Optimal. Leaf size=153 \[ -\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^3}+\frac {x}{3 a^2 \cosh ^{-1}(a x)^2}-\frac {x^3}{2 \cosh ^{-1}(a x)^2}+\frac {\sqrt {-1+a x} \sqrt {1+a x}}{3 a^3 \cosh ^{-1}(a x)}-\frac {3 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \cosh ^{-1}(a x)}+\frac {\text {Chi}\left (\cosh ^{-1}(a x)\right )}{24 a^3}+\frac {9 \text {Chi}\left (3 \cosh ^{-1}(a x)\right )}{8 a^3} \]
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Rubi [A]
time = 0.44, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5886, 5951,
5885, 3382, 5880, 5953} \begin {gather*} \frac {\text {Chi}\left (\cosh ^{-1}(a x)\right )}{24 a^3}+\frac {9 \text {Chi}\left (3 \cosh ^{-1}(a x)\right )}{8 a^3}+\frac {\sqrt {a x-1} \sqrt {a x+1}}{3 a^3 \cosh ^{-1}(a x)}+\frac {x}{3 a^2 \cosh ^{-1}(a x)^2}-\frac {x^3}{2 \cosh ^{-1}(a x)^2}-\frac {3 x^2 \sqrt {a x-1} \sqrt {a x+1}}{2 a \cosh ^{-1}(a x)}-\frac {x^2 \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 5880
Rule 5885
Rule 5886
Rule 5951
Rule 5953
Rubi steps
\begin {align*} \int \frac {x^2}{\cosh ^{-1}(a x)^4} \, dx &=-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^3}-\frac {2 \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3} \, dx}{3 a}+a \int \frac {x^3}{\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3} \, dx\\ &=-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^3}+\frac {x}{3 a^2 \cosh ^{-1}(a x)^2}-\frac {x^3}{2 \cosh ^{-1}(a x)^2}+\frac {3}{2} \int \frac {x^2}{\cosh ^{-1}(a x)^2} \, dx-\frac {\int \frac {1}{\cosh ^{-1}(a x)^2} \, dx}{3 a^2}\\ &=-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^3}+\frac {x}{3 a^2 \cosh ^{-1}(a x)^2}-\frac {x^3}{2 \cosh ^{-1}(a x)^2}+\frac {\sqrt {-1+a x} \sqrt {1+a x}}{3 a^3 \cosh ^{-1}(a x)}-\frac {3 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \cosh ^{-1}(a x)}-\frac {3 \text {Subst}\left (\int \left (-\frac {\cosh (x)}{4 x}-\frac {3 \cosh (3 x)}{4 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{2 a^3}-\frac {\int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)} \, dx}{3 a}\\ &=-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^3}+\frac {x}{3 a^2 \cosh ^{-1}(a x)^2}-\frac {x^3}{2 \cosh ^{-1}(a x)^2}+\frac {\sqrt {-1+a x} \sqrt {1+a x}}{3 a^3 \cosh ^{-1}(a x)}-\frac {3 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \cosh ^{-1}(a x)}-\frac {\text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{3 a^3}+\frac {3 \text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^3}+\frac {9 \text {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^3}\\ &=-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \cosh ^{-1}(a x)^3}+\frac {x}{3 a^2 \cosh ^{-1}(a x)^2}-\frac {x^3}{2 \cosh ^{-1}(a x)^2}+\frac {\sqrt {-1+a x} \sqrt {1+a x}}{3 a^3 \cosh ^{-1}(a x)}-\frac {3 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \cosh ^{-1}(a x)}+\frac {\text {Chi}\left (\cosh ^{-1}(a x)\right )}{24 a^3}+\frac {9 \text {Chi}\left (3 \cosh ^{-1}(a x)\right )}{8 a^3}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 183, normalized size = 1.20 \begin {gather*} \frac {\sqrt {-1+a x} \left (-4 \sqrt {\frac {-1+a x}{1+a x}} \left (2 a^2 x^2 \left (-1+a^2 x^2\right )+a x \sqrt {-1+a x} \sqrt {1+a x} \left (-2+3 a^2 x^2\right ) \cosh ^{-1}(a x)+\left (2-11 a^2 x^2+9 a^4 x^4\right ) \cosh ^{-1}(a x)^2\right )+(-1+a x) \cosh ^{-1}(a x)^3 \text {Chi}\left (\cosh ^{-1}(a x)\right )+27 (-1+a x) \cosh ^{-1}(a x)^3 \text {Chi}\left (3 \cosh ^{-1}(a x)\right )\right )}{24 a^3 \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 = 1.82, size = 121, normalized size = 0.79
method | result | size |
derivativedivides | \(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{12 \mathrm {arccosh}\left (a x \right )^{3}}-\frac {a x}{24 \mathrm {arccosh}\left (a x \right )^{2}}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{24 \,\mathrm {arccosh}\left (a x \right )}+\frac {\hyperbolicCosineIntegral \left (\mathrm {arccosh}\left (a x \right )\right )}{24}-\frac {\sinh \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{12 \mathrm {arccosh}\left (a x \right )^{3}}-\frac {\cosh \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{8 \mathrm {arccosh}\left (a x \right )^{2}}-\frac {3 \sinh \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{8 \,\mathrm {arccosh}\left (a x \right )}+\frac {9 \hyperbolicCosineIntegral \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{8}}{a^{3}}\) | \(121\) |
default | \(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{12 \mathrm {arccosh}\left (a x \right )^{3}}-\frac {a x}{24 \mathrm {arccosh}\left (a x \right )^{2}}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{24 \,\mathrm {arccosh}\left (a x \right )}+\frac {\hyperbolicCosineIntegral \left (\mathrm {arccosh}\left (a x \right )\right )}{24}-\frac {\sinh \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{12 \mathrm {arccosh}\left (a x \right )^{3}}-\frac {\cosh \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{8 \mathrm {arccosh}\left (a x \right )^{2}}-\frac {3 \sinh \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{8 \,\mathrm {arccosh}\left (a x \right )}+\frac {9 \hyperbolicCosineIntegral \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{8}}{a^{3}}\) | \(121\) |
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^{2}}{\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]
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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2}{{\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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