Optimal. Leaf size=34 \[ -\frac {1}{4 (\sinh (x)+i)}-\frac {i}{4 (\sinh (x)+i)^2}-\frac {1}{4} \tan ^{-1}(\sinh (x)) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2667, 44, 203} \[ -\frac {1}{4 (\sinh (x)+i)}-\frac {i}{4 (\sinh (x)+i)^2}-\frac {1}{4} \tan ^{-1}(\sinh (x)) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rule 203
Rule 2667
Rubi steps
\begin {align*} \int \frac {\text {sech}(x)}{(i+\sinh (x))^2} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{(i-x) (i+x)^3} \, dx,x,\sinh (x)\right )\\ &=-\operatorname {Subst}\left (\int \left (-\frac {i}{2 (i+x)^3}-\frac {1}{4 (i+x)^2}+\frac {1}{4 \left (1+x^2\right )}\right ) \, dx,x,\sinh (x)\right )\\ &=-\frac {i}{4 (i+\sinh (x))^2}-\frac {1}{4 (i+\sinh (x))}-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (x)\right )\\ &=-\frac {1}{4} \tan ^{-1}(\sinh (x))-\frac {i}{4 (i+\sinh (x))^2}-\frac {1}{4 (i+\sinh (x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 26, normalized size = 0.76 \[ \frac {1}{4} \left (-\tan ^{-1}(\sinh (x))-\frac {\sinh (x)+2 i}{(\sinh (x)+i)^2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.74, size = 104, normalized size = 3.06 \[ \frac {{\left (-i \, e^{\left (4 \, x\right )} + 4 \, e^{\left (3 \, x\right )} + 6 i \, e^{\left (2 \, x\right )} - 4 \, e^{x} - i\right )} \log \left (e^{x} + i\right ) + {\left (i \, e^{\left (4 \, x\right )} - 4 \, e^{\left (3 \, x\right )} - 6 i \, e^{\left (2 \, x\right )} + 4 \, e^{x} + i\right )} \log \left (e^{x} - i\right ) - 2 \, e^{\left (3 \, x\right )} - 8 i \, e^{\left (2 \, x\right )} + 2 \, e^{x}}{4 \, e^{\left (4 \, x\right )} + 16 i \, e^{\left (3 \, x\right )} - 24 \, e^{\left (2 \, x\right )} - 16 i \, e^{x} + 4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.16, size = 70, normalized size = 2.06 \[ \frac {3 i \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 20 \, e^{\left (-x\right )} - 20 \, e^{x} - 44 i}{16 \, {\left (e^{\left (-x\right )} - e^{x} - 2 i\right )}^{2}} - \frac {1}{8} i \, \log \left (i \, e^{\left (-x\right )} - i \, e^{x} + 2\right ) + \frac {1}{8} i \, \log \left (i \, e^{\left (-x\right )} - i \, e^{x} - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.06, size = 70, normalized size = 2.06 \[ \frac {i}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{4}}-\frac {i \ln \left (\tanh \left (\frac {x}{2}\right )+i\right )}{4}-\frac {5 i}{2 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}-\frac {2}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}}+\frac {3}{2 \left (\tanh \left (\frac {x}{2}\right )+i\right )}+\frac {i \ln \left (\tanh \left (\frac {x}{2}\right )-i\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.35, size = 70, normalized size = 2.06 \[ -\frac {2 \, {\left (e^{\left (-x\right )} + 4 i \, e^{\left (-2 \, x\right )} - e^{\left (-3 \, x\right )}\right )}}{16 i \, e^{\left (-x\right )} - 24 \, e^{\left (-2 \, x\right )} - 16 i \, e^{\left (-3 \, x\right )} + 4 \, e^{\left (-4 \, x\right )} + 4} - \frac {1}{4} i \, \log \left (i \, e^{\left (-x\right )} + 1\right ) + \frac {1}{4} i \, \log \left (i \, e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.75, size = 86, normalized size = 2.53 \[ -\frac {\mathrm {atan}\left ({\mathrm {e}}^x\right )}{2}-\frac {1{}\mathrm {i}}{2\,\left ({\mathrm {e}}^{2\,x}-1+{\mathrm {e}}^x\,2{}\mathrm {i}\right )}+\frac {1{}\mathrm {i}}{{\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}+1+{\mathrm {e}}^{3\,x}\,4{}\mathrm {i}-{\mathrm {e}}^x\,4{}\mathrm {i}}-\frac {1}{2\,\left ({\mathrm {e}}^x+1{}\mathrm {i}\right )}-\frac {2}{{\mathrm {e}}^{2\,x}\,3{}\mathrm {i}+{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x-\mathrm {i}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {sech}{\relax (x )}}{\left (\sinh {\relax (x )} + i\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________