Optimal. Leaf size=74 \[ \frac{\sqrt{c x-1} \sqrt{c x+1} \text{Unintegrable}\left (\frac{\log \left (h (f+g x)^m\right )}{\sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )},x\right )}{\sqrt{1-c^2 x^2}} \]
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Rubi [A] time = 1.06267, 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 = {} \[ \int \frac{\log \left (h (f+g x)^m\right )}{\sqrt{1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\log \left (h (f+g x)^m\right )}{\sqrt{1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )} \, dx &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{\log \left (h (f+g x)^m\right )}{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt{1-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.279179, size = 0, normalized size = 0. \[ \int \frac{\log \left (h (f+g x)^m\right )}{\sqrt{1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.925, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( h \left ( gx+f \right ) ^{m} \right ) }{a+b{\rm arccosh} \left (cx\right )}{\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*} \int \frac{\log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt{-c^{2} x^{2} + 1}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}}\,{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} \log \left ({\left (g x + f\right )}^{m} h\right )}{a c^{2} x^{2} +{\left (b c^{2} x^{2} - b\right )} \operatorname{arcosh}\left (c x\right ) - a}, 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{\log{\left (h \left (f + g x\right )^{m} \right )}}{\sqrt{- \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname{acosh}{\left (c x \right )}\right )}\, 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{\log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt{-c^{2} x^{2} + 1}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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