Optimal. Leaf size=600 \[ -\frac{m \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right ) \text{PolyLog}\left (2,-\frac{g e^{\cosh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right )}{c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right ) \text{PolyLog}\left (2,-\frac{g e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )}{c \sqrt{1-c^2 x^2}}+\frac{b m \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (3,-\frac{g e^{\cosh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right )}{c \sqrt{1-c^2 x^2}}+\frac{b m \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (3,-\frac{g e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )}{c \sqrt{1-c^2 x^2}}+\frac{m \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b^2 c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (\frac{g e^{\cosh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}+1\right )}{2 b c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (\frac{g e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}+1\right )}{2 b c \sqrt{1-c^2 x^2}}+\frac{\sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (h (f+g x)^m\right )}{2 b c \sqrt{1-c^2 x^2}} \]
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Rubi [A] time = 1.12927, antiderivative size = 600, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.303, Rules used = {5713, 5676, 5841, 5839, 5800, 5562, 2190, 2531, 2282, 6589} \[ -\frac{m \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right ) \text{PolyLog}\left (2,-\frac{g e^{\cosh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right )}{c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right ) \text{PolyLog}\left (2,-\frac{g e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )}{c \sqrt{1-c^2 x^2}}+\frac{b m \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (3,-\frac{g e^{\cosh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right )}{c \sqrt{1-c^2 x^2}}+\frac{b m \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (3,-\frac{g e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )}{c \sqrt{1-c^2 x^2}}+\frac{m \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b^2 c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (\frac{g e^{\cosh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}+1\right )}{2 b c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (\frac{g e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}+1\right )}{2 b c \sqrt{1-c^2 x^2}}+\frac{\sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (h (f+g x)^m\right )}{2 b c \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 5713
Rule 5676
Rule 5841
Rule 5839
Rule 5800
Rule 5562
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (h (f+g x)^m\right )}{\sqrt{1-c^2 x^2}} \, dx &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (h (f+g x)^m\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (h (f+g x)^m\right )}{2 b c \sqrt{1-c^2 x^2}}-\frac{\left (g m \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{f+g x} \, dx}{2 b c \sqrt{1-c^2 x^2}}\\ &=\frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (h (f+g x)^m\right )}{2 b c \sqrt{1-c^2 x^2}}-\frac{\left (g m \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^2 \sinh (x)}{c f+g \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b c \sqrt{1-c^2 x^2}}\\ &=\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b^2 c \sqrt{1-c^2 x^2}}+\frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (h (f+g x)^m\right )}{2 b c \sqrt{1-c^2 x^2}}-\frac{\left (g m \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{e^x (a+b x)^2}{c f+e^x g-\sqrt{c^2 f^2-g^2}} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b c \sqrt{1-c^2 x^2}}-\frac{\left (g m \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{e^x (a+b x)^2}{c f+e^x g+\sqrt{c^2 f^2-g^2}} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b c \sqrt{1-c^2 x^2}}\\ &=\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b^2 c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{2 b c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{2 b c \sqrt{1-c^2 x^2}}+\frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (h (f+g x)^m\right )}{2 b c \sqrt{1-c^2 x^2}}+\frac{\left (m \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1+\frac{e^x g}{c f-\sqrt{c^2 f^2-g^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c \sqrt{1-c^2 x^2}}+\frac{\left (m \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1+\frac{e^x g}{c f+\sqrt{c^2 f^2-g^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c \sqrt{1-c^2 x^2}}\\ &=\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b^2 c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{2 b c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{2 b c \sqrt{1-c^2 x^2}}+\frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (h (f+g x)^m\right )}{2 b c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (-\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (-\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{c \sqrt{1-c^2 x^2}}+\frac{\left (b m \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{e^x g}{c f-\sqrt{c^2 f^2-g^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c \sqrt{1-c^2 x^2}}+\frac{\left (b m \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{e^x g}{c f+\sqrt{c^2 f^2-g^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c \sqrt{1-c^2 x^2}}\\ &=\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b^2 c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{2 b c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{2 b c \sqrt{1-c^2 x^2}}+\frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (h (f+g x)^m\right )}{2 b c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (-\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (-\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{c \sqrt{1-c^2 x^2}}+\frac{\left (b m \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{g x}{-c f+\sqrt{c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{c \sqrt{1-c^2 x^2}}+\frac{\left (b m \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{g x}{c f+\sqrt{c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{c \sqrt{1-c^2 x^2}}\\ &=\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b^2 c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{2 b c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{2 b c \sqrt{1-c^2 x^2}}+\frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (h (f+g x)^m\right )}{2 b c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (-\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (-\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{c \sqrt{1-c^2 x^2}}+\frac{b m \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_3\left (-\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{c \sqrt{1-c^2 x^2}}+\frac{b m \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_3\left (-\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{c \sqrt{1-c^2 x^2}}\\ \end{align*}
Mathematica [F] time = 1.84202, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (h (f+g x)^m\right )}{\sqrt{1-c^2 x^2}} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 1.412, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b{\rm arccosh} \left (cx\right ) \right ) \ln \left ( h \left ( gx+f \right ) ^{m} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )} \log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt{-c^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-c^{2} x^{2} + 1}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )} \log \left ({\left (g x + f\right )}^{m} h\right )}{c^{2} x^{2} - 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{acosh}{\left (c x \right )}\right ) \log{\left (h \left (f + g x\right )^{m} \right )}}{\sqrt{- \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )} \log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt{-c^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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