Integrand size = 27, antiderivative size = 341 \[ \int \frac {x^2 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {b^2 x}{12 c^2 d^3 \left (1-c^2 x^2\right )}-\frac {b (a+b \arcsin (c x))}{6 c^3 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {b (a+b \arcsin (c x))}{4 c^3 d^3 \sqrt {1-c^2 x^2}}+\frac {x (a+b \arcsin (c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {x (a+b \arcsin (c x))^2}{8 c^2 d^3 \left (1-c^2 x^2\right )}+\frac {i (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{4 c^3 d^3}-\frac {b^2 \text {arctanh}(c x)}{6 c^3 d^3}-\frac {i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{4 c^3 d^3}+\frac {i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{4 c^3 d^3}+\frac {b^2 \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )}{4 c^3 d^3}-\frac {b^2 \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )}{4 c^3 d^3} \]
1/12*b^2*x/c^2/d^3/(-c^2*x^2+1)-1/6*b*(a+b*arcsin(c*x))/c^3/d^3/(-c^2*x^2+ 1)^(3/2)+1/4*x*(a+b*arcsin(c*x))^2/c^2/d^3/(-c^2*x^2+1)^2-1/8*x*(a+b*arcsi n(c*x))^2/c^2/d^3/(-c^2*x^2+1)+1/4*I*(a+b*arcsin(c*x))^2*arctan(I*c*x+(-c^ 2*x^2+1)^(1/2))/c^3/d^3-1/6*b^2*arctanh(c*x)/c^3/d^3-1/4*I*b*(a+b*arcsin(c *x))*polylog(2,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))/c^3/d^3+1/4*I*b*(a+b*arcsin( c*x))*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))/c^3/d^3+1/4*b^2*polylog(3,-I *(I*c*x+(-c^2*x^2+1)^(1/2)))/c^3/d^3-1/4*b^2*polylog(3,I*(I*c*x+(-c^2*x^2+ 1)^(1/2)))/c^3/d^3+1/4*b*(a+b*arcsin(c*x))/c^3/d^3/(-c^2*x^2+1)^(1/2)
Time = 5.33 (sec) , antiderivative size = 670, normalized size of antiderivative = 1.96 \[ \int \frac {x^2 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {\frac {4 a^2 c x}{\left (-1+c^2 x^2\right )^2}+\frac {2 a^2 c x}{-1+c^2 x^2}-\frac {2 a b \left (\sqrt {1-c^2 x^2}-\arcsin (c x)\right )}{-1+c x}+\frac {2 a b \left (\sqrt {1-c^2 x^2}+\arcsin (c x)\right )}{1+c x}+\frac {2 a b \left ((-2+c x) \sqrt {1-c^2 x^2}+3 \arcsin (c x)\right )}{3 (-1+c x)^2}-\frac {2 a b \left ((2+c x) \sqrt {1-c^2 x^2}+3 \arcsin (c x)\right )}{3 (1+c x)^2}+a^2 \log (1-c x)-a^2 \log (1+c x)+a b \left (-i \arcsin (c x)^2+\arcsin (c x) \left (3 i \pi +4 \log \left (1+i e^{i \arcsin (c x)}\right )\right )+2 \pi \left (2 \log \left (1+e^{-i \arcsin (c x)}\right )-\log \left (1+i e^{i \arcsin (c x)}\right )-2 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )\right )+\log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )\right )-4 i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )\right )+a b \left (i \arcsin (c x)^2+\arcsin (c x) \left (-i \pi -4 \log \left (1-i e^{i \arcsin (c x)}\right )\right )+2 \pi \left (-2 \log \left (1+e^{-i \arcsin (c x)}\right )-\log \left (1-i e^{i \arcsin (c x)}\right )+2 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )\right )+\log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )\right )+4 i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )+\frac {4}{3} b^2 \left (3 i \arcsin (c x)^2 \arctan \left (e^{i \arcsin (c x)}\right )-2 \text {arctanh}(c x)-3 i \arcsin (c x) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )+3 i \arcsin (c x) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )+3 \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )-3 \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )\right )+\frac {b^2 \left (2 \arcsin (c x) \left (\sqrt {1-c^2 x^2}+3 \cos (3 \arcsin (c x))\right )-3 \arcsin (c x)^2 (-7 c x+\sin (3 \arcsin (c x)))+2 (c x+\sin (3 \arcsin (c x)))\right )}{6 \left (-1+c^2 x^2\right )^2}}{16 c^3 d^3} \]
