left (1-c^2 x^2\right )}+\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \arccos (c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {(a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {\int \frac {(a+b \arccos (c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {2 b c \sqrt {1-c^2 x^2} \left (-\frac {-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{2 c}+\frac {x (a+b \arccos (c x))}{2 \left (1-c^2 x^2\right )}+\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5209 |
| \(\displaystyle \frac {\frac {2 b c \sqrt {1-c^2 x^2} \int \frac {a+b \arccos (c x)}{1-c^2 x^2}dx}{d \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \arccos (c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}+\frac {(a+b \arccos (c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {2 b c \sqrt {1-c^2 x^2} \left (-\frac {-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{2 c}+\frac {x (a+b \arccos (c x))}{2 \left (1-c^2 x^2\right )}+\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5165 |
| \(\displaystyle \frac {-\frac {2 b \sqrt {1-c^2 x^2} \int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}d\arccos (c x)}{d \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \arccos (c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}+\frac {(a+b \arccos (c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {2 b c \sqrt {1-c^2 x^2} \left (-\frac {-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{2 c}+\frac {x (a+b \arccos (c x))}{2 \left (1-c^2 x^2\right )}+\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {(a+b \arccos (c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}-\frac {2 b \sqrt {1-c^2 x^2} \int (a+b \arccos (c x)) \csc (\arccos (c x))d\arccos (c x)}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \arccos (c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {2 b c \sqrt {1-c^2 x^2} \left (-\frac {-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{2 c}+\frac {x (a+b \arccos (c x))}{2 \left (1-c^2 x^2\right )}+\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle \frac {-\frac {2 b \sqrt {1-c^2 x^2} \left (-b \int \log \left (1-e^{i \arccos (c x)}\right )d\arccos (c x)+b \int \log \left (1+e^{i \arccos (c x)}\right )d\arccos (c x)-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \arccos (c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}+\frac {(a+b \arccos (c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {2 b c \sqrt {1-c^2 x^2} \left (-\frac {-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{2 c}+\frac {x (a+b \arccos (c x))}{2 \left (1-c^2 x^2\right )}+\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {-\frac {2 b \sqrt {1-c^2 x^2} \left (i b \int e^{-i \arccos (c x)} \log \left (1-e^{i \arccos (c x)}\right )de^{i \arccos (c x)}-i b \int e^{-i \arccos (c x)} \log \left (1+e^{i \arccos (c x)}\right )de^{i \arccos (c x)}-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \arccos (c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}+\frac {(a+b \arccos (c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {2 b c \sqrt {1-c^2 x^2} \left (-\frac {-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{2 c}+\frac {x (a+b \arccos (c x))}{2 \left (1-c^2 x^2\right )}+\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {\frac {\int \frac {(a+b \arccos (c x))^2}{x \sqrt {d-c^2 d x^2}}dx}{d}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \arccos (c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {2 b c \sqrt {1-c^2 x^2} \left (-\frac {-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{2 c}+\frac {x (a+b \arccos (c x))}{2 \left (1-c^2 x^2\right )}+\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5219 |
| \(\displaystyle \frac {-\frac {\sqrt {1-c^2 x^2} \int \frac {(a+b \arccos (c x))^2}{c x}d\arccos (c x)}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \arccos (c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {2 b c \sqrt {1-c^2 x^2} \left (-\frac {-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{2 c}+\frac {x (a+b \arccos (c x))}{2 \left (1-c^2 x^2\right )}+\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\sqrt {1-c^2 x^2} \int (a+b \arccos (c x))^2 \csc \left (\arccos (c x)+\frac {\pi }{2}\right )d\arccos (c x)}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \arccos (c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {2 b c \sqrt {1-c^2 x^2} \left (-\frac {-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{2 c}+\frac {x (a+b \arccos (c x))}{2 \left (1-c^2 x^2\right )}+\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle \frac {-\frac {\sqrt {1-c^2 x^2} \left (-2 b \int (a+b \arccos (c x)) \log \left (1-i e^{i \arccos (c x)}\right )d\arccos (c x)+2 b \int (a+b \arccos (c x)) \log \left (1+i e^{i \arccos (c x)}\right )d\arccos (c x)-2 i \arctan \left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \arccos (c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {2 b c \sqrt {1-c^2 x^2} \left (-\frac {-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{2 c}+\frac {x (a+b \arccos (c x))}{2 \left (1-c^2 x^2\right )}+\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {-\frac {\sqrt {1-c^2 x^2} \left (2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-i b \int \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )d\arccos (c x)\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-i b \int \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )d\arccos (c x)\right )-2 i \arctan \left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \arccos (c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {2 b c \sqrt {1-c^2 x^2} \left (-\frac {-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{2 c}+\frac {x (a+b \arccos (c x))}{2 \left (1-c^2 x^2\right )}+\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {-\frac {\sqrt {1-c^2 x^2} \left (2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 i \arctan \left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \arccos (c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {2 b c \sqrt {1-c^2 x^2} \left (-\frac {-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{2 c}+\frac {x (a+b \arccos (c x))}{2 \left (1-c^2 x^2\right )}+\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {-\frac {\sqrt {1-c^2 x^2} \left (-2 i \arctan \left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2+2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \operatorname {PolyLog}\left (3,-i e^{i \arccos (c x)}\right )\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \operatorname {PolyLog}\left (3,i e^{i \arccos (c x)}\right )\right )\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {(a+b \arccos (c x))^2}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {2 b c \sqrt {1-c^2 x^2} \left (-\frac {-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{2 c}+\frac {x (a+b \arccos (c x))}{2 \left (1-c^2 x^2\right )}+\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \arccos (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
Input:
Int[(a + b*ArcCos[c*x])^2/(x*(d - c^2*d*x^2)^(5/2)),x]
Output:
(a + b*ArcCos[c*x])^2/(3*d*(d - c^2*d*x^2)^(3/2)) + (2*b*c*Sqrt[1 - c^2*x^ 2]*(b/(2*c*Sqrt[1 - c^2*x^2]) + (x*(a + b*ArcCos[c*x]))/(2*(1 - c^2*x^2)) - (-2*(a + b*ArcCos[c*x])*ArcTanh[E^(I*ArcCos[c*x])] + I*b*PolyLog[2, -E^( I*ArcCos[c*x])] - I*b*PolyLog[2, E^(I*ArcCos[c*x])])/(2*c)))/(3*d^2*Sqrt[d - c^2*d*x^2]) + ((a + b*ArcCos[c*x])^2/(d*Sqrt[d - c^2*d*x^2]) - (2*b*Sqr t[1 - c^2*x^2]*(-2*(a + b*ArcCos[c*x])*ArcTanh[E^(I*ArcCos[c*x])] + I*b*Po lyLog[2, -E^(I*ArcCos[c*x])] - I*b*PolyLog[2, E^(I*ArcCos[c*x])]))/(d*Sqrt [d - c^2*d*x^2]) - (Sqrt[1 - c^2*x^2]*((-2*I)*(a + b*ArcCos[c*x])^2*ArcTan [E^(I*ArcCos[c*x])] + 2*b*(I*(a + b*ArcCos[c*x])*PolyLog[2, (-I)*E^(I*ArcC os[c*x])] - b*PolyLog[3, (-I)*E^(I*ArcCos[c*x])]) - 2*b*(I*(a + b*ArcCos[c *x])*PolyLog[2, I*E^(I*ArcCos[c*x])] - b*PolyLog[3, I*E^(I*ArcCos[c*x])])) )/(d*Sqrt[d - c^2*d*x^2]))/d
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
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_.)*((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 = 0.87 (sec) , antiderivative size = 741, normalized size of antiderivative = 1.28
| method | result | size |
| default | \(\frac {a^{2}}{3 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {a^{2}}{d^{2} \sqrt {-c^{2} d \,x^{2}+d}}-\frac {a^{2} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {5}{2}}}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (3 \arccos \left (c x \right )^{2} x^{2} c^{2}-\sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) x c +c^{2} x^{2}-4 \arccos \left (c x \right )^{2}-1\right )}{3 \left (c^{2} x^{2}-1\right )^{2} d^{3}}-\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (3 \arccos \left (c x \right )^{2} \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-3 \arccos \left (c x \right )^{2} \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-6 