Optimal. Leaf size=79 \[ -\frac{3 x}{2 \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}-\frac{1}{2 a \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2}+\frac{3 \text{Chi}\left (\tanh ^{-1}(a x)\right )}{8 a}+\frac{9 \text{Chi}\left (3 \tanh ^{-1}(a x)\right )}{8 a} \]
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Rubi [A] time = 0.361985, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5966, 6032, 6034, 5448, 3301, 5968, 3312} \[ -\frac{3 x}{2 \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}-\frac{1}{2 a \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2}+\frac{3 \text{Chi}\left (\tanh ^{-1}(a x)\right )}{8 a}+\frac{9 \text{Chi}\left (3 \tanh ^{-1}(a x)\right )}{8 a} \]
Antiderivative was successfully verified.
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Rule 5966
Rule 6032
Rule 6034
Rule 5448
Rule 3301
Rule 5968
Rule 3312
Rubi steps
\begin{align*} \int \frac{1}{\left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)^3} \, dx &=-\frac{1}{2 a \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2}+\frac{1}{2} (3 a) \int \frac{x}{\left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)^2} \, dx\\ &=-\frac{1}{2 a \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2}-\frac{3 x}{2 \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}+\frac{3}{2} \int \frac{1}{\left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)} \, dx+\left (3 a^2\right ) \int \frac{x^2}{\left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)} \, dx\\ &=-\frac{1}{2 a \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2}-\frac{3 x}{2 \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}+\frac{3 \operatorname{Subst}\left (\int \frac{\cosh ^3(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{2 a}+\frac{3 \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac{1}{2 a \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2}-\frac{3 x}{2 \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}+\frac{3 \operatorname{Subst}\left (\int \left (\frac{3 \cosh (x)}{4 x}+\frac{\cosh (3 x)}{4 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{2 a}+\frac{3 \operatorname{Subst}\left (\int \left (-\frac{\cosh (x)}{4 x}+\frac{\cosh (3 x)}{4 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac{1}{2 a \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2}-\frac{3 x}{2 \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}+\frac{3 \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{8 a}-\frac{3 \operatorname{Subst}\left (\int \frac{\cosh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{4 a}+\frac{3 \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{4 a}+\frac{9 \operatorname{Subst}\left (\int \frac{\cosh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{8 a}\\ &=-\frac{1}{2 a \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2}-\frac{3 x}{2 \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}+\frac{3 \text{Chi}\left (\tanh ^{-1}(a x)\right )}{8 a}+\frac{9 \text{Chi}\left (3 \tanh ^{-1}(a x)\right )}{8 a}\\ \end{align*}
Mathematica [A] time = 0.206677, size = 56, normalized size = 0.71 \[ \frac{-\frac{4 \left (3 a x \tanh ^{-1}(a x)+1\right )}{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2}+3 \text{Chi}\left (\tanh ^{-1}(a x)\right )+9 \text{Chi}\left (3 \tanh ^{-1}(a x)\right )}{8 a} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.163, size = 180, normalized size = 2.3 \begin{align*}{\frac{1}{8\,a \left ({\it Artanh} \left ( ax \right ) \right ) ^{2} \left ({a}^{2}{x}^{2}-1 \right ) } \left ( 9\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}{\it Chi} \left ( 3\,{\it Artanh} \left ( ax \right ) \right ){x}^{2}{a}^{2}+3\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}{\it Chi} \left ({\it Artanh} \left ( ax \right ) \right ){x}^{2}{a}^{2}-3\,{\it Artanh} \left ( ax \right ) \sinh \left ( 3\,{\it Artanh} \left ( ax \right ) \right ){x}^{2}{a}^{2}-\cosh \left ( 3\,{\it Artanh} \left ( ax \right ) \right ){x}^{2}{a}^{2}+3\,\sqrt{-{a}^{2}{x}^{2}+1}ax{\it Artanh} \left ( ax \right ) -9\,{\it Chi} \left ( 3\,{\it Artanh} \left ( ax \right ) \right ) \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}-3\,{\it Chi} \left ({\it Artanh} \left ( ax \right ) \right ) \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}+3\,\sinh \left ( 3\,{\it Artanh} \left ( ax \right ) \right ){\it Artanh} \left ( ax \right ) +3\,\sqrt{-{a}^{2}{x}^{2}+1}+\cosh \left ( 3\,{\it Artanh} \left ( ax \right ) \right ) \right ) } \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*} \int \frac{1}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{5}{2}} \operatorname{artanh}\left (a x\right )^{3}}\,{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{\sqrt{-a^{2} x^{2} + 1}}{{\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \operatorname{artanh}\left (a x\right )^{3}}, 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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{5}{2}} \operatorname{artanh}\left (a x\right )^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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