Optimal. Leaf size=67 \[ \frac{35 c^3 \text{Shi}\left (\cosh ^{-1}(a x)\right )}{64 a}-\frac{21 c^3 \text{Shi}\left (3 \cosh ^{-1}(a x)\right )}{64 a}+\frac{7 c^3 \text{Shi}\left (5 \cosh ^{-1}(a x)\right )}{64 a}-\frac{c^3 \text{Shi}\left (7 \cosh ^{-1}(a x)\right )}{64 a} \]
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Rubi [A] time = 0.140096, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {5700, 3312, 3298} \[ \frac{35 c^3 \text{Shi}\left (\cosh ^{-1}(a x)\right )}{64 a}-\frac{21 c^3 \text{Shi}\left (3 \cosh ^{-1}(a x)\right )}{64 a}+\frac{7 c^3 \text{Shi}\left (5 \cosh ^{-1}(a x)\right )}{64 a}-\frac{c^3 \text{Shi}\left (7 \cosh ^{-1}(a x)\right )}{64 a} \]
Antiderivative was successfully verified.
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Rule 5700
Rule 3312
Rule 3298
Rubi steps
\begin{align*} \int \frac{\left (c-a^2 c x^2\right )^3}{\cosh ^{-1}(a x)} \, dx &=-\frac{c^3 \operatorname{Subst}\left (\int \frac{\sinh ^7(x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{a}\\ &=-\frac{\left (i c^3\right ) \operatorname{Subst}\left (\int \left (\frac{35 i \sinh (x)}{64 x}-\frac{21 i \sinh (3 x)}{64 x}+\frac{7 i \sinh (5 x)}{64 x}-\frac{i \sinh (7 x)}{64 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a}\\ &=-\frac{c^3 \operatorname{Subst}\left (\int \frac{\sinh (7 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a}+\frac{\left (7 c^3\right ) \operatorname{Subst}\left (\int \frac{\sinh (5 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a}-\frac{\left (21 c^3\right ) \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a}+\frac{\left (35 c^3\right ) \operatorname{Subst}\left (\int \frac{\sinh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a}\\ &=\frac{35 c^3 \text{Shi}\left (\cosh ^{-1}(a x)\right )}{64 a}-\frac{21 c^3 \text{Shi}\left (3 \cosh ^{-1}(a x)\right )}{64 a}+\frac{7 c^3 \text{Shi}\left (5 \cosh ^{-1}(a x)\right )}{64 a}-\frac{c^3 \text{Shi}\left (7 \cosh ^{-1}(a x)\right )}{64 a}\\ \end{align*}
Mathematica [A] time = 0.26443, size = 45, normalized size = 0.67 \[ \frac{c^3 \left (35 \text{Shi}\left (\cosh ^{-1}(a x)\right )-21 \text{Shi}\left (3 \cosh ^{-1}(a x)\right )+7 \text{Shi}\left (5 \cosh ^{-1}(a x)\right )-\text{Shi}\left (7 \cosh ^{-1}(a x)\right )\right )}{64 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 44, normalized size = 0.7 \begin{align*}{\frac{{c}^{3} \left ( 35\,{\it Shi} \left ({\rm arccosh} \left (ax\right ) \right ) -21\,{\it Shi} \left ( 3\,{\rm arccosh} \left (ax\right ) \right ) +7\,{\it Shi} \left ( 5\,{\rm arccosh} \left (ax\right ) \right ) -{\it Shi} \left ( 7\,{\rm arccosh} \left (ax\right ) \right ) \right ) }{64\,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{{\left (a^{2} c x^{2} - c\right )}^{3}}{\operatorname{arcosh}\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{a^{6} c^{3} x^{6} - 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} - c^{3}}{\operatorname{arcosh}\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*} - c^{3} \left (\int \frac{3 a^{2} x^{2}}{\operatorname{acosh}{\left (a x \right )}}\, dx + \int - \frac{3 a^{4} x^{4}}{\operatorname{acosh}{\left (a x \right )}}\, dx + \int \frac{a^{6} x^{6}}{\operatorname{acosh}{\left (a x \right )}}\, dx + \int - \frac{1}{\operatorname{acosh}{\left (a x \right )}}\, dx\right ) \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{{\left (a^{2} c x^{2} - c\right )}^{3}}{\operatorname{arcosh}\left (a x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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