Optimal. Leaf size=197 \[ \frac {m \sinh ^{-1}(c x)^2}{2 c}-\frac {m \sinh ^{-1}(c x) \log \left (1+\frac {e^{\sinh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{c}-\frac {m \sinh ^{-1}(c x) \log \left (1+\frac {e^{\sinh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{c}+\frac {\sinh ^{-1}(c x) \log \left (h (f+g x)^m\right )}{c}-\frac {m \text {PolyLog}\left (2,-\frac {e^{\sinh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{c}-\frac {m \text {PolyLog}\left (2,-\frac {e^{\sinh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{c} \]
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Rubi [A]
time = 0.22, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {221, 2451,
5827, 5680, 2221, 2317, 2438} \begin {gather*} -\frac {m \text {Li}_2\left (-\frac {e^{\sinh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{c}-\frac {m \text {Li}_2\left (-\frac {e^{\sinh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{c}-\frac {m \sinh ^{-1}(c x) \log \left (\frac {g e^{\sinh ^{-1}(c x)}}{c f-\sqrt {c^2 f^2+g^2}}+1\right )}{c}-\frac {m \sinh ^{-1}(c x) \log \left (\frac {g e^{\sinh ^{-1}(c x)}}{\sqrt {c^2 f^2+g^2}+c f}+1\right )}{c}+\frac {\sinh ^{-1}(c x) \log \left (h (f+g x)^m\right )}{c}+\frac {m \sinh ^{-1}(c x)^2}{2 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 2221
Rule 2317
Rule 2438
Rule 2451
Rule 5680
Rule 5827
Rubi steps
\begin {align*} \int \frac {\log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx &=\frac {\sinh ^{-1}(c x) \log \left (h (f+g x)^m\right )}{c}-(g m) \int \frac {\sinh ^{-1}(c x)}{c f+c g x} \, dx\\ &=\frac {\sinh ^{-1}(c x) \log \left (h (f+g x)^m\right )}{c}-(g m) \text {Subst}\left (\int \frac {x \cosh (x)}{c^2 f+c g \sinh (x)} \, dx,x,\sinh ^{-1}(c x)\right )\\ &=\frac {m \sinh ^{-1}(c x)^2}{2 c}+\frac {\sinh ^{-1}(c x) \log \left (h (f+g x)^m\right )}{c}-(g m) \text {Subst}\left (\int \frac {e^x x}{c^2 f+c e^x g-c \sqrt {c^2 f^2+g^2}} \, dx,x,\sinh ^{-1}(c x)\right )-(g m) \text {Subst}\left (\int \frac {e^x x}{c^2 f+c e^x g+c \sqrt {c^2 f^2+g^2}} \, dx,x,\sinh ^{-1}(c x)\right )\\ &=\frac {m \sinh ^{-1}(c x)^2}{2 c}-\frac {m \sinh ^{-1}(c x) \log \left (1+\frac {e^{\sinh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{c}-\frac {m \sinh ^{-1}(c x) \log \left (1+\frac {e^{\sinh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{c}+\frac {\sinh ^{-1}(c x) \log \left (h (f+g x)^m\right )}{c}+\frac {m \text {Subst}\left (\int \log \left (1+\frac {c e^x g}{c^2 f-c \sqrt {c^2 f^2+g^2}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c}+\frac {m \text {Subst}\left (\int \log \left (1+\frac {c e^x g}{c^2 f+c \sqrt {c^2 f^2+g^2}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c}\\ &=\frac {m \sinh ^{-1}(c x)^2}{2 c}-\frac {m \sinh ^{-1}(c x) \log \left (1+\frac {e^{\sinh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{c}-\frac {m \sinh ^{-1}(c x) \log \left (1+\frac {e^{\sinh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{c}+\frac {\sinh ^{-1}(c x) \log \left (h (f+g x)^m\right )}{c}+\frac {m \text {Subst}\left (\int \frac {\log \left (1+\frac {c g x}{c^2 f-c \sqrt {c^2 f^2+g^2}}\right )}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{c}+\frac {m \text {Subst}\left (\int \frac {\log \left (1+\frac {c g x}{c^2 f+c \sqrt {c^2 f^2+g^2}}\right )}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{c}\\ &=\frac {m \sinh ^{-1}(c x)^2}{2 c}-\frac {m \sinh ^{-1}(c x) \log \left (1+\frac {e^{\sinh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{c}-\frac {m \sinh ^{-1}(c x) \log \left (1+\frac {e^{\sinh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{c}+\frac {\sinh ^{-1}(c x) \log \left (h (f+g x)^m\right )}{c}-\frac {m \text {Li}_2\left (-\frac {e^{\sinh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{c}-\frac {m \text {Li}_2\left (-\frac {e^{\sinh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{c}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 206, normalized size = 1.05 \begin {gather*} \frac {m \sinh ^{-1}(c x)^2}{2 c}-\frac {m \sinh ^{-1}(c x) \log \left (1+\frac {c e^{\sinh ^{-1}(c x)} g}{c^2 f-c \sqrt {c^2 f^2+g^2}}\right )}{c}-\frac {m \sinh ^{-1}(c x) \log \left (1+\frac {c e^{\sinh ^{-1}(c x)} g}{c^2 f+c \sqrt {c^2 f^2+g^2}}\right )}{c}+\frac {\sinh ^{-1}(c x) \log \left (h (f+g x)^m\right )}{c}-\frac {m \text {PolyLog}\left (2,-\frac {e^{\sinh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{c}-\frac {m \text {PolyLog}\left (2,-\frac {e^{\sinh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{c} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (h \left (g x +f \right )^{m}\right )}{\sqrt {c^{2} x^{2}+1}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\log {\left (h \left (f + g x\right )^{m} \right )}}{\sqrt {c^{2} x^{2} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\ln \left (h\,{\left (f+g\,x\right )}^m\right )}{\sqrt {c^2\,x^2+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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