Optimal. Leaf size=55 \[ -\frac{5 \text{Chi}\left (\sinh ^{-1}(a x)\right )}{64 a^7}+\frac{9 \text{Chi}\left (3 \sinh ^{-1}(a x)\right )}{64 a^7}-\frac{5 \text{Chi}\left (5 \sinh ^{-1}(a x)\right )}{64 a^7}+\frac{\text{Chi}\left (7 \sinh ^{-1}(a x)\right )}{64 a^7} \]
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Rubi [A] time = 0.100261, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {5669, 5448, 3301} \[ -\frac{5 \text{Chi}\left (\sinh ^{-1}(a x)\right )}{64 a^7}+\frac{9 \text{Chi}\left (3 \sinh ^{-1}(a x)\right )}{64 a^7}-\frac{5 \text{Chi}\left (5 \sinh ^{-1}(a x)\right )}{64 a^7}+\frac{\text{Chi}\left (7 \sinh ^{-1}(a x)\right )}{64 a^7} \]
Antiderivative was successfully verified.
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Rule 5669
Rule 5448
Rule 3301
Rubi steps
\begin{align*} \int \frac{x^6}{\sinh ^{-1}(a x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cosh (x) \sinh ^6(x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{a^7}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{5 \cosh (x)}{64 x}+\frac{9 \cosh (3 x)}{64 x}-\frac{5 \cosh (5 x)}{64 x}+\frac{\cosh (7 x)}{64 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a^7}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\cosh (7 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a^7}-\frac{5 \operatorname{Subst}\left (\int \frac{\cosh (x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a^7}-\frac{5 \operatorname{Subst}\left (\int \frac{\cosh (5 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a^7}+\frac{9 \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a^7}\\ &=-\frac{5 \text{Chi}\left (\sinh ^{-1}(a x)\right )}{64 a^7}+\frac{9 \text{Chi}\left (3 \sinh ^{-1}(a x)\right )}{64 a^7}-\frac{5 \text{Chi}\left (5 \sinh ^{-1}(a x)\right )}{64 a^7}+\frac{\text{Chi}\left (7 \sinh ^{-1}(a x)\right )}{64 a^7}\\ \end{align*}
Mathematica [A] time = 0.0118237, size = 40, normalized size = 0.73 \[ \frac{-5 \text{Chi}\left (\sinh ^{-1}(a x)\right )+9 \text{Chi}\left (3 \sinh ^{-1}(a x)\right )-5 \text{Chi}\left (5 \sinh ^{-1}(a x)\right )+\text{Chi}\left (7 \sinh ^{-1}(a x)\right )}{64 a^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 40, normalized size = 0.7 \begin{align*}{\frac{1}{{a}^{7}} \left ( -{\frac{5\,{\it Chi} \left ({\it Arcsinh} \left ( ax \right ) \right ) }{64}}+{\frac{9\,{\it Chi} \left ( 3\,{\it Arcsinh} \left ( ax \right ) \right ) }{64}}-{\frac{5\,{\it Chi} \left ( 5\,{\it Arcsinh} \left ( ax \right ) \right ) }{64}}+{\frac{{\it Chi} \left ( 7\,{\it Arcsinh} \left ( ax \right ) \right ) }{64}} \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{x^{6}}{\operatorname{arsinh}\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{x^{6}}{\operatorname{arsinh}\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{x^{6}}{\operatorname{asinh}{\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{x^{6}}{\operatorname{arsinh}\left (a x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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