Optimal. Leaf size=59 \[ \frac {i \cosh (c+d x)}{3 d (1+i \sinh (c+d x))^2}+\frac {i \cosh (c+d x)}{3 d (1+i \sinh (c+d x))} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2729, 2727}
\begin {gather*} \frac {i \cosh (c+d x)}{3 d (1+i \sinh (c+d x))}+\frac {i \cosh (c+d x)}{3 d (1+i \sinh (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2727
Rule 2729
Rubi steps
\begin {align*} \int \frac {1}{(1+i \sinh (c+d x))^2} \, dx &=\frac {i \cosh (c+d x)}{3 d (1+i \sinh (c+d x))^2}+\frac {1}{3} \int \frac {1}{1+i \sinh (c+d x)} \, dx\\ &=\frac {i \cosh (c+d x)}{3 d (1+i \sinh (c+d x))^2}+\frac {i \cosh (c+d x)}{3 d (1+i \sinh (c+d x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.08, size = 61, normalized size = 1.03 \begin {gather*} \frac {3 i-4 i \cosh (c+d x)-i \cosh (2 (c+d x))-4 \sinh (c+d x)+\sinh (2 (c+d x))}{6 d (-i+\sinh (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 1.27, size = 55, normalized size = 0.93
method | result | size |
risch | \(\frac {-\frac {2 i}{3}+2 \,{\mathrm e}^{d x +c}}{\left ({\mathrm e}^{d x +c}-i\right )^{3} d}\) | \(28\) |
derivativedivides | \(\frac {\frac {2 i}{\left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {2}{-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {4}{3 \left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}}{d}\) | \(55\) |
default | \(\frac {\frac {2 i}{\left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {2}{-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {4}{3 \left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}}{d}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 94, normalized size = 1.59 \begin {gather*} \frac {2 \, e^{\left (-d x - c\right )}}{d {\left (3 \, e^{\left (-d x - c\right )} - 3 i \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-3 \, d x - 3 \, c\right )} + i\right )}} + \frac {2 i}{3 \, d {\left (3 \, e^{\left (-d x - c\right )} - 3 i \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-3 \, d x - 3 \, c\right )} + i\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.44, size = 50, normalized size = 0.85 \begin {gather*} \frac {2 \, {\left (3 \, e^{\left (d x + c\right )} - i\right )}}{3 \, {\left (d e^{\left (3 \, d x + 3 \, c\right )} - 3 i \, d e^{\left (2 \, d x + 2 \, c\right )} - 3 \, d e^{\left (d x + c\right )} + i \, d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.10, size = 61, normalized size = 1.03 \begin {gather*} \frac {6 e^{c} e^{d x} - 2 i}{3 d e^{3 c} e^{3 d x} - 9 i d e^{2 c} e^{2 d x} - 9 d e^{c} e^{d x} + 3 i d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.42, size = 25, normalized size = 0.42 \begin {gather*} \frac {2 \, {\left (3 \, e^{\left (d x + c\right )} - i\right )}}{3 \, d {\left (e^{\left (d x + c\right )} - i\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.53, size = 29, normalized size = 0.49 \begin {gather*} -\frac {\frac {2}{3}+{\mathrm {e}}^{c+d\,x}\,2{}\mathrm {i}}{d\,{\left (1+{\mathrm {e}}^{c+d\,x}\,1{}\mathrm {i}\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________