Integrand size = 27, antiderivative size = 333 \[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )} \, dx=-\frac {b^2 c^2}{3 d x}-\frac {b c \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 d x^2}-\frac {(a+b \arcsin (c x))^2}{3 d x^3}-\frac {c^2 (a+b \arcsin (c x))^2}{d x}-\frac {2 i c^3 (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{d}-\frac {14 b c^3 (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{3 d}+\frac {7 i b^2 c^3 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{3 d}+\frac {2 i b c^3 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{d}-\frac {2 i b c^3 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{d}-\frac {7 i b^2 c^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{3 d}-\frac {2 b^2 c^3 \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )}{d}+\frac {2 b^2 c^3 \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )}{d} \]
-1/3*b^2*c^2/d/x-1/3*(a+b*arcsin(c*x))^2/d/x^3-c^2*(a+b*arcsin(c*x))^2/d/x -2*I*c^3*(a+b*arcsin(c*x))^2*arctan(I*c*x+(-c^2*x^2+1)^(1/2))/d-14/3*b*c^3 *(a+b*arcsin(c*x))*arctanh(I*c*x+(-c^2*x^2+1)^(1/2))/d+7/3*I*b^2*c^3*polyl og(2,-I*c*x-(-c^2*x^2+1)^(1/2))/d+2*I*b*c^3*(a+b*arcsin(c*x))*polylog(2,-I *(I*c*x+(-c^2*x^2+1)^(1/2)))/d-2*I*b*c^3*(a+b*arcsin(c*x))*polylog(2,I*(I* c*x+(-c^2*x^2+1)^(1/2)))/d-7/3*I*b^2*c^3*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2 ))/d-2*b^2*c^3*polylog(3,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))/d+2*b^2*c^3*polylo g(3,I*(I*c*x+(-c^2*x^2+1)^(1/2)))/d-1/3*b*c*(a+b*arcsin(c*x))*(-c^2*x^2+1) ^(1/2)/d/x^2
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(849\) vs. \(2(333)=666\).
Time = 7.44 (sec) , antiderivative size = 849, normalized size of antiderivative = 2.55 \[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )} \, dx=-\frac {a^2}{3 d x^3}-\frac {a^2 c^2}{d x}-\frac {a^2 c^3 \log (1-c x)}{2 d}+\frac {a^2 c^3 \log (1+c x)}{2 d}-\frac {2 a b \left (-c^2 \left (-\frac {\arcsin (c x)}{x}-c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )+\frac {c x \sqrt {1-c^2 x^2}+2 \arcsin (c x)+c^3 x^3 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{6 x^3}+\frac {1}{2} c^4 \left (\frac {3 i \pi \arcsin (c x)}{2 c}-\frac {i \arcsin (c x)^2}{2 c}+\frac {2 \pi \log \left (1+e^{-i \arcsin (c x)}\right )}{c}-\frac {\pi \log \left (1+i e^{i \arcsin (c x)}\right )}{c}+\frac {2 \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right )}{c}-\frac {2 \pi \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )\right )}{c}+\frac {\pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )}{c}-\frac {2 i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{c}\right )-\frac {1}{2} c^4 \left (\frac {i \pi \arcsin (c x)}{2 c}-\frac {i \arcsin (c x)^2}{2 c}+\frac {2 \pi \log \left (1+e^{-i \arcsin (c x)}\right )}{c}+\frac {\pi \log \left (1-i e^{i \arcsin (c x)}\right )}{c}+\frac {2 \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right )}{c}-\frac {2 \pi \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )\right )}{c}-\frac {\pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )}{c}-\frac {2 i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{c}\right )\right )}{d}-\frac {b^2 c^3 \left (4 \cot \left (\frac {1}{2} \arcsin (c x)\right )+14 \arcsin (c x)^2 \cot \left (\frac {1}{2} \arcsin (c x)\right )+2 \arcsin (c x) \csc ^2\left (\frac {1}{2} \arcsin (c x)\right )+\frac {1}{2} c x \arcsin (c x)^2 \csc ^4\left (\frac {1}{2} \arcsin (c x)\right )-56 \arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right )-24 \arcsin (c x)^2 \log \left (1-i e^{i \arcsin (c x)}\right )+24 \arcsin (c x)^2 \log \left (1+i e^{i \arcsin (c x)}\right )+56 \arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right )-56 i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-48 i \arcsin (c x) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )+48 i \arcsin (c x) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )+56 i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )+48 \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )-48 \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )-2 \arcsin (c x) \sec ^2\left (\frac {1}{2} \arcsin (c x)\right )+\frac {8 \arcsin (c x)^2 \sin ^4\left (\frac {1}{2} \arcsin (c x)\right )}{c^3 x^3}+4 \tan \left (\frac {1}{2} \arcsin (c x)\right )+14 \arcsin (c x)^2 \tan \left (\frac {1}{2} \arcsin (c x)\right )\right )}{24 d} \]
-1/3*a^2/(d*x^3) - (a^2*c^2)/(d*x) - (a^2*c^3*Log[1 - c*x])/(2*d) + (a^2*c ^3*Log[1 + c*x])/(2*d) - (2*a*b*(-(c^2*(-(ArcSin[c*x]/x) - c*ArcTanh[Sqrt[ 1 - c^2*x^2]])) + (c*x*Sqrt[1 - c^2*x^2] + 2*ArcSin[c*x] + c^3*x^3*ArcTanh [Sqrt[1 - c^2*x^2]])/(6*x^3) + (c^4*((((3*I)/2)*Pi*ArcSin[c*x])/c - ((I/2) *ArcSin[c*x]^2)/c + (2*Pi*Log[1 + E^((-I)*ArcSin[c*x])])/c - (Pi*Log[1 + I *E^(I*ArcSin[c*x])])/c + (2*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])])/c - (2*Pi*Log[Cos[ArcSin[c*x]/2]])/c + (Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]])/ c - ((2*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/c))/2 - (c^4*(((I/2)*Pi*Arc Sin[c*x])/c - ((I/2)*ArcSin[c*x]^2)/c + (2*Pi*Log[1 + E^((-I)*ArcSin[c*x]) ])/c + (Pi*Log[1 - I*E^(I*ArcSin[c*x])])/c + (2*ArcSin[c*x]*Log[1 - I*E^(I *ArcSin[c*x])])/c - (2*Pi*Log[Cos[ArcSin[c*x]/2]])/c - (Pi*Log[Sin[(Pi + 2 *ArcSin[c*x])/4]])/c - ((2*I)*PolyLog[2, I*E^(I*ArcSin[c*x])])/c))/2))/d - (b^2*c^3*(4*Cot[ArcSin[c*x]/2] + 14*ArcSin[c*x]^2*Cot[ArcSin[c*x]/2] + 2* ArcSin[c*x]*Csc[ArcSin[c*x]/2]^2 + (c*x*ArcSin[c*x]^2*Csc[ArcSin[c*x]/2]^4 )/2 - 56*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] - 24*ArcSin[c*x]^2*Log[1 - I*E^(I*ArcSin[c*x])] + 24*ArcSin[c*x]^2*Log[1 + I*E^(I*ArcSin[c*x])] + 56 *ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] - (56*I)*PolyLog[2, -E^(I*ArcSin[c *x])] - (48*I)*ArcSin[c*x]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (48*I)*Arc Sin[c*x]*PolyLog[2, I*E^(I*ArcSin[c*x])] + (56*I)*PolyLog[2, E^(I*ArcSin[c *x])] + 48*PolyLog[3, (-I)*E^(I*ArcSin[c*x])] - 48*PolyLog[3, I*E^(I*Ar...