((4*a^2*c*x)/(-1 + c^2*x^2)^2 + (2*a^2*c*x)/(-1 + c^2*x^2) - (2*a*b*(Sqrt[ 1 - c^2*x^2] - ArcSin[c*x]))/(-1 + c*x) + (2*a*b*(Sqrt[1 - c^2*x^2] + ArcS in[c*x]))/(1 + c*x) + (2*a*b*((-2 + c*x)*Sqrt[1 - c^2*x^2] + 3*ArcSin[c*x] ))/(3*(-1 + c*x)^2) - (2*a*b*((2 + c*x)*Sqrt[1 - c^2*x^2] + 3*ArcSin[c*x]) )/(3*(1 + c*x)^2) + a^2*Log[1 - c*x] - a^2*Log[1 + c*x] + a*b*((-I)*ArcSin [c*x]^2 + ArcSin[c*x]*((3*I)*Pi + 4*Log[1 + I*E^(I*ArcSin[c*x])]) + 2*Pi*( 2*Log[1 + E^((-I)*ArcSin[c*x])] - Log[1 + I*E^(I*ArcSin[c*x])] - 2*Log[Cos [ArcSin[c*x]/2]] + Log[-Cos[(Pi + 2*ArcSin[c*x])/4]]) - (4*I)*PolyLog[2, ( -I)*E^(I*ArcSin[c*x])]) + a*b*(I*ArcSin[c*x]^2 + ArcSin[c*x]*((-I)*Pi - 4* Log[1 - I*E^(I*ArcSin[c*x])]) + 2*Pi*(-2*Log[1 + E^((-I)*ArcSin[c*x])] - L og[1 - I*E^(I*ArcSin[c*x])] + 2*Log[Cos[ArcSin[c*x]/2]] + Log[Sin[(Pi + 2* ArcSin[c*x])/4]]) + (4*I)*PolyLog[2, I*E^(I*ArcSin[c*x])]) + (4*b^2*((3*I) *ArcSin[c*x]^2*ArcTan[E^(I*ArcSin[c*x])] - 2*ArcTanh[c*x] - (3*I)*ArcSin[c *x]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (3*I)*ArcSin[c*x]*PolyLog[2, I*E^ (I*ArcSin[c*x])] + 3*PolyLog[3, (-I)*E^(I*ArcSin[c*x])] - 3*PolyLog[3, I*E ^(I*ArcSin[c*x])]))/3 + (b^2*(2*ArcSin[c*x]*(Sqrt[1 - c^2*x^2] + 3*Cos[3*A rcSin[c*x]]) - 3*ArcSin[c*x]^2*(-7*c*x + Sin[3*ArcSin[c*x]]) + 2*(c*x + Si n[3*ArcSin[c*x]])))/(6*(-1 + c^2*x^2)^2))/(16*c^3*d^3)
Time = 1.74 (sec) , antiderivative size = 326, normalized size of antiderivative = 0.96, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {5206, 27, 5162, 5164, 3042, 4669, 3011, 2720, 5182, 215, 219, 7143}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^2 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 5206 |
| \(\displaystyle -\frac {b \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx}{2 c d^3}-\frac {\int \frac {(a+b \arcsin (c x))^2}{d^2 \left (1-c^2 x^2\right )^2}dx}{4 c^2 d}+\frac {x (a+b \arcsin (c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx}{2 c d^3}-\frac {\int \frac {(a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^2}dx}{4 c^2 d^3}+\frac {x (a+b \arcsin (c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 5162 |
| \(\displaystyle -\frac {b \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx}{2 c d^3}-\frac {-b c \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {1}{2} \int \frac {(a+b \arcsin (c x))^2}{1-c^2 x^2}dx+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}}{4 c^2 d^3}+\frac {x (a+b \arcsin (c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 5164 |
| \(\displaystyle -\frac {b \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx}{2 c d^3}-\frac {-b c \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {\int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}d\arcsin (c x)}{2 c}+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}}{4 c^2 d^3}+\frac {x (a+b \arcsin (c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx}{2 c d^3}-\frac {-b c \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {\int (a+b \arcsin (c x))^2 \csc \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)}{2 c}+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}}{4 c^2 d^3}+\frac {x (a+b \arcsin (c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle -\frac {\frac {-2 b \int (a+b \arcsin (c x)) \log \left (1-i e^{i \arcsin (c x)}\right )d\arcsin (c x)+2 b \int (a+b \arcsin (c x)) \log \left (1+i e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}}{4 c^2 d^3}-\frac {b \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx}{2 c d^3}+\frac {x (a+b \arcsin (c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-i b \int \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )d\arcsin (c x)\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-i b \int \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )d\arcsin (c x)\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}}{4 c^2 d^3}-\frac {b \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx}{2 c d^3}+\frac {x (a+b \arcsin (c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}}{4 c^2 d^3}-\frac {b \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx}{2 c d^3}+\frac {x (a+b \arcsin (c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle -\frac {\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \left (\frac {a+b \arcsin (c x)}{c^2 \sqrt {1-c^2 x^2}}-\frac {b \int \frac {1}{1-c^2 x^2}dx}{c}\right )+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}}{4 c^2 d^3}-\frac {b \left (\frac {a+b \arcsin (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {b \int \frac {1}{\left (1-c^2 x^2\right )^2}dx}{3 c}\right )}{2 c d^3}+\frac {x (a+b \arcsin (c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle -\frac {\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \left (\frac {a+b \arcsin (c x)}{c^2 \sqrt {1-c^2 x^2}}-\frac {b \int \frac {1}{1-c^2 x^2}dx}{c}\right )+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}}{4 c^2 d^3}-\frac {b \left (\frac {a+b \arcsin (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {b \left (\frac {1}{2} \int \frac {1}{1-c^2 x^2}dx+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c}\right )}{2 c d^3}+\frac {x (a+b \arcsin (c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c}-b c \left (\frac {a+b \arcsin (c x)}{c^2 \sqrt {1-c^2 x^2}}-\frac {b \text {arctanh}(c x)}{c^2}\right )+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}}{4 c^2 d^3}-\frac {b \left (\frac {a+b \arcsin (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {b \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c}\right )}{2 c d^3}+\frac {x (a+b \arcsin (c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -\frac {\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2+2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )\right )}{2 c}-b c \left (\frac {a+b \arcsin (c x)}{c^2 \sqrt {1-c^2 x^2}}-\frac {b \text {arctanh}(c x)}{c^2}\right )+\frac {x (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}}{4 c^2 d^3}-\frac {b \left (\frac {a+b \arcsin (c x)}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {b \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c}\right )}{2 c d^3}+\frac {x (a+b \arcsin (c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\) |
(x*(a + b*ArcSin[c*x])^2)/(4*c^2*d^3*(1 - c^2*x^2)^2) - (b*((a + b*ArcSin[ c*x])/(3*c^2*(1 - c^2*x^2)^(3/2)) - (b*(x/(2*(1 - c^2*x^2)) + ArcTanh[c*x] /(2*c)))/(3*c)))/(2*c*d^3) - ((x*(a + b*ArcSin[c*x])^2)/(2*(1 - c^2*x^2)) - b*c*((a + b*ArcSin[c*x])/(c^2*Sqrt[1 - c^2*x^2]) - (b*ArcTanh[c*x])/c^2) + ((-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])]))/(2*c))/(4*c^2*d^3)
3.3.3.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[csc[(e_.) + 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[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_ Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[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 cSin[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*ArcSin[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_.) + 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_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ .), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 1))), x] + Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] I nt[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 1))), x] + (-Simp[f^2*((m - 1)/(2*e*(p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x], x] + Simp [b*f*(n/(2*c*(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*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{ a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && IG tQ[m, 1]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Time = 0.40 (sec) , antiderivative size = 568, normalized size of antiderivative = 1.67
