Optimal. Leaf size=341 \[ -\frac{i b \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 c^3 d^3}+\frac{i b \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 c^3 d^3}+\frac{b^2 \text{PolyLog}\left (3,-i e^{i \sin ^{-1}(c x)}\right )}{4 c^3 d^3}-\frac{b^2 \text{PolyLog}\left (3,i e^{i \sin ^{-1}(c x)}\right )}{4 c^3 d^3}+\frac{b \left (a+b \sin ^{-1}(c x)\right )}{4 c^3 d^3 \sqrt{1-c^2 x^2}}-\frac{b \left (a+b \sin ^{-1}(c x)\right )}{6 c^3 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2 d^3 \left (1-c^2 x^2\right )}+\frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}+\frac{i \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^3 d^3}+\frac{b^2 x}{12 c^2 d^3 \left (1-c^2 x^2\right )}-\frac{b^2 \tanh ^{-1}(c x)}{6 c^3 d^3} \]
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Rubi [A] time = 0.421745, antiderivative size = 341, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.37, Rules used = {4703, 4655, 4657, 4181, 2531, 2282, 6589, 4677, 206, 199} \[ -\frac{i b \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 c^3 d^3}+\frac{i b \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 c^3 d^3}+\frac{b^2 \text{PolyLog}\left (3,-i e^{i \sin ^{-1}(c x)}\right )}{4 c^3 d^3}-\frac{b^2 \text{PolyLog}\left (3,i e^{i \sin ^{-1}(c x)}\right )}{4 c^3 d^3}+\frac{b \left (a+b \sin ^{-1}(c x)\right )}{4 c^3 d^3 \sqrt{1-c^2 x^2}}-\frac{b \left (a+b \sin ^{-1}(c x)\right )}{6 c^3 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2 d^3 \left (1-c^2 x^2\right )}+\frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}+\frac{i \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^3 d^3}+\frac{b^2 x}{12 c^2 d^3 \left (1-c^2 x^2\right )}-\frac{b^2 \tanh ^{-1}(c x)}{6 c^3 d^3} \]
Antiderivative was successfully verified.
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Rule 4703
Rule 4655
Rule 4657
Rule 4181
Rule 2531
Rule 2282
Rule 6589
Rule 4677
Rule 206
Rule 199
Rubi steps
\begin{align*} \int \frac{x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^3} \, dx &=\frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac{b \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{2 c d^3}-\frac{\int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^2} \, dx}{4 c^2 d}\\ &=-\frac{b \left (a+b \sin ^{-1}(c x)\right )}{6 c^3 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2 d^3 \left (1-c^2 x^2\right )}+\frac{b^2 \int \frac{1}{\left (1-c^2 x^2\right )^2} \, dx}{6 c^2 d^3}+\frac{b \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{4 c d^3}-\frac{\int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{d-c^2 d x^2} \, dx}{8 c^2 d^2}\\ &=\frac{b^2 x}{12 c^2 d^3 \left (1-c^2 x^2\right )}-\frac{b \left (a+b \sin ^{-1}(c x)\right )}{6 c^3 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac{b \left (a+b \sin ^{-1}(c x)\right )}{4 c^3 d^3 \sqrt{1-c^2 x^2}}+\frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2 d^3 \left (1-c^2 x^2\right )}-\frac{\operatorname{Subst}\left (\int (a+b x)^2 \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{8 c^3 d^3}+\frac{b^2 \int \frac{1}{1-c^2 x^2} \, dx}{12 c^2 d^3}-\frac{b^2 \int \frac{1}{1-c^2 x^2} \, dx}{4 c^2 d^3}\\ &=\frac{b^2 x}{12 c^2 d^3 \left (1-c^2 x^2\right )}-\frac{b \left (a+b \sin ^{-1}(c x)\right )}{6 c^3 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac{b \left (a+b \sin ^{-1}(c x)\right )}{4 c^3 d^3 \sqrt{1-c^2 x^2}}+\frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2 d^3 \left (1-c^2 x^2\right )}+\frac{i \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 c^3 d^3}-\frac{b^2 \tanh ^{-1}(c x)}{6 c^3 d^3}+\frac{b \operatorname{Subst}\left (\int (a+b x) \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{4 c^3 d^3}-\frac{b \operatorname{Subst}\left (\int (a+b x) \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{4 c^3 d^3}\\ &=\frac{b^2 x}{12 c^2 d^3 \left (1-c^2 x^2\right )}-\frac{b \left (a+b \sin ^{-1}(c x)\right )}{6 c^3 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac{b \left (a+b \sin ^{-1}(c x)\right )}{4 c^3 d^3 \sqrt{1-c^2 x^2}}+\frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2 d^3 \left (1-c^2 x^2\right )}+\frac{i \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 c^3 d^3}-\frac{b^2 \tanh ^{-1}(c x)}{6 c^3 d^3}-\frac{i b \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{4 c^3 d^3}+\frac{i b \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{4 c^3 d^3}+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{4 