Optimal. Leaf size=555 \[ \frac{2 i a h (g+h x) \cos (e+f x) \text{PolyLog}\left (2,-i e^{i (e+f x)}\right )}{f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{2 i a h (g+h x) \cos (e+f x) \text{PolyLog}\left (2,i e^{i (e+f x)}\right )}{f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{i a h (g+h x) \cos (e+f x) \text{PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{2 a h^2 \cos (e+f x) \text{PolyLog}\left (3,-i e^{i (e+f x)}\right )}{f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{2 a h^2 \cos (e+f x) \text{PolyLog}\left (3,i e^{i (e+f x)}\right )}{f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{a h^2 \cos (e+f x) \text{PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{i a (g+h x)^3 \cos (e+f x)}{3 h \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{a (g+h x)^2 \log \left (1+e^{2 i (e+f x)}\right ) \cos (e+f x)}{f \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{2 i a (g+h x)^2 \cos (e+f x) \tan ^{-1}\left (e^{i (e+f x)}\right )}{f \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}} \]
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Rubi [A] time = 0.876905, antiderivative size = 555, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 10, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.27, Rules used = {4604, 6741, 12, 6742, 4181, 2531, 2282, 6589, 3719, 2190} \[ \frac{2 i a h (g+h x) \cos (e+f x) \text{PolyLog}\left (2,-i e^{i (e+f x)}\right )}{f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{2 i a h (g+h x) \cos (e+f x) \text{PolyLog}\left (2,i e^{i (e+f x)}\right )}{f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{i a h (g+h x) \cos (e+f x) \text{PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{2 a h^2 \cos (e+f x) \text{PolyLog}\left (3,-i e^{i (e+f x)}\right )}{f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{2 a h^2 \cos (e+f x) \text{PolyLog}\left (3,i e^{i (e+f x)}\right )}{f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{a h^2 \cos (e+f x) \text{PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{i a (g+h x)^3 \cos (e+f x)}{3 h \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{a (g+h x)^2 \log \left (1+e^{2 i (e+f x)}\right ) \cos (e+f x)}{f \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{2 i a (g+h x)^2 \cos (e+f x) \tan ^{-1}\left (e^{i (e+f x)}\right )}{f \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}} \]
Antiderivative was successfully verified.
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Rule 4604
Rule 6741
Rule 12
Rule 6742
Rule 4181
Rule 2531
Rule 2282
Rule 6589
Rule 3719
Rule 2190
Rubi steps
\begin{align*} \int \frac{(g+h x)^2 \sqrt{a-a \sin (e+f x)}}{\sqrt{c+c \sin (e+f x)}} \, dx &=\frac{\cos (e+f x) \int (g+h x)^2 \sec (e+f x) (a-a \sin (e+f x)) \, dx}{\sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}\\ &=\frac{\cos (e+f x) \int a (g+h x)^2 \sec (e+f x) (1-\sin (e+f x)) \, dx}{\sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}\\ &=\frac{(a \cos (e+f x)) \int (g+h x)^2 \sec (e+f x) (1-\sin (e+f x)) \, dx}{\sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}\\ &=\frac{(a \cos (e+f x)) \int \left ((g+h x)^2 \sec (e+f x)-(g+h x)^2 \tan (e+f x)\right ) \, dx}{\sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}\\ &=\frac{(a \cos (e+f x)) \int (g+h x)^2 \sec (e+f x) \, dx}{\sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}-\frac{(a \cos (e+f x)) \int (g+h x)^2 \tan (e+f x) \, dx}{\sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}\\ &=-\frac{i a (g+h x)^3 \cos (e+f x)}{3 h \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}-\frac{2 i a (g+h x)^2 \tan ^{-1}\left (e^{i (e+f x)}\right ) \cos (e+f x)}{f \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{(2 i a \cos (e+f x)) \int \frac{e^{2 i (e+f x)} (g+h x)^2}{1+e^{2 i (e+f x)}} \, dx}{\sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}-\frac{(2 a h \cos (e+f x)) \int (g+h x) \log \left (1-i e^{i (e+f x)}\right ) \, dx}{f \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{(2 a h \cos (e+f x)) \int (g+h x) \log \left (1+i e^{i (e+f x)}\right ) \, dx}{f \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}\\ &=-\frac{i a (g+h x)^3 \cos (e+f x)}{3 h \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}-\frac{2 i a (g+h x)^2 \tan ^{-1}\left (e^{i (e+f x)}\right ) \cos (e+f x)}{f \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{a (g+h x)^2 \cos (e+f x) \log \left (1+e^{2 i (e+f x)}\right )}{f \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{2 i a h (g+h x) \cos (e+f x) \text{Li}_2\left (-i e^{i (e+f x)}\right )}{f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}-\frac{2 i a h (g+h x) \cos (e+f x) \text{Li}_2\left (i e^{i (e+f x)}\right )}{f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}-\frac{(2 a h \cos (e+f x)) \int (g+h x) \log \left (1+e^{2 i (e+f x)}\right ) \, dx}{f \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}-\frac{\left (2 i a h^2 \cos (e+f x)\right ) \int \text{Li}_2\left (-i e^{i (e+f x)}\right ) \, dx}{f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{\left (2 i a h^2 \cos (e+f x)\right ) \int \text{Li}_2\left (i e^{i (e+f x)}\right ) \, dx}{f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}\\ &=-\frac{i a (g+h x)^3 \cos (e+f x)}{3 h \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}-\frac{2 i a (g+h x)^2 \tan ^{-1}\left (e^{i (e+f x)}\right ) \cos (e+f x)}{f \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{a (g+h x)^2 \cos (e+f x) \log \left (1+e^{2 i (e+f x)}\right )}{f \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{2 i a h (g+h x) \cos (e+f x) \text{Li}_2\left (-i e^{i (e+f x)}\right )}{f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}-\frac{2 i a h (g+h x) \cos (e+f x) \text{Li}_2\left (i e^{i (e+f x)}\right )}{f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}-\frac{i a h (g+h x) \cos (e+f x) \text{Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}-\frac{\left (2 a h^2 \cos (e+f x)\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{\left (2 a h^2 \cos (e+f x)\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{\left (i a h^2 \cos (e+f x)\right ) \int \text{Li}_2\left (-e^{2 i (e+f x)}\right ) \, dx}{f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}\\ &=-\frac{i a (g+h x)^3 \cos (e+f x)}{3 h \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}-\frac{2 i a (g+h x)^2 \tan ^{-1}\left (e^{i (e+f x)}\right ) \cos (e+f x)}{f \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{a (g+h x)^2 \cos (e+f x) \log \left (1+e^{2 i (e+f x)}\right )}{f \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{2 i a h (g+h x) \cos (e+f x) \text{Li}_2\left (-i e^{i (e+f x)}\right )}{f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}-\frac{2 i a h (g+h x) \cos (e+f x) \text{Li}_2\left (i e^{i (e+f x)}\right )}{f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}-\frac{i a h (g+h x) \cos (e+f x) \text{Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}-\frac{2 a h^2 \cos (e+f x) \text{Li}_3\left (-i e^{i (e+f x)}\right )}{f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{2 a h^2 \cos (e+f x) \text{Li}_3\left (i e^{i (e+f x)}\right )}{f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{\left (a h^2 \cos (e+f x)\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{2 f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}\\ &=-\frac{i a (g+h x)^3 \cos (e+f x)}{3 h \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}-\frac{2 i a (g+h x)^2 \tan ^{-1}\left (e^{i (e+f x)}\right ) \cos (e+f x)}{f \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{a (g+h x)^2 \cos (e+f x) \log \left (1+e^{2 i (e+f x)}\right )}{f \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{2 i a h (g+h x) \cos (e+f x) \text{Li}_2\left (-i e^{i (e+f x)}\right )}{f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}-\frac{2 i a h (g+h x) \cos (e+f x) \text{Li}_2\left (i e^{i (e+f x)}\right )}{f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}-\frac{i a h (g+h x) \cos (e+f x) \text{Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}-\frac{2 a h^2 \cos (e+f x) \text{Li}_3\left (-i e^{i (e+f x)}\right )}{f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{2 a h^2 \cos (e+f x) \text{Li}_3\left (i e^{i (e+f x)}\right )}{f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}+\frac{a h^2 \cos (e+f x) \text{Li}_3\left (-e^{2 i (e+f x)}\right )}{2 f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 1.92293, size = 194, normalized size = 0.35 \[ \frac{\sqrt{2} \left (e^{i (e+f x)}+i\right ) \sqrt{a-a \sin (e+f x)} \left (12 f h^2 (g+h x) \text{PolyLog}\left (2,-i e^{-i (e+f x)}\right )-12 i h^3 \text{PolyLog}\left (3,-i e^{-i (e+f x)}\right )+f^2 (g+h x)^2 \left (f (g+h x)-6 i h \log \left (1+i e^{-i (e+f x)}\right )\right )\right )}{3 f^3 h \left (e^{i (e+f x)}-i\right ) \sqrt{-i c e^{-i (e+f x)} \left (e^{i (e+f x)}+i\right )^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.092, size = 0, normalized size = 0. \begin{align*} \int{ \left ( hx+g \right ) ^{2}\sqrt{a-a\sin \left ( fx+e \right ) }{\frac{1}{\sqrt{c+c\sin \left ( fx+e \right ) }}}}\, 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 (h x + g\right )}^{2} \sqrt{-a \sin \left (f x + e\right ) + a}}{\sqrt{c \sin \left (f x + e\right ) + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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{\sqrt{- a \left (\sin{\left (e + f x \right )} - 1\right )} \left (g + h x\right )^{2}}{\sqrt{c \left (\sin{\left (e + f x \right )} + 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 (h x + g\right )}^{2} \sqrt{-a \sin \left (f x + e\right ) + a}}{\sqrt{c \sin \left (f x + e\right ) + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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