Integrand size = 27, antiderivative size = 270 \[ \int \frac {(a+b \arccos (c x))^2}{x^3 \left (d-c^2 d x^2\right )^2} \, dx=-\frac {b c (a+b \arccos (c x))}{d^2 x \sqrt {1-c^2 x^2}}+\frac {c^2 (a+b \arccos (c x))^2}{d^2 \left (1-c^2 x^2\right )}-\frac {(a+b \arccos (c x))^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}-\frac {4 c^2 (a+b \arccos (c x))^2 \text {arctanh}\left (e^{2 i \arccos (c x)}\right )}{d^2}+\frac {b^2 c^2 \log (x)}{d^2}-\frac {b^2 c^2 \log \left (1-c^2 x^2\right )}{2 d^2}+\frac {2 i b c^2 (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )}{d^2}-\frac {2 i b c^2 (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )}{d^2}-\frac {b^2 c^2 \operatorname {PolyLog}\left (3,-e^{2 i \arccos (c x)}\right )}{d^2}+\frac {b^2 c^2 \operatorname {PolyLog}\left (3,e^{2 i \arccos (c x)}\right )}{d^2} \] Output:
-b*c*(a+b*arccos(c*x))/d^2/x/(-c^2*x^2+1)^(1/2)+c^2*(a+b*arccos(c*x))^2/d^ 2/(-c^2*x^2+1)-1/2*(a+b*arccos(c*x))^2/d^2/x^2/(-c^2*x^2+1)-4*c^2*(a+b*arc cos(c*x))^2*arctanh((c*x+I*(-c^2*x^2+1)^(1/2))^2)/d^2+b^2*c^2*ln(x)/d^2-1/ 2*b^2*c^2*ln(-c^2*x^2+1)/d^2+2*I*b*c^2*(a+b*arccos(c*x))*polylog(2,-(c*x+I *(-c^2*x^2+1)^(1/2))^2)/d^2-2*I*b*c^2*(a+b*arccos(c*x))*polylog(2,(c*x+I*( -c^2*x^2+1)^(1/2))^2)/d^2-b^2*c^2*polylog(3,-(c*x+I*(-c^2*x^2+1)^(1/2))^2) /d^2+b^2*c^2*polylog(3,(c*x+I*(-c^2*x^2+1)^(1/2))^2)/d^2
Time = 1.49 (sec) , antiderivative size = 529, normalized size of antiderivative = 1.96 \[ \int \frac {(a+b \arccos (c x))^2}{x^3 \left (d-c^2 d x^2\right )^2} \, dx=\frac {-\frac {a^2}{x^2}+\frac {a^2 c^2}{1-c^2 x^2}+4 a^2 c^2 \log (x)-2 a^2 c^2 \log \left (1-c^2 x^2\right )+a b \left (\frac {2 c \sqrt {1-c^2 x^2}}{x}+\frac {c^2 \sqrt {1-c^2 x^2}}{1-c x}-\frac {c^2 \sqrt {1-c^2 x^2}}{1+c x}-\frac {2 \arccos (c x)}{x^2}+\frac {c^2 \arccos (c x)}{1-c x}+\frac {c^2 \arccos (c x)}{1+c x}-8 c^2 \arccos (c x) \log \left (1-e^{i \arccos (c x)}\right )-8 c^2 \arccos (c x) \log \left (1+e^{i \arccos (c x)}\right )+8 c^2 \arccos (c x) \log \left (1+e^{2 i \arccos (c x)}\right )+8 i c^2 \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )+8 i c^2 \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )-4 i c^2 \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )\right )+b^2 c^2 \left (\frac {2 c x \arccos (c x)}{\sqrt {1-c^2 x^2}}+\frac {2 \sqrt {1-c^2 x^2} \arccos (c x)}{c x}-\frac {\arccos (c x)^2}{c^2 x^2}+\frac {\arccos (c x)^2}{1-c^2 x^2}-4 \arccos (c x)^2 \left (\log \left (1-e^{2 i \arccos (c x)}\right )-\log \left (1+e^{2 i \arccos (c x)}\right )\right )-2 \log \left (\frac {\sqrt {1-c^2 x^2}}{c x}\right )-4 i \arccos (c x) \left (\operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )-\operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )\right )+2 \left (\operatorname {PolyLog}\left (3,-e^{2 i \arccos (c x)}\right )-\operatorname {PolyLog}\left (3,e^{2 i \arccos (c x)}\right )\right )\right )}{2 d^2} \] Input:
Integrate[(a + b*ArcCos[c*x])^2/(x^3*(d - c^2*d*x^2)^2),x]
Output:
(-(a^2/x^2) + (a^2*c^2)/(1 - c^2*x^2) + 4*a^2*c^2*Log[x] - 2*a^2*c^2*Log[1 - c^2*x^2] + a*b*((2*c*Sqrt[1 - c^2*x^2])/x + (c^2*Sqrt[1 - c^2*x^2])/(1 - c*x) - (c^2*Sqrt[1 - c^2*x^2])/(1 + c*x) - (2*ArcCos[c*x])/x^2 + (c^2*Ar cCos[c*x])/(1 - c*x) + (c^2*ArcCos[c*x])/(1 + c*x) - 8*c^2*ArcCos[c*x]*Log [1 - E^(I*ArcCos[c*x])] - 8*c^2*ArcCos[c*x]*Log[1 + E^(I*ArcCos[c*x])] + 8 *c^2*ArcCos[c*x]*Log[1 + E^((2*I)*ArcCos[c*x])] + (8*I)*c^2*PolyLog[2, -E^ (I*ArcCos[c*x])] + (8*I)*c^2*PolyLog[2, E^(I*ArcCos[c*x])] - (4*I)*c^2*Pol yLog[2, -E^((2*I)*ArcCos[c*x])]) + b^2*c^2*((2*c*x*ArcCos[c*x])/Sqrt[1 - c ^2*x^2] + (2*Sqrt[1 - c^2*x^2]*ArcCos[c*x])/(c*x) - ArcCos[c*x]^2/(c^2*x^2 ) + ArcCos[c*x]^2/(1 - c^2*x^2) - 4*ArcCos[c*x]^2*(Log[1 - E^((2*I)*ArcCos [c*x])] - Log[1 + E^((2*I)*ArcCos[c*x])]) - 2*Log[Sqrt[1 - c^2*x^2]/(c*x)] - (4*I)*ArcCos[c*x]*(PolyLog[2, -E^((2*I)*ArcCos[c*x])] - PolyLog[2, E^(( 2*I)*ArcCos[c*x])]) + 2*(PolyLog[3, -E^((2*I)*ArcCos[c*x])] - PolyLog[3, E ^((2*I)*ArcCos[c*x])])))/(2*d^2)
Time = 1.99 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.20, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.630, Rules used = {5205, 27, 5195, 25, 354, 86, 2009, 5209, 5161, 240, 5185, 4919, 3042, 4671, 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.
