Optimal. Leaf size=84 \[ -\frac {2 \sqrt {a^2 x^2+1}}{3 a \sinh ^{-1}(a x)^{3/2}}+\frac {2 \sqrt {\pi } \text {erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{3 a}+\frac {2 \sqrt {\pi } \text {erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{3 a}-\frac {4 x}{3 \sqrt {\sinh ^{-1}(a x)}} \]
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Rubi [A] time = 0.12, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {5655, 5774, 5657, 3307, 2180, 2204, 2205} \[ -\frac {2 \sqrt {a^2 x^2+1}}{3 a \sinh ^{-1}(a x)^{3/2}}+\frac {2 \sqrt {\pi } \text {Erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{3 a}+\frac {2 \sqrt {\pi } \text {Erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{3 a}-\frac {4 x}{3 \sqrt {\sinh ^{-1}(a x)}} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 5655
Rule 5657
Rule 5774
Rubi steps
\begin {align*} \int \frac {1}{\sinh ^{-1}(a x)^{5/2}} \, dx &=-\frac {2 \sqrt {1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}+\frac {1}{3} (2 a) \int \frac {x}{\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac {2 \sqrt {1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac {4 x}{3 \sqrt {\sinh ^{-1}(a x)}}+\frac {4}{3} \int \frac {1}{\sqrt {\sinh ^{-1}(a x)}} \, dx\\ &=-\frac {2 \sqrt {1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac {4 x}{3 \sqrt {\sinh ^{-1}(a x)}}+\frac {4 \operatorname {Subst}\left (\int \frac {\cosh (x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a}\\ &=-\frac {2 \sqrt {1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac {4 x}{3 \sqrt {\sinh ^{-1}(a x)}}+\frac {2 \operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a}+\frac {2 \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a}\\ &=-\frac {2 \sqrt {1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac {4 x}{3 \sqrt {\sinh ^{-1}(a x)}}+\frac {4 \operatorname {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{3 a}+\frac {4 \operatorname {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{3 a}\\ &=-\frac {2 \sqrt {1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac {4 x}{3 \sqrt {\sinh ^{-1}(a x)}}+\frac {2 \sqrt {\pi } \text {erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{3 a}+\frac {2 \sqrt {\pi } \text {erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{3 a}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 105, normalized size = 1.25 \[ -\frac {e^{-\sinh ^{-1}(a x)} \left (e^{2 \sinh ^{-1}(a x)}+2 e^{2 \sinh ^{-1}(a x)} \sinh ^{-1}(a x)-2 \sinh ^{-1}(a x)+2 e^{\sinh ^{-1}(a x)} \left (-\sinh ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},-\sinh ^{-1}(a x)\right )+2 e^{\sinh ^{-1}(a x)} \sinh ^{-1}(a x)^{3/2} \Gamma \left (\frac {1}{2},\sinh ^{-1}(a x)\right )+1\right )}{3 a \sinh ^{-1}(a x)^{3/2}} \]
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 \[ \int \frac {1}{\operatorname {arsinh}\left (a x\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 81, normalized size = 0.96 \[ \frac {-\frac {4 \arcsinh \left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, x a}{3}+\frac {2 \arcsinh \left (a x \right )^{2} \pi \erf \left (\sqrt {\arcsinh \left (a x \right )}\right )}{3}+\frac {2 \arcsinh \left (a x \right )^{2} \pi \erfi \left (\sqrt {\arcsinh \left (a x \right )}\right )}{3}-\frac {2 \sqrt {\arcsinh \left (a x \right )}\, \sqrt {\pi }\, \sqrt {a^{2} x^{2}+1}}{3}}{\sqrt {\pi }\, a \arcsinh \left (a x \right )^{2}} \]
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 \[ \int \frac {1}{\operatorname {arsinh}\left (a x\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\mathrm {asinh}\left (a\,x\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{\operatorname {asinh}^{\frac {5}{2}}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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