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )+6 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )+7 \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )-7 i \operatorname {dilog}\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )-7 i \operatorname {dilog}\left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+6 \operatorname {polylog}\left (3, -i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-6 \operatorname {polylog}\left (3, i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{3 d^{3} \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (6 c^{2} x^{2} \arccos \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-8 \arccos \left (c x \right )\right )}{6 \left (c^{2} x^{2}-1\right )^{2} d^{3}}+\frac {i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (6 i \arccos \left (c x \right ) \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-6 i \arccos \left (c x \right ) \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-7 i \ln \left (i \sqrt {-c^{2} x^{2}+1}+c x -1\right )+7 i \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+6 \operatorname {dilog}\left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-6 \operatorname {dilog}\left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{6 d^{3} \left (c^{2} x^{2}-1\right )}\right )\) | \(741\) |
| parts | \(\frac {a^{2}}{3 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {a^{2}}{d^{2} \sqrt {-c^{2} d \,x^{2}+d}}-\frac {a^{2} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {5}{2}}}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (3 \arccos \left (c x \right )^{2} x^{2} c^{2}-\sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) x c +c^{2} x^{2}-4 \arccos \left (c x \right )^{2}-1\right )}{3 \left (c^{2} x^{2}-1\right )^{2} d^{3}}-\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (3 \arccos \left (c x \right )^{2} \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-3 \arccos \left (c x \right )^{2} \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-6 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )+6 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )+7 \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )-7 i \operatorname {dilog}\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )-7 i \operatorname {dilog}\left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+6 \operatorname {polylog}\left (3, -i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-6 \operatorname {polylog}\left (3, i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{3 d^{3} \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (6 c^{2} x^{2} \arccos \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-8 \arccos \left (c x \right )\right )}{6 \left (c^{2} x^{2}-1\right )^{2} d^{3}}+\frac {i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (6 i \arccos \left (c x \right ) \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-6 i \arccos \left (c x \right ) \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-7 i \ln \left (i \sqrt {-c^{2} x^{2}+1}+c x -1\right )+7 i \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+6 \operatorname {dilog}\left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-6 \operatorname {dilog}\left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{6 d^{3} \left (c^{2} x^{2}-1\right )}\right )\) | \(741\) |
Input:
int((a+b*arccos(c*x))^2/x/(-c^2*d*x^2+d)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/3*a^2/d/(-c^2*d*x^2+d)^(3/2)+a^2/d^2/(-c^2*d*x^2+d)^(1/2)-a^2/d^(5/2)*ln ((2*d+2*d^(1/2)*(-c^2*d*x^2+d)^(1/2))/x)+b^2*(-1/3*(-d*(c^2*x^2-1))^(1/2)* (3*arccos(c*x)^2*x^2*c^2-(-c^2*x^2+1)^(1/2)*arccos(c*x)*x*c+c^2*x^2-4*arcc os(c*x)^2-1)/(c^2*x^2-1)^2/d^3-1/3*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/ 2)*(3*arccos(c*x)^2*ln(1+I*(c*x+I*(-c^2*x^2+1)^(1/2)))-3*arccos(c*x)^2*ln( 1-I*(c*x+I*(-c^2*x^2+1)^(1/2)))-6*I*arccos(c*x)*polylog(2,-I*(c*x+I*(-c^2* x^2+1)^(1/2)))+6*I*arccos(c*x)*polylog(2,I*(c*x+I*(-c^2*x^2+1)^(1/2)))+7*a rccos(c*x)*ln(1+c*x+I*(-c^2*x^2+1)^(1/2))-7*I*dilog(c*x+I*(-c^2*x^2+1)^(1/ 2))-7*I*dilog(1+c*x+I*(-c^2*x^2+1)^(1/2))+6*polylog(3,-I*(c*x+I*(-c^2*x^2+ 1)^(1/2)))-6*polylog(3,I*(c*x+I*(-c^2*x^2+1)^(1/2))))/d^3/(c^2*x^2-1))+2*a *b*(-1/6*(-d*(c^2*x^2-1))^(1/2)*(6*c^2*x^2*arccos(c*x)-c*x*(-c^2*x^2+1)^(1 /2)-8*arccos(c*x))/(c^2*x^2-1)^2/d^3+1/6*I*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2 -1))^(1/2)*(6*I*arccos(c*x)*ln(1+I*(c*x+I*(-c^2*x^2+1)^(1/2)))-6*I*arccos( c*x)*ln(1-I*(c*x+I*(-c^2*x^2+1)^(1/2)))-7*I*ln(I*(-c^2*x^2+1)^(1/2)+c*x-1) +7*I*ln(1+c*x+I*(-c^2*x^2+1)^(1/2))+6*dilog(1+I*(c*x+I*(-c^2*x^2+1)^(1/2)) )-6*dilog(1-I*(c*x+I*(-c^2*x^2+1)^(1/2))))/d^3/(c^2*x^2-1))
