Optimal. Leaf size=127 \[ -\frac {3 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{64 a^2}+\frac {3 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{64 a^2}-\frac {\cosh ^{-1}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \cosh ^{-1}(a x)^{3/2}-\frac {3 x \sqrt {a x-1} \sqrt {a x+1} \sqrt {\cosh ^{-1}(a x)}}{8 a} \]
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Rubi [A] time = 0.44, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5664, 5759, 5676, 5670, 5448, 12, 3308, 2180, 2204, 2205} \[ -\frac {3 \sqrt {\frac {\pi }{2}} \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{64 a^2}+\frac {3 \sqrt {\frac {\pi }{2}} \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{64 a^2}-\frac {\cosh ^{-1}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \cosh ^{-1}(a x)^{3/2}-\frac {3 x \sqrt {a x-1} \sqrt {a x+1} \sqrt {\cosh ^{-1}(a x)}}{8 a} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2180
Rule 2204
Rule 2205
Rule 3308
Rule 5448
Rule 5664
Rule 5670
Rule 5676
Rule 5759
Rubi steps
\begin {align*} \int x \cosh ^{-1}(a x)^{3/2} \, dx &=\frac {1}{2} x^2 \cosh ^{-1}(a x)^{3/2}-\frac {1}{4} (3 a) \int \frac {x^2 \sqrt {\cosh ^{-1}(a x)}}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=-\frac {3 x \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\cosh ^{-1}(a x)}}{8 a}+\frac {1}{2} x^2 \cosh ^{-1}(a x)^{3/2}+\frac {3}{16} \int \frac {x}{\sqrt {\cosh ^{-1}(a x)}} \, dx-\frac {3 \int \frac {\sqrt {\cosh ^{-1}(a x)}}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{8 a}\\ &=-\frac {3 x \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\cosh ^{-1}(a x)}}{8 a}-\frac {\cosh ^{-1}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \cosh ^{-1}(a x)^{3/2}+\frac {3 \operatorname {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a^2}\\ &=-\frac {3 x \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\cosh ^{-1}(a x)}}{8 a}-\frac {\cosh ^{-1}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \cosh ^{-1}(a x)^{3/2}+\frac {3 \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{2 \sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a^2}\\ &=-\frac {3 x \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\cosh ^{-1}(a x)}}{8 a}-\frac {\cosh ^{-1}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \cosh ^{-1}(a x)^{3/2}+\frac {3 \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{32 a^2}\\ &=-\frac {3 x \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\cosh ^{-1}(a x)}}{8 a}-\frac {\cosh ^{-1}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \cosh ^{-1}(a x)^{3/2}-\frac {3 \operatorname {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a^2}+\frac {3 \operatorname {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a^2}\\ &=-\frac {3 x \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\cosh ^{-1}(a x)}}{8 a}-\frac {\cosh ^{-1}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \cosh ^{-1}(a x)^{3/2}-\frac {3 \operatorname {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{32 a^2}+\frac {3 \operatorname {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{32 a^2}\\ &=-\frac {3 x \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\cosh ^{-1}(a x)}}{8 a}-\frac {\cosh ^{-1}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \cosh ^{-1}(a x)^{3/2}-\frac {3 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{64 a^2}+\frac {3 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{64 a^2}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 84, normalized size = 0.66 \[ \frac {3 \sqrt {2 \pi } \left (\text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )-\text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )\right )+32 \cosh \left (2 \cosh ^{-1}(a x)\right ) \cosh ^{-1}(a x)^{3/2}-24 \sqrt {\cosh ^{-1}(a x)} \sinh \left (2 \cosh ^{-1}(a x)\right )}{128 a^2} \]
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 \[ \int x \operatorname {arcosh}\left (a x\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 105, normalized size = 0.83 \[ \frac {\sqrt {2}\, \left (32 \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, x^{2} a^{2}-24 \sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, x a -16 \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }-3 \pi \erf \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right )+3 \pi \erfi \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right )\right )}{128 \sqrt {\pi }\, a^{2}} \]
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 \[ \int x \operatorname {arcosh}\left (a x\right )^{\frac {3}{2}}\,{d x} \]
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 \[ \int x\,{\mathrm {acosh}\left (a\,x\right )}^{3/2} \,d x \]
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 \[ \int x \operatorname {acosh}^{\frac {3}{2}}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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