Time = 2.37 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.05, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {5204, 27, 5204, 15, 5164, 3042, 4669, 3011, 2720, 5218, 3042, 4671, 2715, 2838, 7143}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )} \, dx\) |
| \(\Big \downarrow \) 5204 |
| \(\displaystyle c^2 \int \frac {(a+b \arcsin (c x))^2}{d x^2 \left (1-c^2 x^2\right )}dx+\frac {2 b c \int \frac {a+b \arcsin (c x)}{x^3 \sqrt {1-c^2 x^2}}dx}{3 d}-\frac {(a+b \arcsin (c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c^2 \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (1-c^2 x^2\right )}dx}{d}+\frac {2 b c \int \frac {a+b \arcsin (c x)}{x^3 \sqrt {1-c^2 x^2}}dx}{3 d}-\frac {(a+b \arcsin (c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 5204 |
| \(\displaystyle \frac {c^2 \left (c^2 \int \frac {(a+b \arcsin (c x))^2}{1-c^2 x^2}dx+2 b c \int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx-\frac {(a+b \arcsin (c x))^2}{x}\right )}{d}+\frac {2 b c \left (\frac {1}{2} c^2 \int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx+\frac {1}{2} b c \int \frac {1}{x^2}dx-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 x^2}\right )}{3 d}-\frac {(a+b \arcsin (c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {c^2 \left (c^2 \int \frac {(a+b \arcsin (c x))^2}{1-c^2 x^2}dx+2 b c \int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx-\frac {(a+b \arcsin (c x))^2}{x}\right )}{d}+\frac {2 b c \left (\frac {1}{2} c^2 \int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 x^2}-\frac {b c}{2 x}\right )}{3 d}-\frac {(a+b \arcsin (c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 5164 |
| \(\displaystyle \frac {c^2 \left (2 b c \int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx+c \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}d\arcsin (c x)-\frac {(a+b \arcsin (c x))^2}{x}\right )}{d}+\frac {2 b c \left (\frac {1}{2} c^2 \int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 x^2}-\frac {b c}{2 x}\right )}{3 d}-\frac {(a+b \arcsin (c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 b c \left (\frac {1}{2} c^2 \int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 x^2}-\frac {b c}{2 x}\right )}{3 d}+\frac {c^2 \left (2 b c \int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx+c \int (a+b \arcsin (c x))^2 \csc \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)-\frac {(a+b \arcsin (c x))^2}{x}\right )}{d}-\frac {(a+b \arcsin (c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle \frac {c^2 \left (c \left (-2 b \int (a+b \arcsin (c x)) \log \left (1-i e^{i \arcsin (c x)}\right )d\arcsin (c x)+2 b \int (a+b \arcsin (c x)) \log \left (1+i e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )+2 b c \int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx-\frac {(a+b \arcsin (c x))^2}{x}\right )}{d}+\frac {2 b c \left (\frac {1}{2} c^2 \int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 x^2}-\frac {b c}{2 x}\right )}{3 d}-\frac {(a+b \arcsin (c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {c^2 \left (c \left (2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-i b \int \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )d\arcsin (c x)\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-i b \int \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )d\arcsin (c x)\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )+2 b c \int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx-\frac {(a+b \arcsin (c x))^2}{x}\right )}{d}+\frac {2 b c \left (\frac {1}{2} c^2 \int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 x^2}-\frac {b c}{2 x}\right )}{3 d}-\frac {(a+b \arcsin (c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {c^2 \left (c \left (2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )+2 b c \int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx-\frac {(a+b \arcsin (c x))^2}{x}\right )}{d}+\frac {2 b c \left (\frac {1}{2} c^2 \int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 x^2}-\frac {b c}{2 x}\right )}{3 d}-\frac {(a+b \arcsin (c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 5218 |
| \(\displaystyle \frac {c^2 \left (c \left (2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )+2 b c \int \frac {a+b \arcsin (c x)}{c x}d\arcsin (c x)-\frac {(a+b \arcsin (c x))^2}{x}\right )}{d}+\frac {2 b c \left (\frac {1}{2} c^2 \int \frac {a+b \arcsin (c x)}{c x}d\arcsin (c x)-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 x^2}-\frac {b c}{2 x}\right )}{3 d}-\frac {(a+b \arcsin (c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {c^2 \left (c \left (2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )+2 b c \int (a+b \arcsin (c x)) \csc (\arcsin (c x))d\arcsin (c x)-\frac {(a+b \arcsin (c x))^2}{x}\right )}{d}+\frac {2 b c \left (\frac {1}{2} c^2 \int (a+b \arcsin (c x)) \csc (\arcsin (c x))d\arcsin (c x)-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 x^2}-\frac {b c}{2 x}\right )}{3 