Optimal. Leaf size=188 \[ -\frac{4 i f^2 \text{PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}+\frac{4 f (e+f x) \log \left (1-i e^{i (c+d x)}\right )}{a d^2}+\frac{2 f (e+f x) \sin (c+d x)}{a d^2}+\frac{2 f^2 \cos (c+d x)}{a d^3}-\frac{(e+f x)^2 \cos (c+d x)}{a d}-\frac{(e+f x)^2 \cot \left (\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right )}{a d}-\frac{i (e+f x)^2}{a d}-\frac{(e+f x)^3}{3 a f} \]
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Rubi [A] time = 0.347819, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {4515, 3296, 2638, 32, 3318, 4184, 3717, 2190, 2279, 2391} \[ -\frac{4 i f^2 \text{PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}+\frac{4 f (e+f x) \log \left (1-i e^{i (c+d x)}\right )}{a d^2}+\frac{2 f (e+f x) \sin (c+d x)}{a d^2}+\frac{2 f^2 \cos (c+d x)}{a d^3}-\frac{(e+f x)^2 \cos (c+d x)}{a d}-\frac{(e+f x)^2 \cot \left (\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right )}{a d}-\frac{i (e+f x)^2}{a d}-\frac{(e+f x)^3}{3 a f} \]
Antiderivative was successfully verified.
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Rule 4515
Rule 3296
Rule 2638
Rule 32
Rule 3318
Rule 4184
Rule 3717
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{(e+f x)^2 \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\int (e+f x)^2 \sin (c+d x) \, dx}{a}-\int \frac{(e+f x)^2 \sin (c+d x)}{a+a \sin (c+d x)} \, dx\\ &=-\frac{(e+f x)^2 \cos (c+d x)}{a d}-\frac{\int (e+f x)^2 \, dx}{a}+\frac{(2 f) \int (e+f x) \cos (c+d x) \, dx}{a d}+\int \frac{(e+f x)^2}{a+a \sin (c+d x)} \, dx\\ &=-\frac{(e+f x)^3}{3 a f}-\frac{(e+f x)^2 \cos (c+d x)}{a d}+\frac{2 f (e+f x) \sin (c+d x)}{a d^2}+\frac{\int (e+f x)^2 \csc ^2\left (\frac{1}{2} \left (c+\frac{\pi }{2}\right )+\frac{d x}{2}\right ) \, dx}{2 a}-\frac{\left (2 f^2\right ) \int \sin (c+d x) \, dx}{a d^2}\\ &=-\frac{(e+f x)^3}{3 a f}+\frac{2 f^2 \cos (c+d x)}{a d^3}-\frac{(e+f x)^2 \cos (c+d x)}{a d}-\frac{(e+f x)^2 \cot \left (\frac{c}{2}+\frac{\pi }{4}+\frac{d x}{2}\right )}{a d}+\frac{2 f (e+f x) \sin (c+d x)}{a d^2}+\frac{(2 f) \int (e+f x) \cot \left (\frac{c}{2}+\frac{\pi }{4}+\frac{d x}{2}\right ) \, dx}{a d}\\ &=-\frac{i (e+f x)^2}{a d}-\frac{(e+f x)^3}{3 a f}+\frac{2 f^2 \cos (c+d x)}{a d^3}-\frac{(e+f x)^2 \cos (c+d x)}{a d}-\frac{(e+f x)^2 \cot \left (\frac{c}{2}+\frac{\pi }{4}+\frac{d x}{2}\right )}{a d}+\frac{2 f (e+f x) \sin (c+d x)}{a d^2}+\frac{(4 f) \int \frac{e^{2 i \left (\frac{c}{2}+\frac{d x}{2}\right )} (e+f x)}{1-i e^{2 i \left (\frac{c}{2}+\frac{d x}{2}\right )}} \, dx}{a d}\\ &=-\frac{i (e+f x)^2}{a d}-\frac{(e+f x)^3}{3 a f}+\frac{2 f^2 \cos (c+d x)}{a d^3}-\frac{(e+f x)^2 \cos (c+d x)}{a d}-\frac{(e+f x)^2 \cot \left (\frac{c}{2}+\frac{\pi }{4}+\frac{d x}{2}\right )}{a d}+\frac{4 f (e+f x) \log \left (1-i e^{i (c+d x)}\right )}{a d^2}+\frac{2 f (e+f x) \sin (c+d x)}{a d^2}-\frac{\left (4 f^2\right ) \int \log \left (1-i e^{2 i \left (\frac{c}{2}+\frac{d x}{2}\right )}\right ) \, dx}{a d^2}\\ &=-\frac{i (e+f x)^2}{a d}-\frac{(e+f x)^3}{3 a f}+\frac{2 f^2 \cos (c+d x)}{a d^3}-\frac{(e+f x)^2 \cos (c+d x)}{a d}-\frac{(e+f x)^2 \cot \left (\frac{c}{2}+\frac{\pi }{4}+\frac{d x}{2}\right )}{a d}+\frac{4 f (e+f x) \log \left (1-i e^{i (c+d x)}\right )}{a d^2}+\frac{2 f (e+f x) \sin (c+d x)}{a d^2}+\frac{\left (4 i f^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{2 i \left (\frac{c}{2}+\frac{d x}{2}\right )}\right )}{a d^3}\\ &=-\frac{i (e+f x)^2}{a d}-\frac{(e+f x)^3}{3 a f}+\frac{2 f^2 \cos (c+d x)}{a d^3}-\frac{(e+f x)^2 \cos (c+d x)}{a d}-\frac{(e+f x)^2 \cot \left (\frac{c}{2}+\frac{\pi }{4}+\frac{d x}{2}\right )}{a d}+\frac{4 f (e+f x) \log \left (1-i e^{i (c+d x)}\right )}{a d^2}-\frac{4 i f^2 \text{Li}_2\left (i e^{i (c+d x)}\right )}{a d^3}+\frac{2 f (e+f x) \sin (c+d x)}{a d^2}\\ \end{align*}
