Optimal. Leaf size=23 \[ -\frac {1}{3} i \tanh ^3(x)+\frac {\text {sech}^3(x)}{3}-\text {sech}(x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2706, 2607, 30, 2606} \[ -\frac {1}{3} i \tanh ^3(x)+\frac {\text {sech}^3(x)}{3}-\text {sech}(x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 2606
Rule 2607
Rule 2706
Rubi steps
\begin {align*} \int \frac {\tanh ^2(x)}{i+\sinh (x)} \, dx &=-\left (i \int \text {sech}^2(x) \tanh ^2(x) \, dx\right )+\int \text {sech}(x) \tanh ^3(x) \, dx\\ &=\operatorname {Subst}\left (\int x^2 \, dx,x,i \tanh (x)\right )+\operatorname {Subst}\left (\int \left (-1+x^2\right ) \, dx,x,\text {sech}(x)\right )\\ &=-\text {sech}(x)+\frac {\text {sech}^3(x)}{3}-\frac {1}{3} i \tanh ^3(x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.06, size = 67, normalized size = 2.91 \[ \frac {4 i \sinh (x)-\cosh (2 x)+(5-5 i \sinh (x)) \cosh (x)-3}{6 \left (\cosh \left (\frac {x}{2}\right )-i \sinh \left (\frac {x}{2}\right )\right )^3 \left (\cosh \left (\frac {x}{2}\right )+i \sinh \left (\frac {x}{2}\right )\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.61, size = 40, normalized size = 1.74 \[ -\frac {6 \, e^{\left (3 \, x\right )} + 6 i \, e^{\left (2 \, x\right )} - 2 \, e^{x} + 2 i}{3 \, e^{\left (4 \, x\right )} + 6 i \, e^{\left (3 \, x\right )} + 6 i \, e^{x} - 3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.20, size = 29, normalized size = 1.26 \[ -\frac {1}{2 \, {\left (e^{x} - i\right )}} - \frac {9 \, e^{\left (2 \, x\right )} + 12 i \, e^{x} - 7}{6 \, {\left (e^{x} + i\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.06, size = 47, normalized size = 2.04 \[ -\frac {2 i}{3 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}}-\frac {i}{2 \left (\tanh \left (\frac {x}{2}\right )+i\right )}+\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}+\frac {i}{2 \tanh \left (\frac {x}{2}\right )-2 i} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.31, size = 109, normalized size = 4.74 \[ \frac {2 \, e^{\left (-x\right )}}{-6 i \, e^{\left (-x\right )} - 6 i \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} - 3} + \frac {6 i \, e^{\left (-2 \, x\right )}}{-6 i \, e^{\left (-x\right )} - 6 i \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} - 3} - \frac {6 \, e^{\left (-3 \, x\right )}}{-6 i \, e^{\left (-x\right )} - 6 i \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} - 3} + \frac {2 i}{-6 i \, e^{\left (-x\right )} - 6 i \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} - 3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.67, size = 80, normalized size = 3.48 \[ -\frac {\frac {{\mathrm {e}}^x}{2}+\frac {1}{6}{}\mathrm {i}}{{\mathrm {e}}^{2\,x}-1+{\mathrm {e}}^x\,2{}\mathrm {i}}-\frac {\frac {{\mathrm {e}}^{2\,x}}{2}-\frac {1}{2}+\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{3}}{{\mathrm {e}}^{2\,x}\,3{}\mathrm {i}+{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x-\mathrm {i}}-\frac {1}{2\,\left ({\mathrm {e}}^x-\mathrm {i}\right )}-\frac {1}{2\,\left ({\mathrm {e}}^x+1{}\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.17, size = 46, normalized size = 2.00 \[ \frac {6 e^{3 x} + 6 i e^{2 x} - 2 e^{x} + 2 i}{- 3 e^{4 x} - 6 i e^{3 x} - 6 i e^{x} + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________