Optimal. Leaf size=41 \[ \frac{5 \text{Chi}\left (\tanh ^{-1}(a x)\right )}{8 a}+\frac{5 \text{Chi}\left (3 \tanh ^{-1}(a x)\right )}{16 a}+\frac{\text{Chi}\left (5 \tanh ^{-1}(a x)\right )}{16 a} \]
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Rubi [A] time = 0.108834, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {5968, 3312, 3301} \[ \frac{5 \text{Chi}\left (\tanh ^{-1}(a x)\right )}{8 a}+\frac{5 \text{Chi}\left (3 \tanh ^{-1}(a x)\right )}{16 a}+\frac{\text{Chi}\left (5 \tanh ^{-1}(a x)\right )}{16 a} \]
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 )^{7/2} \tanh ^{-1}(a x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cosh ^5(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{5 \cosh (x)}{8 x}+\frac{5 \cosh (3 x)}{16 x}+\frac{\cosh (5 x)}{16 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\cosh (5 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{16 a}+\frac{5 \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{16 a}+\frac{5 \operatorname{Subst}\left (\int \frac{\cosh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{8 a}\\ &=\frac{5 \text{Chi}\left (\tanh ^{-1}(a x)\right )}{8 a}+\frac{5 \text{Chi}\left (3 \tanh ^{-1}(a x)\right )}{16 a}+\frac{\text{Chi}\left (5 \tanh ^{-1}(a x)\right )}{16 a}\\ \end{align*}
Mathematica [A] time = 0.0627063, size = 31, normalized size = 0.76 \[ \frac{10 \text{Chi}\left (\tanh ^{-1}(a x)\right )+5 \text{Chi}\left (3 \tanh ^{-1}(a x)\right )+\text{Chi}\left (5 \tanh ^{-1}(a x)\right )}{16 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.154, size = 30, normalized size = 0.7 \begin{align*}{\frac{10\,{\it Chi} \left ({\it Artanh} \left ( ax \right ) \right ) +5\,{\it Chi} \left ( 3\,{\it Artanh} \left ( ax \right ) \right ) +{\it Chi} \left ( 5\,{\it Artanh} \left ( ax \right ) \right ) }{16\,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{7}{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^{8} x^{8} - 4 \, a^{6} x^{6} + 6 \, a^{4} x^{4} - 4 \, 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(-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{7}{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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