d}-\frac {(a+b \arcsin (c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle \frac {c^2 \left (c \left (2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )+2 b c \left (-b \int \log \left (1-e^{i \arcsin (c x)}\right )d\arcsin (c x)+b \int \log \left (1+e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )-\frac {(a+b \arcsin (c x))^2}{x}\right )}{d}+\frac {2 b c \left (\frac {1}{2} c^2 \left (-b \int \log \left (1-e^{i \arcsin (c x)}\right )d\arcsin (c x)+b \int \log \left (1+e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 x^2}-\frac {b c}{2 x}\right )}{3 d}-\frac {(a+b \arcsin (c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {c^2 \left (c \left (2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )+2 b c \left (i b \int e^{-i \arcsin (c x)} \log \left (1-e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-i b \int e^{-i \arcsin (c x)} \log \left (1+e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )-\frac {(a+b \arcsin (c x))^2}{x}\right )}{d}+\frac {2 b c \left (\frac {1}{2} c^2 \left (i b \int e^{-i \arcsin (c x)} \log \left (1-e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-i b \int e^{-i \arcsin (c x)} \log \left (1+e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 x^2}-\frac {b c}{2 x}\right )}{3 d}-\frac {(a+b \arcsin (c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {c^2 \left (c \left (2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )+2 b c \left (-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right )-\frac {(a+b \arcsin (c x))^2}{x}\right )}{d}+\frac {2 b c \left (\frac {1}{2} c^2 \left (-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right )-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 x^2}-\frac {b c}{2 x}\right )}{3 d}-\frac {(a+b \arcsin (c x))^2}{3 d x^3}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {c^2 \left (c \left (-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2+2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )\right )\right )+2 b c \left (-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right )-\frac {(a+b \arcsin (c x))^2}{x}\right )}{d}+\frac {2 b c \left (\frac {1}{2} c^2 \left (-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right )-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 x^2}-\frac {b c}{2 x}\right )}{3 d}-\frac {(a+b \arcsin (c x))^2}{3 d x^3}\) |
-1/3*(a + b*ArcSin[c*x])^2/(d*x^3) + (2*b*c*(-1/2*(b*c)/x - (Sqrt[1 - c^2* x^2]*(a + b*ArcSin[c*x]))/(2*x^2) + (c^2*(-2*(a + b*ArcSin[c*x])*ArcTanh[E ^(I*ArcSin[c*x])] + I*b*PolyLog[2, -E^(I*ArcSin[c*x])] - I*b*PolyLog[2, E^ (I*ArcSin[c*x])]))/2))/(3*d) + (c^2*(-((a + b*ArcSin[c*x])^2/x) + 2*b*c*(- 2*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])] + I*b*PolyLog[2, -E^(I*Ar cSin[c*x])] - I*b*PolyLog[2, E^(I*ArcSin[c*x])]) + c*((-2*I)*(a + b*ArcSin [c*x])^2*ArcTan[E^(I*ArcSin[c*x])] + 2*b*(I*(a + b*ArcSin[c*x])*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - b*PolyLog[3, (-I)*E^(I*ArcSin[c*x])]) - 2*b*(I* (a + b*ArcSin[c*x])*PolyLog[2, I*E^(I*ArcSin[c*x])] - b*PolyLog[3, I*E^(I* ArcSin[c*x])]))))/d
3.2.91.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol ] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Si mp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x ))], x], x]) /; FreeQ[{c, d, e, f}, x] && IntegerQ[2*k] && IGtQ[m, 0]
Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x )^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG tQ[m, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbo l] :> Simp[1/(c*d) Subst[Int[(a + b*x)^n*Sec[x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
Int[((a_.) + ArcSin[(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 *ArcSin[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*ArcSin[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*ArcSin[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_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)* (x_)^2], x_Symbol] :> Simp[(1/c^(m + 1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e* x^2]] Subst[Int[(a + b*x)^n*Sin[x]^m, x], x, ArcSin[c*x]], x] /; FreeQ[{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 = 0.49 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.68
| method | result | size |
| derivativedivides | \(c^{3} \left (-\frac {a^{2} \left (\frac {1}{3 c^{3} x^{3}}+\frac {1}{c x}+\frac {\ln \left (c x -1\right )}{2}-\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b^{2} \left (\frac {3 \arcsin \left (c x \right )^{2} x^{2} c^{2}+\sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c +\arcsin \left (c x \right )^{2}+c^{2} x^{2}}{3 c^{3} x^{3}}-\frac {7 i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {7 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {7 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+2 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-2 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{d}-\frac {2 a b \left (\frac {6 c^{2} x^{2} \arcsin \left (c x \right )+c x \sqrt {-c^{2} x^{2}+1}+2 \arcsin \left (c x \right )}{6 c^{3} x^{3}}-\frac {7 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )}{6}+\frac {7 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{6}+\arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{d}\right )\) | \(561\) |
