Optimal. Leaf size=237 \[ \frac {i m \text {ArcSin}(c x)^2}{2 c}-\frac {m \text {ArcSin}(c x) \log \left (1-\frac {i e^{i \text {ArcSin}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c}-\frac {m \text {ArcSin}(c x) \log \left (1-\frac {i e^{i \text {ArcSin}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c}+\frac {\text {ArcSin}(c x) \log \left (h (f+g x)^m\right )}{c}+\frac {i m \text {PolyLog}\left (2,\frac {i e^{i \text {ArcSin}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c}+\frac {i m \text {PolyLog}\left (2,\frac {i e^{i \text {ArcSin}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c} \]
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Rubi [A]
time = 0.24, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {222, 2451,
4825, 4615, 2221, 2317, 2438} \begin {gather*} \frac {i m \text {Li}_2\left (\frac {i e^{i \text {ArcSin}(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 \text {ArcSin}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c}-\frac {m \text {ArcSin}(c x) \log \left (1-\frac {i g e^{i \text {ArcSin}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c}-\frac {m \text {ArcSin}(c x) \log \left (1-\frac {i g e^{i \text {ArcSin}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{c}+\frac {\text {ArcSin}(c x) \log \left (h (f+g x)^m\right )}{c}+\frac {i m \text {ArcSin}(c x)^2}{2 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 2221
Rule 2317
Rule 2438
Rule 2451
Rule 4615
Rule 4825
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) \text {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) \text {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) \text {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 \text {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 \text {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) \text {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) \text {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*}
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Mathematica [A]
time = 0.02, size = 246, normalized size = 1.04 \begin {gather*} \frac {i m \text {ArcSin}(c x)^2}{2 c}-\frac {m \text {ArcSin}(c x) \log \left (1-\frac {i c e^{i \text {ArcSin}(c x)} g}{c^2 f-c \sqrt {c^2 f^2-g^2}}\right )}{c}-\frac {m \text {ArcSin}(c x) \log \left (1-\frac {i c e^{i \text {ArcSin}(c x)} g}{c^2 f+c \sqrt {c^2 f^2-g^2}}\right )}{c}+\frac {\text {ArcSin}(c x) \log \left (h (f+g x)^m\right )}{c}+\frac {i m \text {PolyLog}\left (2,\frac {i e^{i \text {ArcSin}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c}+\frac {i m \text {PolyLog}\left (2,\frac {i e^{i \text {ArcSin}(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 = 0.02, 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 {- \left (c x - 1\right ) \left (c x + 1\right )}}\, 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 = -0.00 \begin {gather*} \int \frac {\ln \left (h\,{\left (f+g\,x\right )}^m\right )}{\sqrt {1-c^2\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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