| \(\displaystyle \int \frac {(a+b \arccos (c x))^2}{x^3 \left (d-c^2 d x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 5205 |
| \(\displaystyle 2 c^2 \int \frac {(a+b \arccos (c x))^2}{d^2 x \left (1-c^2 x^2\right )^2}dx-\frac {b c \int \frac {a+b \arccos (c x)}{x^2 \left (1-c^2 x^2\right )^{3/2}}dx}{d^2}-\frac {(a+b \arccos (c x))^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 c^2 \int \frac {(a+b \arccos (c x))^2}{x \left (1-c^2 x^2\right )^2}dx}{d^2}-\frac {b c \int \frac {a+b \arccos (c x)}{x^2 \left (1-c^2 x^2\right )^{3/2}}dx}{d^2}-\frac {(a+b \arccos (c x))^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 5195 |
| \(\displaystyle \frac {2 c^2 \int \frac {(a+b \arccos (c x))^2}{x \left (1-c^2 x^2\right )^2}dx}{d^2}-\frac {b c \left (b c \int -\frac {1-2 c^2 x^2}{x \left (1-c^2 x^2\right )}dx+\frac {2 c^2 x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}-\frac {a+b \arccos (c x)}{x \sqrt {1-c^2 x^2}}\right )}{d^2}-\frac {(a+b \arccos (c x))^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 c^2 \int \frac {(a+b \arccos (c x))^2}{x \left (1-c^2 x^2\right )^2}dx}{d^2}-\frac {b c \left (-b c \int \frac {1-2 c^2 x^2}{x \left (1-c^2 x^2\right )}dx+\frac {2 c^2 x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}-\frac {a+b \arccos (c x)}{x \sqrt {1-c^2 x^2}}\right )}{d^2}-\frac {(a+b \arccos (c x))^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {2 c^2 \int \frac {(a+b \arccos (c x))^2}{x \left (1-c^2 x^2\right )^2}dx}{d^2}-\frac {b c \left (-\frac {1}{2} b c \int \frac {1-2 c^2 x^2}{x^2 \left (1-c^2 x^2\right )}dx^2+\frac {2 c^2 x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}-\frac {a+b \arccos (c x)}{x \sqrt {1-c^2 x^2}}\right )}{d^2}-\frac {(a+b \arccos (c x))^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {2 c^2 \int \frac {(a+b \arccos (c x))^2}{x \left (1-c^2 x^2\right )^2}dx}{d^2}-\frac {b c \left (-\frac {1}{2} b c \int \left (\frac {c^2}{c^2 x^2-1}+\frac {1}{x^2}\right )dx^2+\frac {2 c^2 x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}-\frac {a+b \arccos (c x)}{x \sqrt {1-c^2 x^2}}\right )}{d^2}-\frac {(a+b \arccos (c x))^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 c^2 \int \frac {(a+b \arccos (c x))^2}{x \left (1-c^2 x^2\right )^2}dx}{d^2}-\frac {(a+b \arccos (c x))^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}-\frac {b c \left (\frac {2 c^2 x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}-\frac {a+b \arccos (c x)}{x \sqrt {1-c^2 x^2}}-\frac {1}{2} b c \left (\log \left (1-c^2 x^2\right )+\log \left (x^2\right )\right )\right )}{d^2}\) |
| \(\Big \downarrow \) 5209 |
| \(\displaystyle \frac {2 c^2 \left (b c \int \frac {a+b \arccos (c x)}{\left (1-c^2 x^2\right )^{3/2}}dx+\int \frac {(a+b \arccos (c x))^2}{x \left (1-c^2 x^2\right )}dx+\frac {(a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{d^2}-\frac {(a+b \arccos (c x))^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}-\frac {b c \left (\frac {2 c^2 x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}-\frac {a+b \arccos (c x)}{x \sqrt {1-c^2 x^2}}-\frac {1}{2} b c \left (\log \left (1-c^2 x^2\right )+\log \left (x^2\right )\right )\right )}{d^2}\) |
| \(\Big \downarrow \) 5161 |
| \(\displaystyle \frac {2 c^2 \left (b c \left (b c \int \frac {x}{1-c^2 x^2}dx+\frac {x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}\right )+\int \frac {(a+b \arccos (c x))^2}{x \left (1-c^2 x^2\right )}dx+\frac {(a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{d^2}-\frac {(a+b \arccos (c x))^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}-\frac {b c \left (\frac {2 c^2 x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}-\frac {a+b \arccos (c x)}{x \sqrt {1-c^2 x^2}}-\frac {1}{2} b c \left (\log \left (1-c^2 x^2\right )+\log \left (x^2\right )\right )\right )}{d^2}\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle \frac {2 c^2 \left (\int \frac {(a+b \arccos (c x))^2}{x \left (1-c^2 x^2\right )}dx+\frac {(a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}+b c \left (\frac {x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}-\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )\right )}{d^2}-\frac {(a+b \arccos (c x))^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}-\frac {b c \left (\frac {2 c^2 x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}-\frac {a+b \arccos (c x)}{x \sqrt {1-c^2 x^2}}-\frac {1}{2} b c \left (\log \left (1-c^2 x^2\right )+\log \left (x^2\right )\right )\right )}{d^2}\) |
