Optimal. Leaf size=130 \[ \frac {\sqrt {\pi } \text {erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{4 a^3}-\frac {\sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{4 a^3}-\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{4 a^3}+\frac {\sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{4 a^3}-\frac {2 x^2 \sqrt {a^2 x^2+1}}{a \sqrt {\sinh ^{-1}(a x)}} \]
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Rubi [A] time = 0.12, antiderivative size = 130, 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 = {5665, 3308, 2180, 2204, 2205} \[ \frac {\sqrt {\pi } \text {Erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{4 a^3}-\frac {\sqrt {3 \pi } \text {Erf}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{4 a^3}-\frac {\sqrt {\pi } \text {Erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{4 a^3}+\frac {\sqrt {3 \pi } \text {Erfi}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{4 a^3}-\frac {2 x^2 \sqrt {a^2 x^2+1}}{a \sqrt {\sinh ^{-1}(a x)}} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3308
Rule 5665
Rubi steps
\begin {align*} \int \frac {x^2}{\sinh ^{-1}(a x)^{3/2}} \, dx &=-\frac {2 x^2 \sqrt {1+a^2 x^2}}{a \sqrt {\sinh ^{-1}(a x)}}+\frac {2 \operatorname {Subst}\left (\int \left (-\frac {\sinh (x)}{4 \sqrt {x}}+\frac {3 \sinh (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}}{a \sqrt {\sinh ^{-1}(a x)}}-\frac {\operatorname {Subst}\left (\int \frac {\sinh (x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^3}+\frac {3 \operatorname {Subst}\left (\int \frac {\sinh (3 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^3}\\ &=-\frac {2 x^2 \sqrt {1+a^2 x^2}}{a \sqrt {\sinh ^{-1}(a x)}}+\frac {\operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^3}-\frac {\operatorname {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^3}-\frac {3 \operatorname {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^3}+\frac {3 \operatorname {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^3}\\ &=-\frac {2 x^2 \sqrt {1+a^2 x^2}}{a \sqrt {\sinh ^{-1}(a x)}}+\frac {\operatorname {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{2 a^3}-\frac {\operatorname {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{2 a^3}-\frac {3 \operatorname {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{2 a^3}+\frac {3 \operatorname {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{2 a^3}\\ &=-\frac {2 x^2 \sqrt {1+a^2 x^2}}{a \sqrt {\sinh ^{-1}(a x)}}+\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{4 a^3}-\frac {\sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{4 a^3}-\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{4 a^3}+\frac {\sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{4 a^3}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 140, normalized size = 1.08 \[ \frac {-e^{-3 \sinh ^{-1}(a x)}+e^{-\sinh ^{-1}(a x)}+e^{\sinh ^{-1}(a x)}-e^{3 \sinh ^{-1}(a x)}+\sqrt {3} \sqrt {-\sinh ^{-1}(a x)} \Gamma \left (\frac {1}{2},-3 \sinh ^{-1}(a x)\right )-\sqrt {-\sinh ^{-1}(a x)} \Gamma \left (\frac {1}{2},-\sinh ^{-1}(a x)\right )-\sqrt {\sinh ^{-1}(a x)} \Gamma \left (\frac {1}{2},\sinh ^{-1}(a x)\right )+\sqrt {3} \sqrt {\sinh ^{-1}(a x)} \Gamma \left (\frac {1}{2},3 \sinh ^{-1}(a x)\right )}{4 a^3 \sqrt {\sinh ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {x^{2}}{\operatorname {arsinh}\left (a x\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\arcsinh \left (a x \right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {x^{2}}{\operatorname {arsinh}\left (a x\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^2}{{\mathrm {asinh}\left (a\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{2}}{\operatorname {asinh}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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