c^3 d^3}-\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{4 c^3 d^3}\\ &=\frac{b^2 x}{12 c^2 d^3 \left (1-c^2 x^2\right )}-\frac{b \left (a+b \sin ^{-1}(c x)\right )}{6 c^3 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac{b \left (a+b \sin ^{-1}(c x)\right )}{4 c^3 d^3 \sqrt{1-c^2 x^2}}+\frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2 d^3 \left (1-c^2 x^2\right )}+\frac{i \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 c^3 d^3}-\frac{b^2 \tanh ^{-1}(c x)}{6 c^3 d^3}-\frac{i b \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{4 c^3 d^3}+\frac{i b \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{4 c^3 d^3}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{4 c^3 d^3}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{4 c^3 d^3}\\ &=\frac{b^2 x}{12 c^2 d^3 \left (1-c^2 x^2\right )}-\frac{b \left (a+b \sin ^{-1}(c x)\right )}{6 c^3 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac{b \left (a+b \sin ^{-1}(c x)\right )}{4 c^3 d^3 \sqrt{1-c^2 x^2}}+\frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2 d^3 \left (1-c^2 x^2\right )}+\frac{i \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 c^3 d^3}-\frac{b^2 \tanh ^{-1}(c x)}{6 c^3 d^3}-\frac{i b \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{4 c^3 d^3}+\frac{i b \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{4 c^3 d^3}+\frac{b^2 \text{Li}_3\left (-i e^{i \sin ^{-1}(c x)}\right )}{4 c^3 d^3}-\frac{b^2 \text{Li}_3\left (i e^{i \sin ^{-1}(c x)}\right )}{4 c^3 d^3}\\ \end{align*}
Mathematica [A] time = 4.37925, size = 446, normalized size = 1.31 \[ \frac{-12 i a b \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )+12 i a b \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )+4 b^2 \left (-3 i \sin ^{-1}(c x) \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )+3 i \sin ^{-1}(c x) \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )+3 \text{PolyLog}\left (3,-i e^{i \sin ^{-1}(c x)}\right )-3 \text{PolyLog}\left (3,i e^{i \sin ^{-1}(c x)}\right )-2 \tanh ^{-1}(c x)+3 i \sin ^{-1}(c x)^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )\right )+\frac{6 a^2 c x}{c^2 x^2-1}+\frac{12 a^2 c x}{\left (c^2 x^2-1\right )^2}+3 a^2 \log (1-c x)-3 a^2 \log (c x+1)+\frac{a b \left (\sqrt{1-c^2 x^2}+12 \sin ^{-1}(c x) \left (c^3 x^3-\left (c^2 x^2-1\right )^2 \log \left (1-i e^{i \sin ^{-1}(c x)}\right )+\left (c^2 x^2-1\right )^2 \log \left (1+i e^{i \sin ^{-1}(c x)}\right )+c x\right )-4 \cos \left (2 \sin ^{-1}(c x)\right )+3 \cos \left (3 \sin ^{-1}(c x)\right )-\cos \left (4 \sin ^{-1}(c x)\right )-3\right )}{\left (c^2 x^2-1\right )^2}+\frac{b^2 \left (2 \sin ^{-1}(c x) \left (\sqrt{1-c^2 x^2}+3 \cos \left (3 \sin ^{-1}(c x)\right )\right )-3 \left (\sin \left (3 \sin ^{-1}(c x)\right )-7 c x\right ) \sin ^{-1}(c x)^2+2 \left (c x+\sin \left (3 \sin ^{-1}(c x)\right )\right )\right )}{2 \left (c^2 x^2-1\right )^2}}{48 c^3 d^3} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.437, size = 894, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{16} \, a^{2}{\left (\frac{2 \,{\left (c^{2} x^{3} + x\right )}}{c^{6} d^{3} x^{4} - 2 \, c^{4} d^{3} x^{2} + c^{2} d^{3}} - \frac{\log \left (c x + 1\right )}{c^{3} d^{3}} + \frac{\log \left (c x - 1\right )}{c^{3} d^{3}}\right )} - \frac{{\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2} \log \left (c x + 1\right ) -{\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2} \log \left (-c x + 1\right ) - 2 \,{\left (b^{2} c^{3} x^{3} + b^{2} c x\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2} + 2 \,{\left (c^{7} d^{3} x^{4} - 2 \, c^{5} d^{3} x^{2} + c^{3} d^{3}\right )} \int \frac{16 \, a b c^{2} x^{2} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) +{\left ({\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) \log \left (c x + 1\right ) -{\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) \log \left (-c x + 1\right ) - 2 \,{\left (b^{2} c^{3} x^{3} + b^{2} c x\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )\right )} \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}}\,{d x}}{16 \,{\left (c^{7} d^{3} x^{4} - 2 \, c^{5} d^{3} x^{2} + c^{3} d^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{b^{2} x^{2} \arcsin \left (c x\right )^{2} + 2 \, a b x^{2} \arcsin \left (c x\right ) + 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\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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