Optimal. Leaf size=45 \[ \frac {i \text {Li}_2\left (-i e^{a+b x}\right )}{2 b}-\frac {i \text {Li}_2\left (i e^{a+b x}\right )}{2 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2282, 4848, 2391} \[ \frac {i \text {PolyLog}\left (2,-i e^{a+b x}\right )}{2 b}-\frac {i \text {PolyLog}\left (2,i e^{a+b x}\right )}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2282
Rule 2391
Rule 4848
Rubi steps
\begin {align*} \int \tan ^{-1}\left (e^{a+b x}\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\tan ^{-1}(x)}{x} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {i \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{a+b x}\right )}{2 b}-\frac {i \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{a+b x}\right )}{2 b}\\ &=\frac {i \text {Li}_2\left (-i e^{a+b x}\right )}{2 b}-\frac {i \text {Li}_2\left (i e^{a+b x}\right )}{2 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.11, size = 83, normalized size = 1.84 \[ x \tan ^{-1}\left (e^{a+b x}\right )-\frac {i \left (-\text {Li}_2\left (-i e^{a+b x}\right )+\text {Li}_2\left (i e^{a+b x}\right )+b x \left (\log \left (1-i e^{a+b x}\right )-\log \left (1+i e^{a+b x}\right )\right )\right )}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.50, size = 103, normalized size = 2.29 \[ \frac {2 \, b x \arctan \left (e^{\left (b x + a\right )}\right ) + i \, a \log \left (e^{\left (b x + a\right )} + i\right ) - i \, a \log \left (e^{\left (b x + a\right )} - i\right ) + {\left (i \, b x + i \, a\right )} \log \left (i \, e^{\left (b x + a\right )} + 1\right ) + {\left (-i \, b x - i \, a\right )} \log \left (-i \, e^{\left (b x + a\right )} + 1\right ) - i \, {\rm Li}_2\left (i \, e^{\left (b x + a\right )}\right ) + i \, {\rm Li}_2\left (-i \, e^{\left (b x + a\right )}\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \arctan \left (e^{\left (b x + a\right )}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.06, size = 106, normalized size = 2.36 \[ \frac {\ln \left ({\mathrm e}^{b x +a}\right ) \arctan \left ({\mathrm e}^{b x +a}\right )}{b}+\frac {i \ln \left ({\mathrm e}^{b x +a}\right ) \ln \left (1+i {\mathrm e}^{b x +a}\right )}{2 b}-\frac {i \ln \left ({\mathrm e}^{b x +a}\right ) \ln \left (1-i {\mathrm e}^{b x +a}\right )}{2 b}+\frac {i \dilog \left (1+i {\mathrm e}^{b x +a}\right )}{2 b}-\frac {i \dilog \left (1-i {\mathrm e}^{b x +a}\right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.44, size = 63, normalized size = 1.40 \[ \frac {{\left (b x + a\right )} \arctan \left (e^{\left (b x + a\right )}\right )}{b} - \frac {\pi \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) + 2 i \, {\rm Li}_2\left (i \, e^{\left (b x + a\right )} + 1\right ) - 2 i \, {\rm Li}_2\left (-i \, e^{\left (b x + a\right )} + 1\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.75, size = 37, normalized size = 0.82 \[ -\frac {{\mathrm {Li}}_{\mathrm {2}}\left (1-{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,b}+\frac {{\mathrm {Li}}_{\mathrm {2}}\left (1+{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {atan}{\left (e^{a + b x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________