Optimal. Leaf size=68 \[ \frac {\sqrt {\pi } \text {erf}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{a}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{a}-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\cosh ^{-1}(a x)}} \]
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Rubi [A] time = 0.22, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5656, 5781, 3307, 2180, 2204, 2205} \[ \frac {\sqrt {\pi } \text {Erf}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{a}+\frac {\sqrt {\pi } \text {Erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{a}-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\cosh ^{-1}(a x)}} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 5656
Rule 5781
Rubi steps
\begin {align*} \int \frac {1}{\cosh ^{-1}(a x)^{3/2}} \, dx &=-\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{a \sqrt {\cosh ^{-1}(a x)}}+(2 a) \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x} \sqrt {\cosh ^{-1}(a x)}} \, dx\\ &=-\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{a \sqrt {\cosh ^{-1}(a x)}}+\frac {2 \operatorname {Subst}\left (\int \frac {\cosh (x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a}\\ &=-\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{a \sqrt {\cosh ^{-1}(a x)}}+\frac {\operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a}+\frac {\operatorname {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a}\\ &=-\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{a \sqrt {\cosh ^{-1}(a x)}}+\frac {2 \operatorname {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{a}+\frac {2 \operatorname {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{a}\\ &=-\frac {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 )}{a}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 76, normalized size = 1.12 \[ \frac {-2 \sqrt {\frac {a x-1}{a x+1}} (a x+1)+\sqrt {-\cosh ^{-1}(a x)} \Gamma \left (\frac {1}{2},-\cosh ^{-1}(a x)\right )-\sqrt {\cosh ^{-1}(a x)} \Gamma \left (\frac {1}{2},\cosh ^{-1}(a x)\right )}{a \sqrt {\cosh ^{-1}(a x)}} \]
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 \[ \int \frac {1}{\operatorname {arcosh}\left (a x\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 66, normalized size = 0.97 \[ \frac {-2 \sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}+\mathrm {arccosh}\left (a x \right ) \pi \erf \left (\sqrt {\mathrm {arccosh}\left (a x \right )}\right )+\mathrm {arccosh}\left (a x \right ) \pi \erfi \left (\sqrt {\mathrm {arccosh}\left (a x \right )}\right )}{\sqrt {\pi }\, a \,\mathrm {arccosh}\left (a x \right )} \]
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 \[ \int \frac {1}{\operatorname {arcosh}\left (a x\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\mathrm {acosh}\left (a\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\operatorname {acosh}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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