| default | \(c^{3} \left (-\frac {a^{2} \left (\frac {1}{3 c^{3} x^{3}}+\frac {1}{c x}+\frac {\ln \left (c x -1\right )}{2}-\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b^{2} \left (\frac {3 \arcsin \left (c x \right )^{2} x^{2} c^{2}+\sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c +\arcsin \left (c x \right )^{2}+c^{2} x^{2}}{3 c^{3} x^{3}}-\frac {7 i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {7 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {7 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+2 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-2 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{d}-\frac {2 a b \left (\frac {6 c^{2} x^{2} \arcsin \left (c x \right )+c x \sqrt {-c^{2} x^{2}+1}+2 \arcsin \left (c x \right )}{6 c^{3} x^{3}}-\frac {7 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )}{6}+\frac {7 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{6}+\arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{d}\right )\) | \(561\) |
| parts | \(-\frac {a^{2} \left (\frac {1}{3 x^{3}}+\frac {c^{2}}{x}+\frac {c^{3} \ln \left (c x -1\right )}{2}-\frac {c^{3} \ln \left (c x +1\right )}{2}\right )}{d}-\frac {b^{2} c^{3} \left (\frac {3 \arcsin \left (c x \right )^{2} x^{2} c^{2}+\sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c +\arcsin \left (c x \right )^{2}+c^{2} x^{2}}{3 c^{3} x^{3}}-\frac {7 i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {7 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {7 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+2 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-2 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{d}-\frac {2 a b \,c^{3} \left (\frac {6 c^{2} x^{2} \arcsin \left (c x \right )+c x \sqrt {-c^{2} x^{2}+1}+2 \arcsin \left (c x \right )}{6 c^{3} x^{3}}-\frac {7 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )}{6}+\frac {7 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{6}+\arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{d}\) | \(566\) |
c^3*(-a^2/d*(1/3/c^3/x^3+1/c/x+1/2*ln(c*x-1)-1/2*ln(c*x+1))-b^2/d*(1/3*(3* arcsin(c*x)^2*x^2*c^2+(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x*c+arcsin(c*x)^2+c^2 *x^2)/c^3/x^3-7/3*I*dilog(I*c*x+(-c^2*x^2+1)^(1/2))+7/3*arcsin(c*x)*ln(1+I *c*x+(-c^2*x^2+1)^(1/2))-7/3*I*dilog(1+I*c*x+(-c^2*x^2+1)^(1/2))+arcsin(c* x)^2*ln(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))-2*I*arcsin(c*x)*polylog(2,-I*(I*c* x+(-c^2*x^2+1)^(1/2)))+2*polylog(3,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-arcsin(c *x)^2*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+2*I*arcsin(c*x)*polylog(2,I*(I*c* x+(-c^2*x^2+1)^(1/2)))-2*polylog(3,I*(I*c*x+(-c^2*x^2+1)^(1/2))))-2*a*b/d* (1/6*(6*c^2*x^2*arcsin(c*x)+c*x*(-c^2*x^2+1)^(1/2)+2*arcsin(c*x))/c^3/x^3+ arcsin(c*x)*ln(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))-7/6*ln(I*c*x+(-c^2*x^2+1)^( 1/2)-1)-arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+7/6*ln(1+I*c*x+(-c^ 2*x^2+1)^(1/2))-I*dilog(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))+I*dilog(1-I*(I*c*x +(-c^2*x^2+1)^(1/2)))))
\[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )} x^{4}} \,d x } \]
\[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )} \, dx=- \frac {\int \frac {a^{2}}{c^{2} x^{6} - x^{4}}\, dx + \int \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{c^{2} x^{6} - x^{4}}\, dx + \int \frac {2 a b \operatorname {asin}{\left (c x \right )}}{c^{2} x^{6} - x^{4}}\, dx}{d} \]
-(Integral(a**2/(c**2*x**6 - x**4), x) + Integral(b**2*asin(c*x)**2/(c**2* x**6 - x**4), x) + Integral(2*a*b*asin(c*x)/(c**2*x**6 - x**4), x))/d
\[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )} x^{4}} \,d x } \]
1/6*(3*c^3*log(c*x + 1)/d - 3*c^3*log(c*x - 1)/d - 2*(3*c^2*x^2 + 1)/(d*x^ 3))*a^2 + 1/6*(3*b^2*c^3*x^3*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2* log(c*x + 1) - 3*b^2*c^3*x^3*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2* log(-c*x + 1) + 6*d*x^3*integrate(-1/3*(6*a*b*arctan2(c*x, sqrt(c*x + 1)*s qrt(-c*x + 1)) - (3*b^2*c^4*x^4*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) *log(c*x + 1) - 3*b^2*c^4*x^4*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))*l og(-c*x + 1) - 2*(3*b^2*c^3*x^3 + b^2*c*x)*arctan2(c*x, sqrt(c*x + 1)*sqrt (-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1))/(c^2*d*x^6 - d*x^4), x) - 2*(3* b^2*c^2*x^2 + b^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2)/(d*x^3)
\[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )} x^{4}} \,d x } \]
Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x^4\,\left (d-c^2\,d\,x^2\right )} \,d x \]