Optimal. Leaf size=30 \[ \text {Int}\left ((f x)^m \left (d-c^2 d x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )^n,x\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int (f x)^m \left (d-c^2 d x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )^n \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int (f x)^m \left (d-c^2 d x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )^n \, dx &=\int (f x)^m \left (d-c^2 d x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )^n \, dx\\ \end {align*}
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Mathematica [A]
time = 0.57, size = 0, normalized size = 0.00 \begin {gather*} \int (f x)^m \left (d-c^2 d x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )^n \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (f x \right )^{m} \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{n}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
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 [A]
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 [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - d \left (\int \left (- \left (f x\right )^{m} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{n}\right )\, dx + \int c^{2} x^{2} \left (f x\right )^{m} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{n}\, dx\right ) \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^n\,\left (d-c^2\,d\,x^2\right )\,{\left (f\,x\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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