Optimal. Leaf size=41 \[ -\frac {\text {Chi}\left (2 \sinh ^{-1}(a x)\right )}{2 a^5}+\frac {\text {Chi}\left (4 \sinh ^{-1}(a x)\right )}{8 a^5}+\frac {3 \log \left (\sinh ^{-1}(a x)\right )}{8 a^5} \]
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Rubi [A]
time = 0.10, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {5819, 3393,
3382} \begin {gather*} -\frac {\text {Chi}\left (2 \sinh ^{-1}(a x)\right )}{2 a^5}+\frac {\text {Chi}\left (4 \sinh ^{-1}(a x)\right )}{8 a^5}+\frac {3 \log \left (\sinh ^{-1}(a x)\right )}{8 a^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 3382
Rule 3393
Rule 5819
Rubi steps
\begin {align*} \int \frac {x^4}{\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)} \, dx &=\frac {\text {Subst}\left (\int \frac {\sinh ^4(x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{a^5}\\ &=\frac {\text {Subst}\left (\int \left (\frac {3}{8 x}-\frac {\cosh (2 x)}{2 x}+\frac {\cosh (4 x)}{8 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a^5}\\ &=\frac {3 \log \left (\sinh ^{-1}(a x)\right )}{8 a^5}+\frac {\text {Subst}\left (\int \frac {\cosh (4 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^5}-\frac {\text {Subst}\left (\int \frac {\cosh (2 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^5}\\ &=-\frac {\text {Chi}\left (2 \sinh ^{-1}(a x)\right )}{2 a^5}+\frac {\text {Chi}\left (4 \sinh ^{-1}(a x)\right )}{8 a^5}+\frac {3 \log \left (\sinh ^{-1}(a x)\right )}{8 a^5}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 31, normalized size = 0.76 \begin {gather*} \frac {-4 \text {Chi}\left (2 \sinh ^{-1}(a x)\right )+\text {Chi}\left (4 \sinh ^{-1}(a x)\right )+3 \log \left (\sinh ^{-1}(a x)\right )}{8 a^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.95, size = 30, normalized size = 0.73
method | result | size |
default | \(\frac {3 \ln \left (\arcsinh \left (a x \right )\right )-4 \hyperbolicCosineIntegral \left (2 \arcsinh \left (a x \right )\right )+\hyperbolicCosineIntegral \left (4 \arcsinh \left (a x \right )\right )}{8 a^{5}}\) | \(30\) |
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^{4}}{\sqrt {a^{2} x^{2} + 1} \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]
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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^4}{\mathrm {asinh}\left (a\,x\right )\,\sqrt {a^2\,x^2+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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