Optimal. Leaf size=132 \[ -\frac {16 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)}{75 a^5}+\frac {16 x}{75 a^4}-\frac {8 x^2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)}{75 a^3}+\frac {8 x^3}{225 a^2}+\frac {1}{5} x^5 \cosh ^{-1}(a x)^2-\frac {2 x^4 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)}{25 a}+\frac {2 x^5}{125} \]
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Rubi [A] time = 0.49, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5662, 5759, 5718, 8, 30} \[ \frac {8 x^3}{225 a^2}-\frac {8 x^2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)}{75 a^3}+\frac {16 x}{75 a^4}-\frac {16 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)}{75 a^5}+\frac {1}{5} x^5 \cosh ^{-1}(a x)^2-\frac {2 x^4 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)}{25 a}+\frac {2 x^5}{125} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 5662
Rule 5718
Rule 5759
Rubi steps
\begin {align*} \int x^4 \cosh ^{-1}(a x)^2 \, dx &=\frac {1}{5} x^5 \cosh ^{-1}(a x)^2-\frac {1}{5} (2 a) \int \frac {x^5 \cosh ^{-1}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=-\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{25 a}+\frac {1}{5} x^5 \cosh ^{-1}(a x)^2+\frac {2 \int x^4 \, dx}{25}-\frac {8 \int \frac {x^3 \cosh ^{-1}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{25 a}\\ &=\frac {2 x^5}{125}-\frac {8 x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{75 a^3}-\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{25 a}+\frac {1}{5} x^5 \cosh ^{-1}(a x)^2-\frac {16 \int \frac {x \cosh ^{-1}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{75 a^3}+\frac {8 \int x^2 \, dx}{75 a^2}\\ &=\frac {8 x^3}{225 a^2}+\frac {2 x^5}{125}-\frac {16 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{75 a^5}-\frac {8 x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{75 a^3}-\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{25 a}+\frac {1}{5} x^5 \cosh ^{-1}(a x)^2+\frac {16 \int 1 \, dx}{75 a^4}\\ &=\frac {16 x}{75 a^4}+\frac {8 x^3}{225 a^2}+\frac {2 x^5}{125}-\frac {16 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{75 a^5}-\frac {8 x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{75 a^3}-\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{25 a}+\frac {1}{5} x^5 \cosh ^{-1}(a x)^2\\ \end {align*}
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Mathematica [A] time = 0.12, size = 80, normalized size = 0.61 \[ \frac {\frac {240 x}{a^4}+\frac {40 x^3}{a^2}-\frac {30 \sqrt {a x-1} \sqrt {a x+1} \left (3 a^4 x^4+4 a^2 x^2+8\right ) \cosh ^{-1}(a x)}{a^5}+225 x^5 \cosh ^{-1}(a x)^2+18 x^5}{1125} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 99, normalized size = 0.75 \[ \frac {225 \, a^{5} x^{5} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} + 18 \, a^{5} x^{5} + 40 \, a^{3} x^{3} - 30 \, {\left (3 \, a^{4} x^{4} + 4 \, a^{2} x^{2} + 8\right )} \sqrt {a^{2} x^{2} - 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) + 240 \, a x}{1125 \, a^{5}} \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 112, normalized size = 0.85 \[ \frac {\frac {\mathrm {arccosh}\left (a x \right )^{2} a^{5} x^{5}}{5}-\frac {16 \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right )}{75}-\frac {2 \,\mathrm {arccosh}\left (a x \right ) a^{4} x^{4} \sqrt {a x -1}\, \sqrt {a x +1}}{25}-\frac {8 \,\mathrm {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}\, a^{2} x^{2}}{75}+\frac {16 a x}{75}+\frac {2 x^{5} a^{5}}{125}+\frac {8 x^{3} a^{3}}{225}}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.68, size = 99, normalized size = 0.75 \[ \frac {1}{5} \, x^{5} \operatorname {arcosh}\left (a x\right )^{2} - \frac {2}{75} \, {\left (\frac {3 \, \sqrt {a^{2} x^{2} - 1} x^{4}}{a^{2}} + \frac {4 \, \sqrt {a^{2} x^{2} - 1} x^{2}}{a^{4}} + \frac {8 \, \sqrt {a^{2} x^{2} - 1}}{a^{6}}\right )} a \operatorname {arcosh}\left (a x\right ) + \frac {2 \, {\left (9 \, a^{4} x^{5} + 20 \, a^{2} x^{3} + 120 \, x\right )}}{1125 \, a^{4}} \]
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 \[ \int x^4\,{\mathrm {acosh}\left (a\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.42, size = 122, normalized size = 0.92 \[ \begin {cases} \frac {x^{5} \operatorname {acosh}^{2}{\left (a x \right )}}{5} + \frac {2 x^{5}}{125} - \frac {2 x^{4} \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}{\left (a x \right )}}{25 a} + \frac {8 x^{3}}{225 a^{2}} - \frac {8 x^{2} \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}{\left (a x \right )}}{75 a^{3}} + \frac {16 x}{75 a^{4}} - \frac {16 \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}{\left (a x \right )}}{75 a^{5}} & \text {for}\: a \neq 0 \\- \frac {\pi ^{2} x^{5}}{20} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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