Optimal. Leaf size=155 \[ -\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^3}+\frac {x^2}{2 a^2 \cosh ^{-1}(a x)^2}-\frac {2 x^4}{3 \cosh ^{-1}(a x)^2}+\frac {x \sqrt {-1+a x} \sqrt {1+a x}}{a^3 \cosh ^{-1}(a x)}-\frac {8 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)}+\frac {\text {Chi}\left (2 \cosh ^{-1}(a x)\right )}{3 a^4}+\frac {4 \text {Chi}\left (4 \cosh ^{-1}(a x)\right )}{3 a^4} \]
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Rubi [A]
time = 0.39, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5886, 5951,
5885, 3382} \begin {gather*} \frac {\text {Chi}\left (2 \cosh ^{-1}(a x)\right )}{3 a^4}+\frac {4 \text {Chi}\left (4 \cosh ^{-1}(a x)\right )}{3 a^4}+\frac {x \sqrt {a x-1} \sqrt {a x+1}}{a^3 \cosh ^{-1}(a x)}+\frac {x^2}{2 a^2 \cosh ^{-1}(a x)^2}-\frac {2 x^4}{3 \cosh ^{-1}(a x)^2}-\frac {8 x^3 \sqrt {a x-1} \sqrt {a x+1}}{3 a \cosh ^{-1}(a x)}-\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{3 a \cosh ^{-1}(a x)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 3382
Rule 5885
Rule 5886
Rule 5951
Rubi steps
\begin {align*} \int \frac {x^3}{\cosh ^{-1}(a x)^4} \, dx &=-\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^3}-\frac {\int \frac {x^2}{\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3} \, dx}{a}+\frac {1}{3} (4 a) \int \frac {x^4}{\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3} \, dx\\ &=-\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^3}+\frac {x^2}{2 a^2 \cosh ^{-1}(a x)^2}-\frac {2 x^4}{3 \cosh ^{-1}(a x)^2}+\frac {8}{3} \int \frac {x^3}{\cosh ^{-1}(a x)^2} \, dx-\frac {\int \frac {x}{\cosh ^{-1}(a x)^2} \, dx}{a^2}\\ &=-\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^3}+\frac {x^2}{2 a^2 \cosh ^{-1}(a x)^2}-\frac {2 x^4}{3 \cosh ^{-1}(a x)^2}+\frac {x \sqrt {-1+a x} \sqrt {1+a x}}{a^3 \cosh ^{-1}(a x)}-\frac {8 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)}-\frac {\text {Subst}\left (\int \frac {\cosh (2 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{a^4}-\frac {8 \text {Subst}\left (\int \left (-\frac {\cosh (2 x)}{2 x}-\frac {\cosh (4 x)}{2 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{3 a^4}\\ &=-\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^3}+\frac {x^2}{2 a^2 \cosh ^{-1}(a x)^2}-\frac {2 x^4}{3 \cosh ^{-1}(a x)^2}+\frac {x \sqrt {-1+a x} \sqrt {1+a x}}{a^3 \cosh ^{-1}(a x)}-\frac {8 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)}-\frac {\text {Chi}\left (2 \cosh ^{-1}(a x)\right )}{a^4}+\frac {4 \text {Subst}\left (\int \frac {\cosh (2 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{3 a^4}+\frac {4 \text {Subst}\left (\int \frac {\cosh (4 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{3 a^4}\\ &=-\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^3}+\frac {x^2}{2 a^2 \cosh ^{-1}(a x)^2}-\frac {2 x^4}{3 \cosh ^{-1}(a x)^2}+\frac {x \sqrt {-1+a x} \sqrt {1+a x}}{a^3 \cosh ^{-1}(a x)}-\frac {8 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)}+\frac {\text {Chi}\left (2 \cosh ^{-1}(a x)\right )}{3 a^4}+\frac {4 \text {Chi}\left (4 \cosh ^{-1}(a x)\right )}{3 a^4}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 188, normalized size = 1.21 \begin {gather*} \frac {\sqrt {-1+a x} \left (a x \sqrt {\frac {-1+a x}{1+a x}} \left (2 a^2 x^2-2 a^4 x^4-a x \sqrt {-1+a x} \sqrt {1+a x} \left (-3+4 a^2 x^2\right ) \cosh ^{-1}(a x)-2 \left (3-11 a^2 x^2+8 a^4 x^4\right ) \cosh ^{-1}(a x)^2\right )+2 (-1+a x) \cosh ^{-1}(a x)^3 \text {Chi}\left (2 \cosh ^{-1}(a x)\right )+8 (-1+a x) \cosh ^{-1}(a x)^3 \text {Chi}\left (4 \cosh ^{-1}(a x)\right )\right )}{6 a^4 \left (\frac {-1+a x}{1+a x}\right )^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^3} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 2.05, size = 114, normalized size = 0.74
method | result | size |
derivativedivides | \(\frac {-\frac {\sinh \left (2 \,\mathrm {arccosh}\left (a x \right )\right )}{12 \mathrm {arccosh}\left (a x \right )^{3}}-\frac {\cosh \left (2 \,\mathrm {arccosh}\left (a x \right )\right )}{12 \mathrm {arccosh}\left (a x \right )^{2}}-\frac {\sinh \left (2 \,\mathrm {arccosh}\left (a x \right )\right )}{6 \,\mathrm {arccosh}\left (a x \right )}+\frac {\hyperbolicCosineIntegral \left (2 \,\mathrm {arccosh}\left (a x \right )\right )}{3}-\frac {\sinh \left (4 \,\mathrm {arccosh}\left (a x \right )\right )}{24 \mathrm {arccosh}\left (a x \right )^{3}}-\frac {\cosh \left (4 \,\mathrm {arccosh}\left (a x \right )\right )}{12 \mathrm {arccosh}\left (a x \right )^{2}}-\frac {\sinh \left (4 \,\mathrm {arccosh}\left (a x \right )\right )}{3 \,\mathrm {arccosh}\left (a x \right )}+\frac {4 \hyperbolicCosineIntegral \left (4 \,\mathrm {arccosh}\left (a x \right )\right )}{3}}{a^{4}}\) | \(114\) |
default | \(\frac {-\frac {\sinh \left (2 \,\mathrm {arccosh}\left (a x \right )\right )}{12 \mathrm {arccosh}\left (a x \right )^{3}}-\frac {\cosh \left (2 \,\mathrm {arccosh}\left (a x \right )\right )}{12 \mathrm {arccosh}\left (a x \right )^{2}}-\frac {\sinh \left (2 \,\mathrm {arccosh}\left (a x \right )\right )}{6 \,\mathrm {arccosh}\left (a x \right )}+\frac {\hyperbolicCosineIntegral \left (2 \,\mathrm {arccosh}\left (a x \right )\right )}{3}-\frac {\sinh \left (4 \,\mathrm {arccosh}\left (a x \right )\right )}{24 \mathrm {arccosh}\left (a x \right )^{3}}-\frac {\cosh \left (4 \,\mathrm {arccosh}\left (a x \right )\right )}{12 \mathrm {arccosh}\left (a x \right )^{2}}-\frac {\sinh \left (4 \,\mathrm {arccosh}\left (a x \right )\right )}{3 \,\mathrm {arccosh}\left (a x \right )}+\frac {4 \hyperbolicCosineIntegral \left (4 \,\mathrm {arccosh}\left (a x \right )\right )}{3}}{a^{4}}\) | \(114\) |
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 {x^{3}}{\operatorname {acosh}^{4}{\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}{{\mathrm {acosh}\left (a\,x\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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