Optimal. Leaf size=91 \[ \frac{f m \sqrt{c x-1} \text{Unintegrable}\left (\frac{(f x)^{m-1}}{a+b \cosh ^{-1}(c x)},x\right )}{b c \sqrt{1-c x}}-\frac{\sqrt{c x-1} (f x)^m}{b c \sqrt{1-c x} \left (a+b \cosh ^{-1}(c x)\right )} \]
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Rubi [A] time = 0.52198, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \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^2 x^2}}\\ &=-\frac{(f x)^m \sqrt{-1+c x} \sqrt{1+c x}}{b c \sqrt{1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )}+\frac{\left (f m \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{(f x)^{-1+m}}{a+b \cosh ^{-1}(c x)} \, dx}{b c \sqrt{1-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.635352, size = 0, normalized size = 0. \[ \int \frac{(f x)^m}{\sqrt{1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.358, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx \right ) ^{m}}{ \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{2}}{\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (c^{2} f^{m} x^{2} - f^{m}\right )} \sqrt{c x + 1} \sqrt{c x - 1} x^{m} +{\left (c^{3} f^{m} x^{3} - c f^{m} x\right )} x^{m}}{{\left ({\left (c x + 1\right )} \sqrt{c x - 1} b^{2} c^{2} x +{\left (b^{2} c^{3} x^{2} - b^{2} c\right )} \sqrt{c x + 1}\right )} \sqrt{-c x + 1} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right ) +{\left ({\left (c x + 1\right )} \sqrt{c x - 1} a b c^{2} x +{\left (a b c^{3} x^{2} - a b c\right )} \sqrt{c x + 1}\right )} \sqrt{-c x + 1}} + \int \frac{{\left (c^{3} f^{m} m x^{3} - c f^{m}{\left (m - 1\right )} x\right )}{\left (c x + 1\right )}{\left (c x - 1\right )} x^{m} +{\left (2 \, c^{4} f^{m} m x^{4} - 3 \, c^{2} f^{m} m x^{2} + f^{m} m\right )} \sqrt{c x + 1} \sqrt{c x - 1} x^{m} +{\left (c^{5} f^{m} m x^{5} - c^{3} f^{m}{\left (2 \, m + 1\right )} x^{3} + c f^{m}{\left (m + 1\right )} x\right )} x^{m}}{{\left ({\left (c x + 1\right )}^{\frac{3}{2}}{\left (c x - 1\right )} b^{2} c^{3} x^{3} + 2 \,{\left (b^{2} c^{4} x^{4} - b^{2} c^{2} x^{2}\right )}{\left (c x + 1\right )} \sqrt{c x - 1} +{\left (b^{2} c^{5} x^{5} - 2 \, b^{2} c^{3} x^{3} + b^{2} c x\right )} \sqrt{c x + 1}\right )} \sqrt{-c x + 1} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right ) +{\left ({\left (c x + 1\right )}^{\frac{3}{2}}{\left (c x - 1\right )} a b c^{3} x^{3} + 2 \,{\left (a b c^{4} x^{4} - a b c^{2} x^{2}\right )}{\left (c x + 1\right )} \sqrt{c x - 1} +{\left (a b c^{5} x^{5} - 2 \, a b c^{3} x^{3} + a b c x\right )} \sqrt{c x + 1}\right )} \sqrt{-c x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-c^{2} x^{2} + 1} \left (f x\right )^{m}}{a^{2} c^{2} x^{2} +{\left (b^{2} c^{2} x^{2} - b^{2}\right )} \operatorname{arcosh}\left (c x\right )^{2} - a^{2} + 2 \,{\left (a b c^{2} x^{2} - a b\right )} \operatorname{arcosh}\left (c x\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (f x\right )^{m}}{\sqrt{-c^{2} x^{2} + 1}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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