Optimal. Leaf size=388 \[ -\frac{2 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(a x)}\right )}{a c^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \text{PolyLog}\left (3,-e^{2 i \sin ^{-1}(a x)}\right )}{a c^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt{c-a^2 c x^2}}+\frac{2 x \sin ^{-1}(a x)^3}{3 c^2 \sqrt{c-a^2 c x^2}}-\frac{2 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a c^2 \sqrt{c-a^2 c x^2}}-\frac{\sin ^{-1}(a x)^2}{2 a c^2 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)}{c^2 \sqrt{c-a^2 c x^2}}+\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2 \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{a c^2 \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.305563, antiderivative size = 388, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 11, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4655, 4653, 4675, 3719, 2190, 2531, 2282, 6589, 4677, 4651, 260} \[ -\frac{2 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(a x)}\right )}{a c^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \text{PolyLog}\left (3,-e^{2 i \sin ^{-1}(a x)}\right )}{a c^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt{c-a^2 c x^2}}+\frac{2 x \sin ^{-1}(a x)^3}{3 c^2 \sqrt{c-a^2 c x^2}}-\frac{2 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a c^2 \sqrt{c-a^2 c x^2}}-\frac{\sin ^{-1}(a x)^2}{2 a c^2 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)}{c^2 \sqrt{c-a^2 c x^2}}+\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2 \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{a c^2 \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4655
Rule 4653
Rule 4675
Rule 3719
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rule 4677
Rule 4651
Rule 260
Rubi steps
\begin{align*} \int \frac{\sin ^{-1}(a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=\frac{x \sin ^{-1}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac{2 \int \frac{\sin ^{-1}(a x)^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{3 c}-\frac{\left (a \sqrt{1-a^2 x^2}\right ) \int \frac{x \sin ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx}{c^2 \sqrt{c-a^2 c x^2}}\\ &=-\frac{\sin ^{-1}(a x)^2}{2 a c^2 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac{2 x \sin ^{-1}(a x)^3}{3 c^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \int \frac{\sin ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c^2 \sqrt{c-a^2 c x^2}}-\frac{\left (2 a \sqrt{1-a^2 x^2}\right ) \int \frac{x \sin ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{c^2 \sqrt{c-a^2 c x^2}}\\ &=\frac{x \sin ^{-1}(a x)}{c^2 \sqrt{c-a^2 c x^2}}-\frac{\sin ^{-1}(a x)^2}{2 a c^2 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac{2 x \sin ^{-1}(a x)^3}{3 c^2 \sqrt{c-a^2 c x^2}}-\frac{\left (2 \sqrt{1-a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \tan (x) \, dx,x,\sin ^{-1}(a x)\right )}{a c^2 \sqrt{c-a^2 c x^2}}-\frac{\left (a \sqrt{1-a^2 x^2}\right ) \int \frac{x}{1-a^2 x^2} \, dx}{c^2 \sqrt{c-a^2 c x^2}}\\ &=\frac{x \sin ^{-1}(a x)}{c^2 \sqrt{c-a^2 c x^2}}-\frac{\sin ^{-1}(a x)^2}{2 a c^2 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac{2 x \sin ^{-1}(a x)^3}{3 c^2 \sqrt{c-a^2 c x^2}}-\frac{2 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a c^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt{c-a^2 c x^2}}+\frac{\left (4 i \sqrt{1-a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 i x} x^2}{1+e^{2 i x}} \, dx,x,\sin ^{-1}(a x)\right )}{a c^2 \sqrt{c-a^2 c x^2}}\\ &=\frac{x \sin ^{-1}(a x)}{c^2 \sqrt{c-a^2 c x^2}}-\frac{\sin ^{-1}(a x)^2}{2 a c^2 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac{2 x \sin ^{-1}(a x)^3}{3 c^2 \sqrt{c-a^2 c x^2}}-\frac{2 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a c^2 \sqrt{c-a^2 c x^2}}+\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2 \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{a c^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt{c-a^2 