Optimal. Leaf size=91 \[ \frac {x^{1+m} \cosh ^{-1}(a x)}{1+m}-\frac {a x^{2+m} \sqrt {1-a^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};a^2 x^2\right )}{\left (2+3 m+m^2\right ) \sqrt {-1+a x} \sqrt {1+a x}} \]
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Rubi [A]
time = 0.03, 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 = {5883, 127, 372,
371} \begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 127
Rule 371
Rule 372
Rule 5883
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 = 0.07, size = 82, normalized size = 0.90 \begin {gather*} \frac {x^{1+m} \left (\cosh ^{-1}(a x)-\frac {a x \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 )}{(2+m) \sqrt {-1+a x} \sqrt {1+a x}}\right )}{1+m} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 4.65, 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 \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 x^{m} \operatorname {acosh}{\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 x^m\,\mathrm {acosh}\left (a\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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