| method | result | size |
| derivativedivides | \(\frac {-\frac {a^{2} \left (-\frac {1}{16 \left (c x -1\right )^{2}}-\frac {1}{16 \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{16}+\frac {1}{16 \left (c x +1\right )^{2}}-\frac {1}{16 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{16}\right )}{d^{3}}-\frac {b^{2} \left (-\frac {3 c^{3} x^{3} \arcsin \left (c x \right )^{2}-6 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{2} c^{2}+3 c x \arcsin \left (c x \right )^{2}-2 c^{3} x^{3}+2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}+2 c x}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {\arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {\operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}+\frac {\arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}+\frac {\operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}\right )}{d^{3}}-\frac {2 a b \left (-\frac {3 c^{3} x^{3} \arcsin \left (c x \right )-3 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}+3 c x \arcsin \left (c x \right )+\sqrt {-c^{2} x^{2}+1}}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {\arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {\arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}\right )}{d^{3}}}{c^{3}}\) | \(568\) |
| default | \(\frac {-\frac {a^{2} \left (-\frac {1}{16 \left (c x -1\right )^{2}}-\frac {1}{16 \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{16}+\frac {1}{16 \left (c x +1\right )^{2}}-\frac {1}{16 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{16}\right )}{d^{3}}-\frac {b^{2} \left (-\frac {3 c^{3} x^{3} \arcsin \left (c x \right )^{2}-6 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{2} c^{2}+3 c x \arcsin \left (c x \right )^{2}-2 c^{3} x^{3}+2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}+2 c x}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {\arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {\operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}+\frac {\arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}+\frac {\operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}\right )}{d^{3}}-\frac {2 a b \left (-\frac {3 c^{3} x^{3} \arcsin \left (c x \right )-3 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}+3 c x \arcsin \left (c x \right )+\sqrt {-c^{2} x^{2}+1}}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {\arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {\arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}\right )}{d^{3}}}{c^{3}}\) | \(568\) |
| parts | \(-\frac {a^{2} \left (-\frac {1}{16 c^{3} \left (c x -1\right )^{2}}-\frac {1}{16 c^{3} \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{16 c^{3}}+\frac {1}{16 c^{3} \left (c x +1\right )^{2}}-\frac {1}{16 c^{3} \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{16 c^{3}}\right )}{d^{3}}-\frac {b^{2} \left (-\frac {3 c^{3} x^{3} \arcsin \left (c x \right )^{2}-6 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{2} c^{2}+3 c x \arcsin \left (c x \right )^{2}-2 c^{3} x^{3}+2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}+2 c x}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {\arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {\operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}+\frac {\arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}+\frac {\operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}\right )}{d^{3} c^{3}}-\frac {2 a b \left (-\frac {3 c^{3} x^{3} \arcsin \left (c x \right )-3 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}+3 c x \arcsin \left (c x \right )+\sqrt {-c^{2} x^{2}+1}}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {\arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {\arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}\right )}{d^{3} c^{3}}\) | \(588\) |
