Optimal. Leaf size=27 \[ \frac{3 \text{Chi}\left (\tanh ^{-1}(a x)\right )}{4 a}+\frac{\text{Chi}\left (3 \tanh ^{-1}(a x)\right )}{4 a} \]
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Rubi [A] time = 0.0930111, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {5968, 3312, 3301} \[ \frac{3 \text{Chi}\left (\tanh ^{-1}(a x)\right )}{4 a}+\frac{\text{Chi}\left (3 \tanh ^{-1}(a x)\right )}{4 a} \]
Antiderivative was successfully verified.
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Rule 5968
Rule 3312
Rule 3301
Rubi steps
\begin{align*} \int \frac{1}{\left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cosh ^3(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=\frac{\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 )}{a}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\cosh (3 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{4 a}+\frac{3 \operatorname{Subst}\left (\int \frac{\cosh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{4 a}\\ &=\frac{3 \text{Chi}\left (\tanh ^{-1}(a x)\right )}{4 a}+\frac{\text{Chi}\left (3 \tanh ^{-1}(a x)\right )}{4 a}\\ \end{align*}
Mathematica [A] time = 0.0602268, size = 22, normalized size = 0.81 \[ \frac{3 \text{Chi}\left (\tanh ^{-1}(a x)\right )+\text{Chi}\left (3 \tanh ^{-1}(a x)\right )}{4 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.155, size = 21, normalized size = 0.8 \begin{align*}{\frac{3\,{\it Chi} \left ({\it Artanh} \left ( ax \right ) \right ) +{\it Chi} \left ( 3\,{\it Artanh} \left ( ax \right ) \right ) }{4\,a}} \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 )}\,{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 )}, 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{1}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{5}{2}} \operatorname{atanh}{\left (a x \right )}}\, 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{1}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{5}{2}} \operatorname{artanh}\left (a x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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