Optimal. Leaf size=66 \[ -\frac{1}{a \left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)}+\frac{5 \text{Shi}\left (\tanh ^{-1}(a x)\right )}{8 a}+\frac{15 \text{Shi}\left (3 \tanh ^{-1}(a x)\right )}{16 a}+\frac{5 \text{Shi}\left (5 \tanh ^{-1}(a x)\right )}{16 a} \]
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Rubi [A] time = 0.176986, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {5966, 6034, 5448, 3298} \[ -\frac{1}{a \left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)}+\frac{5 \text{Shi}\left (\tanh ^{-1}(a x)\right )}{8 a}+\frac{15 \text{Shi}\left (3 \tanh ^{-1}(a x)\right )}{16 a}+\frac{5 \text{Shi}\left (5 \tanh ^{-1}(a x)\right )}{16 a} \]
Antiderivative was successfully verified.
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Rule 5966
Rule 6034
Rule 5448
Rule 3298
Rubi steps
\begin{align*} \int \frac{1}{\left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)^2} \, dx &=-\frac{1}{a \left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)}+(5 a) \int \frac{x}{\left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)} \, dx\\ &=-\frac{1}{a \left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)}+\frac{5 \operatorname{Subst}\left (\int \frac{\cosh ^4(x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac{1}{a \left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)}+\frac{5 \operatorname{Subst}\left (\int \left (\frac{\sinh (x)}{8 x}+\frac{3 \sinh (3 x)}{16 x}+\frac{\sinh (5 x)}{16 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac{1}{a \left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)}+\frac{5 \operatorname{Subst}\left (\int \frac{\sinh (5 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{16 a}+\frac{5 \operatorname{Subst}\left (\int \frac{\sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{8 a}+\frac{15 \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{16 a}\\ &=-\frac{1}{a \left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)}+\frac{5 \text{Shi}\left (\tanh ^{-1}(a x)\right )}{8 a}+\frac{15 \text{Shi}\left (3 \tanh ^{-1}(a x)\right )}{16 a}+\frac{5 \text{Shi}\left (5 \tanh ^{-1}(a x)\right )}{16 a}\\ \end{align*}
Mathematica [A] time = 0.166656, size = 56, normalized size = 0.85 \[ \frac{5 \left (2 \text{Shi}\left (\tanh ^{-1}(a x)\right )+3 \text{Shi}\left (3 \tanh ^{-1}(a x)\right )+\text{Shi}\left (5 \tanh ^{-1}(a x)\right )\right )-\frac{16}{\left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)}}{16 a} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.171, size = 176, normalized size = 2.7 \begin{align*}{\frac{1}{16\,a{\it Artanh} \left ( ax \right ) \left ({a}^{2}{x}^{2}-1 \right ) } \left ( 5\,{\it Artanh} \left ( ax \right ){\it Shi} \left ( 5\,{\it Artanh} \left ( ax \right ) \right ){x}^{2}{a}^{2}+10\,{\it Artanh} \left ( ax \right ){\it Shi} \left ({\it Artanh} \left ( ax \right ) \right ){x}^{2}{a}^{2}+15\,{\it Artanh} \left ( ax \right ){\it Shi} \left ( 3\,{\it Artanh} \left ( ax \right ) \right ){x}^{2}{a}^{2}-5\,\cosh \left ( 3\,{\it Artanh} \left ( ax \right ) \right ){x}^{2}{a}^{2}-\cosh \left ( 5\,{\it Artanh} \left ( ax \right ) \right ){x}^{2}{a}^{2}-5\,{\it Shi} \left ( 5\,{\it Artanh} \left ( ax \right ) \right ){\it Artanh} \left ( ax \right ) -10\,{\it Shi} \left ({\it Artanh} \left ( ax \right ) \right ){\it Artanh} \left ( ax \right ) -15\,{\it Shi} \left ( 3\,{\it Artanh} \left ( ax \right ) \right ){\it Artanh} \left ( ax \right ) +10\,\sqrt{-{a}^{2}{x}^{2}+1}+5\,\cosh \left ( 3\,{\it Artanh} \left ( ax \right ) \right ) +\cosh \left ( 5\,{\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{7}{2}} \operatorname{artanh}\left (a x\right )^{2}}\,{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 )^{2}}, 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 )^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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