Integrand size = 27, antiderivative size = 300 \[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=-\frac {2 b^2 x}{c^4 d^2}-\frac {b (a+b \arcsin (c x))}{c^5 d^2 \sqrt {1-c^2 x^2}}+\frac {2 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^5 d^2}+\frac {3 x (a+b \arcsin (c x))^2}{2 c^4 d^2}+\frac {x^3 (a+b \arcsin (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {3 i (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{c^5 d^2}+\frac {b^2 \text {arctanh}(c x)}{c^5 d^2}-\frac {3 i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{c^5 d^2}+\frac {3 i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{c^5 d^2}+\frac {3 b^2 \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )}{c^5 d^2}-\frac {3 b^2 \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )}{c^5 d^2} \]
-2*b^2*x/c^4/d^2+3/2*x*(a+b*arcsin(c*x))^2/c^4/d^2+1/2*x^3*(a+b*arcsin(c*x ))^2/c^2/d^2/(-c^2*x^2+1)+3*I*(a+b*arcsin(c*x))^2*arctan(I*c*x+(-c^2*x^2+1 )^(1/2))/c^5/d^2+b^2*arctanh(c*x)/c^5/d^2-3*I*b*(a+b*arcsin(c*x))*polylog( 2,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))/c^5/d^2+3*I*b*(a+b*arcsin(c*x))*polylog(2 ,I*(I*c*x+(-c^2*x^2+1)^(1/2)))/c^5/d^2+3*b^2*polylog(3,-I*(I*c*x+(-c^2*x^2 +1)^(1/2)))/c^5/d^2-3*b^2*polylog(3,I*(I*c*x+(-c^2*x^2+1)^(1/2)))/c^5/d^2- b*(a+b*arcsin(c*x))/c^5/d^2/(-c^2*x^2+1)^(1/2)+2*b*(a+b*arcsin(c*x))*(-c^2 *x^2+1)^(1/2)/c^5/d^2
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(614\) vs. \(2(300)=600\).
Time = 2.82 (sec) , antiderivative size = 614, normalized size of antiderivative = 2.05 \[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {4 a^2 c x+\frac {8 b^2 c^3 x^3}{1-c^2 x^2}+8 a b \sqrt {1-c^2 x^2}+\frac {2 a b \sqrt {1-c^2 x^2}}{-1+c x}-\frac {2 a b \sqrt {1-c^2 x^2}}{1+c x}-\frac {2 a^2 c x}{-1+c^2 x^2}+\frac {8 b^2 c x}{-1+c^2 x^2}+6 i a b \pi \arcsin (c x)+8 a b c x \arcsin (c x)-\frac {2 a b \arcsin (c x)}{-1+c x}-\frac {2 a b \arcsin (c x)}{1+c x}+\frac {2 b^2 \arcsin (c x)}{\sqrt {1-c^2 x^2}}-\frac {6 b^2 c^2 x^2 \arcsin (c x)}{\sqrt {1-c^2 x^2}}+2 b^2 \sqrt {1-c^2 x^2} \arcsin (c x)+\frac {6 b^2 c x \arcsin (c x)^2}{1-c^2 x^2}+\frac {4 b^2 c^3 x^3 \arcsin (c x)^2}{-1+c^2 x^2}+12 i b^2 \arcsin (c x)^2 \arctan \left (e^{i \arcsin (c x)}\right )+4 b^2 \text {arctanh}(c x)-6 a b \pi \log \left (1-i e^{i \arcsin (c x)}\right )-12 a b \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right )-6 a b \pi \log \left (1+i e^{i \arcsin (c x)}\right )+12 a b \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right )+3 a^2 \log (1-c x)-3 a^2 \log (1+c x)+6 a b \pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+6 a b \pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-12 i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )+12 i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )+12 b^2 \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )-12 b^2 \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )}{4 c^5 d^2} \]
(4*a^2*c*x + (8*b^2*c^3*x^3)/(1 - c^2*x^2) + 8*a*b*Sqrt[1 - c^2*x^2] + (2* a*b*Sqrt[1 - c^2*x^2])/(-1 + c*x) - (2*a*b*Sqrt[1 - c^2*x^2])/(1 + c*x) - (2*a^2*c*x)/(-1 + c^2*x^2) + (8*b^2*c*x)/(-1 + c^2*x^2) + (6*I)*a*b*Pi*Arc Sin[c*x] + 8*a*b*c*x*ArcSin[c*x] - (2*a*b*ArcSin[c*x])/(-1 + c*x) - (2*a*b *ArcSin[c*x])/(1 + c*x) + (2*b^2*ArcSin[c*x])/Sqrt[1 - c^2*x^2] - (6*b^2*c ^2*x^2*ArcSin[c*x])/Sqrt[1 - c^2*x^2] + 2*b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x ] + (6*b^2*c*x*ArcSin[c*x]^2)/(1 - c^2*x^2) + (4*b^2*c^3*x^3*ArcSin[c*x]^2 )/(-1 + c^2*x^2) + (12*I)*b^2*ArcSin[c*x]^2*ArcTan[E^(I*ArcSin[c*x])] + 4* b^2*ArcTanh[c*x] - 6*a*b*Pi*Log[1 - I*E^(I*ArcSin[c*x])] - 12*a*b*ArcSin[c *x]*Log[1 - I*E^(I*ArcSin[c*x])] - 6*a*b*Pi*Log[1 + I*E^(I*ArcSin[c*x])] + 12*a*b*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + 3*a^2*Log[1 - c*x] - 3* a^2*Log[1 + c*x] + 6*a*b*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + 6*a*b*Pi*L og[Sin[(Pi + 2*ArcSin[c*x])/4]] - (12*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (12*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, I*E^(I*A rcSin[c*x])] + 12*b^2*PolyLog[3, (-I)*E^(I*ArcSin[c*x])] - 12*b^2*PolyLog[ 3, I*E^(I*ArcSin[c*x])])/(4*c^5*d^2)
