Optimal. Leaf size=240 \[ -\frac {2 b c (f x)^{2+m} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{f^2 (2+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {(f x)^{1+m} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{f (2+m)}-\frac {2 b^2 c^2 (f x)^{3+m} \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \, _2F_1\left (\frac {1}{2},\frac {3+m}{2};\frac {5+m}{2};c^2 x^2\right )}{f^3 (2+m)^2 (3+m) (1-c x) (1+c x)}+\frac {d \text {Int}\left (\frac {(f x)^m \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {d-c^2 d x^2}},x\right )}{2+m} \]
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Rubi [A]
time = 0.28, 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 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx \end {gather*}
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 )^2 \, 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 )^2 \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 0, normalized size = 0.00 \begin {gather*} \int (f x)^m \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, 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 \[\int \left (f x \right )^{m} \sqrt {-c^{2} d \,x^{2}+d}\, \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{2}\, dx\]
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*} \int \left (f x\right )^{m} \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}\, dx \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 = -0.00 \begin {gather*} \int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,\sqrt {d-c^2\,d\,x^2}\,{\left (f\,x\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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