| \(\Big \downarrow \) 5185 |
| \(\displaystyle \frac {2 c^2 \left (-\int \frac {(a+b \arccos (c x))^2}{c x \sqrt {1-c^2 x^2}}d\arccos (c x)+\frac {(a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}+b c \left (\frac {x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}-\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )\right )}{d^2}-\frac {(a+b \arccos (c x))^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}-\frac {b c \left (\frac {2 c^2 x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}-\frac {a+b \arccos (c x)}{x \sqrt {1-c^2 x^2}}-\frac {1}{2} b c \left (\log \left (1-c^2 x^2\right )+\log \left (x^2\right )\right )\right )}{d^2}\) |
| \(\Big \downarrow \) 4919 |
| \(\displaystyle \frac {2 c^2 \left (-2 \int (a+b \arccos (c x))^2 \csc (2 \arccos (c x))d\arccos (c x)+\frac {(a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}+b c \left (\frac {x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}-\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )\right )}{d^2}-\frac {(a+b \arccos (c x))^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}-\frac {b c \left (\frac {2 c^2 x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}-\frac {a+b \arccos (c x)}{x \sqrt {1-c^2 x^2}}-\frac {1}{2} b c \left (\log \left (1-c^2 x^2\right )+\log \left (x^2\right )\right )\right )}{d^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 c^2 \left (-2 \int (a+b \arccos (c x))^2 \csc (2 \arccos (c x))d\arccos (c x)+\frac {(a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}+b c \left (\frac {x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}-\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )\right )}{d^2}-\frac {(a+b \arccos (c x))^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}-\frac {b c \left (\frac {2 c^2 x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}-\frac {a+b \arccos (c x)}{x \sqrt {1-c^2 x^2}}-\frac {1}{2} b c \left (\log \left (1-c^2 x^2\right )+\log \left (x^2\right )\right )\right )}{d^2}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle \frac {2 c^2 \left (-2 \left (-b \int (a+b \arccos (c x)) \log \left (1-e^{2 i \arccos (c x)}\right )d\arccos (c x)+b \int (a+b \arccos (c x)) \log \left (1+e^{2 i \arccos (c x)}\right )d\arccos (c x)-\text {arctanh}\left (e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))^2\right )+\frac {(a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}+b c \left (\frac {x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}-\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )\right )}{d^2}-\frac {(a+b \arccos (c x))^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}-\frac {b c \left (\frac {2 c^2 x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}-\frac {a+b \arccos (c x)}{x \sqrt {1-c^2 x^2}}-\frac {1}{2} b c \left (\log \left (1-c^2 x^2\right )+\log \left (x^2\right )\right )\right )}{d^2}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {2 c^2 \left (-2 \left (b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))-\frac {1}{2} i b \int \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )d\arccos (c x)\right )-b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))-\frac {1}{2} i b \int \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )d\arccos (c x)\right )-\text {arctanh}\left (e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))^2\right )+\frac {(a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}+b c \left (\frac {x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}-\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )\right )}{d^2}-\frac {(a+b \arccos (c x))^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}-\frac {b c \left (\frac {2 c^2 