Integrand size = 15, antiderivative size = 43 \[ \int \frac {A+B \sinh (x)}{(i+\sinh (x))^2} \, dx=-\frac {(i A+B) \cosh (x)}{3 (i+\sinh (x))^2}-\frac {(A+2 i B) \cosh (x)}{3 (i+\sinh (x))} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2829, 2727} \[ \int \frac {A+B \sinh (x)}{(i+\sinh (x))^2} \, dx=-\frac {(A+2 i B) \cosh (x)}{3 (\sinh (x)+i)}-\frac {(B+i A) \cosh (x)}{3 (\sinh (x)+i)^2} \]
[In]
[Out]
Rule 2727
Rule 2829
Rubi steps \begin{align*} \text {integral}& = -\frac {(i A+B) \cosh (x)}{3 (i+\sinh (x))^2}+\frac {1}{3} (-i A+2 B) \int \frac {1}{i+\sinh (x)} \, dx \\ & = -\frac {(i A+B) \cosh (x)}{3 (i+\sinh (x))^2}-\frac {(A+2 i B) \cosh (x)}{3 (i+\sinh (x))} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.74 \[ \int \frac {A+B \sinh (x)}{(i+\sinh (x))^2} \, dx=\frac {\cosh (x) (-2 i A+B-(A+2 i B) \sinh (x))}{3 (i+\sinh (x))^2} \]
[In]
[Out]
Time = 0.86 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.84
method | result | size |
risch | \(-\frac {2 \left (3 A \,{\mathrm e}^{x}+3 i B \,{\mathrm e}^{x}+3 B \,{\mathrm e}^{2 x}+i A -2 B \right )}{3 \left ({\mathrm e}^{x}+i\right )^{3}}\) | \(36\) |
default | \(-\frac {-2 i A -2 B}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}-\frac {2 A}{\tanh \left (\frac {x}{2}\right )+i}-\frac {2 \left (2 i B -2 A \right )}{3 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}}\) | \(52\) |
parallelrisch | \(\frac {\left (3 i A -3 B \right ) \cosh \left (2 x \right )+\left (-i B +A \right ) \sinh \left (2 x \right )+\left (-2 i B -10 A \right ) \sinh \left (x \right )-3 i A +3 B}{-3 i \sinh \left (2 x \right )+12 i \sinh \left (x \right )+6 \cosh \left (x \right )+3 \cosh \left (2 x \right )-9}\) | \(71\) |
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int \frac {A+B \sinh (x)}{(i+\sinh (x))^2} \, dx=-\frac {2 \, {\left (3 \, B e^{\left (2 \, x\right )} + 3 \, {\left (A + i \, B\right )} e^{x} + i \, A - 2 \, B\right )}}{3 \, {\left (e^{\left (3 \, x\right )} + 3 i \, e^{\left (2 \, x\right )} - 3 \, e^{x} - i\right )}} \]
[In]
[Out]
Time = 0.13 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.23 \[ \int \frac {A+B \sinh (x)}{(i+\sinh (x))^2} \, dx=\frac {- 2 i A - 6 B e^{2 x} + 4 B + \left (- 6 A - 6 i B\right ) e^{x}}{3 e^{3 x} + 9 i e^{2 x} - 9 e^{x} - 3 i} \]
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (31) = 62\).
Time = 0.20 (sec) , antiderivative size = 141, normalized size of antiderivative = 3.28 \[ \int \frac {A+B \sinh (x)}{(i+\sinh (x))^2} \, dx=-\frac {2}{3} \, A {\left (\frac {3 \, e^{\left (-x\right )}}{3 \, e^{\left (-x\right )} + 3 i \, e^{\left (-2 \, x\right )} - e^{\left (-3 \, x\right )} - i} - \frac {i}{3 \, e^{\left (-x\right )} + 3 i \, e^{\left (-2 \, x\right )} - e^{\left (-3 \, x\right )} - i}\right )} - \frac {2}{3} \, B {\left (\frac {3 i \, e^{\left (-x\right )}}{3 \, e^{\left (-x\right )} + 3 i \, e^{\left (-2 \, x\right )} - e^{\left (-3 \, x\right )} - i} - \frac {3 \, e^{\left (-2 \, x\right )}}{3 \, e^{\left (-x\right )} + 3 i \, e^{\left (-2 \, x\right )} - e^{\left (-3 \, x\right )} - i} + \frac {2}{3 \, e^{\left (-x\right )} + 3 i \, e^{\left (-2 \, x\right )} - e^{\left (-3 \, x\right )} - i}\right )} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.74 \[ \int \frac {A+B \sinh (x)}{(i+\sinh (x))^2} \, dx=-\frac {2 \, {\left (3 \, B e^{\left (2 \, x\right )} + 3 \, A e^{x} + 3 i \, B e^{x} + i \, A - 2 \, B\right )}}{3 \, {\left (e^{x} + i\right )}^{3}} \]
[In]
[Out]
Time = 1.32 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.91 \[ \int \frac {A+B \sinh (x)}{(i+\sinh (x))^2} \, dx=-\frac {\frac {2\,A}{3}+\frac {B\,4{}\mathrm {i}}{3}-{\mathrm {e}}^x\,\left (-2\,B+A\,2{}\mathrm {i}\right )-B\,{\mathrm {e}}^{2\,x}\,2{}\mathrm {i}}{{\left (-1+{\mathrm {e}}^x\,1{}\mathrm {i}\right )}^3} \]
[In]
[Out]