Optimal. Leaf size=199 \[ -\frac {3 \sqrt {\pi } \text {erf}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{2048 a^4}+\frac {3 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{128 a^4}+\frac {3 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{2048 a^4}-\frac {3 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{128 a^4}-\frac {3 \sinh ^{-1}(a x)^{3/2}}{32 a^4}-\frac {3 x^3 \sqrt {a^2 x^2+1} \sqrt {\sinh ^{-1}(a x)}}{32 a}+\frac {9 x \sqrt {a^2 x^2+1} \sqrt {\sinh ^{-1}(a x)}}{64 a^3}+\frac {1}{4} x^4 \sinh ^{-1}(a x)^{3/2} \]
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Rubi [A] time = 0.49, antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 10, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {5663, 5758, 5675, 5669, 5448, 12, 3308, 2180, 2204, 2205} \[ -\frac {3 \sqrt {\pi } \text {Erf}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{2048 a^4}+\frac {3 \sqrt {\frac {\pi }{2}} \text {Erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{128 a^4}+\frac {3 \sqrt {\pi } \text {Erfi}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{2048 a^4}-\frac {3 \sqrt {\frac {\pi }{2}} \text {Erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{128 a^4}-\frac {3 x^3 \sqrt {a^2 x^2+1} \sqrt {\sinh ^{-1}(a x)}}{32 a}+\frac {9 x \sqrt {a^2 x^2+1} \sqrt {\sinh ^{-1}(a x)}}{64 a^3}-\frac {3 \sinh ^{-1}(a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \sinh ^{-1}(a x)^{3/2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2180
Rule 2204
Rule 2205
Rule 3308
Rule 5448
Rule 5663
Rule 5669
Rule 5675
Rule 5758
Rubi steps
\begin {align*} \int x^3 \sinh ^{-1}(a x)^{3/2} \, dx &=\frac {1}{4} x^4 \sinh ^{-1}(a x)^{3/2}-\frac {1}{8} (3 a) \int \frac {x^4 \sqrt {\sinh ^{-1}(a x)}}{\sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {3 x^3 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{32 a}+\frac {1}{4} x^4 \sinh ^{-1}(a x)^{3/2}+\frac {3}{64} \int \frac {x^3}{\sqrt {\sinh ^{-1}(a x)}} \, dx+\frac {9 \int \frac {x^2 \sqrt {\sinh ^{-1}(a x)}}{\sqrt {1+a^2 x^2}} \, dx}{32 a}\\ &=\frac {9 x \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{64 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{32 a}+\frac {1}{4} x^4 \sinh ^{-1}(a x)^{3/2}+\frac {3 \operatorname {Subst}\left (\int \frac {\cosh (x) \sinh ^3(x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a^4}-\frac {9 \int \frac {\sqrt {\sinh ^{-1}(a x)}}{\sqrt {1+a^2 x^2}} \, dx}{64 a^3}-\frac {9 \int \frac {x}{\sqrt {\sinh ^{-1}(a x)}} \, dx}{128 a^2}\\ &=\frac {9 x \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{64 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{32 a}-\frac {3 \sinh ^{-1}(a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \sinh ^{-1}(a x)^{3/2}+\frac {3 \operatorname {Subst}\left (\int \left (-\frac {\sinh (2 x)}{4 \sqrt {x}}+\frac {\sinh (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{64 a^4}-\frac {9 \operatorname {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{128 a^4}\\ &=\frac {9 x \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{64 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{32 a}-\frac {3 \sinh ^{-1}(a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \sinh ^{-1}(a x)^{3/2}+\frac {3 \operatorname {Subst}\left (\int \frac {\sinh (4 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{512 a^4}-\frac {3 \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{256 a^4}-\frac {9 \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{2 \sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{128 a^4}\\ &=\frac {9 x \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{64 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{32 a}-\frac {3 \sinh ^{-1}(a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \sinh ^{-1}(a x)^{3/2}-\frac {3 \operatorname {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{1024 a^4}+\frac {3 \operatorname {Subst}\left (\int \frac {e^{4 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{1024 a^4}+\frac {3 \operatorname {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{512 a^4}-\frac {3 \operatorname {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{512 a^4}-\frac {9 \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{256 a^4}\\ &=\frac {9 x \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{64 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{32 a}-\frac {3 \sinh ^{-1}(a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \sinh ^{-1}(a x)^{3/2}-\frac {3 \operatorname {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{512 a^4}+\frac {3 \operatorname {Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{512 a^4}+\frac {3 \operatorname {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{256 a^4}-\frac {3 \operatorname {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{256 a^4}+\frac {9 \operatorname {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{512 a^4}-\frac {9 \operatorname {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{512 a^4}\\ &=\frac {9 x \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{64 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{32 a}-\frac {3 \sinh ^{-1}(a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \sinh ^{-1}(a x)^{3/2}-\frac {3 \sqrt {\pi } \text {erf}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{2048 a^4}+\frac {3 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{512 a^4}+\frac {3 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{2048 a^4}-\frac {3 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{512 a^4}+\frac {9 \operatorname {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{256 a^4}-\frac {9 \operatorname {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{256 a^4}\\ &=\frac {9 x \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{64 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{32 a}-\frac {3 \sinh ^{-1}(a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \sinh ^{-1}(a x)^{3/2}-\frac {3 \sqrt {\pi } \text {erf}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{2048 a^4}+\frac {3 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{128 a^4}+\frac {3 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{2048 a^4}-\frac {3 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{128 a^4}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 102, normalized size = 0.51 \[ \frac {-\sqrt {\sinh ^{-1}(a x)} \Gamma \left (\frac {5}{2},-4 \sinh ^{-1}(a x)\right )+8 \sqrt {2} \sqrt {\sinh ^{-1}(a x)} \Gamma \left (\frac {5}{2},-2 \sinh ^{-1}(a x)\right )+\sqrt {-\sinh ^{-1}(a x)} \left (\Gamma \left (\frac {5}{2},4 \sinh ^{-1}(a x)\right )-8 \sqrt {2} \Gamma \left (\frac {5}{2},2 \sinh ^{-1}(a x)\right )\right )}{512 a^4 \sqrt {-\sinh ^{-1}(a x)}} \]
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 \[ \int x^{3} \arcsinh \left (a x \right )^{\frac {3}{2}}\, dx \]
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 \[ \int x^{3} \operatorname {arsinh}\left (a x\right )^{\frac {3}{2}}\,{d x} \]
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 \[ \int x^3\,{\mathrm {asinh}\left (a\,x\right )}^{3/2} \,d x \]
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 \[ \int x^{3} \operatorname {asinh}^{\frac {3}{2}}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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