x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}-\frac {a+b \arccos (c x)}{x \sqrt {1-c^2 x^2}}-\frac {1}{2} b c \left (\log \left (1-c^2 x^2\right )+\log \left (x^2\right )\right )\right )}{d^2}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {2 c^2 \left (-2 \left (b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))-\frac {1}{4} b \int e^{-2 i \arccos (c x)} \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )de^{2 i \arccos (c x)}\right )-b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))-\frac {1}{4} b \int e^{-2 i \arccos (c x)} \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )de^{2 i \arccos (c x)}\right )-\text {arctanh}\left (e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))^2\right )+\frac {(a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}+b c \left (\frac {x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}-\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )\right )}{d^2}-\frac {(a+b \arccos (c x))^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}-\frac {b c \left (\frac {2 c^2 x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}-\frac {a+b \arccos (c x)}{x \sqrt {1-c^2 x^2}}-\frac {1}{2} b c \left (\log \left (1-c^2 x^2\right )+\log \left (x^2\right )\right )\right )}{d^2}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {2 c^2 \left (-2 \left (-\text {arctanh}\left (e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))^2+b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (3,-e^{2 i \arccos (c x)}\right )\right )-b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (3,e^{2 i \arccos (c x)}\right )\right )\right )+\frac {(a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}+b c \left (\frac {x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}-\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )\right )}{d^2}-\frac {(a+b \arccos (c x))^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}-\frac {b c \left (\frac {2 c^2 x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}-\frac {a+b \arccos (c x)}{x \sqrt {1-c^2 x^2}}-\frac {1}{2} b c \left (\log \left (1-c^2 x^2\right )+\log \left (x^2\right )\right )\right )}{d^2}\) |
Input:
Int[(a + b*ArcCos[c*x])^2/(x^3*(d - c^2*d*x^2)^2),x]
Output:
-1/2*(a + b*ArcCos[c*x])^2/(d^2*x^2*(1 - c^2*x^2)) - (b*c*(-((a + b*ArcCos [c*x])/(x*Sqrt[1 - c^2*x^2])) + (2*c^2*x*(a + b*ArcCos[c*x]))/Sqrt[1 - c^2 *x^2] - (b*c*(Log[x^2] + Log[1 - c^2*x^2]))/2))/d^2 + (2*c^2*((a + b*ArcCo s[c*x])^2/(2*(1 - c^2*x^2)) + b*c*((x*(a + b*ArcCos[c*x]))/Sqrt[1 - c^2*x^ 2] - (b*Log[1 - c^2*x^2])/(2*c)) - 2*(-((a + b*ArcCos[c*x])^2*ArcTanh[E^(( 2*I)*ArcCos[c*x])]) + b*((I/2)*(a + b*ArcCos[c*x])*PolyLog[2, -E^((2*I)*Ar cCos[c*x])] - (b*PolyLog[3, -E^((2*I)*ArcCos[c*x])])/4) - b*((I/2)*(a + b* ArcCos[c*x])*PolyLog[2, E^((2*I)*ArcCos[c*x])] - (b*PolyLog[3, E^((2*I)*Ar cCos[c*x])])/4))))/d^2
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_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
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[Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n Int[(c + d*x)^m*Csc[2*a + 2*b*x]^n , x], x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[n] && RationalQ[m]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x _Symbol] :> Simp[x*((a + b*ArcCos[c*x])^n/(d*Sqrt[d + e*x^2])), x] + Simp[b *c*(n/d)*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]] Int[x*((a + b*ArcCos[c*x ])^(n - 1)/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[-d^(-1) Subst[Int[(a + b*x)^n/(Cos[x]*Sin[x]), x], x, A rcCos[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_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_) , x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcCos [c*x]) u, x] + Simp[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int[Sim plifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 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/(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[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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 672 vs. \(2 (317 ) = 634\).