Time = 1.80 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.04, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {5206, 27, 5194, 27, 299, 219, 5210, 5164, 3042, 4669, 3011, 2720, 5182, 24, 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.
\(\displaystyle \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 5206 |
\(\displaystyle -\frac {b \int \frac {x^3 (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx}{c d^2}-\frac {3 \int \frac {x^2 (a+b \arcsin (c x))^2}{d \left (1-c^2 x^2\right )}dx}{2 c^2 d}+\frac {x^3 (a+b \arcsin (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {3 \int \frac {x^2 (a+b \arcsin (c x))^2}{1-c^2 x^2}dx}{2 c^2 d^2}-\frac {b \int \frac {x^3 (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx}{c d^2}+\frac {x^3 (a+b \arcsin (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}\) |
\(\Big \downarrow \) 5194 |
\(\displaystyle -\frac {3 \int \frac {x^2 (a+b \arcsin (c x))^2}{1-c^2 x^2}dx}{2 c^2 d^2}-\frac {b \left (-b c \int \frac {2-c^2 x^2}{c^4 \left (1-c^2 x^2\right )}dx+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^4}+\frac {a+b \arcsin (c x)}{c^4 \sqrt {1-c^2 x^2}}\right )}{c d^2}+\frac {x^3 (a+b \arcsin (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {3 \int \frac {x^2 (a+b \arcsin (c x))^2}{1-c^2 x^2}dx}{2 c^2 d^2}-\frac {b \left (-\frac {b \int \frac {2-c^2 x^2}{1-c^2 x^2}dx}{c^3}+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^4}+\frac {a+b \arcsin (c x)}{c^4 \sqrt {1-c^2 x^2}}\right )}{c d^2}+\frac {x^3 (a+b \arcsin (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}\) |
\(\Big \downarrow \) 299 |
\(\displaystyle -\frac {3 \int \frac {x^2 (a+b \arcsin (c x))^2}{1-c^2 x^2}dx}{2 c^2 d^2}-\frac {b \left (-\frac {b \left (\int \frac {1}{1-c^2 x^2}dx+x\right )}{c^3}+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^4}+\frac {a+b \arcsin (c x)}{c^4 \sqrt {1-c^2 x^2}}\right )}{c d^2}+\frac {x^3 (a+b \arcsin (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {3 \int \frac {x^2 (a+b \arcsin (c x))^2}{1-c^2 x^2}dx}{2 c^2 d^2}-\frac {b \left (\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^4}+\frac {a+b \arcsin (c x)}{c^4 \sqrt {1-c^2 x^2}}-\frac {b \left (\frac {\text {arctanh}(c x)}{c}+x\right )}{c^3}\right )}{c d^2}+\frac {x^3 (a+b \arcsin (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}\) |
\(\Big \downarrow \) 5210 |
\(\displaystyle -\frac {3 \left (\frac {2 b \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{c}+\frac {\int \frac {(a+b \arcsin (c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {x (a+b \arcsin (c x))^2}{c^2}\right )}{2 c^2 d^2}-\frac {b \left (\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^4}+\frac {a+b \arcsin (c x)}{c^4 \sqrt {1-c^2 x^2}}-\frac {b \left (\frac {\text {arctanh}(c x)}{c}+x\right )}{c^3}\right )}{c d^2}+\frac {x^3 (a+b \arcsin (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}\) |
\(\Big \downarrow \) 5164 |
\(\displaystyle -\frac {3 \left (\frac {2 b \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{c}+\frac {\int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}d\arcsin (c x)}{c^3}-\frac {x (a+b \arcsin (c x))^2}{c^2}\right )}{2 c^2 d^2}-\frac {b \left (\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^4}+\frac {a+b \arcsin (c x)}{c^4 \sqrt {1-c^2 x^2}}-\frac {b \left (\frac {\text {arctanh}(c x)}{c}+x\right )}{c^3}\right )}{c d^2}+\frac {x^3 (a+b \arcsin (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {3 \left (\frac {\int (a+b \arcsin (c x))^2 \csc \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)}{c^3}+\frac {2 b \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{c}-\frac {x (a+b \arcsin (c x))^2}{c^2}\right )}{2 c^2 d^2}-\frac {b \left (\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^4}+\frac {a+b \arcsin (c x)}{c^4 \sqrt {1-c^2 x^2}}-\frac {b \left (\frac {\text {arctanh}(c x)}{c}+x\right )}{c^3}\right )}{c d^2}+\frac {x^3 (a+b \arcsin (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}\) |
\(\Big \downarrow \) 4669 |
