Optimal. Leaf size=94 \[ -\frac {5 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^{3/2}}{2 a}+\frac {15 \sqrt {\pi } \text {erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{16 a}-\frac {15 \sqrt {\pi } \text {erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{16 a}+x \sinh ^{-1}(a x)^{5/2}+\frac {15}{4} x \sqrt {\sinh ^{-1}(a x)} \]
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Rubi [A] time = 0.18, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {5653, 5717, 5779, 3308, 2180, 2204, 2205} \[ -\frac {5 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^{3/2}}{2 a}+\frac {15 \sqrt {\pi } \text {Erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{16 a}-\frac {15 \sqrt {\pi } \text {Erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{16 a}+x \sinh ^{-1}(a x)^{5/2}+\frac {15}{4} x \sqrt {\sinh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3308
Rule 5653
Rule 5717
Rule 5779
Rubi steps
\begin {align*} \int \sinh ^{-1}(a x)^{5/2} \, dx &=x \sinh ^{-1}(a x)^{5/2}-\frac {1}{2} (5 a) \int \frac {x \sinh ^{-1}(a x)^{3/2}}{\sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {5 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{2 a}+x \sinh ^{-1}(a x)^{5/2}+\frac {15}{4} \int \sqrt {\sinh ^{-1}(a x)} \, dx\\ &=\frac {15}{4} x \sqrt {\sinh ^{-1}(a x)}-\frac {5 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{2 a}+x \sinh ^{-1}(a x)^{5/2}-\frac {1}{8} (15 a) \int \frac {x}{\sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}} \, dx\\ &=\frac {15}{4} x \sqrt {\sinh ^{-1}(a x)}-\frac {5 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{2 a}+x \sinh ^{-1}(a x)^{5/2}-\frac {15 \operatorname {Subst}\left (\int \frac {\sinh (x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{8 a}\\ &=\frac {15}{4} x \sqrt {\sinh ^{-1}(a x)}-\frac {5 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{2 a}+x \sinh ^{-1}(a x)^{5/2}+\frac {15 \operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a}-\frac {15 \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a}\\ &=\frac {15}{4} x \sqrt {\sinh ^{-1}(a x)}-\frac {5 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{2 a}+x \sinh ^{-1}(a x)^{5/2}+\frac {15 \operatorname {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{8 a}-\frac {15 \operatorname {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{8 a}\\ &=\frac {15}{4} x \sqrt {\sinh ^{-1}(a x)}-\frac {5 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{2 a}+x \sinh ^{-1}(a x)^{5/2}+\frac {15 \sqrt {\pi } \text {erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{16 a}-\frac {15 \sqrt {\pi } \text {erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{16 a}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 45, normalized size = 0.48 \[ -\frac {\frac {\sqrt {-\sinh ^{-1}(a x)} \Gamma \left (\frac {7}{2},-\sinh ^{-1}(a x)\right )}{\sqrt {\sinh ^{-1}(a x)}}+\Gamma \left (\frac {7}{2},\sinh ^{-1}(a x)\right )}{2 a} \]
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 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 78, normalized size = 0.83 \[ \frac {16 \arcsinh \left (a x \right )^{\frac {5}{2}} \sqrt {\pi }\, x a -40 \arcsinh \left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, \sqrt {a^{2} x^{2}+1}+60 \sqrt {\arcsinh \left (a x \right )}\, \sqrt {\pi }\, x a +15 \pi \erf \left (\sqrt {\arcsinh \left (a x \right )}\right )-15 \pi \erfi \left (\sqrt {\arcsinh \left (a x \right )}\right )}{16 \sqrt {\pi }\, a} \]
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 \[ \int \operatorname {arsinh}\left (a x\right )^{\frac {5}{2}}\,{d x} \]
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 \[ \int {\mathrm {asinh}\left (a\,x\right )}^{5/2} \,d x \]
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 \[ \int \operatorname {asinh}^{\frac {5}{2}}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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