Time = 0.71 (sec) , antiderivative size = 673, normalized size of antiderivative = 2.49
| method | result | size |
| derivativedivides | \(c^{2} \left (\frac {a^{2} \left (-\frac {1}{4 \left (c x -1\right )}-\ln \left (c x -1\right )-\frac {1}{2 c^{2} x^{2}}+2 \ln \left (c x \right )+\frac {1}{4 c x +4}-\ln \left (c x +1\right )\right )}{d^{2}}+\frac {b^{2} \left (-\frac {\arccos \left (c x \right ) \left (2 c^{2} x^{2} \arccos \left (c x \right )+2 c x \sqrt {-c^{2} x^{2}+1}-\arccos \left (c x \right )\right )}{2 \left (c^{2} x^{2}-1\right ) c^{2} x^{2}}+\ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )-\ln \left (i \sqrt {-c^{2} x^{2}+1}+c x -1\right )+2 \arccos \left (c x \right )^{2} \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-2 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\operatorname {polylog}\left (3, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-2 \arccos \left (c x \right )^{2} \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+4 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )-4 \operatorname {polylog}\left (3, -c x -i \sqrt {-c^{2} x^{2}+1}\right )-2 \arccos \left (c x \right )^{2} \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )+4 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )-4 \operatorname {polylog}\left (3, c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {2 a b \left (-\frac {2 c^{2} x^{2} \arccos \left (c x \right )+c x \sqrt {-c^{2} x^{2}+1}-\arccos \left (c x \right )}{2 \left (c^{2} x^{2}-1\right ) c^{2} x^{2}}+2 \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-2 \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )-2 \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+2 i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )+2 i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}\right )\) | \(673\) |
| default | \(c^{2} \left (\frac {a^{2} \left (-\frac {1}{4 \left (c x -1\right )}-\ln \left (c x -1\right )-\frac {1}{2 c^{2} x^{2}}+2 \ln \left (c x \right )+\frac {1}{4 c x +4}-\ln \left (c x +1\right )\right )}{d^{2}}+\frac {b^{2} \left (-\frac {\arccos \left (c x \right ) \left (2 c^{2} x^{2} \arccos \left (c x \right )+2 c x \sqrt {-c^{2} x^{2}+1}-\arccos \left (c x \right )\right )}{2 \left (c^{2} x^{2}-1\right ) c^{2} x^{2}}+\ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )-\ln \left (i \sqrt {-c^{2} x^{2}+1}+c x -1\right )+2 \arccos \left (c x \right )^{2} \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-2 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\operatorname {polylog}\left (3, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-2 \arccos \left (c x \right )^{2} \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+4 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )-4 \operatorname {polylog}\left (3, -c x -i \sqrt {-c^{2} x^{2}+1}\right )-2 \arccos \left (c x \right )^{2} \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )+4 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )-4 \operatorname {polylog}\left (3, c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {2 a b \left (-\frac {2 c^{2} x^{2} \arccos \left (c x \right )+c x \sqrt {-c^{2} x^{2}+1}-\arccos \left (c x \right )}{2 \left (c^{2} x^{2}-1\right ) c^{2} x^{2}}+2 \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-2 \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )-2 \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+2 i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )+2 i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}\right )\) | \(673\) |
