Optimal. Leaf size=138 \[ -\frac {2 x^3 \sqrt {1+a^2 x^2}}{a \sqrt {\sinh ^{-1}(a x)}}+\frac {\sqrt {\pi } \text {Erf}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{4 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {Erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{2 a^4}+\frac {\sqrt {\pi } \text {Erfi}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{4 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {Erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{2 a^4} \]
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Rubi [A]
time = 0.10, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5778, 3388,
2211, 2235, 2236} \begin {gather*} \frac {\sqrt {\pi } \text {Erf}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{4 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {Erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{2 a^4}+\frac {\sqrt {\pi } \text {Erfi}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{4 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {Erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{2 a^4}-\frac {2 x^3 \sqrt {a^2 x^2+1}}{a \sqrt {\sinh ^{-1}(a x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 5778
Rubi steps
\begin {align*} \int \frac {x^3}{\sinh ^{-1}(a x)^{3/2}} \, dx &=-\frac {2 x^3 \sqrt {1+a^2 x^2}}{a \sqrt {\sinh ^{-1}(a x)}}+\frac {2 \text {Subst}\left (\int \left (-\frac {\cosh (2 x)}{2 \sqrt {x}}+\frac {\cosh (4 x)}{2 \sqrt {x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a^4}\\ &=-\frac {2 x^3 \sqrt {1+a^2 x^2}}{a \sqrt {\sinh ^{-1}(a x)}}-\frac {\text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{a^4}+\frac {\text {Subst}\left (\int \frac {\cosh (4 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{a^4}\\ &=-\frac {2 x^3 \sqrt {1+a^2 x^2}}{a \sqrt {\sinh ^{-1}(a x)}}+\frac {\text {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^4}-\frac {\text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^4}-\frac {\text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^4}+\frac {\text {Subst}\left (\int \frac {e^{4 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^4}\\ &=-\frac {2 x^3 \sqrt {1+a^2 x^2}}{a \sqrt {\sinh ^{-1}(a x)}}+\frac {\text {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{a^4}-\frac {\text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{a^4}-\frac {\text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{a^4}+\frac {\text {Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{a^4}\\ &=-\frac {2 x^3 \sqrt {1+a^2 x^2}}{a \sqrt {\sinh ^{-1}(a x)}}+\frac {\sqrt {\pi } \text {erf}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{4 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{2 a^4}+\frac {\sqrt {\pi } \text {erfi}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{4 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{2 a^4}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 126, normalized size = 0.91 \begin {gather*} \frac {\sqrt {-\sinh ^{-1}(a x)} \Gamma \left (\frac {1}{2},-4 \sinh ^{-1}(a x)\right )-\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 )-\sqrt {\sinh ^{-1}(a x)} \Gamma \left (\frac {1}{2},4 \sinh ^{-1}(a x)\right )+2 \sinh \left (2 \sinh ^{-1}(a x)\right )-\sinh \left (4 \sinh ^{-1}(a x)\right )}{4 a^4 \sqrt {\sinh ^{-1}(a x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 3.90, size = 0, normalized size = 0.00 \[\int \frac {x^{3}}{\arcsinh \left (a x \right )^{\frac {3}{2}}}\, dx\]
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^{3}}{\operatorname {asinh}^{\frac {3}{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^3}{{\mathrm {asinh}\left (a\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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