\[ \int \frac {(a+b \arccos (c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x} \,d x } \] Input:
integrate((a+b*arccos(c*x))^2/x/(-c^2*d*x^2+d)^(5/2),x, algorithm="fricas" )
Output:
integral(-sqrt(-c^2*d*x^2 + d)*(b^2*arccos(c*x)^2 + 2*a*b*arccos(c*x) + a^ 2)/(c^6*d^3*x^7 - 3*c^4*d^3*x^5 + 3*c^2*d^3*x^3 - d^3*x), x)
\[ \int \frac {(a+b \arccos (c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {\left (a + b \operatorname {acos}{\left (c x \right )}\right )^{2}}{x \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((a+b*acos(c*x))**2/x/(-c**2*d*x**2+d)**(5/2),x)
Output:
Integral((a + b*acos(c*x))**2/(x*(-d*(c*x - 1)*(c*x + 1))**(5/2)), x)
\[ \int \frac {(a+b \arccos (c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x} \,d x } \] Input:
integrate((a+b*arccos(c*x))^2/x/(-c^2*d*x^2+d)^(5/2),x, algorithm="maxima" )
Output:
-1/3*a^2*(3*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x))/d^(5/2 ) - 3/(sqrt(-c^2*d*x^2 + d)*d^2) - 1/((-c^2*d*x^2 + d)^(3/2)*d)) - sqrt(d) *integrate((b^2*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^2 + 2*a*b*arcta n2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x))*sqrt(c*x + 1)*sqrt(-c*x + 1)/(c^6*d ^3*x^7 - 3*c^4*d^3*x^5 + 3*c^2*d^3*x^3 - d^3*x), x)
Exception generated. \[ \int \frac {(a+b \arccos (c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((a+b*arccos(c*x))^2/x/(-c^2*d*x^2+d)^(5/2),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. \[ \int \frac {(a+b \arccos (c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2}{x\,{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \] Input:
int((a + b*acos(c*x))^2/(x*(d - c^2*d*x^2)^(5/2)),x)
Output:
int((a + b*acos(c*x))^2/(x*(d - c^2*d*x^2)^(5/2)), x)
\[ \int \frac {(a+b \arccos (c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {6 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acos} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{5}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{3}+\sqrt {-c^{2} x^{2}+1}\, x}d x \right ) a b \,c^{2} x^{2}-6 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acos} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{5}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{3}+\sqrt {-c^{2} x^{2}+1}\, x}d x \right ) a b +3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acos} \left (c x \right )^{2}}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{5}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{3}+\sqrt {-c^{2} x^{2}+1}\, x}d x \right ) b^{2} c^{2} x^{2}-3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acos} \left (c x \right )^{2}}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{5}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{3}+\sqrt {-c^{2} x^{2}+1}\, x}d x \right ) b^{2}+3 \sqrt {-c^{2} x^{2}+1}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right )\right ) a^{2} c^{2} x^{2}-3 \sqrt {-c^{2} x^{2}+1}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right )\right ) a^{2}-4 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{2} x^{2}+4 \sqrt {-c^{2} x^{2}+1}\, a^{2}+3 a^{2} c^{2} x^{2}-4 a^{2}}{3 \sqrt {d}\, \sqrt {-c^{2} x^{2}+1}\, d^{2} \left (c^{2} x^{2}-1\right )} \] Input:
int((a+b*acos(c*x))^2/x/(-c^2*d*x^2+d)^(5/2),x)
Output:
(6*sqrt( - c**2*x**2 + 1)*int(acos(c*x)/(sqrt( - c**2*x**2 + 1)*c**4*x**5 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**3 + sqrt( - c**2*x**2 + 1)*x),x)*a*b*c* *2*x**2 - 6*sqrt( - c**2*x**2 + 1)*int(acos(c*x)/(sqrt( - c**2*x**2 + 1)*c **4*x**5 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**3 + sqrt( - c**2*x**2 + 1)*x), x)*a*b + 3*sqrt( - c**2*x**2 + 1)*int(acos(c*x)**2/(sqrt( - c**2*x**2 + 1) *c**4*x**5 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**3 + sqrt( - c**2*x**2 + 1)*x ),x)*b**2*c**2*x**2 - 3*sqrt( - c**2*x**2 + 1)*int(acos(c*x)**2/(sqrt( - c **2*x**2 + 1)*c**4*x**5 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**3 + sqrt( - c** 2*x**2 + 1)*x),x)*b**2 + 3*sqrt( - c**2*x**2 + 1)*log(tan(asin(c*x)/2))*a* *2*c**2*x**2 - 3*sqrt( - c**2*x**2 + 1)*log(tan(asin(c*x)/2))*a**2 - 4*sqr t( - c**2*x**2 + 1)*a**2*c**2*x**2 + 4*sqrt( - c**2*x**2 + 1)*a**2 + 3*a** 2*c**2*x**2 - 4*a**2)/(3*sqrt(d)*sqrt( - c**2*x**2 + 1)*d**2*(c**2*x**2 - 1))