| parts | \(\frac {a^{2} \left (-\frac {1}{2 x^{2}}+2 c^{2} \ln \left (x \right )-\frac {c^{2}}{4 \left (c x -1\right )}-c^{2} \ln \left (c x -1\right )+\frac {c^{2}}{4 c x +4}-c^{2} \ln \left (c x +1\right )\right )}{d^{2}}+\frac {b^{2} c^{2} \left (-\frac {\arccos \left (c x \right ) \left (2 c^{2} x^{2} \arccos \left (c x \right )+2 c x \sqrt {-c^{2} x^{2}+1}-\arccos \left (c x \right )\right )}{2 \left (c^{2} x^{2}-1\right ) c^{2} x^{2}}+\ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )-\ln \left (i \sqrt {-c^{2} x^{2}+1}+c x -1\right )+2 \arccos \left (c x \right )^{2} \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-2 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\operatorname {polylog}\left (3, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-2 \arccos \left (c x \right )^{2} \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+4 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )-4 \operatorname {polylog}\left (3, -c x -i \sqrt {-c^{2} x^{2}+1}\right )-2 \arccos \left (c x \right )^{2} \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )+4 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )-4 \operatorname {polylog}\left (3, c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {2 a b \,c^{2} \left (-\frac {2 c^{2} x^{2} \arccos \left (c x \right )+c x \sqrt {-c^{2} x^{2}+1}-\arccos \left (c x \right )}{2 \left (c^{2} x^{2}-1\right ) c^{2} x^{2}}+2 \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-2 \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )-2 \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+2 i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )+2 i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}\) | \(685\) |
Input:
int((a+b*arccos(c*x))^2/x^3/(-c^2*d*x^2+d)^2,x,method=_RETURNVERBOSE)
Output:
c^2*(a^2/d^2*(-1/4/(c*x-1)-ln(c*x-1)-1/2/c^2/x^2+2*ln(c*x)+1/4/(c*x+1)-ln( c*x+1))+b^2/d^2*(-1/2/(c^2*x^2-1)/c^2/x^2*arccos(c*x)*(2*c^2*x^2*arccos(c* x)+2*c*x*(-c^2*x^2+1)^(1/2)-arccos(c*x))+ln(1+(c*x+I*(-c^2*x^2+1)^(1/2))^2 )-ln(1+c*x+I*(-c^2*x^2+1)^(1/2))-ln(I*(-c^2*x^2+1)^(1/2)+c*x-1)+2*arccos(c *x)^2*ln(1+(c*x+I*(-c^2*x^2+1)^(1/2))^2)-2*I*arccos(c*x)*polylog(2,-(c*x+I *(-c^2*x^2+1)^(1/2))^2)+polylog(3,-(c*x+I*(-c^2*x^2+1)^(1/2))^2)-2*arccos( c*x)^2*ln(1+c*x+I*(-c^2*x^2+1)^(1/2))+4*I*arccos(c*x)*polylog(2,-c*x-I*(-c ^2*x^2+1)^(1/2))-4*polylog(3,-c*x-I*(-c^2*x^2+1)^(1/2))-2*arccos(c*x)^2*ln (1-c*x-I*(-c^2*x^2+1)^(1/2))+4*I*arccos(c*x)*polylog(2,c*x+I*(-c^2*x^2+1)^ (1/2))-4*polylog(3,c*x+I*(-c^2*x^2+1)^(1/2)))+2*a*b/d^2*(-1/2*(2*c^2*x^2*a rccos(c*x)+c*x*(-c^2*x^2+1)^(1/2)-arccos(c*x))/(c^2*x^2-1)/c^2/x^2+2*arcco s(c*x)*ln(1+(c*x+I*(-c^2*x^2+1)^(1/2))^2)-2*arccos(c*x)*ln(1+c*x+I*(-c^2*x ^2+1)^(1/2))-2*arccos(c*x)*ln(1-c*x-I*(-c^2*x^2+1)^(1/2))-I*polylog(2,-(c* x+I*(-c^2*x^2+1)^(1/2))^2)+2*I*polylog(2,-c*x-I*(-c^2*x^2+1)^(1/2))+2*I*po lylog(2,c*x+I*(-c^2*x^2+1)^(1/2))))
\[ \int \frac {(a+b \arccos (c x))^2}{x^3 \left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{2} x^{3}} \,d x } \] Input:
integrate((a+b*arccos(c*x))^2/x^3/(-c^2*d*x^2+d)^2,x, algorithm="fricas")
Output:
