Optimal. Leaf size=237 \[ \frac{i m \text{PolyLog}\left (2,\frac{i g e^{i \sin ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right )}{c}+\frac{i m \text{PolyLog}\left (2,\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )}{c}-\frac{m \sin ^{-1}(c x) \log \left (1-\frac{i g e^{i \sin ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right )}{c}-\frac{m \sin ^{-1}(c x) \log \left (1-\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )}{c}+\frac{\sin ^{-1}(c x) \log \left (h (f+g x)^m\right )}{c}+\frac{i m \sin ^{-1}(c x)^2}{2 c} \]
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Rubi [A] time = 0.33166, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {216, 2404, 4741, 4519, 2190, 2279, 2391} \[ \frac{i m \text{PolyLog}\left (2,\frac{i g e^{i \sin ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right )}{c}+\frac{i m \text{PolyLog}\left (2,\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )}{c}-\frac{m \sin ^{-1}(c x) \log \left (1-\frac{i g e^{i \sin ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right )}{c}-\frac{m \sin ^{-1}(c x) \log \left (1-\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )}{c}+\frac{\sin ^{-1}(c x) \log \left (h (f+g x)^m\right )}{c}+\frac{i m \sin ^{-1}(c x)^2}{2 c} \]
Antiderivative was successfully verified.
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Rule 216
Rule 2404
Rule 4741
Rule 4519
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\log \left (h (f+g x)^m\right )}{\sqrt{1-c^2 x^2}} \, dx &=\frac{\sin ^{-1}(c x) \log \left (h (f+g x)^m\right )}{c}-(g m) \int \frac{\sin ^{-1}(c x)}{c f+c g x} \, dx\\ &=\frac{\sin ^{-1}(c x) \log \left (h (f+g x)^m\right )}{c}-(g m) \operatorname{Subst}\left (\int \frac{x \cos (x)}{c^2 f+c g \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )\\ &=\frac{i m \sin ^{-1}(c x)^2}{2 c}+\frac{\sin ^{-1}(c x) \log \left (h (f+g x)^m\right )}{c}-(g m) \operatorname{Subst}\left (\int \frac{e^{i x} x}{c^2 f-i c e^{i x} g-c \sqrt{c^2 f^2-g^2}} \, dx,x,\sin ^{-1}(c x)\right )-(g m) \operatorname{Subst}\left (\int \frac{e^{i x} x}{c^2 f-i c e^{i x} g+c \sqrt{c^2 f^2-g^2}} \, dx,x,\sin ^{-1}(c x)\right )\\ &=\frac{i m \sin ^{-1}(c x)^2}{2 c}-\frac{m \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{c}-\frac{m \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{c}+\frac{\sin ^{-1}(c x) \log \left (h (f+g x)^m\right )}{c}+\frac{m \operatorname{Subst}\left (\int \log \left (1-\frac{i c e^{i x} g}{c^2 f-c \sqrt{c^2 f^2-g^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c}+\frac{m \operatorname{Subst}\left (\int \log \left (1-\frac{i c e^{i x} g}{c^2 f+c \sqrt{c^2 f^2-g^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c}\\ &=\frac{i m \sin ^{-1}(c x)^2}{2 c}-\frac{m \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{c}-\frac{m \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{c}+\frac{\sin ^{-1}(c x) \log \left (h (f+g x)^m\right )}{c}-\frac{(i m) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{i c g x}{c^2 f-c \sqrt{c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{c}-\frac{(i m) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{i c g x}{c^2 f+c \sqrt{c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{c}\\ &=\frac{i m \sin ^{-1}(c x)^2}{2 c}-\frac{m \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{c}-\frac{m \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{c}+\frac{\sin ^{-1}(c x) \log \left (h (f+g x)^m\right )}{c}+\frac{i m \text{Li}_2\left (\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{c}+\frac{i m \text{Li}_2\left (\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{c}\\ \end{align*}
Mathematica [A] time = 0.0236326, size = 246, normalized size = 1.04 \[ \frac{i m \text{PolyLog}\left (2,\frac{i g e^{i \sin ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right )}{c}+\frac{i m \text{PolyLog}\left (2,\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )}{c}-\frac{m \sin ^{-1}(c x) \log \left (1-\frac{i c g e^{i \sin ^{-1}(c x)}}{c^2 f-c \sqrt{c^2 f^2-g^2}}\right )}{c}-\frac{m \sin ^{-1}(c x) \log \left (1-\frac{i c g e^{i \sin ^{-1}(c x)}}{c \sqrt{c^2 f^2-g^2}+c^2 f}\right )}{c}+\frac{\sin ^{-1}(c x) \log \left (h (f+g x)^m\right )}{c}+\frac{i m \sin ^{-1}(c x)^2}{2 c} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.165, size = 0, normalized size = 0. \begin{align*} \int{\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{\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} \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{\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{\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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