Optimal. Leaf size=412 \[ -\frac{2 i b^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{2 i b^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d^2}-\frac{4 a b x \sqrt{1-c^2 x^2}}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{2 b x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{4 i b \sqrt{1-c^2 x^2} \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{2 b^2 \left (1-c^2 x^2\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{4 b^2 x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c^3 d \sqrt{d-c^2 d x^2}} \]
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Rubi [A] time = 0.454065, antiderivative size = 412, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.31, Rules used = {4703, 4677, 4619, 261, 4715, 4657, 4181, 2279, 2391} \[ -\frac{2 i b^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{2 i b^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d^2}-\frac{4 a b x \sqrt{1-c^2 x^2}}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{2 b x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{4 i b \sqrt{1-c^2 x^2} \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{2 b^2 \left (1-c^2 x^2\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{4 b^2 x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c^3 d \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 4703
Rule 4677
Rule 4619
Rule 261
Rule 4715
Rule 4657
Rule 4181
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx &=\frac{x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{2 \int \frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{d-c^2 d x^2}} \, dx}{c^2 d}-\frac{\left (2 b \sqrt{1-c^2 x^2}\right ) \int \frac{x^2 \left (a+b \sin ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{c d \sqrt{d-c^2 d x^2}}\\ &=\frac{2 b x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d^2}-\frac{\left (2 b \sqrt{1-c^2 x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{1-c^2 x^2} \, dx}{c^3 d \sqrt{d-c^2 d x^2}}-\frac{\left (4 b \sqrt{1-c^2 x^2}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{c^3 d \sqrt{d-c^2 d x^2}}-\frac{\left (2 b^2 \sqrt{1-c^2 x^2}\right ) \int \frac{x}{\sqrt{1-c^2 x^2}} \, dx}{c^2 d \sqrt{d-c^2 d x^2}}\\ &=-\frac{4 a b x \sqrt{1-c^2 x^2}}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{2 b^2 \left (1-c^2 x^2\right )}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{2 b x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d^2}-\frac{\left (2 b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{\left (4 b^2 \sqrt{1-c^2 x^2}\right ) \int \sin ^{-1}(c x) \, dx}{c^3 d \sqrt{d-c^2 d x^2}}\\ &=-\frac{4 a b x \sqrt{1-c^2 x^2}}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{2 b^2 \left (1-c^2 x^2\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{4 b^2 x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{2 b x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d^2}+\frac{4 i b \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{\left (2 b^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{\left (2 b^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{\left (4 b^2 \sqrt{1-c^2 x^2}\right ) \int \frac{x}{\sqrt{1-c^2 x^2}} \, dx}{c^2 d \sqrt{d-c^2 d x^2}}\\ &=-\frac{4 a b x \sqrt{1-c^2 x^2}}{c^3 d \sqrt{d-c^2 d x^2}}-\frac{2 b^2 \left (1-c^2 x^2\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{4 b^2 x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{2 b x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d^2}+\frac{4 i b \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{\left (2 i b^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{\left (2 i b^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt{d-c^2 d x^2}}\\ &=-\frac{4 a b x \sqrt{1-c^2 x^2}}{c^3 d \sqrt{d-c^2 d x^2}}-\frac{2 b^2 \left (1-c^2 x^2\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{4 b^2 x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{2 b x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d^2}+\frac{4 i b \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{2 i b^2 \sqrt{1-c^2 x^2} \text{Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{2 i b^2 \sqrt{1-c^2 x^2} \text{Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt{d-c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 0.534469, size = 369, normalized size = 0.9 \[ \frac{-4 i b^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )+4 i b^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )-2 a^2 c^2 x^2+4 a^2+4 a b \sqrt{1-c^2 x^2} \log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )-4 a b \sqrt{1-c^2 x^2} \log \left (\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )+6 a b \sin ^{-1}(c x)-2 a b \sin \left (2 \sin ^{-1}(c x)\right )+2 a b \sin ^{-1}(c x) \cos \left (2 \sin ^{-1}(c x)\right )-4 b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )+4 b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left (1+i e^{i \sin ^{-1}(c x)}\right )+3 b^2 \sin ^{-1}(c x)^2-2 b^2 \sin ^{-1}(c x) \sin \left (2 \sin ^{-1}(c x)\right )-2 b^2 \cos \left (2 \sin ^{-1}(c x)\right )+b^2 \sin ^{-1}(c x)^2 \cos \left (2 \sin ^{-1}(c x)\right )-2 b^2}{2 c^4 d \sqrt{d-c^2 d x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.346, size = 830, normalized size = 2. \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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{{\left (b^{2} x^{3} \arcsin \left (c x\right )^{2} + 2 \, a b x^{3} \arcsin \left (c x\right ) + a^{2} x^{3}\right )} \sqrt{-c^{2} d x^{2} + d}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \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^{3}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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