\(\displaystyle -\frac {3 \left (\frac {-2 b \int (a+b \arcsin (c x)) \log \left (1-i e^{i \arcsin (c x)}\right )d\arcsin (c x)+2 b \int (a+b \arcsin (c x)) \log \left (1+i e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{c^3}+\frac {2 b \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{c}-\frac {x (a+b \arcsin (c x))^2}{c^2}\right )}{2 c^2 d^2}-\frac {b \left (\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^4}+\frac {a+b \arcsin (c x)}{c^4 \sqrt {1-c^2 x^2}}-\frac {b \left (\frac {\text {arctanh}(c x)}{c}+x\right )}{c^3}\right )}{c d^2}+\frac {x^3 (a+b \arcsin (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle -\frac {3 \left (\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-i b \int \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )d\arcsin (c x)\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-i b \int \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )d\arcsin (c x)\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{c^3}+\frac {2 b \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{c}-\frac {x (a+b \arcsin (c x))^2}{c^2}\right )}{2 c^2 d^2}-\frac {b \left (\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^4}+\frac {a+b \arcsin (c x)}{c^4 \sqrt {1-c^2 x^2}}-\frac {b \left (\frac {\text {arctanh}(c x)}{c}+x\right )}{c^3}\right )}{c d^2}+\frac {x^3 (a+b \arcsin (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle -\frac {3 \left (\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{c^3}+\frac {2 b \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{c}-\frac {x (a+b \arcsin (c x))^2}{c^2}\right )}{2 c^2 d^2}-\frac {b \left (\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^4}+\frac {a+b \arcsin (c x)}{c^4 \sqrt {1-c^2 x^2}}-\frac {b \left (\frac {\text {arctanh}(c x)}{c}+x\right )}{c^3}\right )}{c d^2}+\frac {x^3 (a+b \arcsin (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}\) |
\(\Big \downarrow \) 5182 |
\(\displaystyle -\frac {3 \left (\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{c^3}+\frac {2 b \left (\frac {b \int 1dx}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )}{c}-\frac {x (a+b \arcsin (c x))^2}{c^2}\right )}{2 c^2 d^2}-\frac {b \left (\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^4}+\frac {a+b \arcsin (c x)}{c^4 \sqrt {1-c^2 x^2}}-\frac {b \left (\frac {\text {arctanh}(c x)}{c}+x\right )}{c^3}\right )}{c d^2}+\frac {x^3 (a+b \arcsin (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle -\frac {3 \left (\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{c^3}+\frac {2 b \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )}{c}-\frac {x (a+b \arcsin (c x))^2}{c^2}\right )}{2 c^2 d^2}-\frac {b \left (\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^4}+\frac {a+b \arcsin (c x)}{c^4 \sqrt {1-c^2 x^2}}-\frac {b \left (\frac {\text {arctanh}(c x)}{c}+x\right )}{c^3}\right )}{c d^2}+\frac {x^3 (a+b \arcsin (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle -\frac {3 \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2+2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )\right )}{c^3}+\frac {2 b \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )}{c}-\frac {x (a+b \arcsin (c x))^2}{c^2}\right )}{2 c^2 d^2}-\frac {b \left (\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^4}+\frac {a+b \arcsin (c x)}{c^4 \sqrt {1-c^2 x^2}}-\frac {b \left (\frac {\text {arctanh}(c x)}{c}+x\right )}{c^3}\right )}{c d^2}+\frac {x^3 (a+b \arcsin (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}\) |
(x^3*(a + b*ArcSin[c*x])^2)/(2*c^2*d^2*(1 - c^2*x^2)) - (b*((a + b*ArcSin[ c*x])/(c^4*Sqrt[1 - c^2*x^2]) + (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c^ 4 - (b*(x + ArcTanh[c*x]/c))/c^3))/(c*d^2) - (3*(-((x*(a + b*ArcSin[c*x])^ 2)/c^2) + (2*b*((b*x)/c - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c^2))/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*A rcSin[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])]))/c^3))/(2*c^2*d^2)