c x^2}}-\frac{\left (4 \sqrt{1-a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a c^2 \sqrt{c-a^2 c x^2}}\\ &=\frac{x \sin ^{-1}(a x)}{c^2 \sqrt{c-a^2 c x^2}}-\frac{\sin ^{-1}(a x)^2}{2 a c^2 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac{2 x \sin ^{-1}(a x)^3}{3 c^2 \sqrt{c-a^2 c x^2}}-\frac{2 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a c^2 \sqrt{c-a^2 c x^2}}+\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2 \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{a c^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt{c-a^2 c x^2}}-\frac{2 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \text{Li}_2\left (-e^{2 i \sin ^{-1}(a x)}\right )}{a c^2 \sqrt{c-a^2 c x^2}}+\frac{\left (2 i \sqrt{1-a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a c^2 \sqrt{c-a^2 c x^2}}\\ &=\frac{x \sin ^{-1}(a x)}{c^2 \sqrt{c-a^2 c x^2}}-\frac{\sin ^{-1}(a x)^2}{2 a c^2 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac{2 x \sin ^{-1}(a x)^3}{3 c^2 \sqrt{c-a^2 c x^2}}-\frac{2 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a c^2 \sqrt{c-a^2 c x^2}}+\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2 \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{a c^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt{c-a^2 c x^2}}-\frac{2 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \text{Li}_2\left (-e^{2 i \sin ^{-1}(a x)}\right )}{a c^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(a x)}\right )}{a c^2 \sqrt{c-a^2 c x^2}}\\ &=\frac{x \sin ^{-1}(a x)}{c^2 \sqrt{c-a^2 c x^2}}-\frac{\sin ^{-1}(a x)^2}{2 a c^2 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac{2 x \sin ^{-1}(a x)^3}{3 c^2 \sqrt{c-a^2 c x^2}}-\frac{2 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a c^2 \sqrt{c-a^2 c x^2}}+\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2 \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{a c^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt{c-a^2 c x^2}}-\frac{2 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \text{Li}_2\left (-e^{2 i \sin ^{-1}(a x)}\right )}{a c^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \text{Li}_3\left (-e^{2 i \sin ^{-1}(a x)}\right )}{a c^2 \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.558723, size = 211, normalized size = 0.54 \[ \frac{\left (1-a^2 x^2\right )^{3/2} \left (-12 i \sin ^{-1}(a x) \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(a x)}\right )+6 \text{PolyLog}\left (3,-e^{2 i \sin ^{-1}(a x)}\right )+3 \log \left (1-a^2 x^2\right )+\frac{4 a x \sin ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}+\frac{2 a x \sin ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}}+\frac{3 \sin ^{-1}(a x)^2}{a^2 x^2-1}+\frac{6 a x \sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-4 i \sin ^{-1}(a x)^3+12 \sin ^{-1}(a x)^2 \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )\right )}{6 a c \left (c-a^2 c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.207, size = 661, normalized size = 1.7 \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 = 2.70521, size = 143, normalized size = 0.37 \begin{align*} \frac{1}{2} \, a{\left (\frac{1}{a^{4} c^{\frac{5}{2}} x^{2} - a^{2} c^{\frac{5}{2}}} + \frac{2 \, \log \left (a x + 1\right )}{a^{2} c^{\frac{5}{2}}} + \frac{2 \, \log \left (a x - 1\right )}{a^{2} c^{\frac{5}{2}}}\right )} \arcsin \left (a x\right )^{2} + \frac{1}{3} \,{\left (\frac{2 \, x}{\sqrt{-a^{2} c x^{2} + c} c^{2}} + \frac{x}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}} c}\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{\sqrt{-a^{2} c x^{2} + c} \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*} \int \frac{\operatorname{asin}^{3}{\left (a x \right )}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{5}{2}}}\, dx \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 )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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