Optimal. Leaf size=161 \[ -\frac {2 x^2 \sqrt {1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac {8 x}{3 a^2 \sqrt {\sinh ^{-1}(a x)}}-\frac {4 x^3}{\sqrt {\sinh ^{-1}(a x)}}-\frac {\sqrt {\pi } \text {Erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{6 a^3}+\frac {\sqrt {3 \pi } \text {Erf}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{2 a^3}-\frac {\sqrt {\pi } \text {Erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{6 a^3}+\frac {\sqrt {3 \pi } \text {Erfi}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{2 a^3} \]
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Rubi [A]
time = 0.28, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps
used = 22, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5779, 5818,
5780, 5556, 3388, 2211, 2235, 2236, 5774} \begin {gather*} -\frac {\sqrt {\pi } \text {Erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{6 a^3}+\frac {\sqrt {3 \pi } \text {Erf}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{2 a^3}-\frac {\sqrt {\pi } \text {Erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{6 a^3}+\frac {\sqrt {3 \pi } \text {Erfi}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{2 a^3}-\frac {2 x^2 \sqrt {a^2 x^2+1}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac {8 x}{3 a^2 \sqrt {\sinh ^{-1}(a x)}}-\frac {4 x^3}{\sqrt {\sinh ^{-1}(a x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 5556
Rule 5774
Rule 5779
Rule 5780
Rule 5818
Rubi steps
\begin {align*} \int \frac {x^2}{\sinh ^{-1}(a x)^{5/2}} \, dx &=-\frac {2 x^2 \sqrt {1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}+\frac {4 \int \frac {x}{\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}} \, dx}{3 a}+(2 a) \int \frac {x^3}{\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac {2 x^2 \sqrt {1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac {8 x}{3 a^2 \sqrt {\sinh ^{-1}(a x)}}-\frac {4 x^3}{\sqrt {\sinh ^{-1}(a x)}}+12 \int \frac {x^2}{\sqrt {\sinh ^{-1}(a x)}} \, dx+\frac {8 \int \frac {1}{\sqrt {\sinh ^{-1}(a x)}} \, dx}{3 a^2}\\ &=-\frac {2 x^2 \sqrt {1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac {8 x}{3 a^2 \sqrt {\sinh ^{-1}(a x)}}-\frac {4 x^3}{\sqrt {\sinh ^{-1}(a x)}}+\frac {8 \text {Subst}\left (\int \frac {\cosh (x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a^3}+\frac {12 \text {Subst}\left (\int \frac {\cosh (x) \sinh ^2(x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{a^3}\\ &=-\frac {2 x^2 \sqrt {1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac {8 x}{3 a^2 \sqrt {\sinh ^{-1}(a x)}}-\frac {4 x^3}{\sqrt {\sinh ^{-1}(a x)}}+\frac {4 \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a^3}+\frac {4 \text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a^3}+\frac {12 \text {Subst}\left (\int \left (-\frac {\cosh (x)}{4 \sqrt {x}}+\frac {\cosh (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a^3}\\ &=-\frac {2 x^2 \sqrt {1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac {8 x}{3 a^2 \sqrt {\sinh ^{-1}(a x)}}-\frac {4 x^3}{\sqrt {\sinh ^{-1}(a x)}}+\frac {8 \text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{3 a^3}+\frac {8 \text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{3 a^3}-\frac {3 \text {Subst}\left (\int \frac {\cosh (x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{a^3}+\frac {3 \text {Subst}\left (\int \frac {\cosh (3 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{a^3}\\ &=-\frac {2 x^2 \sqrt {1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac {8 x}{3 a^2 \sqrt {\sinh ^{-1}(a x)}}-\frac {4 x^3}{\sqrt {\sinh ^{-1}(a x)}}+\frac {4 \sqrt {\pi } \text {erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{3 a^3}+\frac {4 \sqrt {\pi } \text {erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{3 a^3}+\frac {3 \text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^3}-\frac {3 \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^3}-\frac {3 \text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^3}+\frac {3 \text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^3}\\ &=-\frac {2 x^2 \sqrt {1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac {8 x}{3 a^2 \sqrt {\sinh ^{-1}(a x)}}-\frac {4 x^3}{\sqrt {\sinh ^{-1}(a x)}}+\frac {4 \sqrt {\pi } \text {erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{3 a^3}+\frac {4 \sqrt {\pi } \text {erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{3 a^3}+\frac {3 \text {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{a^3}-\frac {3 \text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{a^3}-\frac {3 \text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{a^3}+\frac {3 \text {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{a^3}\\ &=-\frac {2 x^2 \sqrt {1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac {8 x}{3 a^2 \sqrt {\sinh ^{-1}(a x)}}-\frac {4 x^3}{\sqrt {\sinh ^{-1}(a x)}}-\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{6 a^3}+\frac {\sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{2 a^3}-\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{6 a^3}+\frac {\sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{2 a^3}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 225, normalized size = 1.40 \begin {gather*} \frac {-\frac {e^{3 \sinh ^{-1}(a x)} \left (1+6 \sinh ^{-1}(a x)\right )+6 \sqrt {3} \left (-\sinh ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},-3 \sinh ^{-1}(a x)\right )}{12 \sinh ^{-1}(a x)^{3/2}}+\frac {e^{\sinh ^{-1}(a x)} \left (1+2 \sinh ^{-1}(a x)\right )+2 \left (-\sinh ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},-\sinh ^{-1}(a x)\right )}{12 \sinh ^{-1}(a x)^{3/2}}+\frac {e^{-\sinh ^{-1}(a x)} \left (1-2 \sinh ^{-1}(a x)+2 e^{\sinh ^{-1}(a x)} \sinh ^{-1}(a x)^{3/2} \Gamma \left (\frac {1}{2},\sinh ^{-1}(a x)\right )\right )}{12 \sinh ^{-1}(a x)^{3/2}}-\frac {e^{-3 \sinh ^{-1}(a x)} \left (1-6 \sinh ^{-1}(a x)+6 \sqrt {3} e^{3 \sinh ^{-1}(a x)} \sinh ^{-1}(a x)^{3/2} \Gamma \left (\frac {1}{2},3 \sinh ^{-1}(a x)\right )\right )}{12 \sinh ^{-1}(a x)^{3/2}}}{a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 4.44, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{\arcsinh \left (a x \right )^{\frac {5}{2}}}\, dx\]
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^{2}}{\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^2}{{\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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