3.2.92.3.1 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)^(-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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 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), 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_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_) , x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcSin [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_.) + 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[((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/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] + S imp[b*f*(n/(c*(m + 2*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, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m , 1] && NeQ[m + 2*p + 1, 0]
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 = 0.35 (sec) , antiderivative size = 618, normalized size of antiderivative = 2.06
method | result | size |
derivativedivides | \(\frac {\frac {a^{2} \left (c x -\frac {1}{4 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{4}\right )}{d^{2}}+\frac {2 b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{d^{2}}+\frac {b^{2} \arcsin \left (c x \right )^{2} c x}{d^{2}}-\frac {2 b^{2} c x}{d^{2}}-\frac {b^{2} \arcsin \left (c x \right )^{2} c x}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {3 b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2 d^{2}}-\frac {2 i b^{2} \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {3 b^{2} \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}-\frac {3 b^{2} \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2 d^{2}}-\frac {3 i a b \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}-\frac {3 b^{2} \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {3 i b^{2} \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {2 a b \sqrt {-c^{2} x^{2}+1}}{d^{2}}+\frac {2 a b \arcsin \left (c x \right ) c x}{d^{2}}-\frac {a b \arcsin \left (c x \right ) c x}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {a b \sqrt {-c^{2} x^{2}+1}}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {3 a b \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}-\frac {3 a b \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {3 i a b \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}-\frac {3 i b^{2} \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}}{c^{5}}\) | \(618\) |
default | \(\frac {\frac {a^{2} \left (c x -\frac {1}{4 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{4}\right )}{d^{2}}+\frac {2 b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{d^{2}}+\frac {b^{2} \arcsin \left (c x \right )^{2} c x}{d^{2}}-\frac {2 b^{2} c x}{d^{2}}-\frac {b^{2} \arcsin \left (c x \right )^{2} c x}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {3 b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2 d^{2}}-\frac {2 i b^{2} \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {3 b^{2} \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}-\frac {3 b^{2} \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2 d^{2}}-\frac {3 i a b \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}-\frac {3 b^{2} \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {3 i b^{2} \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {2 a b \sqrt {-c^{2} x^{2}+1}}{d^{2}}+\frac {2 a b \arcsin \left (c x \right ) c x}{d^{2}}-\frac {a b \arcsin \left (c x \right ) c x}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {a b \sqrt {-c^{2} x^{2}+1}}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {3 a b \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}-\frac {3 a b \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {3 i a b \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}-\frac {3 i b^{2} \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}}{c^{5}}\) | \(618\) |
