Optimal. Leaf size=237 \[ \frac {i m \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c}+\frac {i m \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c}-\frac {m \sin ^{-1}(c x) \log \left (1-\frac {i g e^{i \sin ^{-1}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c}-\frac {m \sin ^{-1}(c x) \log \left (1-\frac {i g e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{c}+\frac {\sin ^{-1}(c x) \log \left (h (f+g x)^m\right )}{c}+\frac {i m \sin ^{-1}(c x)^2}{2 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.35, 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 = {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.
[In]
[Out]
Rule 216
Rule 2190
Rule 2279
Rule 2391
Rule 2404
Rule 4519
Rule 4741
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 = 0.02, size = 246, normalized size = 1.04 \[ \frac {i m \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c}+\frac {i m \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c}-\frac {m \sin ^{-1}(c x) \log \left (1-\frac {i c g e^{i \sin ^{-1}(c x)}}{c^2 f-c \sqrt {c^2 f^2-g^2}}\right )}{c}-\frac {m \sin ^{-1}(c x) \log \left (1-\frac {i c g e^{i \sin ^{-1}(c x)}}{c \sqrt {c^2 f^2-g^2}+c^2 f}\right )}{c}+\frac {\sin ^{-1}(c x) \log \left (h (f+g x)^m\right )}{c}+\frac {i m \sin ^{-1}(c x)^2}{2 c} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-c^{2} x^{2} + 1} \log \left ({\left (g x + f\right )}^{m} h\right )}{c^{2} x^{2} - 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {-c^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.22, 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.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {-c^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\ln \left (h\,{\left (f+g\,x\right )}^m\right )}{\sqrt {1-c^2\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log {\left (h \left (f + g x\right )^{m} \right )}}{\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________