Optimal. Leaf size=70 \[ -\frac {x^5 \sqrt {1+a^2 x^2}}{a \sinh ^{-1}(a x)}+\frac {5 \text {Chi}\left (2 \sinh ^{-1}(a x)\right )}{16 a^6}-\frac {\text {Chi}\left (4 \sinh ^{-1}(a x)\right )}{2 a^6}+\frac {3 \text {Chi}\left (6 \sinh ^{-1}(a x)\right )}{16 a^6} \]
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Rubi [A]
time = 0.05, antiderivative size = 70, 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, 3382}
\begin {gather*} \frac {5 \text {Chi}\left (2 \sinh ^{-1}(a x)\right )}{16 a^6}-\frac {\text {Chi}\left (4 \sinh ^{-1}(a x)\right )}{2 a^6}+\frac {3 \text {Chi}\left (6 \sinh ^{-1}(a x)\right )}{16 a^6}-\frac {x^5 \sqrt {a^2 x^2+1}}{a \sinh ^{-1}(a x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 3382
Rule 5778
Rubi steps
\begin {align*} \int \frac {x^5}{\sinh ^{-1}(a x)^2} \, dx &=-\frac {x^5 \sqrt {1+a^2 x^2}}{a \sinh ^{-1}(a x)}+\frac {\text {Subst}\left (\int \left (\frac {5 \cosh (2 x)}{16 x}-\frac {\cosh (4 x)}{2 x}+\frac {3 \cosh (6 x)}{16 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a^6}\\ &=-\frac {x^5 \sqrt {1+a^2 x^2}}{a \sinh ^{-1}(a x)}+\frac {3 \text {Subst}\left (\int \frac {\cosh (6 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^6}+\frac {5 \text {Subst}\left (\int \frac {\cosh (2 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^6}-\frac {\text {Subst}\left (\int \frac {\cosh (4 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^6}\\ &=-\frac {x^5 \sqrt {1+a^2 x^2}}{a \sinh ^{-1}(a x)}+\frac {5 \text {Chi}\left (2 \sinh ^{-1}(a x)\right )}{16 a^6}-\frac {\text {Chi}\left (4 \sinh ^{-1}(a x)\right )}{2 a^6}+\frac {3 \text {Chi}\left (6 \sinh ^{-1}(a x)\right )}{16 a^6}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 78, normalized size = 1.11 \begin {gather*} -\frac {-10 \sinh ^{-1}(a x) \text {Chi}\left (2 \sinh ^{-1}(a x)\right )+16 \sinh ^{-1}(a x) \text {Chi}\left (4 \sinh ^{-1}(a x)\right )-6 \sinh ^{-1}(a x) \text {Chi}\left (6 \sinh ^{-1}(a x)\right )+5 \sinh \left (2 \sinh ^{-1}(a x)\right )-4 \sinh \left (4 \sinh ^{-1}(a x)\right )+\sinh \left (6 \sinh ^{-1}(a x)\right )}{32 a^6 \sinh ^{-1}(a x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.62, size = 78, normalized size = 1.11
method | result | size |
derivativedivides | \(\frac {-\frac {5 \sinh \left (2 \arcsinh \left (a x \right )\right )}{32 \arcsinh \left (a x \right )}+\frac {5 \hyperbolicCosineIntegral \left (2 \arcsinh \left (a x \right )\right )}{16}+\frac {\sinh \left (4 \arcsinh \left (a x \right )\right )}{8 \arcsinh \left (a x \right )}-\frac {\hyperbolicCosineIntegral \left (4 \arcsinh \left (a x \right )\right )}{2}-\frac {\sinh \left (6 \arcsinh \left (a x \right )\right )}{32 \arcsinh \left (a x \right )}+\frac {3 \hyperbolicCosineIntegral \left (6 \arcsinh \left (a x \right )\right )}{16}}{a^{6}}\) | \(78\) |
default | \(\frac {-\frac {5 \sinh \left (2 \arcsinh \left (a x \right )\right )}{32 \arcsinh \left (a x \right )}+\frac {5 \hyperbolicCosineIntegral \left (2 \arcsinh \left (a x \right )\right )}{16}+\frac {\sinh \left (4 \arcsinh \left (a x \right )\right )}{8 \arcsinh \left (a x \right )}-\frac {\hyperbolicCosineIntegral \left (4 \arcsinh \left (a x \right )\right )}{2}-\frac {\sinh \left (6 \arcsinh \left (a x \right )\right )}{32 \arcsinh \left (a x \right )}+\frac {3 \hyperbolicCosineIntegral \left (6 \arcsinh \left (a x \right )\right )}{16}}{a^{6}}\) | \(78\) |
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}^{2}{\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 = -0.01 \begin {gather*} \int \frac {x^5}{{\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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