Optimal. Leaf size=85 \[ \frac {i \text {Li}_2\left (e^{2 i \csc ^{-1}\left (c e^{a+b x}\right )}\right )}{2 b}+\frac {i \csc ^{-1}\left (c e^{a+b x}\right )^2}{2 b}-\frac {\csc ^{-1}\left (c e^{a+b x}\right ) \log \left (1-e^{2 i \csc ^{-1}\left (c e^{a+b x}\right )}\right )}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, 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.
[In]
[Out]
Rule 2190
Rule 2279
Rule 2282
Rule 2391
Rule 3717
Rule 4625
Rule 5219
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 = 0.73, size = 280, normalized size = 3.29 \[ \frac {e^{-a-b x} \left (-4 \sqrt {1-c^2 e^{2 (a+b x)}} \text {Li}_2\left (\frac {1}{2} \left (1-\sqrt {1-c^2 e^{2 (a+b x)}}\right )\right )+\sqrt {1-c^2 e^{2 (a+b x)}} \left (\log ^2\left (c^2 e^{2 (a+b x)}\right )+2 \log ^2\left (\frac {1}{2} \left (\sqrt {1-c^2 e^{2 (a+b x)}}+1\right )\right )-4 \log \left (\frac {1}{2} \left (\sqrt {1-c^2 e^{2 (a+b x)}}+1\right )\right ) \log \left (c^2 e^{2 (a+b x)}\right )\right )+4 \sqrt {c^2 e^{2 (a+b x)}-1} \left (2 b x-\log \left (c^2 e^{2 (a+b x)}\right )\right ) \tan ^{-1}\left (\sqrt {c^2 e^{2 (a+b x)}-1}\right )\right )}{8 b c \sqrt {1-\frac {e^{-2 (a+b x)}}{c^2}}}+x \csc ^{-1}\left (c e^{a+b x}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {arccsc}\left (c e^{\left (b x + a\right )}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.12, size = 199, normalized size = 2.34 \[ \frac {i \mathrm {arccsc}\left (c \,{\mathrm e}^{b x +a}\right )^{2}}{2 b}-\frac {\mathrm {arccsc}\left (c \,{\mathrm e}^{b x +a}\right ) \ln \left (1+\frac {i {\mathrm e}^{-b x -a}}{c}+\sqrt {1-\frac {{\mathrm e}^{-2 b x -2 a}}{c^{2}}}\right )}{b}-\frac {\mathrm {arccsc}\left (c \,{\mathrm e}^{b x +a}\right ) \ln \left (1-\frac {i {\mathrm e}^{-b x -a}}{c}-\sqrt {1-\frac {{\mathrm e}^{-2 b x -2 a}}{c^{2}}}\right )}{b}+\frac {i \polylog \left (2, -\frac {i {\mathrm e}^{-b x -a}}{c}-\sqrt {1-\frac {{\mathrm e}^{-2 b x -2 a}}{c^{2}}}\right )}{b}+\frac {i \polylog \left (2, \frac {i {\mathrm e}^{-b x -a}}{c}+\sqrt {1-\frac {{\mathrm e}^{-2 b x -2 a}}{c^{2}}}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.25, size = 91, normalized size = 1.07 \[ \frac {\mathrm {polylog}\left (2,{\mathrm {e}}^{\mathrm {asin}\left (\frac {{\mathrm {e}}^{-a-b\,x}}{c}\right )\,2{}\mathrm {i}}\right )\,1{}\mathrm {i}}{2\,b}+\frac {{\mathrm {asin}\left (\frac {{\mathrm {e}}^{-a-b\,x}}{c}\right )}^2\,1{}\mathrm {i}}{2\,b}-\frac {\ln \left (1-{\mathrm {e}}^{\mathrm {asin}\left (\frac {{\mathrm {e}}^{-a-b\,x}}{c}\right )\,2{}\mathrm {i}}\right )\,\mathrm {asin}\left (\frac {{\mathrm {e}}^{-a-b\,x}}{c}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {acsc}{\left (c e^{a + b x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________