Optimal. Leaf size=455 \[ \frac{9 i \sin ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(a x)}\right )}{8 a c^3}-\frac{9 i \sin ^{-1}(a x)^2 \text{PolyLog}\left (2,i e^{i \sin ^{-1}(a x)}\right )}{8 a c^3}-\frac{9 \sin ^{-1}(a x) \text{PolyLog}\left (3,-i e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}+\frac{9 \sin ^{-1}(a x) \text{PolyLog}\left (3,i e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}+\frac{5 i \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(a x)}\right )}{2 a c^3}-\frac{5 i \text{PolyLog}\left (2,i e^{i \sin ^{-1}(a x)}\right )}{2 a c^3}-\frac{9 i \text{PolyLog}\left (4,-i e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}+\frac{9 i \text{PolyLog}\left (4,i e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}-\frac{1}{4 a c^3 \sqrt{1-a^2 x^2}}+\frac{3 x \sin ^{-1}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}+\frac{x \sin ^{-1}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}-\frac{9 \sin ^{-1}(a x)^2}{8 a c^3 \sqrt{1-a^2 x^2}}-\frac{\sin ^{-1}(a x)^2}{4 a c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{x \sin ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )}-\frac{3 i \sin ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}-\frac{5 i \sin ^{-1}(a x) \tan ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )}{a c^3} \]
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Rubi [A] time = 0.507124, antiderivative size = 455, normalized size of antiderivative = 1., number of steps used = 28, number of rules used = 11, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.55, Rules used = {4655, 4657, 4181, 2531, 6609, 2282, 6589, 4677, 2279, 2391, 261} \[ \frac{9 i \sin ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(a x)}\right )}{8 a c^3}-\frac{9 i \sin ^{-1}(a x)^2 \text{PolyLog}\left (2,i e^{i \sin ^{-1}(a x)}\right )}{8 a c^3}-\frac{9 \sin ^{-1}(a x) \text{PolyLog}\left (3,-i e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}+\frac{9 \sin ^{-1}(a x) \text{PolyLog}\left (3,i e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}+\frac{5 i \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(a x)}\right )}{2 a c^3}-\frac{5 i \text{PolyLog}\left (2,i e^{i \sin ^{-1}(a x)}\right )}{2 a c^3}-\frac{9 i \text{PolyLog}\left (4,-i e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}+\frac{9 i \text{PolyLog}\left (4,i e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}-\frac{1}{4 a c^3 \sqrt{1-a^2 x^2}}+\frac{3 x \sin ^{-1}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}+\frac{x \sin ^{-1}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}-\frac{9 \sin ^{-1}(a x)^2}{8 a c^3 \sqrt{1-a^2 x^2}}-\frac{\sin ^{-1}(a x)^2}{4 a c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{x \sin ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )}-\frac{3 i \sin ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}-\frac{5 i \sin ^{-1}(a x) \tan ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )}{a c^3} \]
Antiderivative was successfully verified.
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Rule 4655
Rule 4657
Rule 4181
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rule 4677
Rule 2279
Rule 2391
Rule 261
Rubi steps
\begin{align*} \int \frac{\sin ^{-1}(a x)^3}{\left (c-a^2 c x^2\right )^3} \, dx &=\frac{x \sin ^{-1}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}-\frac{(3 a) \int \frac{x \sin ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{4 c^3}+\frac{3 \int \frac{\sin ^{-1}(a x)^3}{\left (c-a^2 c x^2\right )^2} \, dx}{4 c}\\ &=-\frac{\sin ^{-1}(a x)^2}{4 a c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{x \sin ^{-1}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac{3 x \sin ^{-1}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}+\frac{\int \frac{\sin ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx}{2 c^3}-\frac{(9 a) \int \frac{x \sin ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{8 c^3}+\frac{3 \int \frac{\sin ^{-1}(a x)^3}{c-a^2 c x^2} \, dx}{8 c^2}\\ &=\frac{x \sin ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )}-\frac{\sin ^{-1}(a x)^2}{4 a c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac{9 \sin ^{-1}(a x)^2}{8 a c^3 \sqrt{1-a^2 x^2}}+\frac{x \sin ^{-1}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac{3 x \sin ^{-1}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}+\frac{\int \frac{\sin ^{-1}(a x)}{1-a^2 x^2} \, dx}{4 c^3}+\frac{9 \int \frac{\sin ^{-1}(a x)}{1-a^2 x^2} \, dx}{4 c^3}+\frac{3 \operatorname{Subst}\left (\int x^3 \sec (x) \, dx,x,\sin ^{-1}(a x)\right )}{8 a c^3}-\frac{a \int \frac{x}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{4 c^3}\\ &=-\frac{1}{4 a c^3 \sqrt{1-a^2 x^2}}+\frac{x \sin ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )}-\frac{\sin ^{-1}(a x)^2}{4 a c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac{9 \sin ^{-1}(a x)^2}{8 a c^3 \sqrt{1-a^2 x^2}}+\frac{x \sin ^{-1}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac{3 x \sin ^{-1}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}-\frac{3 i \sin ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}+\frac{\operatorname{Subst}\left (\int x \sec (x) \, dx,x,\sin ^{-1}(a x)\right )}{4 a c^3}-\frac{9 \operatorname{Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{8 a c^3}+\frac{9 \operatorname{Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{8 a c^3}+\frac{9 \operatorname{Subst}\left (\int x \sec (x) \, dx,x,\sin ^{-1}(a x)\right )}{4 a c^3}\\ &=-\frac{1}{4 a c^3 \sqrt{1-a^2 x^2}}+\frac{x \sin ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )}-\frac{\sin ^{-1}(a x)^2}{4 a c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac{9 \sin ^{-1}(a x)^2}{8 a c^3 \sqrt{1-a^2 x^2}}+\frac{x \sin ^{-1}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac{3 x \sin ^{-1}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}-\frac{5 i \sin ^{-1}(a x) \tan ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )}{a c^3}-\frac{3 i \sin ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}+\frac{9 