Optimal. Leaf size=91 \[ \frac {x^{m+1} \cosh ^{-1}(a x)}{m+1}-\frac {a \sqrt {1-a^2 x^2} x^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};a^2 x^2\right )}{\left (m^2+3 m+2\right ) \sqrt {a x-1} \sqrt {a x+1}} \]
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Rubi [A] time = 0.05, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5662, 126, 365, 364} \[ \frac {x^{m+1} \cosh ^{-1}(a x)}{m+1}-\frac {a \sqrt {1-a^2 x^2} x^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};a^2 x^2\right )}{\left (m^2+3 m+2\right ) \sqrt {a x-1} \sqrt {a x+1}} \]
Antiderivative was successfully verified.
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Rule 126
Rule 364
Rule 365
Rule 5662
Rubi steps
\begin {align*} \int x^m \cosh ^{-1}(a x) \, dx &=\frac {x^{1+m} \cosh ^{-1}(a x)}{1+m}-\frac {a \int \frac {x^{1+m}}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{1+m}\\ &=\frac {x^{1+m} \cosh ^{-1}(a x)}{1+m}-\frac {\left (a \sqrt {-1+a^2 x^2}\right ) \int \frac {x^{1+m}}{\sqrt {-1+a^2 x^2}} \, dx}{(1+m) \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {x^{1+m} \cosh ^{-1}(a x)}{1+m}-\frac {\left (a \sqrt {1-a^2 x^2}\right ) \int \frac {x^{1+m}}{\sqrt {1-a^2 x^2}} \, dx}{(1+m) \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {x^{1+m} \cosh ^{-1}(a x)}{1+m}-\frac {a x^{2+m} \sqrt {1-a^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};a^2 x^2\right )}{\left (2+3 m+m^2\right ) \sqrt {-1+a x} \sqrt {1+a x}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 82, normalized size = 0.90 \[ \frac {x^{m+1} \left (\cosh ^{-1}(a x)-\frac {a x \sqrt {1-a^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};a^2 x^2\right )}{(m+2) \sqrt {a x-1} \sqrt {a x+1}}\right )}{m+1} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{m} \operatorname {arcosh}\left (a x\right ), x\right ) \]
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 \[ \int x^{m} \operatorname {arcosh}\left (a x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.04, size = 0, normalized size = 0.00 \[ \int x^{m} \mathrm {arccosh}\left (a x \right )\, dx \]
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 \[ -a^{2} \int \frac {x^{2} x^{m}}{a^{2} {\left (m + 1\right )} x^{2} - m - 1}\,{d x} + a \int \frac {x x^{m}}{a^{3} {\left (m + 1\right )} x^{3} - a {\left (m + 1\right )} x + {\left (a^{2} {\left (m + 1\right )} x^{2} - m - 1\right )} \sqrt {a x + 1} \sqrt {a x - 1}}\,{d x} + \frac {x x^{m} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )}{m + 1} \]
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 \[ \int x^m\,\mathrm {acosh}\left (a\,x\right ) \,d x \]
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 \[ \int x^{m} \operatorname {acosh}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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