Optimal. Leaf size=68 \[ -\frac {x^4 \sqrt {1+a^2 x^2}}{a \sinh ^{-1}(a x)}+\frac {\text {Shi}\left (\sinh ^{-1}(a x)\right )}{8 a^5}-\frac {9 \text {Shi}\left (3 \sinh ^{-1}(a x)\right )}{16 a^5}+\frac {5 \text {Shi}\left (5 \sinh ^{-1}(a x)\right )}{16 a^5} \]
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Rubi [A]
time = 0.05, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5778, 3379}
\begin {gather*} \frac {\text {Shi}\left (\sinh ^{-1}(a x)\right )}{8 a^5}-\frac {9 \text {Shi}\left (3 \sinh ^{-1}(a x)\right )}{16 a^5}+\frac {5 \text {Shi}\left (5 \sinh ^{-1}(a x)\right )}{16 a^5}-\frac {x^4 \sqrt {a^2 x^2+1}}{a \sinh ^{-1}(a x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 3379
Rule 5778
Rubi steps
\begin {align*} \int \frac {x^4}{\sinh ^{-1}(a x)^2} \, dx &=-\frac {x^4 \sqrt {1+a^2 x^2}}{a \sinh ^{-1}(a x)}+\frac {\text {Subst}\left (\int \left (\frac {\sinh (x)}{8 x}-\frac {9 \sinh (3 x)}{16 x}+\frac {5 \sinh (5 x)}{16 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a^5}\\ &=-\frac {x^4 \sqrt {1+a^2 x^2}}{a \sinh ^{-1}(a x)}+\frac {\text {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^5}+\frac {5 \text {Subst}\left (\int \frac {\sinh (5 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^5}-\frac {9 \text {Subst}\left (\int \frac {\sinh (3 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^5}\\ &=-\frac {x^4 \sqrt {1+a^2 x^2}}{a \sinh ^{-1}(a x)}+\frac {\text {Shi}\left (\sinh ^{-1}(a x)\right )}{8 a^5}-\frac {9 \text {Shi}\left (3 \sinh ^{-1}(a x)\right )}{16 a^5}+\frac {5 \text {Shi}\left (5 \sinh ^{-1}(a x)\right )}{16 a^5}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 60, normalized size = 0.88 \begin {gather*} \frac {-\frac {16 a^4 x^4 \sqrt {1+a^2 x^2}}{\sinh ^{-1}(a x)}+2 \text {Shi}\left (\sinh ^{-1}(a x)\right )-9 \text {Shi}\left (3 \sinh ^{-1}(a x)\right )+5 \text {Shi}\left (5 \sinh ^{-1}(a x)\right )}{16 a^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.31, size = 80, normalized size = 1.18
method | result | size |
derivativedivides | \(\frac {-\frac {\sqrt {a^{2} x^{2}+1}}{8 \arcsinh \left (a x \right )}+\frac {\hyperbolicSineIntegral \left (\arcsinh \left (a x \right )\right )}{8}+\frac {3 \cosh \left (3 \arcsinh \left (a x \right )\right )}{16 \arcsinh \left (a x \right )}-\frac {9 \hyperbolicSineIntegral \left (3 \arcsinh \left (a x \right )\right )}{16}-\frac {\cosh \left (5 \arcsinh \left (a x \right )\right )}{16 \arcsinh \left (a x \right )}+\frac {5 \hyperbolicSineIntegral \left (5 \arcsinh \left (a x \right )\right )}{16}}{a^{5}}\) | \(80\) |
default | \(\frac {-\frac {\sqrt {a^{2} x^{2}+1}}{8 \arcsinh \left (a x \right )}+\frac {\hyperbolicSineIntegral \left (\arcsinh \left (a x \right )\right )}{8}+\frac {3 \cosh \left (3 \arcsinh \left (a x \right )\right )}{16 \arcsinh \left (a x \right )}-\frac {9 \hyperbolicSineIntegral \left (3 \arcsinh \left (a x \right )\right )}{16}-\frac {\cosh \left (5 \arcsinh \left (a x \right )\right )}{16 \arcsinh \left (a x \right )}+\frac {5 \hyperbolicSineIntegral \left (5 \arcsinh \left (a x \right )\right )}{16}}{a^{5}}\) | \(80\) |
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}}{\operatorname {asinh}^{2}{\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 = -0.01 \begin {gather*} \int \frac {x^4}{{\mathrm {asinh}\left (a\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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