Optimal. Leaf size=134 \[ \frac{i f \text{PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}-\frac{i f \text{PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2}-\frac{2 f \log \left (\sin \left (\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right )\right )}{a d^2}+\frac{(e+f x) \cot \left (\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right )}{a d}-\frac{2 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d} \]
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Rubi [A] time = 0.156537, antiderivative size = 134, normalized size of antiderivative = 1., 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 = {4535, 4183, 2279, 2391, 3318, 4184, 3475} \[ \frac{i f \text{PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}-\frac{i f \text{PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2}-\frac{2 f \log \left (\sin \left (\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right )\right )}{a d^2}+\frac{(e+f x) \cot \left (\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right )}{a d}-\frac{2 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d} \]
Antiderivative was successfully verified.
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Rule 4535
Rule 4183
Rule 2279
Rule 2391
Rule 3318
Rule 4184
Rule 3475
Rubi steps
\begin{align*} \int \frac{(e+f x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\int (e+f x) \csc (c+d x) \, dx}{a}-\int \frac{e+f x}{a+a \sin (c+d x)} \, dx\\ &=-\frac{2 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}-\frac{\int (e+f x) \csc ^2\left (\frac{1}{2} \left (c+\frac{\pi }{2}\right )+\frac{d x}{2}\right ) \, dx}{2 a}-\frac{f \int \log \left (1-e^{i (c+d x)}\right ) \, dx}{a d}+\frac{f \int \log \left (1+e^{i (c+d x)}\right ) \, dx}{a d}\\ &=-\frac{2 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac{(e+f x) \cot \left (\frac{c}{2}+\frac{\pi }{4}+\frac{d x}{2}\right )}{a d}+\frac{(i f) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^2}-\frac{(i f) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^2}-\frac{f \int \cot \left (\frac{c}{2}+\frac{\pi }{4}+\frac{d x}{2}\right ) \, dx}{a d}\\ &=-\frac{2 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac{(e+f x) \cot \left (\frac{c}{2}+\frac{\pi }{4}+\frac{d x}{2}\right )}{a d}-\frac{2 f \log \left (\sin \left (\frac{c}{2}+\frac{\pi }{4}+\frac{d x}{2}\right )\right )}{a d^2}+\frac{i f \text{Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}-\frac{i f \text{Li}_2\left (e^{i (c+d x)}\right )}{a d^2}\\ \end{align*}
Mathematica [B] time = 1.06422, size = 300, normalized size = 2.24 \[ \frac{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right ) \left (f \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right ) \left (i \left (\text{PolyLog}\left (2,-e^{i (c+d x)}\right )-\text{PolyLog}\left (2,e^{i (c+d x)}\right )\right )+(c+d x) \left (\log \left (1-e^{i (c+d x)}\right )-\log \left (1+e^{i (c+d x)}\right )\right )\right )-2 d (e+f x) \sin \left (\frac{1}{2} (c+d x)\right )+d e \log \left (\tan \left (\frac{1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+f (c+d x) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )-2 f \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )-c f \log \left (\tan \left (\frac{1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )}{a d^2 (\sin (c+d x)+1)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.144, size = 245, normalized size = 1.8 \begin{align*} 2\,{\frac{fx+e}{da \left ({{\rm e}^{i \left ( dx+c \right ) }}+i \right ) }}-2\,{\frac{f\ln \left ({{\rm e}^{i \left ( dx+c \right ) }}+i \right ) }{a{d}^{2}}}+{\frac{e\ln \left ({{\rm e}^{i \left ( dx+c \right ) }}-1 \right ) }{da}}-{\frac{e\ln \left ({{\rm e}^{i \left ( dx+c \right ) }}+1 \right ) }{da}}-{\frac{fc\ln \left ({{\rm e}^{i \left ( dx+c \right ) }}-1 \right ) }{a{d}^{2}}}-{\frac{if{\it polylog} \left ( 2,{{\rm e}^{i \left ( dx+c \right ) }} \right ) }{a{d}^{2}}}+{\frac{if{\it polylog} \left ( 2,-{{\rm e}^{i \left ( dx+c \right ) }} \right ) }{a{d}^{2}}}+2\,{\frac{f\ln \left ({{\rm e}^{i \left ( dx+c \right ) }} \right ) }{a{d}^{2}}}+{\frac{\ln \left ( 1-{{\rm e}^{i \left ( dx+c \right ) }} \right ) fx}{da}}+{\frac{\ln \left ( 1-{{\rm e}^{i \left ( dx+c \right ) }} \right ) cf}{a{d}^{2}}}-{\frac{\ln \left ({{\rm e}^{i \left ( dx+c \right ) }}+1 \right ) fx}{da}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.45336, size = 698, normalized size = 5.21 \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 = 2.10089, size = 1654, normalized size = 12.34 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{e \csc{\left (c + d x \right )}}{\sin{\left (c + d x \right )} + 1}\, dx + \int \frac{f x \csc{\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 )} \csc \left (d x + c\right )}{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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