integral((b^2*arccos(c*x)^2 + 2*a*b*arccos(c*x) + a^2)/(c^4*d^2*x^7 - 2*c^ 2*d^2*x^5 + d^2*x^3), x)
\[ \int \frac {(a+b \arccos (c x))^2}{x^3 \left (d-c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a^{2}}{c^{4} x^{7} - 2 c^{2} x^{5} + x^{3}}\, dx + \int \frac {b^{2} \operatorname {acos}^{2}{\left (c x \right )}}{c^{4} x^{7} - 2 c^{2} x^{5} + x^{3}}\, dx + \int \frac {2 a b \operatorname {acos}{\left (c x \right )}}{c^{4} x^{7} - 2 c^{2} x^{5} + x^{3}}\, dx}{d^{2}} \] Input:
integrate((a+b*acos(c*x))**2/x**3/(-c**2*d*x**2+d)**2,x)
Output:
(Integral(a**2/(c**4*x**7 - 2*c**2*x**5 + x**3), x) + Integral(b**2*acos(c *x)**2/(c**4*x**7 - 2*c**2*x**5 + x**3), x) + Integral(2*a*b*acos(c*x)/(c* *4*x**7 - 2*c**2*x**5 + x**3), x))/d**2
\[ \int \frac {(a+b \arccos (c x))^2}{x^3 \left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{2} x^{3}} \,d x } \] Input:
integrate((a+b*arccos(c*x))^2/x^3/(-c^2*d*x^2+d)^2,x, algorithm="maxima")
Output:
-1/2*a^2*(2*c^2*log(c*x + 1)/d^2 + 2*c^2*log(c*x - 1)/d^2 - 4*c^2*log(x)/d ^2 + (2*c^2*x^2 - 1)/(c^2*d^2*x^4 - d^2*x^2)) + integrate((b^2*arctan2(sqr t(c*x + 1)*sqrt(-c*x + 1), c*x)^2 + 2*a*b*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x))/(c^4*d^2*x^7 - 2*c^2*d^2*x^5 + d^2*x^3), x)
Exception generated. \[ \int \frac {(a+b \arccos (c x))^2}{x^3 \left (d-c^2 d x^2\right )^2} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((a+b*arccos(c*x))^2/x^3/(-c^2*d*x^2+d)^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. \[ \int \frac {(a+b \arccos (c x))^2}{x^3 \left (d-c^2 d x^2\right )^2} \, dx=\int \frac {{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2}{x^3\,{\left (d-c^2\,d\,x^2\right )}^2} \,d x \] Input:
int((a + b*acos(c*x))^2/(x^3*(d - c^2*d*x^2)^2),x)
Output:
int((a + b*acos(c*x))^2/(x^3*(d - c^2*d*x^2)^2), x)
\[ \int \frac {(a+b \arccos (c x))^2}{x^3 \left (d-c^2 d x^2\right )^2} \, dx=\frac {4 \left (\int \frac {\mathit {acos} \left (c x \right )}{c^{4} x^{7}-2 c^{2} x^{5}+x^{3}}d x \right ) a b \,c^{2} x^{4}-4 \left (\int \frac {\mathit {acos} \left (c x \right )}{c^{4} x^{7}-2 c^{2} x^{5}+x^{3}}d x \right ) a b \,x^{2}+2 \left (\int \frac {\mathit {acos} \left (c x \right )^{2}}{c^{4} x^{7}-2 c^{2} x^{5}+x^{3}}d x \right ) b^{2} c^{2} x^{4}-2 \left (\int \frac {\mathit {acos} \left (c x \right )^{2}}{c^{4} x^{7}-2 c^{2} x^{5}+x^{3}}d x \right ) b^{2} x^{2}-2 \,\mathrm {log}\left (c^{2} x -c \right ) a^{2} c^{4} x^{4}+2 \,\mathrm {log}\left (c^{2} x -c \right ) a^{2} c^{2} x^{2}-2 \,\mathrm {log}\left (c^{2} x +c \right ) a^{2} c^{4} x^{4}+2 \,\mathrm {log}\left (c^{2} x +c \right ) a^{2} c^{2} x^{2}+4 \,\mathrm {log}\left (x \right ) a^{2} c^{4} x^{4}-4 \,\mathrm {log}\left (x \right ) a^{2} c^{2} x^{2}-2 a^{2} c^{4} x^{4}+a^{2}}{2 d^{2} x^{2} \left (c^{2} x^{2}-1\right )} \] Input:
int((a+b*acos(c*x))^2/x^3/(-c^2*d*x^2+d)^2,x)
Output:
(4*int(acos(c*x)/(c**4*x**7 - 2*c**2*x**5 + x**3),x)*a*b*c**2*x**4 - 4*int (acos(c*x)/(c**4*x**7 - 2*c**2*x**5 + x**3),x)*a*b*x**2 + 2*int(acos(c*x)* *2/(c**4*x**7 - 2*c**2*x**5 + x**3),x)*b**2*c**2*x**4 - 2*int(acos(c*x)**2 /(c**4*x**7 - 2*c**2*x**5 + x**3),x)*b**2*x**2 - 2*log(c**2*x - c)*a**2*c* *4*x**4 + 2*log(c**2*x - c)*a**2*c**2*x**2 - 2*log(c**2*x + c)*a**2*c**4*x **4 + 2*log(c**2*x + c)*a**2*c**2*x**2 + 4*log(x)*a**2*c**4*x**4 - 4*log(x )*a**2*c**2*x**2 - 2*a**2*c**4*x**4 + a**2)/(2*d**2*x**2*(c**2*x**2 - 1))