Optimal. Leaf size=67 \[ \frac{i d \text{PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^2}-\frac{i d \text{PolyLog}\left (2,e^{i (a+b x)}\right )}{b^2}-\frac{2 (c+d x) \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b} \]
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Rubi [A] time = 0.0394014, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4183, 2279, 2391} \[ \frac{i d \text{PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^2}-\frac{i d \text{PolyLog}\left (2,e^{i (a+b x)}\right )}{b^2}-\frac{2 (c+d x) \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 4183
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int (c+d x) \csc (a+b x) \, dx &=-\frac{2 (c+d x) \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}-\frac{d \int \log \left (1-e^{i (a+b x)}\right ) \, dx}{b}+\frac{d \int \log \left (1+e^{i (a+b x)}\right ) \, dx}{b}\\ &=-\frac{2 (c+d x) \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}+\frac{(i d) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^2}-\frac{(i d) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^2}\\ &=-\frac{2 (c+d x) \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}+\frac{i d \text{Li}_2\left (-e^{i (a+b x)}\right )}{b^2}-\frac{i d \text{Li}_2\left (e^{i (a+b x)}\right )}{b^2}\\ \end{align*}
Mathematica [A] time = 0.0751643, size = 134, normalized size = 2. \[ \frac{d \left (i \left (\text{PolyLog}\left (2,-e^{i (a+b x)}\right )-\text{PolyLog}\left (2,e^{i (a+b x)}\right )\right )+(a+b x) \left (\log \left (1-e^{i (a+b x)}\right )-\log \left (1+e^{i (a+b x)}\right )\right )-a \log \left (\tan \left (\frac{1}{2} (a+b x)\right )\right )\right )}{b^2}+\frac{c \log \left (\sin \left (\frac{a}{2}+\frac{b x}{2}\right )\right )}{b}-\frac{c \log \left (\cos \left (\frac{a}{2}+\frac{b x}{2}\right )\right )}{b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.032, size = 151, normalized size = 2.3 \begin{align*} -2\,{\frac{c{\it Artanh} \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) }{b}}+{\frac{d\ln \left ( 1-{{\rm e}^{i \left ( bx+a \right ) }} \right ) x}{b}}+{\frac{d\ln \left ( 1-{{\rm e}^{i \left ( bx+a \right ) }} \right ) a}{{b}^{2}}}-{\frac{id{\it polylog} \left ( 2,{{\rm e}^{i \left ( bx+a \right ) }} \right ) }{{b}^{2}}}-{\frac{d\ln \left ({{\rm e}^{i \left ( bx+a \right ) }}+1 \right ) x}{b}}-{\frac{d\ln \left ({{\rm e}^{i \left ( bx+a \right ) }}+1 \right ) a}{{b}^{2}}}+{\frac{id{\it polylog} \left ( 2,-{{\rm e}^{i \left ( bx+a \right ) }} \right ) }{{b}^{2}}}+2\,{\frac{da{\it Artanh} \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) }{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.32574, size = 235, normalized size = 3.51 \begin{align*} -\frac{2 i \, b d x \arctan \left (\sin \left (b x + a\right ), -\cos \left (b x + a\right ) + 1\right ) - 2 i \, b c \arctan \left (\sin \left (b x + a\right ), \cos \left (b x + a\right ) - 1\right ) +{\left (2 i \, b d x + 2 i \, b c\right )} \arctan \left (\sin \left (b x + a\right ), \cos \left (b x + a\right ) + 1\right ) - 2 i \, d{\rm Li}_2\left (-e^{\left (i \, b x + i \, a\right )}\right ) + 2 i \, d{\rm Li}_2\left (e^{\left (i \, b x + i \, a\right )}\right ) +{\left (b d x + b c\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1\right ) -{\left (b d x + b c\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1\right )}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.86689, size = 721, normalized size = 10.76 \begin{align*} \frac{-i \, d{\rm Li}_2\left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + i \, d{\rm Li}_2\left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) - i \, d{\rm Li}_2\left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + i \, d{\rm Li}_2\left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) -{\left (b d x + b c\right )} \log \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + 1\right ) -{\left (b d x + b c\right )} \log \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + 1\right ) +{\left (b c - a d\right )} \log \left (-\frac{1}{2} \, \cos \left (b x + a\right ) + \frac{1}{2} i \, \sin \left (b x + a\right ) + \frac{1}{2}\right ) +{\left (b c - a d\right )} \log \left (-\frac{1}{2} \, \cos \left (b x + a\right ) - \frac{1}{2} i \, \sin \left (b x + a\right ) + \frac{1}{2}\right ) +{\left (b d x + a d\right )} \log \left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + 1\right ) +{\left (b d x + a d\right )} \log \left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + 1\right )}{2 \, b^{2}} \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 \left (c + d x\right ) \csc{\left (a + b x \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{\left (d x + c\right )} \csc \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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