Optimal. Leaf size=73 \[ \frac {\text {Chi}\left (\cosh ^{-1}(a x)\right )}{8 a^5}+\frac {9 \text {Chi}\left (3 \cosh ^{-1}(a x)\right )}{16 a^5}+\frac {5 \text {Chi}\left (5 \cosh ^{-1}(a x)\right )}{16 a^5}-\frac {x^4 \sqrt {a x-1} \sqrt {a x+1}}{a \cosh ^{-1}(a x)} \]
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Rubi [A] time = 0.06, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5666, 3301} \[ \frac {\text {Chi}\left (\cosh ^{-1}(a x)\right )}{8 a^5}+\frac {9 \text {Chi}\left (3 \cosh ^{-1}(a x)\right )}{16 a^5}+\frac {5 \text {Chi}\left (5 \cosh ^{-1}(a x)\right )}{16 a^5}-\frac {x^4 \sqrt {a x-1} \sqrt {a x+1}}{a \cosh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 3301
Rule 5666
Rubi steps
\begin {align*} \int \frac {x^4}{\cosh ^{-1}(a x)^2} \, dx &=-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{a \cosh ^{-1}(a x)}-\frac {\operatorname {Subst}\left (\int \left (-\frac {\cosh (x)}{8 x}-\frac {9 \cosh (3 x)}{16 x}-\frac {5 \cosh (5 x)}{16 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}\\ &=-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{a \cosh ^{-1}(a x)}+\frac {\operatorname {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^5}+\frac {5 \operatorname {Subst}\left (\int \frac {\cosh (5 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a^5}+\frac {9 \operatorname {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a^5}\\ &=-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{a \cosh ^{-1}(a x)}+\frac {\text {Chi}\left (\cosh ^{-1}(a x)\right )}{8 a^5}+\frac {9 \text {Chi}\left (3 \cosh ^{-1}(a x)\right )}{16 a^5}+\frac {5 \text {Chi}\left (5 \cosh ^{-1}(a x)\right )}{16 a^5}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 101, normalized size = 1.38 \[ \frac {-16 a^5 x^5 \sqrt {\frac {a x-1}{a x+1}}-16 a^4 x^4 \sqrt {\frac {a x-1}{a x+1}}+2 \cosh ^{-1}(a x) \text {Chi}\left (\cosh ^{-1}(a x)\right )+9 \cosh ^{-1}(a x) \text {Chi}\left (3 \cosh ^{-1}(a x)\right )+5 \cosh ^{-1}(a x) \text {Chi}\left (5 \cosh ^{-1}(a x)\right )}{16 a^5 \cosh ^{-1}(a x)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{4}}{\operatorname {arcosh}\left (a x\right )^{2}}, x\right ) \]
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 \[ \int \frac {x^{4}}{\operatorname {arcosh}\left (a x\right )^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 83, normalized size = 1.14 \[ \frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{8 \,\mathrm {arccosh}\left (a x \right )}+\frac {\Chi \left (\mathrm {arccosh}\left (a x \right )\right )}{8}-\frac {3 \sinh \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{16 \,\mathrm {arccosh}\left (a x \right )}+\frac {9 \Chi \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{16}-\frac {\sinh \left (5 \,\mathrm {arccosh}\left (a x \right )\right )}{16 \,\mathrm {arccosh}\left (a x \right )}+\frac {5 \Chi \left (5 \,\mathrm {arccosh}\left (a x \right )\right )}{16}}{a^{5}} \]
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 \[ -\frac {a^{3} x^{7} - a x^{5} + {\left (a^{2} x^{6} - x^{4}\right )} \sqrt {a x + 1} \sqrt {a x - 1}}{{\left (a^{3} x^{2} + \sqrt {a x + 1} \sqrt {a x - 1} a^{2} x - a\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )} + \int \frac {5 \, a^{5} x^{8} - 10 \, a^{3} x^{6} + 5 \, a x^{4} + {\left (5 \, a^{3} x^{6} - 3 \, a x^{4}\right )} {\left (a x + 1\right )} {\left (a x - 1\right )} + {\left (10 \, a^{4} x^{7} - 13 \, a^{2} x^{5} + 4 \, x^{3}\right )} \sqrt {a x + 1} \sqrt {a x - 1}}{{\left (a^{5} x^{4} + {\left (a x + 1\right )} {\left (a x - 1\right )} a^{3} x^{2} - 2 \, a^{3} x^{2} + 2 \, {\left (a^{4} x^{3} - a^{2} x\right )} \sqrt {a x + 1} \sqrt {a x - 1} + a\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )}\,{d x} \]
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 \[ \int \frac {x^4}{{\mathrm {acosh}\left (a\,x\right )}^2} \,d x \]
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 \[ \int \frac {x^{4}}{\operatorname {acosh}^{2}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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