Optimal. Leaf size=33 \[ \text {Int}\left (\frac {\sqrt {1-c^2 x^2} (f x)^m}{\left (a+b \cosh ^{-1}(c x)\right )^2},x\right ) \]
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Rubi [A] time = 0.45, 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 = {} \[ \int \frac {(f x)^m \sqrt {1-c^2 x^2}}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {(f x)^m \sqrt {1-c^2 x^2}}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {(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 = 0.20, size = 0, normalized size = 0.00 \[ \int \frac {(f x)^m \sqrt {1-c^2 x^2}}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-c^{2} x^{2} + 1} \left (f x\right )^{m}}{b^{2} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname {arcosh}\left (c x\right ) + a^{2}}, x\right ) \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.26, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x \right )^{m} \sqrt {-c^{2} x^{2}+1}}{\left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{2}}\, dx \]
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 \[ -\frac {{\left ({\left (c^{2} f^{m} x^{2} - f^{m}\right )} {\left (c x + 1\right )} \sqrt {c x - 1} x^{m} + {\left (c^{3} f^{m} x^{3} - c f^{m} x\right )} \sqrt {c x + 1} x^{m}\right )} \sqrt {-c x + 1}}{a b c^{3} x^{2} + \sqrt {c x + 1} \sqrt {c x - 1} a b c^{2} x - a b c + {\left (b^{2} c^{3} x^{2} + \sqrt {c x + 1} \sqrt {c x - 1} b^{2} c^{2} x - b^{2} c\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )} + \int \frac {{\left ({\left (c^{3} f^{m} {\left (m + 2\right )} x^{3} - c f^{m} {\left (m - 1\right )} x\right )} {\left (c x + 1\right )}^{\frac {3}{2}} {\left (c x - 1\right )} x^{m} + {\left (2 \, c^{4} f^{m} {\left (m + 2\right )} x^{4} - c^{2} f^{m} {\left (3 \, m + 2\right )} x^{2} + f^{m} m\right )} {\left (c x + 1\right )} \sqrt {c x - 1} x^{m} + {\left (c^{5} f^{m} {\left (m + 2\right )} x^{5} - c^{3} f^{m} {\left (2 \, m + 3\right )} x^{3} + c f^{m} {\left (m + 1\right )} x\right )} \sqrt {c x + 1} x^{m}\right )} \sqrt {-c x + 1}}{a b c^{5} x^{5} + {\left (c x + 1\right )} {\left (c x - 1\right )} a b c^{3} x^{3} - 2 \, a b c^{3} x^{3} + a b c x + 2 \, {\left (a b c^{4} x^{4} - a b c^{2} x^{2}\right )} \sqrt {c x + 1} \sqrt {c x - 1} + {\left (b^{2} c^{5} x^{5} + {\left (c x + 1\right )} {\left (c x - 1\right )} b^{2} c^{3} x^{3} - 2 \, b^{2} c^{3} x^{3} + b^{2} c x + 2 \, {\left (b^{2} c^{4} x^{4} - b^{2} c^{2} x^{2}\right )} \sqrt {c x + 1} \sqrt {c x - 1}\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}\,{d x} \]
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 \[ \int \frac {{\left (f\,x\right )}^m\,\sqrt {1-c^2\,x^2}}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2} \,d x \]
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 \[ \int \frac {\left (f x\right )^{m} \sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}{\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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