Optimal. Leaf size=34 \[ \text {Int}\left (\sqrt {d-c^2 d x^2} (f x)^m \left (a+b \cosh ^{-1}(c x)\right )^n,x\right ) \]
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Rubi [A] time = 0.40, 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 = {} \[ \int (f x)^m \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int (f x)^m \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \, dx &=\frac {\sqrt {d-c^2 d x^2} \int (f x)^m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^n \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 0, normalized size = 0.00 \[ \int (f x)^m \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-c^{2} d x^{2} + d} \left (f x\right )^{m} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.54, size = 0, normalized size = 0.00 \[ \int \left (f x \right )^{m} \sqrt {-c^{2} d \,x^{2}+d}\, \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{n}\, dx \]
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 \[ \int \sqrt {-c^{2} d x^{2} + d} \left (f x\right )^{m} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}\,{d x} \]
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 \[ \int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^n\,\sqrt {d-c^2\,d\,x^2}\,{\left (f\,x\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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