Optimal. Leaf size=189 \[ -\frac {3 \sqrt {\pi } \text {erf}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{32 a^3}-\frac {\sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{96 a^3}+\frac {3 \sqrt {\pi } \text {erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{32 a^3}+\frac {\sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{96 a^3}-\frac {\sqrt {a x-1} \sqrt {a x+1} \sqrt {\cosh ^{-1}(a x)}}{3 a^3}+\frac {1}{3} x^3 \cosh ^{-1}(a x)^{3/2}-\frac {x^2 \sqrt {a x-1} \sqrt {a x+1} \sqrt {\cosh ^{-1}(a x)}}{6 a} \]
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Rubi [A] time = 0.64, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 10, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {5664, 5759, 5718, 5658, 3308, 2180, 2204, 2205, 5670, 5448} \[ -\frac {3 \sqrt {\pi } \text {Erf}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{32 a^3}-\frac {\sqrt {\frac {\pi }{3}} \text {Erf}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{96 a^3}+\frac {3 \sqrt {\pi } \text {Erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{32 a^3}+\frac {\sqrt {\frac {\pi }{3}} \text {Erfi}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{96 a^3}-\frac {\sqrt {a x-1} \sqrt {a x+1} \sqrt {\cosh ^{-1}(a x)}}{3 a^3}+\frac {1}{3} x^3 \cosh ^{-1}(a x)^{3/2}-\frac {x^2 \sqrt {a x-1} \sqrt {a x+1} \sqrt {\cosh ^{-1}(a x)}}{6 a} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3308
Rule 5448
Rule 5658
Rule 5664
Rule 5670
Rule 5718
Rule 5759
Rubi steps
\begin {align*} \int x^2 \cosh ^{-1}(a x)^{3/2} \, dx &=\frac {1}{3} x^3 \cosh ^{-1}(a x)^{3/2}-\frac {1}{2} a \int \frac {x^3 \sqrt {\cosh ^{-1}(a x)}}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\cosh ^{-1}(a x)}}{6 a}+\frac {1}{3} x^3 \cosh ^{-1}(a x)^{3/2}+\frac {1}{12} \int \frac {x^2}{\sqrt {\cosh ^{-1}(a x)}} \, dx-\frac {\int \frac {x \sqrt {\cosh ^{-1}(a x)}}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{3 a}\\ &=-\frac {\sqrt {-1+a x} \sqrt {1+a x} \sqrt {\cosh ^{-1}(a x)}}{3 a^3}-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\cosh ^{-1}(a x)}}{6 a}+\frac {1}{3} x^3 \cosh ^{-1}(a x)^{3/2}+\frac {\operatorname {Subst}\left (\int \frac {\cosh ^2(x) \sinh (x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{12 a^3}+\frac {\int \frac {1}{\sqrt {\cosh ^{-1}(a x)}} \, dx}{6 a^2}\\ &=-\frac {\sqrt {-1+a x} \sqrt {1+a x} \sqrt {\cosh ^{-1}(a x)}}{3 a^3}-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\cosh ^{-1}(a x)}}{6 a}+\frac {1}{3} x^3 \cosh ^{-1}(a x)^{3/2}+\frac {\operatorname {Subst}\left (\int \left (\frac {\sinh (x)}{4 \sqrt {x}}+\frac {\sinh (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{12 a^3}+\frac {\operatorname {Subst}\left (\int \frac {\sinh (x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{6 a^3}\\ &=-\frac {\sqrt {-1+a x} \sqrt {1+a x} \sqrt {\cosh ^{-1}(a x)}}{3 a^3}-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\cosh ^{-1}(a x)}}{6 a}+\frac {1}{3} x^3 \cosh ^{-1}(a x)^{3/2}+\frac {\operatorname {Subst}\left (\int \frac {\sinh (x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{48 a^3}+\frac {\operatorname {Subst}\left (\int \frac {\sinh (3 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{48 a^3}-\frac {\operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{12 a^3}+\frac {\operatorname {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{12 a^3}\\ &=-\frac {\sqrt {-1+a x} \sqrt {1+a x} \sqrt {\cosh ^{-1}(a x)}}{3 a^3}-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\cosh ^{-1}(a x)}}{6 a}+\frac {1}{3} x^3 \cosh ^{-1}(a x)^{3/2}-\frac {\operatorname {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{96 a^3}-\frac {\operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{96 a^3}+\frac {\operatorname {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{96 a^3}+\frac {\operatorname {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{96 a^3}-\frac {\operatorname {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{6 a^3}+\frac {\operatorname {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{6 a^3}\\ &=-\frac {\sqrt {-1+a x} \sqrt {1+a x} \sqrt {\cosh ^{-1}(a x)}}{3 a^3}-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\cosh ^{-1}(a x)}}{6 a}+\frac {1}{3} x^3 \cosh ^{-1}(a x)^{3/2}-\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{12 a^3}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{12 a^3}-\frac {\operatorname {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{48 a^3}-\frac {\operatorname {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{48 a^3}+\frac {\operatorname {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{48 a^3}+\frac {\operatorname {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{48 a^3}\\ &=-\frac {\sqrt {-1+a x} \sqrt {1+a x} \sqrt {\cosh ^{-1}(a x)}}{3 a^3}-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {\cosh ^{-1}(a x)}}{6 a}+\frac {1}{3} x^3 \cosh ^{-1}(a x)^{3/2}-\frac {3 \sqrt {\pi } \text {erf}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{32 a^3}-\frac {\sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{96 a^3}+\frac {3 \sqrt {\pi } \text {erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{32 a^3}+\frac {\sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{96 a^3}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 100, normalized size = 0.53 \[ \frac {\sqrt {3} \sqrt {-\cosh ^{-1}(a x)} \Gamma \left (\frac {5}{2},-3 \cosh ^{-1}(a x)\right )+27 \sqrt {-\cosh ^{-1}(a x)} \Gamma \left (\frac {5}{2},-\cosh ^{-1}(a x)\right )+\sqrt {\cosh ^{-1}(a x)} \left (27 \Gamma \left (\frac {5}{2},\cosh ^{-1}(a x)\right )+\sqrt {3} \Gamma \left (\frac {5}{2},3 \cosh ^{-1}(a x)\right )\right )}{216 a^3 \sqrt {\cosh ^{-1}(a x)}} \]
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] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \operatorname {arcosh}\left (a x\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int x^{2} \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}}\, dx \]
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 \[ \int x^{2} \operatorname {arcosh}\left (a x\right )^{\frac {3}{2}}\,{d x} \]
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 \[ \int x^2\,{\mathrm {acosh}\left (a\,x\right )}^{3/2} \,d x \]
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 \[ \int x^{2} \operatorname {acosh}^{\frac {3}{2}}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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