1/c^3*(-a^2/d^3*(-1/16/(c*x-1)^2-1/16/(c*x-1)-1/16*ln(c*x-1)+1/16/(c*x+1)^ 2-1/16/(c*x+1)+1/16*ln(c*x+1))-b^2/d^3*(-1/24*(3*c^3*x^3*arcsin(c*x)^2-6*( -c^2*x^2+1)^(1/2)*arcsin(c*x)*x^2*c^2+3*c*x*arcsin(c*x)^2-2*c^3*x^3+2*arcs in(c*x)*(-c^2*x^2+1)^(1/2)+2*c*x)/(c^4*x^4-2*c^2*x^2+1)-1/8*arcsin(c*x)^2* ln(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))+1/4*I*arcsin(c*x)*polylog(2,-I*(I*c*x+( -c^2*x^2+1)^(1/2)))-1/4*polylog(3,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+1/8*arcsi n(c*x)^2*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-1/4*I*arcsin(c*x)*polylog(2,I* (I*c*x+(-c^2*x^2+1)^(1/2)))+1/4*polylog(3,I*(I*c*x+(-c^2*x^2+1)^(1/2)))-1/ 3*I*arctan(I*c*x+(-c^2*x^2+1)^(1/2)))-2*a*b/d^3*(-1/24*(3*c^3*x^3*arcsin(c *x)-3*c^2*x^2*(-c^2*x^2+1)^(1/2)+3*c*x*arcsin(c*x)+(-c^2*x^2+1)^(1/2))/(c^ 4*x^4-2*c^2*x^2+1)-1/8*arcsin(c*x)*ln(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))+1/8* arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+1/8*I*dilog(1+I*(I*c*x+(-c^ 2*x^2+1)^(1/2)))-1/8*I*dilog(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))))
\[ \int \frac {x^2 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{2}}{{\left (c^{2} d x^{2} - d\right )}^{3}} \,d x } \]
integral(-(b^2*x^2*arcsin(c*x)^2 + 2*a*b*x^2*arcsin(c*x) + a^2*x^2)/(c^6*d ^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x)
\[ \int \frac {x^2 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=- \frac {\int \frac {a^{2} x^{2}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx + \int \frac {b^{2} x^{2} \operatorname {asin}^{2}{\left (c x \right )}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx + \int \frac {2 a b x^{2} \operatorname {asin}{\left (c x \right )}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx}{d^{3}} \]
-(Integral(a**2*x**2/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), x) + Int egral(b**2*x**2*asin(c*x)**2/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), x) + Integral(2*a*b*x**2*asin(c*x)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), x))/d**3
\[ \int \frac {x^2 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{2}}{{\left (c^{2} d x^{2} - d\right )}^{3}} \,d x } \]
1/16*a^2*(2*(c^2*x^3 + x)/(c^6*d^3*x^4 - 2*c^4*d^3*x^2 + c^2*d^3) - log(c* x + 1)/(c^3*d^3) + log(c*x - 1)/(c^3*d^3)) - 1/16*((b^2*c^4*x^4 - 2*b^2*c^ 2*x^2 + b^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2*log(c*x + 1) - ( b^2*c^4*x^4 - 2*b^2*c^2*x^2 + b^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2*log(-c*x + 1) - 2*(b^2*c^3*x^3 + b^2*c*x)*arctan2(c*x, sqrt(c*x + 1) *sqrt(-c*x + 1))^2 + 16*(c^7*d^3*x^4 - 2*c^5*d^3*x^2 + c^3*d^3)*integrate( 1/8*(16*a*b*c^2*x^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + ((b^2*c^4 *x^4 - 2*b^2*c^2*x^2 + b^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))*log (c*x + 1) - (b^2*c^4*x^4 - 2*b^2*c^2*x^2 + b^2)*arctan2(c*x, sqrt(c*x + 1) *sqrt(-c*x + 1))*log(-c*x + 1) - 2*(b^2*c^3*x^3 + b^2*c*x)*arctan2(c*x, sq rt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1))/(c^8*d^3*x^6 - 3*c^6*d^3*x^4 + 3*c^4*d^3*x^2 - c^2*d^3), x))/(c^7*d^3*x^4 - 2*c^5*d^3*x^2 + c^3*d^3)
\[ \int \frac {x^2 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{2}}{{\left (c^{2} d x^{2} - d\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {x^2 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\int \frac {x^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^3} \,d x \]