Optimal. Leaf size=85 \[ \frac{i \text{PolyLog}\left (2,e^{2 i \csc ^{-1}\left (c e^{a+b x}\right )}\right )}{2 b}+\frac{i \csc ^{-1}\left (c e^{a+b x}\right )^2}{2 b}-\frac{\csc ^{-1}\left (c e^{a+b x}\right ) \log \left (1-e^{2 i \csc ^{-1}\left (c e^{a+b x}\right )}\right )}{b} \]
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Rubi [A] time = 0.0857296, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {2282, 5219, 4625, 3717, 2190, 2279, 2391} \[ \frac{i \text{PolyLog}\left (2,e^{2 i \csc ^{-1}\left (c e^{a+b x}\right )}\right )}{2 b}+\frac{i \csc ^{-1}\left (c e^{a+b x}\right )^2}{2 b}-\frac{\csc ^{-1}\left (c e^{a+b x}\right ) \log \left (1-e^{2 i \csc ^{-1}\left (c e^{a+b x}\right )}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 5219
Rule 4625
Rule 3717
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \csc ^{-1}\left (c e^{a+b x}\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\csc ^{-1}(c x)}{x} \, dx,x,e^{a+b x}\right )}{b}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\sin ^{-1}\left (\frac{x}{c}\right )}{x} \, dx,x,e^{-a-b x}\right )}{b}\\ &=-\frac{\operatorname{Subst}\left (\int x \cot (x) \, dx,x,\sin ^{-1}\left (\frac{e^{-a-b x}}{c}\right )\right )}{b}\\ &=\frac{i \sin ^{-1}\left (\frac{e^{-a-b x}}{c}\right )^2}{2 b}+\frac{(2 i) \operatorname{Subst}\left (\int \frac{e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\sin ^{-1}\left (\frac{e^{-a-b x}}{c}\right )\right )}{b}\\ &=\frac{i \sin ^{-1}\left (\frac{e^{-a-b x}}{c}\right )^2}{2 b}-\frac{\sin ^{-1}\left (\frac{e^{-a-b x}}{c}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (\frac{e^{-a-b x}}{c}\right )}\right )}{b}+\frac{\operatorname{Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}\left (\frac{e^{-a-b x}}{c}\right )\right )}{b}\\ &=\frac{i \sin ^{-1}\left (\frac{e^{-a-b x}}{c}\right )^2}{2 b}-\frac{\sin ^{-1}\left (\frac{e^{-a-b x}}{c}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (\frac{e^{-a-b x}}{c}\right )}\right )}{b}-\frac{i \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}\left (\frac{e^{-a-b x}}{c}\right )}\right )}{2 b}\\ &=\frac{i \sin ^{-1}\left (\frac{e^{-a-b x}}{c}\right )^2}{2 b}-\frac{\sin ^{-1}\left (\frac{e^{-a-b x}}{c}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (\frac{e^{-a-b x}}{c}\right )}\right )}{b}+\frac{i \text{Li}_2\left (e^{2 i \sin ^{-1}\left (\frac{e^{-a-b x}}{c}\right )}\right )}{2 b}\\ \end{align*}
Mathematica [B] time = 0.751185, size = 280, normalized size = 3.29 \[ \frac{e^{-a-b x} \left (-4 \sqrt{1-c^2 e^{2 (a+b x)}} \text{PolyLog}\left (2,\frac{1}{2} \left (1-\sqrt{1-c^2 e^{2 (a+b x)}}\right )\right )+\sqrt{1-c^2 e^{2 (a+b x)}} \left (\log ^2\left (c^2 e^{2 (a+b x)}\right )+2 \log ^2\left (\frac{1}{2} \left (\sqrt{1-c^2 e^{2 (a+b x)}}+1\right )\right )-4 \log \left (\frac{1}{2} \left (\sqrt{1-c^2 e^{2 (a+b x)}}+1\right )\right ) \log \left (c^2 e^{2 (a+b x)}\right )\right )+4 \sqrt{c^2 e^{2 (a+b x)}-1} \left (2 b x-\log \left (c^2 e^{2 (a+b x)}\right )\right ) \tan ^{-1}\left (\sqrt{c^2 e^{2 (a+b x)}-1}\right )\right )}{8 b c \sqrt{1-\frac{e^{-2 (a+b x)}}{c^2}}}+x \csc ^{-1}\left (c e^{a+b x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.318, size = 199, normalized size = 2.3 \begin{align*}{\frac{{\frac{i}{2}} \left ({\rm arccsc} \left (c{{\rm e}^{bx+a}}\right ) \right ) ^{2}}{b}}-{\frac{{\rm arccsc} \left (c{{\rm e}^{bx+a}}\right )}{b}\ln \left ( 1-{\frac{i}{c{{\rm e}^{bx+a}}}}-\sqrt{1-{\frac{1}{{c}^{2} \left ({{\rm e}^{bx+a}} \right ) ^{2}}}} \right ) }-{\frac{{\rm arccsc} \left (c{{\rm e}^{bx+a}}\right )}{b}\ln \left ( 1+{\frac{i}{c{{\rm e}^{bx+a}}}}+\sqrt{1-{\frac{1}{{c}^{2} \left ({{\rm e}^{bx+a}} \right ) ^{2}}}} \right ) }+{\frac{i}{b}{\it polylog} \left ( 2,{\frac{-i}{c{{\rm e}^{bx+a}}}}-\sqrt{1-{\frac{1}{{c}^{2} \left ({{\rm e}^{bx+a}} \right ) ^{2}}}} \right ) }+{\frac{i}{b}{\it polylog} \left ( 2,{\frac{i}{c{{\rm e}^{bx+a}}}}+\sqrt{1-{\frac{1}{{c}^{2} \left ({{\rm e}^{bx+a}} \right ) ^{2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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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 \operatorname{acsc}{\left (c e^{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 \operatorname{arccsc}\left (c e^{\left (b x + a\right )}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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