Optimal. Leaf size=135 \[ -\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{a \sqrt {\cosh ^{-1}(a x)}}+\frac {\sqrt {\pi } \text {Erf}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{4 a^3}+\frac {\sqrt {3 \pi } \text {Erf}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{4 a^3}+\frac {\sqrt {\pi } \text {Erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{4 a^3}+\frac {\sqrt {3 \pi } \text {Erfi}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{4 a^3} \]
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Rubi [A]
time = 0.09, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5885, 3388,
2211, 2235, 2236} \begin {gather*} \frac {\sqrt {\pi } \text {Erf}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{4 a^3}+\frac {\sqrt {3 \pi } \text {Erf}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{4 a^3}+\frac {\sqrt {\pi } \text {Erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{4 a^3}+\frac {\sqrt {3 \pi } \text {Erfi}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{4 a^3}-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\cosh ^{-1}(a x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 5885
Rubi steps
\begin {align*} \int \frac {x^2}{\cosh ^{-1}(a x)^{3/2}} \, dx &=-\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{a \sqrt {\cosh ^{-1}(a x)}}-\frac {2 \text {Subst}\left (\int \left (-\frac {\cosh (x)}{4 \sqrt {x}}-\frac {3 \cosh (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a^3}\\ &=-\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{a \sqrt {\cosh ^{-1}(a x)}}+\frac {\text {Subst}\left (\int \frac {\cosh (x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{2 a^3}+\frac {3 \text {Subst}\left (\int \frac {\cosh (3 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{2 a^3}\\ &=-\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{a \sqrt {\cosh ^{-1}(a x)}}+\frac {\text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{4 a^3}+\frac {\text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{4 a^3}+\frac {3 \text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{4 a^3}+\frac {3 \text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{4 a^3}\\ &=-\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{a \sqrt {\cosh ^{-1}(a x)}}+\frac {\text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{2 a^3}+\frac {\text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{2 a^3}+\frac {3 \text {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{2 a^3}+\frac {3 \text {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{2 a^3}\\ &=-\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{a \sqrt {\cosh ^{-1}(a x)}}+\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{4 a^3}+\frac {\sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{4 a^3}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{4 a^3}+\frac {\sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{4 a^3}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 139, normalized size = 1.03 \begin {gather*} -\frac {2 \sqrt {\frac {-1+a x}{1+a x}} (1+a x)-\sqrt {3} \sqrt {-\cosh ^{-1}(a x)} \Gamma \left (\frac {1}{2},-3 \cosh ^{-1}(a x)\right )-\sqrt {-\cosh ^{-1}(a x)} \Gamma \left (\frac {1}{2},-\cosh ^{-1}(a x)\right )+\sqrt {\cosh ^{-1}(a x)} \Gamma \left (\frac {1}{2},\cosh ^{-1}(a x)\right )+\sqrt {3} \sqrt {\cosh ^{-1}(a x)} \Gamma \left (\frac {1}{2},3 \cosh ^{-1}(a x)\right )+2 \sinh \left (3 \cosh ^{-1}(a x)\right )}{4 a^3 \sqrt {\cosh ^{-1}(a x)}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 3.99, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{\mathrm {arccosh}\left (a x \right )^{\frac {3}{2}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {x^{2}}{\operatorname {acosh}^{\frac {3}{2}}{\left (a x \right )}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^2}{{\mathrm {acosh}\left (a\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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