Optimal. Leaf size=139 \[ -\frac {3 \sqrt {\cosh ^{-1}(a x)}}{32 a^4}+\frac {1}{4} x^4 \sqrt {\cosh ^{-1}(a x)}-\frac {\sqrt {\pi } \text {Erf}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{256 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{32 a^4}-\frac {\sqrt {\pi } \text {Erfi}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{256 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{32 a^4} \]
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Rubi [A]
time = 0.28, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5884, 5953,
3393, 3388, 2211, 2235, 2236} \begin {gather*} -\frac {\sqrt {\pi } \text {Erf}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{256 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{32 a^4}-\frac {\sqrt {\pi } \text {Erfi}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{256 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{32 a^4}-\frac {3 \sqrt {\cosh ^{-1}(a x)}}{32 a^4}+\frac {1}{4} x^4 \sqrt {\cosh ^{-1}(a x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 3393
Rule 5884
Rule 5953
Rubi steps
\begin {align*} \int x^3 \sqrt {\cosh ^{-1}(a x)} \, dx &=\frac {1}{4} x^4 \sqrt {\cosh ^{-1}(a x)}-\frac {1}{8} a \int \frac {x^4}{\sqrt {-1+a x} \sqrt {1+a x} \sqrt {\cosh ^{-1}(a x)}} \, dx\\ &=\frac {1}{4} x^4 \sqrt {\cosh ^{-1}(a x)}-\frac {\text {Subst}\left (\int \frac {\cosh ^4(x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^4}\\ &=\frac {1}{4} x^4 \sqrt {\cosh ^{-1}(a x)}-\frac {\text {Subst}\left (\int \left (\frac {3}{8 \sqrt {x}}+\frac {\cosh (2 x)}{2 \sqrt {x}}+\frac {\cosh (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^4}\\ &=-\frac {3 \sqrt {\cosh ^{-1}(a x)}}{32 a^4}+\frac {1}{4} x^4 \sqrt {\cosh ^{-1}(a x)}-\frac {\text {Subst}\left (\int \frac {\cosh (4 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a^4}-\frac {\text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a^4}\\ &=-\frac {3 \sqrt {\cosh ^{-1}(a x)}}{32 a^4}+\frac {1}{4} x^4 \sqrt {\cosh ^{-1}(a x)}-\frac {\text {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{128 a^4}-\frac {\text {Subst}\left (\int \frac {e^{4 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{128 a^4}-\frac {\text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{32 a^4}-\frac {\text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{32 a^4}\\ &=-\frac {3 \sqrt {\cosh ^{-1}(a x)}}{32 a^4}+\frac {1}{4} x^4 \sqrt {\cosh ^{-1}(a x)}-\frac {\text {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{64 a^4}-\frac {\text {Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{64 a^4}-\frac {\text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{16 a^4}-\frac {\text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{16 a^4}\\ &=-\frac {3 \sqrt {\cosh ^{-1}(a x)}}{32 a^4}+\frac {1}{4} x^4 \sqrt {\cosh ^{-1}(a x)}-\frac {\sqrt {\pi } \text {erf}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{256 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{32 a^4}-\frac {\sqrt {\pi } \text {erfi}\left (2 \sqrt {\cosh ^{-1}(a x)}\right )}{256 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{32 a^4}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 101, normalized size = 0.73 \begin {gather*} \frac {\sqrt {\cosh ^{-1}(a x)} \Gamma \left (\frac {3}{2},-4 \cosh ^{-1}(a x)\right )+4 \sqrt {2} \sqrt {\cosh ^{-1}(a x)} \Gamma \left (\frac {3}{2},-2 \cosh ^{-1}(a x)\right )+\sqrt {-\cosh ^{-1}(a x)} \left (4 \sqrt {2} \Gamma \left (\frac {3}{2},2 \cosh ^{-1}(a x)\right )+\Gamma \left (\frac {3}{2},4 \cosh ^{-1}(a x)\right )\right )}{128 a^4 \sqrt {-\cosh ^{-1}(a x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 9.00, size = 150, normalized size = 1.08
method | result | size |
default | \(-\frac {\sqrt {2}\, \left (-8 \sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }\, a^{2} x^{2}+4 \sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }+\pi \erf \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right )+\pi \erfi \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right )\right )}{64 \sqrt {\pi }\, a^{4}}-\frac {-64 \sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }\, a^{4} x^{4}+64 \sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }\, a^{2} x^{2}+\pi \erf \left (2 \sqrt {\mathrm {arccosh}\left (a x \right )}\right )+\pi \erfi \left (2 \sqrt {\mathrm {arccosh}\left (a x \right )}\right )-8 \sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }}{256 \sqrt {\pi }\, a^{4}}\) | \(150\) |
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 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 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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