Mathematica [A] time = 2.56786, size = 295, normalized size = 1.57 \[ -\frac{\frac{12 f (\cos (c)+i \sin (c)) \left (\frac{f (\cos (c)-i (\sin (c)+1)) \text{PolyLog}(2,-\sin (c+d x)-i \cos (c+d x))}{d^2}-\frac{(\sin (c)+i \cos (c)+1) (e+f x) \log (\sin (c+d x)+i \cos (c+d x)+1)}{d}+\frac{(\cos (c)-i \sin (c)) (e+f x)^2}{2 f}\right )}{d (\cos (c)+i (\sin (c)+1))}+\frac{3 \cos (d x) \left (\cos (c) \left (d^2 (e+f x)^2-2 f^2\right )-2 d f \sin (c) (e+f x)\right )}{d^3}-\frac{3 \sin (d x) \left (\sin (c) \left (d^2 (e+f x)^2-2 f^2\right )+2 d f \cos (c) (e+f x)\right )}{d^3}-\frac{6 \sin \left (\frac{d x}{2}\right ) (e+f x)^2}{d \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}+x \left (3 e^2+3 e f x+f^2 x^2\right )}{3 a} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.336, size = 408, normalized size = 2.2 \begin{align*} -{\frac{{f}^{2}{x}^{3}}{3\,a}}-{\frac{fe{x}^{2}}{a}}-{\frac{{e}^{2}x}{a}}-{\frac{ \left ({f}^{2}{x}^{2}{d}^{2}+2\,id{f}^{2}x+2\,{d}^{2}efx+2\,idef+{d}^{2}{e}^{2}-2\,{f}^{2} \right ){{\rm e}^{i \left ( dx+c \right ) }}}{2\,{d}^{3}a}}-{\frac{ \left ({f}^{2}{x}^{2}{d}^{2}-2\,id{f}^{2}x+2\,{d}^{2}efx-2\,idef+{d}^{2}{e}^{2}-2\,{f}^{2} \right ){{\rm e}^{-i \left ( dx+c \right ) }}}{2\,{d}^{3}a}}-2\,{\frac{{f}^{2}{x}^{2}+2\,fex+{e}^{2}}{da \left ({{\rm e}^{i \left ( dx+c \right ) }}+i \right ) }}+4\,{\frac{\ln \left ({{\rm e}^{i \left ( dx+c \right ) }}+i \right ) ef}{a{d}^{2}}}-4\,{\frac{\ln \left ({{\rm e}^{i \left ( dx+c \right ) }} \right ) ef}{a{d}^{2}}}-{\frac{4\,i{f}^{2}cx}{a{d}^{2}}}-{\frac{2\,i{f}^{2}{x}^{2}}{da}}-{\frac{4\,i{f}^{2}{\it polylog} \left ( 2,i{{\rm e}^{i \left ( dx+c \right ) }} \right ) }{{d}^{3}a}}+4\,{\frac{{f}^{2}\ln \left ( 1-i{{\rm e}^{i \left ( dx+c \right ) }} \right ) x}{a{d}^{2}}}+4\,{\frac{{f}^{2}\ln \left ( 1-i{{\rm e}^{i \left ( dx+c \right ) }} \right ) c}{{d}^{3}a}}-{\frac{2\,i{f}^{2}{c}^{2}}{{d}^{3}a}}-4\,{\frac{c{f}^{2}\ln \left ({{\rm e}^{i \left ( dx+c \right ) }}+i \right ) }{{d}^{3}a}}+4\,{\frac{c{f}^{2}\ln \left ({{\rm e}^{i \left ( dx+c \right ) }} \right ) }{{d}^{3}a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.45847, size = 815, normalized size = 4.34 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.18301, size = 1674, normalized size = 8.9 \begin{align*} \text{result too large to display} \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*} \frac{\int \frac{e^{2} \sin ^{2}{\left (c + d x \right )}}{\sin{\left (c + d x \right )} + 1}\, dx + \int \frac{f^{2} x^{2} \sin ^{2}{\left (c + d x \right )}}{\sin{\left (c + d x \right )} + 1}\, dx + \int \frac{2 e f x \sin ^{2}{\left (c + d x \right )}}{\sin{\left (c + d x \right )} + 1}\, dx}{a} \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 (f x + e\right )}^{2} \sin \left (d x + c\right )^{2}}{a \sin \left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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