Integrand size = 15, antiderivative size = 26 \[ \int \frac {\cosh ^3(x)}{(1-i \sinh (x))^3} \, dx=-i \log (i+\sinh (x))-\frac {2 i}{1-i \sinh (x)} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2746, 45} \[ \int \frac {\cosh ^3(x)}{(1-i \sinh (x))^3} \, dx=-\frac {2 i}{1-i \sinh (x)}-i \log (\sinh (x)+i) \]
[In]
[Out]
Rule 45
Rule 2746
Rubi steps \begin{align*} \text {integral}& = i \text {Subst}\left (\int \frac {1-x}{(1+x)^2} \, dx,x,-i \sinh (x)\right ) \\ & = i \text {Subst}\left (\int \left (\frac {1}{-1-x}+\frac {2}{(1+x)^2}\right ) \, dx,x,-i \sinh (x)\right ) \\ & = -i \log (i+\sinh (x))-\frac {2 i}{1-i \sinh (x)} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65 \[ \int \frac {\cosh ^3(x)}{(1-i \sinh (x))^3} \, dx=\frac {\cosh ^4(x) (2+\log (i+\sinh (x))-i \log (i+\sinh (x)) \sinh (x))}{(-i+\sinh (x))^2 (i+\sinh (x))^3} \]
[In]
[Out]
Time = 59.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(\frac {2}{i+\sinh \left (x \right )}-\frac {i \ln \left (\sinh \left (x \right )^{2}+1\right )}{2}-\arctan \left (\sinh \left (x \right )\right )\) | \(26\) |
default | \(\frac {2}{i+\sinh \left (x \right )}-\frac {i \ln \left (\sinh \left (x \right )^{2}+1\right )}{2}-\arctan \left (\sinh \left (x \right )\right )\) | \(26\) |
risch | \(i x +\frac {4 \,{\mathrm e}^{x}}{\left ({\mathrm e}^{x}+i\right )^{2}}-2 i \ln \left ({\mathrm e}^{x}+i\right )\) | \(26\) |
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (18) = 36\).
Time = 0.29 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.92 \[ \int \frac {\cosh ^3(x)}{(1-i \sinh (x))^3} \, dx=\frac {i \, x e^{\left (2 \, x\right )} - 2 \, {\left (x - 2\right )} e^{x} - 2 \, {\left (i \, e^{\left (2 \, x\right )} - 2 \, e^{x} - i\right )} \log \left (e^{x} + i\right ) - i \, x}{e^{\left (2 \, x\right )} + 2 i \, e^{x} - 1} \]
[In]
[Out]
Time = 0.09 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {\cosh ^3(x)}{(1-i \sinh (x))^3} \, dx=i x - 2 i \log {\left (e^{x} + i \right )} + \frac {4 e^{x}}{e^{2 x} + 2 i e^{x} - 1} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.27 \[ \int \frac {\cosh ^3(x)}{(1-i \sinh (x))^3} \, dx=-i \, x - \frac {4 \, e^{\left (-x\right )}}{-2 i \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1} - 2 i \, \log \left (e^{\left (-x\right )} - i\right ) \]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81 \[ \int \frac {\cosh ^3(x)}{(1-i \sinh (x))^3} \, dx=i \, x + \frac {4 \, e^{x}}{{\left (e^{x} + i\right )}^{2}} - 2 i \, \log \left (e^{x} + i\right ) \]
[In]
[Out]
Time = 0.17 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.50 \[ \int \frac {\cosh ^3(x)}{(1-i \sinh (x))^3} \, dx=x\,1{}\mathrm {i}-\ln \left ({\mathrm {e}}^x+1{}\mathrm {i}\right )\,2{}\mathrm {i}-\frac {4{}\mathrm {i}}{{\mathrm {e}}^{2\,x}-1+{\mathrm {e}}^x\,2{}\mathrm {i}}+\frac {4}{{\mathrm {e}}^x+1{}\mathrm {i}} \]
[In]
[Out]