Optimal. Leaf size=275 \[ -\frac{3 b^2 \text{PolyLog}\left (3,e^{2 i \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )}\right ) \left (a+b \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )}{2 c}+\frac{3 i b \text{PolyLog}\left (2,e^{2 i \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )}\right ) \left (a+b \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^2}{2 c}-\frac{3 i b^3 \text{PolyLog}\left (4,e^{2 i \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )}\right )}{4 c}+\frac{i \left (a+b \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^4}{4 b c}-\frac{\log \left (1-e^{2 i \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )}\right ) \left (a+b \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^3}{c} \]
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Rubi [A] time = 0.223743, antiderivative size = 275, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6681, 4625, 3717, 2190, 2531, 6609, 2282, 6589} \[ -\frac{3 b^2 \text{PolyLog}\left (3,e^{2 i \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )}\right ) \left (a+b \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )}{2 c}+\frac{3 i b \text{PolyLog}\left (2,e^{2 i \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )}\right ) \left (a+b \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^2}{2 c}-\frac{3 i b^3 \text{PolyLog}\left (4,e^{2 i \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )}\right )}{4 c}+\frac{i \left (a+b \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^4}{4 b c}-\frac{\log \left (1-e^{2 i \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )}\right ) \left (a+b \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^3}{c} \]
Antiderivative was successfully verified.
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Rule 6681
Rule 4625
Rule 3717
Rule 2190
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{\left (a+b \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \sin ^{-1}(x)\right )^3}{x} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{c}\\ &=-\frac{\operatorname{Subst}\left (\int (a+b x)^3 \cot (x) \, dx,x,\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )}{c}\\ &=\frac{i \left (a+b \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^4}{4 b c}+\frac{(2 i) \operatorname{Subst}\left (\int \frac{e^{2 i x} (a+b x)^3}{1-e^{2 i x}} \, dx,x,\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )}{c}\\ &=\frac{i \left (a+b \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^4}{4 b c}-\frac{\left (a+b \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^3 \log \left (1-e^{2 i \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{c}+\frac{(3 b) \operatorname{Subst}\left (\int (a+b x)^2 \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )}{c}\\ &=\frac{i \left (a+b \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^4}{4 b c}-\frac{\left (a+b \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^3 \log \left (1-e^{2 i \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{c}+\frac{3 i b \left (a+b \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \text{Li}_2\left (e^{2 i \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{2 c}-\frac{\left (3 i b^2\right ) \operatorname{Subst}\left (\int (a+b x) \text{Li}_2\left (e^{2 i x}\right ) \, dx,x,\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )}{c}\\ &=\frac{i \left (a+b \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^4}{4 b c}-\frac{\left (a+b \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^3 \log \left (1-e^{2 i \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{c}+\frac{3 i b \left (a+b \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \text{Li}_2\left (e^{2 i \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{2 c}-\frac{3 b^2 \left (a+b \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right ) \text{Li}_3\left (e^{2 i \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{2 c}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (e^{2 i x}\right ) \, dx,x,\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )}{2 c}\\ &=\frac{i \left (a+b \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^4}{4 b c}-\frac{\left (a+b \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^3 \log \left (1-e^{2 i \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{c}+\frac{3 i b \left (a+b \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \text{Li}_2\left (e^{2 i \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{2 c}-\frac{3 b^2 \left (a+b \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right ) \text{Li}_3\left (e^{2 i \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{2 c}-\frac{\left (3 i b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{2 i \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{4 c}\\ &=\frac{i \left (a+b \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^4}{4 b c}-\frac{\left (a+b \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^3 \log \left (1-e^{2 i \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{c}+\frac{3 i b \left (a+b \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \text{Li}_2\left (e^{2 i \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{2 c}-\frac{3 b^2 \left (a+b \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right ) \text{Li}_3\left (e^{2 i \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{2 c}-\frac{3 i b^3 \text{Li}_4\left (e^{2 i \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{4 c}\\ \end{align*}
Mathematica [F] time = 0.297938, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx \]
Verification is Not applicable to the result.
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Maple [B] time = 0.799, size = 1232, normalized size = 4.5 \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}{2} \, a^{3}{\left (\frac{\log \left (c x + 1\right )}{c} - \frac{\log \left (c x - 1\right )}{c}\right )} - \int \frac{b^{3} \arctan \left (\sqrt{-c x + 1}, \sqrt{2} \sqrt{c} \sqrt{x}\right )^{3} + 3 \, a b^{2} \arctan \left (\sqrt{-c x + 1}, \sqrt{2} \sqrt{c} \sqrt{x}\right )^{2} + 3 \, a^{2} b \arctan \left (\sqrt{-c x + 1}, \sqrt{2} \sqrt{c} \sqrt{x}\right )}{c^{2} x^{2} - 1}\,{d x} \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^{3} \arcsin \left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right )^{3} + 3 \, a b^{2} \arcsin \left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right )^{2} + 3 \, a^{2} b \arcsin \left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right ) + a^{3}}{c^{2} x^{2} - 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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