i \sin ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \sin ^{-1}(a x)}\right )}{8 a c^3}-\frac{9 i \sin ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \sin ^{-1}(a x)}\right )}{8 a c^3}-\frac{(9 i) \operatorname{Subst}\left (\int x \text{Li}_2\left (-i e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{4 a c^3}+\frac{(9 i) \operatorname{Subst}\left (\int x \text{Li}_2\left (i e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{4 a c^3}-\frac{\operatorname{Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{4 a c^3}+\frac{\operatorname{Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{4 a c^3}-\frac{9 \operatorname{Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{4 a c^3}+\frac{9 \operatorname{Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{4 a c^3}\\ &=-\frac{1}{4 a c^3 \sqrt{1-a^2 x^2}}+\frac{x \sin ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )}-\frac{\sin ^{-1}(a x)^2}{4 a c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac{9 \sin ^{-1}(a x)^2}{8 a c^3 \sqrt{1-a^2 x^2}}+\frac{x \sin ^{-1}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac{3 x \sin ^{-1}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}-\frac{5 i \sin ^{-1}(a x) \tan ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )}{a c^3}-\frac{3 i \sin ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}+\frac{9 i \sin ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \sin ^{-1}(a x)}\right )}{8 a c^3}-\frac{9 i \sin ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \sin ^{-1}(a x)}\right )}{8 a c^3}-\frac{9 \sin ^{-1}(a x) \text{Li}_3\left (-i e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}+\frac{9 \sin ^{-1}(a x) \text{Li}_3\left (i e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}+\frac{i \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}-\frac{i \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}+\frac{(9 i) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}-\frac{(9 i) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}+\frac{9 \operatorname{Subst}\left (\int \text{Li}_3\left (-i e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{4 a c^3}-\frac{9 \operatorname{Subst}\left (\int \text{Li}_3\left (i e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{4 a c^3}\\ &=-\frac{1}{4 a c^3 \sqrt{1-a^2 x^2}}+\frac{x \sin ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )}-\frac{\sin ^{-1}(a x)^2}{4 a c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac{9 \sin ^{-1}(a x)^2}{8 a c^3 \sqrt{1-a^2 x^2}}+\frac{x \sin ^{-1}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac{3 x \sin ^{-1}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}-\frac{5 i \sin ^{-1}(a x) \tan ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )}{a c^3}-\frac{3 i \sin ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}+\frac{5 i \text{Li}_2\left (-i e^{i \sin ^{-1}(a x)}\right )}{2 a c^3}+\frac{9 i \sin ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \sin ^{-1}(a x)}\right )}{8 a c^3}-\frac{5 i \text{Li}_2\left (i e^{i \sin ^{-1}(a x)}\right )}{2 a c^3}-\frac{9 i \sin ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \sin ^{-1}(a x)}\right )}{8 a c^3}-\frac{9 \sin ^{-1}(a x) \text{Li}_3\left (-i e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}+\frac{9 \sin ^{-1}(a x) \text{Li}_3\left (i e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}-\frac{(9 i) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-i x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}+\frac{(9 i) \operatorname{Subst}\left (\int \frac{\text{Li}_3(i x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}\\ &=-\frac{1}{4 a c^3 \sqrt{1-a^2 x^2}}+\frac{x \sin ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )}-\frac{\sin ^{-1}(a x)^2}{4 a c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac{9 \sin ^{-1}(a x)^2}{8 a c^3 \sqrt{1-a^2 x^2}}+\frac{x \sin ^{-1}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac{3 x \sin ^{-1}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}-\frac{5 i \sin ^{-1}(a x) \tan ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )}{a c^3}-\frac{3 i \sin ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}+\frac{5 i \text{Li}_2\left (-i e^{i \sin ^{-1}(a x)}\right )}{2 a c^3}+\frac{9 i \sin ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \sin ^{-1}(a x)}\right )}{8 a c^3}-\frac{5 i \text{Li}_2\left (i e^{i \sin ^{-1}(a x)}\right )}{2 a c^3}-\frac{9 i \sin ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \sin ^{-1}(a x)}\right )}{8 a c^3}-\frac{9 \sin ^{-1}(a x) \text{Li}_3\left (-i e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}+\frac{9 \sin ^{-1}(a x) \text{Li}_3\left (i e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}-\frac{9 i \text{Li}_4\left (-i e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}+\frac{9 i \text{Li}_4\left (i e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}\\ \end{align*}
Mathematica [B] time = 12.4414, size = 1544, normalized size = 3.39 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.224, size = 726, normalized size = 1.6 \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 [A] time = 3.31889, size = 105, normalized size = 0.23 \begin{align*} -\frac{1}{16} \,{\left (\frac{2 \,{\left (3 \, a^{2} x^{3} - 5 \, x\right )}}{a^{4} c^{3} x^{4} - 2 \, a^{2} c^{3} x^{2} + c^{3}} - \frac{3 \, \log \left (a x + 1\right )}{a c^{3}} + \frac{3 \, \log \left (a x - 1\right )}{a c^{3}}\right )} \arcsin \left (a x\right )^{3} \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{\arcsin \left (a x\right )^{3}}{a^{6} c^{3} x^{6} - 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} - c^{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{\operatorname{asin}^{3}{\left (a x \right )}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, dx}{c^{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{\arcsin \left (a x\right )^{3}}{{\left (a^{2} c x^{2} - c\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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