Optimal. Leaf size=109 \[ -\frac {\sqrt {\pi } \text {Erf}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{32 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{8 a^4}+\frac {\sqrt {\pi } \text {Erfi}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{32 a^4}+\frac {\sqrt {\frac {\pi }{2}} \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{8 a^4} \]
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Rubi [A]
time = 0.10, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5887, 5556,
3389, 2211, 2235, 2236} \begin {gather*} -\frac {\sqrt {\pi } \text {Erf}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{32 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{8 a^4}+\frac {\sqrt {\pi } \text {Erfi}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{32 a^4}+\frac {\sqrt {\frac {\pi }{2}} \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{8 a^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 5556
Rule 5887
Rubi steps
\begin {align*} \int \frac {x^3}{\sqrt {\cosh ^{-1}(a x)}} \, dx &=\frac {\text {Subst}\left (\int \frac {\cosh ^3(x) \sinh (x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a^4}\\ &=\frac {\text {Subst}\left (\int \left (\frac {\sinh (2 x)}{4 \sqrt {x}}+\frac {\sinh (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a^4}\\ &=\frac {\text {Subst}\left (\int \frac {\sinh (4 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^4}+\frac {\text {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{4 a^4}\\ &=-\frac {\text {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a^4}+\frac {\text {Subst}\left (\int \frac {e^{4 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a^4}-\frac {\text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^4}+\frac {\text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^4}\\ &=-\frac {\text {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{8 a^4}+\frac {\text {Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{8 a^4}-\frac {\text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{4 a^4}+\frac {\text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{4 a^4}\\ &=-\frac {\sqrt {\pi } \text {erf}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{32 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{8 a^4}+\frac {\sqrt {\pi } \text {erfi}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{32 a^4}+\frac {\sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{8 a^4}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 101, normalized size = 0.93 \begin {gather*} \frac {\sqrt {-\cosh ^{-1}(a x)} \Gamma \left (\frac {1}{2},-4 \cosh ^{-1}(a x)\right )+2 \sqrt {2} \sqrt {-\cosh ^{-1}(a x)} \Gamma \left (\frac {1}{2},-2 \cosh ^{-1}(a x)\right )+\sqrt {\cosh ^{-1}(a x)} \left (2 \sqrt {2} \Gamma \left (\frac {1}{2},2 \cosh ^{-1}(a x)\right )+\Gamma \left (\frac {1}{2},4 \cosh ^{-1}(a x)\right )\right )}{32 a^4 \sqrt {\cosh ^{-1}(a x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 9.33, size = 67, normalized size = 0.61
method | result | size |
default | \(-\frac {\sqrt {\pi }\, \sqrt {2}\, \left (\erf \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right )-\erfi \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right )\right )}{16 a^{4}}-\frac {\sqrt {\pi }\, \left (\erf \left (2 \sqrt {\mathrm {arccosh}\left (a x \right )}\right )-\erfi \left (2 \sqrt {\mathrm {arccosh}\left (a x \right )}\right )\right )}{32 a^{4}}\) | \(67\) |
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^{3}}{\sqrt {\operatorname {acosh}{\left (a x \right )}}}\, dx \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \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^3}{\sqrt {\mathrm {acosh}\left (a\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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