Optimal. Leaf size=115 \[ 2 a^2 \cosh ^{-1}(a x)^3+\frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{x}-\frac {\cosh ^{-1}(a x)^4}{2 x^2}-6 a^2 \cosh ^{-1}(a x)^2 \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )-6 a^2 \cosh ^{-1}(a x) \text {PolyLog}\left (2,-e^{2 \cosh ^{-1}(a x)}\right )+3 a^2 \text {PolyLog}\left (3,-e^{2 \cosh ^{-1}(a x)}\right ) \]
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Rubi [A]
time = 0.24, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5883, 5918,
5882, 3799, 2221, 2611, 2320, 6724} \begin {gather*} -6 a^2 \cosh ^{-1}(a x) \text {Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )+3 a^2 \text {Li}_3\left (-e^{2 \cosh ^{-1}(a x)}\right )+2 a^2 \cosh ^{-1}(a x)^3-6 a^2 \cosh ^{-1}(a x)^2 \log \left (e^{2 \cosh ^{-1}(a x)}+1\right )-\frac {\cosh ^{-1}(a x)^4}{2 x^2}+\frac {2 a \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^3}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2320
Rule 2611
Rule 3799
Rule 5882
Rule 5883
Rule 5918
Rule 6724
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(a x)^4}{x^3} \, dx &=-\frac {\cosh ^{-1}(a x)^4}{2 x^2}+(2 a) \int \frac {\cosh ^{-1}(a x)^3}{x^2 \sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=\frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{x}-\frac {\cosh ^{-1}(a x)^4}{2 x^2}-\left (6 a^2\right ) \int \frac {\cosh ^{-1}(a x)^2}{x} \, dx\\ &=\frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{x}-\frac {\cosh ^{-1}(a x)^4}{2 x^2}-\left (6 a^2\right ) \text {Subst}\left (\int x^2 \tanh (x) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=2 a^2 \cosh ^{-1}(a x)^3+\frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{x}-\frac {\cosh ^{-1}(a x)^4}{2 x^2}-\left (12 a^2\right ) \text {Subst}\left (\int \frac {e^{2 x} x^2}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(a x)\right )\\ &=2 a^2 \cosh ^{-1}(a x)^3+\frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{x}-\frac {\cosh ^{-1}(a x)^4}{2 x^2}-6 a^2 \cosh ^{-1}(a x)^2 \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )+\left (12 a^2\right ) \text {Subst}\left (\int x \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=2 a^2 \cosh ^{-1}(a x)^3+\frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{x}-\frac {\cosh ^{-1}(a x)^4}{2 x^2}-6 a^2 \cosh ^{-1}(a x)^2 \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )-6 a^2 \cosh ^{-1}(a x) \text {Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )+\left (6 a^2\right ) \text {Subst}\left (\int \text {Li}_2\left (-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=2 a^2 \cosh ^{-1}(a x)^3+\frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{x}-\frac {\cosh ^{-1}(a x)^4}{2 x^2}-6 a^2 \cosh ^{-1}(a x)^2 \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )-6 a^2 \cosh ^{-1}(a x) \text {Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )+\left (3 a^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(a x)}\right )\\ &=2 a^2 \cosh ^{-1}(a x)^3+\frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{x}-\frac {\cosh ^{-1}(a x)^4}{2 x^2}-6 a^2 \cosh ^{-1}(a x)^2 \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )-6 a^2 \cosh ^{-1}(a x) \text {Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )+3 a^2 \text {Li}_3\left (-e^{2 \cosh ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A]
time = 0.77, size = 112, normalized size = 0.97 \begin {gather*} -\frac {\cosh ^{-1}(a x)^4}{2 x^2}+a^2 \left (2 \cosh ^{-1}(a x)^2 \left (-\cosh ^{-1}(a x)+\frac {\sqrt {\frac {-1+a x}{1+a x}} (1+a x) \cosh ^{-1}(a x)}{a x}-3 \log \left (1+e^{-2 \cosh ^{-1}(a x)}\right )\right )+6 \cosh ^{-1}(a x) \text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(a x)}\right )+3 \text {PolyLog}\left (3,-e^{-2 \cosh ^{-1}(a x)}\right )\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 3.64, size = 149, normalized size = 1.30
method | result | size |
derivativedivides | \(a^{2} \left (-\frac {\mathrm {arccosh}\left (a x \right )^{3} \left (4 a^{2} x^{2}-4 \sqrt {a x +1}\, \sqrt {a x -1}\, a x +\mathrm {arccosh}\left (a x \right )\right )}{2 a^{2} x^{2}}+4 \mathrm {arccosh}\left (a x \right )^{3}-6 \mathrm {arccosh}\left (a x \right )^{2} \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )-6 \,\mathrm {arccosh}\left (a x \right ) \polylog \left (2, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )+3 \polylog \left (3, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )\right )\) | \(149\) |
default | \(a^{2} \left (-\frac {\mathrm {arccosh}\left (a x \right )^{3} \left (4 a^{2} x^{2}-4 \sqrt {a x +1}\, \sqrt {a x -1}\, a x +\mathrm {arccosh}\left (a x \right )\right )}{2 a^{2} x^{2}}+4 \mathrm {arccosh}\left (a x \right )^{3}-6 \mathrm {arccosh}\left (a x \right )^{2} \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )-6 \,\mathrm {arccosh}\left (a x \right ) \polylog \left (2, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )+3 \polylog \left (3, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )\right )\) | \(149\) |
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]
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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {acosh}^{4}{\left (a x \right )}}{x^{3}}\, 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 {{\mathrm {acosh}\left (a\,x\right )}^4}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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