Optimal. Leaf size=43 \[ \frac {5 \text {Shi}\left (2 \sinh ^{-1}(a x)\right )}{32 a^6}-\frac {\text {Shi}\left (4 \sinh ^{-1}(a x)\right )}{8 a^6}+\frac {\text {Shi}\left (6 \sinh ^{-1}(a x)\right )}{32 a^6} \]
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Rubi [A]
time = 0.06, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5780, 5556,
3379} \begin {gather*} \frac {5 \text {Shi}\left (2 \sinh ^{-1}(a x)\right )}{32 a^6}-\frac {\text {Shi}\left (4 \sinh ^{-1}(a x)\right )}{8 a^6}+\frac {\text {Shi}\left (6 \sinh ^{-1}(a x)\right )}{32 a^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 3379
Rule 5556
Rule 5780
Rubi steps
\begin {align*} \int \frac {x^5}{\sinh ^{-1}(a x)} \, dx &=\frac {\text {Subst}\left (\int \frac {\cosh (x) \sinh ^5(x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{a^6}\\ &=\frac {\text {Subst}\left (\int \left (\frac {5 \sinh (2 x)}{32 x}-\frac {\sinh (4 x)}{8 x}+\frac {\sinh (6 x)}{32 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a^6}\\ &=\frac {\text {Subst}\left (\int \frac {\sinh (6 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{32 a^6}-\frac {\text {Subst}\left (\int \frac {\sinh (4 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^6}+\frac {5 \text {Subst}\left (\int \frac {\sinh (2 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{32 a^6}\\ &=\frac {5 \text {Shi}\left (2 \sinh ^{-1}(a x)\right )}{32 a^6}-\frac {\text {Shi}\left (4 \sinh ^{-1}(a x)\right )}{8 a^6}+\frac {\text {Shi}\left (6 \sinh ^{-1}(a x)\right )}{32 a^6}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 33, normalized size = 0.77 \begin {gather*} \frac {5 \text {Shi}\left (2 \sinh ^{-1}(a x)\right )-4 \text {Shi}\left (4 \sinh ^{-1}(a x)\right )+\text {Shi}\left (6 \sinh ^{-1}(a x)\right )}{32 a^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.68, size = 33, normalized size = 0.77
method | result | size |
derivativedivides | \(\frac {\frac {5 \hyperbolicSineIntegral \left (2 \arcsinh \left (a x \right )\right )}{32}-\frac {\hyperbolicSineIntegral \left (4 \arcsinh \left (a x \right )\right )}{8}+\frac {\hyperbolicSineIntegral \left (6 \arcsinh \left (a x \right )\right )}{32}}{a^{6}}\) | \(33\) |
default | \(\frac {\frac {5 \hyperbolicSineIntegral \left (2 \arcsinh \left (a x \right )\right )}{32}-\frac {\hyperbolicSineIntegral \left (4 \arcsinh \left (a x \right )\right )}{8}+\frac {\hyperbolicSineIntegral \left (6 \arcsinh \left (a x \right )\right )}{32}}{a^{6}}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{5}}{\operatorname {asinh}{\left (a x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^5}{\mathrm {asinh}\left (a\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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