Optimal. Leaf size=118 \[ -\frac {2 x \sqrt {1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\sinh ^{-1}(a x)}}-\frac {8 x^2}{3 \sqrt {\sinh ^{-1}(a x)}}-\frac {2 \sqrt {2 \pi } \text {Erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{3 a^2}+\frac {2 \sqrt {2 \pi } \text {Erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{3 a^2} \]
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Rubi [A]
time = 0.15, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 10, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5779, 5818,
5780, 5556, 12, 3389, 2211, 2235, 2236, 5783} \begin {gather*} -\frac {2 \sqrt {2 \pi } \text {Erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{3 a^2}+\frac {2 \sqrt {2 \pi } \text {Erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{3 a^2}-\frac {2 x \sqrt {a^2 x^2+1}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\sinh ^{-1}(a x)}}-\frac {8 x^2}{3 \sqrt {\sinh ^{-1}(a x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 5556
Rule 5779
Rule 5780
Rule 5783
Rule 5818
Rubi steps
\begin {align*} \int \frac {x}{\sinh ^{-1}(a x)^{5/2}} \, dx &=-\frac {2 x \sqrt {1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}+\frac {2 \int \frac {1}{\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}} \, dx}{3 a}+\frac {1}{3} (4 a) \int \frac {x^2}{\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac {2 x \sqrt {1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\sinh ^{-1}(a x)}}-\frac {8 x^2}{3 \sqrt {\sinh ^{-1}(a x)}}+\frac {16}{3} \int \frac {x}{\sqrt {\sinh ^{-1}(a x)}} \, dx\\ &=-\frac {2 x \sqrt {1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\sinh ^{-1}(a x)}}-\frac {8 x^2}{3 \sqrt {\sinh ^{-1}(a x)}}+\frac {16 \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a^2}\\ &=-\frac {2 x \sqrt {1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\sinh ^{-1}(a x)}}-\frac {8 x^2}{3 \sqrt {\sinh ^{-1}(a x)}}+\frac {16 \text {Subst}\left (\int \frac {\sinh (2 x)}{2 \sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a^2}\\ &=-\frac {2 x \sqrt {1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\sinh ^{-1}(a x)}}-\frac {8 x^2}{3 \sqrt {\sinh ^{-1}(a x)}}+\frac {8 \text {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a^2}\\ &=-\frac {2 x \sqrt {1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\sinh ^{-1}(a x)}}-\frac {8 x^2}{3 \sqrt {\sinh ^{-1}(a x)}}-\frac {4 \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a^2}+\frac {4 \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a^2}\\ &=-\frac {2 x \sqrt {1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\sinh ^{-1}(a x)}}-\frac {8 x^2}{3 \sqrt {\sinh ^{-1}(a x)}}-\frac {8 \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{3 a^2}+\frac {8 \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{3 a^2}\\ &=-\frac {2 x \sqrt {1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\sinh ^{-1}(a x)}}-\frac {8 x^2}{3 \sqrt {\sinh ^{-1}(a x)}}-\frac {2 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{3 a^2}+\frac {2 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{3 a^2}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 98, normalized size = 0.83 \begin {gather*} -\frac {2 \sinh ^{-1}(a x) \left (e^{-2 \sinh ^{-1}(a x)}+e^{2 \sinh ^{-1}(a x)}-\sqrt {2} \sqrt {-\sinh ^{-1}(a x)} \Gamma \left (\frac {1}{2},-2 \sinh ^{-1}(a x)\right )-\sqrt {2} \sqrt {\sinh ^{-1}(a x)} \Gamma \left (\frac {1}{2},2 \sinh ^{-1}(a x)\right )\right )+\sinh \left (2 \sinh ^{-1}(a x)\right )}{3 a^2 \sinh ^{-1}(a x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.45, size = 119, normalized size = 1.01
method | result | size |
default | \(-\frac {\sqrt {2}\, \left (4 \arcsinh \left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, \sqrt {2}\, a^{2} x^{2}+\sqrt {\arcsinh \left (a x \right )}\, \sqrt {\pi }\, \sqrt {a^{2} x^{2}+1}\, \sqrt {2}\, a x +2 \arcsinh \left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, \sqrt {2}+2 \arcsinh \left (a x \right )^{2} \pi \erf \left (\sqrt {2}\, \sqrt {\arcsinh \left (a x \right )}\right )-2 \arcsinh \left (a x \right )^{2} \pi \erfi \left (\sqrt {2}\, \sqrt {\arcsinh \left (a x \right )}\right )\right )}{3 \sqrt {\pi }\, a^{2} \arcsinh \left (a x \right )^{2}}\) | \(119\) |
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(-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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\operatorname {asinh}^{\frac {5}{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}{{\mathrm {asinh}\left (a\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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