Optimal. Leaf size=325 \[ \frac{b f \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a d^2 \sqrt{a^2-b^2}}-\frac{b f \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{a d^2 \sqrt{a^2-b^2}}+\frac{i f \text{PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}-\frac{i f \text{PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2}+\frac{i b (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a d \sqrt{a^2-b^2}}-\frac{i b (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{a d \sqrt{a^2-b^2}}-\frac{2 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d} \]
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Rubi [A] time = 0.615057, antiderivative size = 325, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {4535, 4183, 2279, 2391, 3323, 2264, 2190} \[ \frac{b f \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a d^2 \sqrt{a^2-b^2}}-\frac{b f \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{a d^2 \sqrt{a^2-b^2}}+\frac{i f \text{PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}-\frac{i f \text{PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2}+\frac{i b (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a d \sqrt{a^2-b^2}}-\frac{i b (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{a d \sqrt{a^2-b^2}}-\frac{2 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d} \]
Antiderivative was successfully verified.
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Rule 4535
Rule 4183
Rule 2279
Rule 2391
Rule 3323
Rule 2264
Rule 2190
Rubi steps
\begin{align*} \int \frac{(e+f x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx &=\frac{\int (e+f x) \csc (c+d x) \, dx}{a}-\frac{b \int \frac{e+f x}{a+b \sin (c+d x)} \, dx}{a}\\ &=-\frac{2 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}-\frac{(2 b) \int \frac{e^{i (c+d x)} (e+f x)}{i b+2 a e^{i (c+d x)}-i b e^{2 i (c+d x)}} \, dx}{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{\left (2 i b^2\right ) \int \frac{e^{i (c+d x)} (e+f x)}{2 a-2 \sqrt{a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{a \sqrt{a^2-b^2}}-\frac{\left (2 i b^2\right ) \int \frac{e^{i (c+d x)} (e+f x)}{2 a+2 \sqrt{a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{a \sqrt{a^2-b^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{(i f) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^2}\\ &=-\frac{2 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac{i b (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a \sqrt{a^2-b^2} d}-\frac{i b (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{a \sqrt{a^2-b^2} d}+\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}-\frac{(i b f) \int \log \left (1-\frac{2 i b e^{i (c+d x)}}{2 a-2 \sqrt{a^2-b^2}}\right ) \, dx}{a \sqrt{a^2-b^2} d}+\frac{(i b f) \int \log \left (1-\frac{2 i b e^{i (c+d x)}}{2 a+2 \sqrt{a^2-b^2}}\right ) \, dx}{a \sqrt{a^2-b^2} d}\\ &=-\frac{2 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac{i b (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a \sqrt{a^2-b^2} d}-\frac{i b (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{a \sqrt{a^2-b^2} d}+\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}-\frac{(b f) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{2 i b x}{2 a-2 \sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{a \sqrt{a^2-b^2} d^2}+\frac{(b f) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{2 i b x}{2 a+2 \sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{a \sqrt{a^2-b^2} d^2}\\ &=-\frac{2 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac{i b (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a \sqrt{a^2-b^2} d}-\frac{i b (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{a \sqrt{a^2-b^2} d}+\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}+\frac{b f \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a \sqrt{a^2-b^2} d^2}-\frac{b f \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{a \sqrt{a^2-b^2} d^2}\\ \end{align*}
Mathematica [B] time = 6.39248, size = 764, normalized size = 2.35 \[ \frac{-\frac{b d (e+f x) \left (-\frac{i f \left (\text{PolyLog}\left (2,\frac{a \left (1-i \tan \left (\frac{1}{2} (c+d x)\right )\right )}{a+i \left (\sqrt{b^2-a^2}+b\right )}\right )+\log \left (1-i \tan \left (\frac{1}{2} (c+d x)\right )\right ) \log \left (\frac{\sqrt{b^2-a^2}+a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{b^2-a^2}-i a+b}\right )\right )}{\sqrt{b^2-a^2}}+\frac{i f \left (\text{PolyLog}\left (2,\frac{a \left (1+i \tan \left (\frac{1}{2} (c+d x)\right )\right )}{a-i \left (\sqrt{b^2-a^2}+b\right )}\right )+\log \left (1+i \tan \left (\frac{1}{2} (c+d x)\right )\right ) \log \left (\frac{\sqrt{b^2-a^2}+a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{b^2-a^2}+i a+b}\right )\right )}{\sqrt{b^2-a^2}}+\frac{i f \left (\text{PolyLog}\left (2,\frac{a \left (\tan \left (\frac{1}{2} (c+d x)\right )+i\right )}{\sqrt{b^2-a^2}+i a-b}\right )+\log \left (1-i \tan \left (\frac{1}{2} (c+d x)\right )\right ) \log \left (\frac{\sqrt{b^2-a^2}-a \tan \left (\frac{1}{2} (c+d x)\right )-b}{\sqrt{b^2-a^2}+i a-b}\right )\right )}{\sqrt{b^2-a^2}}-\frac{i f \left (\text{PolyLog}\left (2,\frac{a+i a \tan \left (\frac{1}{2} (c+d x)\right )}{a+i \left (\sqrt{b^2-a^2}-b\right )}\right )+\log \left (1+i \tan \left (\frac{1}{2} (c+d x)\right )\right ) \log \left (\frac{-\sqrt{b^2-a^2}+a \tan \left (\frac{1}{2} (c+d x)\right )+b}{-\sqrt{b^2-a^2}+i a+b}\right )\right )}{\sqrt{b^2-a^2}}+\frac{2 (d e-c f) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{\sqrt{a^2-b^2}}\right )}{i f \log \left (1-i \tan \left (\frac{1}{2} (c+d x)\right )\right )-i f \log \left (1+i \tan \left (\frac{1}{2} (c+d x)\right )\right )-c f+d e}+f \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 )+d e \log \left (\tan \left (\frac{1}{2} (c+d x)\right )\right )-c f \log \left (\tan \left (\frac{1}{2} (c+d x)\right )\right )}{a d^2} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.169, size = 660, normalized size = 2. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.9389, size = 3510, normalized size = 10.8 \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*} \int \frac{\left (e + f x\right ) \csc{\left (c + d x \right )}}{a + b \sin{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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