Optimal. Leaf size=138 \[ \frac {\sqrt {\pi } \text {erf}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{4 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{2 a^4}+\frac {\sqrt {\pi } \text {erfi}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{4 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{2 a^4}-\frac {2 x^3 \sqrt {a^2 x^2+1}}{a \sqrt {\sinh ^{-1}(a x)}} \]
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Rubi [A] time = 0.13, 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 = {5665, 3307, 2180, 2204, 2205} \[ \frac {\sqrt {\pi } \text {Erf}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{4 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {Erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{2 a^4}+\frac {\sqrt {\pi } \text {Erfi}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{4 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {Erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{2 a^4}-\frac {2 x^3 \sqrt {a^2 x^2+1}}{a \sqrt {\sinh ^{-1}(a x)}} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 5665
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 \operatorname {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 {\operatorname {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{a^4}+\frac {\operatorname {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 {\operatorname {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^4}-\frac {\operatorname {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^4}-\frac {\operatorname {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^4}+\frac {\operatorname {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 {\operatorname {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{a^4}-\frac {\operatorname {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{a^4}-\frac {\operatorname {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{a^4}+\frac {\operatorname {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 = 0.04, size = 126, normalized size = 0.91 \[ \frac {2 \sinh \left (2 \sinh ^{-1}(a x)\right )-\sinh \left (4 \sinh ^{-1}(a x)\right )+\sqrt {-\sinh ^{-1}(a x)} \Gamma \left (\frac {1}{2},-4 \sinh ^{-1}(a x)\right )-\sqrt {2} \sqrt {-\sinh ^{-1}(a x)} \Gamma \left (\frac {1}{2},-2 \sinh ^{-1}(a x)\right )+\sqrt {2} \sqrt {\sinh ^{-1}(a x)} \Gamma \left (\frac {1}{2},2 \sinh ^{-1}(a x)\right )-\sqrt {\sinh ^{-1}(a x)} \Gamma \left (\frac {1}{2},4 \sinh ^{-1}(a x)\right )}{4 a^4 \sqrt {\sinh ^{-1}(a x)}} \]
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(-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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maple [F(-2)] time = 180.00, 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 \[ \int \frac {x^{3}}{\operatorname {arsinh}\left (a x\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^3}{{\mathrm {asinh}\left (a\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{3}}{\operatorname {asinh}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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