parts | \(\frac {a^{2} \left (\frac {x}{c^{4}}-\frac {1}{4 c^{5} \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{4 c^{5}}-\frac {1}{4 c^{5} \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{4 c^{5}}\right )}{d^{2}}+\frac {2 b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{d^{2} c^{5}}+\frac {b^{2} \arcsin \left (c x \right )^{2} x}{d^{2} c^{4}}-\frac {2 b^{2} x}{c^{4} d^{2}}-\frac {b^{2} \arcsin \left (c x \right )^{2} x}{2 d^{2} c^{4} \left (c^{2} x^{2}-1\right )}+\frac {b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{d^{2} c^{5} \left (c^{2} x^{2}-1\right )}+\frac {3 b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2 d^{2} c^{5}}+\frac {3 i b^{2} \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} c^{5}}+\frac {3 b^{2} \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{c^{5} d^{2}}-\frac {3 b^{2} \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2 d^{2} c^{5}}-\frac {2 i b^{2} \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2} c^{5}}-\frac {3 b^{2} \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{c^{5} d^{2}}-\frac {3 i a b \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} c^{5}}+\frac {2 a b \sqrt {-c^{2} x^{2}+1}}{d^{2} c^{5}}+\frac {2 a b \arcsin \left (c x \right ) x}{d^{2} c^{4}}-\frac {a b \arcsin \left (c x \right ) x}{d^{2} c^{4} \left (c^{2} x^{2}-1\right )}+\frac {a b \sqrt {-c^{2} x^{2}+1}}{d^{2} c^{5} \left (c^{2} x^{2}-1\right )}+\frac {3 a b \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} c^{5}}-\frac {3 a b \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} c^{5}}-\frac {3 i b^{2} \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} c^{5}}+\frac {3 i a b \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} c^{5}}\) | \(683\) |
1/c^5*(a^2/d^2*(c*x-1/4/(c*x-1)+3/4*ln(c*x-1)-1/4/(c*x+1)-3/4*ln(c*x+1))+2 *b^2/d^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)+b^2/d^2*arcsin(c*x)^2*c*x-2*b^2/d^ 2*c*x-1/2*b^2/d^2/(c^2*x^2-1)*arcsin(c*x)^2*c*x+b^2/d^2/(c^2*x^2-1)*arcsin (c*x)*(-c^2*x^2+1)^(1/2)+3/2*b^2/d^2*arcsin(c*x)^2*ln(1+I*(I*c*x+(-c^2*x^2 +1)^(1/2)))-2*I*b^2/d^2*arctan(I*c*x+(-c^2*x^2+1)^(1/2))+3*b^2/d^2*polylog (3,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-3/2*b^2/d^2*arcsin(c*x)^2*ln(1-I*(I*c*x+ (-c^2*x^2+1)^(1/2)))-3*I*a*b/d^2*dilog(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))-3*b ^2/d^2*polylog(3,I*(I*c*x+(-c^2*x^2+1)^(1/2)))+3*I*b^2/d^2*arcsin(c*x)*pol ylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))+2*a*b/d^2*(-c^2*x^2+1)^(1/2)+2*a*b/d^ 2*arcsin(c*x)*c*x-a*b/d^2/(c^2*x^2-1)*arcsin(c*x)*c*x+a*b/d^2/(c^2*x^2-1)* (-c^2*x^2+1)^(1/2)+3*a*b/d^2*arcsin(c*x)*ln(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)) )-3*a*b/d^2*arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+3*I*a*b/d^2*dil og(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-3*I*b^2/d^2*arcsin(c*x)*polylog(2,-I*(I *c*x+(-c^2*x^2+1)^(1/2))))
\[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{4}}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]
integral((b^2*x^4*arcsin(c*x)^2 + 2*a*b*x^4*arcsin(c*x) + a^2*x^4)/(c^4*d^ 2*x^4 - 2*c^2*d^2*x^2 + d^2), x)
\[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a^{2} x^{4}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {b^{2} x^{4} \operatorname {asin}^{2}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {2 a b x^{4} \operatorname {asin}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \]
(Integral(a**2*x**4/(c**4*x**4 - 2*c**2*x**2 + 1), x) + Integral(b**2*x**4 *asin(c*x)**2/(c**4*x**4 - 2*c**2*x**2 + 1), x) + Integral(2*a*b*x**4*asin (c*x)/(c**4*x**4 - 2*c**2*x**2 + 1), x))/d**2
\[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{4}}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]
-1/4*a^2*(2*x/(c^6*d^2*x^2 - c^4*d^2) - 4*x/(c^4*d^2) + 3*log(c*x + 1)/(c^ 5*d^2) - 3*log(c*x - 1)/(c^5*d^2)) - 1/4*(3*(b^2*c^2*x^2 - b^2)*arctan2(c* x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2*log(c*x + 1) - 3*(b^2*c^2*x^2 - b^2)*ar ctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2*log(-c*x + 1) - 2*(2*b^2*c^3*x^ 3 - 3*b^2*c*x)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 4*(c^7*d^2*x ^2 - c^5*d^2)*integrate(-1/2*(4*a*b*c^4*x^4*arctan2(c*x, sqrt(c*x + 1)*sqr t(-c*x + 1)) - (3*(b^2*c^2*x^2 - b^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))*log(c*x + 1) - 3*(b^2*c^2*x^2 - b^2)*arctan2(c*x, sqrt(c*x + 1)*sqr t(-c*x + 1))*log(-c*x + 1) - 2*(2*b^2*c^3*x^3 - 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^2*x^4 - 2*c^6*d^2*x^2 + c^4*d^2), x))/(c^7*d^2*x^2 - c^5*d^2)
\[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{4}}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=\int \frac {x^4\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^2} \,d x \]