Optimal. Leaf size=49 \[ \frac {(A-2 i B) \cosh (x)}{3 (-\sinh (x)+i)}+\frac {(-B+i A) \cosh (x)}{3 (-\sinh (x)+i)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2750, 2648} \[ \frac {(A-2 i B) \cosh (x)}{3 (-\sinh (x)+i)}+\frac {(-B+i A) \cosh (x)}{3 (-\sinh (x)+i)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2648
Rule 2750
Rubi steps
\begin {align*} \int \frac {A+B \sinh (x)}{(i-\sinh (x))^2} \, dx &=\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*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 32, normalized size = 0.65 \[ \frac {\cosh (x) (-(A-2 i B) \sinh (x)+2 i A+B)}{3 (\sinh (x)-i)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.47, size = 46, normalized size = 0.94 \[ -\frac {6 \, B e^{\left (2 \, x\right )} + {\left (6 \, A - 6 i \, B\right )} e^{x} - 2 i \, A - 4 \, B}{3 \, e^{\left (3 \, x\right )} - 9 i \, e^{\left (2 \, x\right )} - 9 \, e^{x} + 3 i} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 32, normalized size = 0.65 \[ -\frac {6 \, B e^{\left (2 \, x\right )} + 6 \, A e^{x} - 6 i \, B e^{x} - 2 i \, A - 4 \, B}{3 \, {\left (e^{x} - i\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 52, normalized size = 1.06 \[ -\frac {2 A}{\tanh \left (\frac {x}{2}\right )-i}-\frac {2 i A -2 B}{\left (\tanh \left (\frac {x}{2}\right )-i\right )^{2}}-\frac {2 \left (-2 i B -2 A \right )}{3 \left (\tanh \left (\frac {x}{2}\right )-i\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.34, size = 141, normalized size = 2.88 \[ -A {\left (\frac {6 \, e^{\left (-x\right )}}{9 \, e^{\left (-x\right )} - 9 i \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-3 \, x\right )} + 3 i} + \frac {2 i}{9 \, e^{\left (-x\right )} - 9 i \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-3 \, x\right )} + 3 i}\right )} + \frac {1}{2} \, B {\left (\frac {12 i \, e^{\left (-x\right )}}{9 \, e^{\left (-x\right )} - 9 i \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-3 \, x\right )} + 3 i} + \frac {12 \, e^{\left (-2 \, x\right )}}{9 \, e^{\left (-x\right )} - 9 i \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-3 \, x\right )} + 3 i} - \frac {8}{9 \, e^{\left (-x\right )} - 9 i \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-3 \, x\right )} + 3 i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.60, size = 37, normalized size = 0.76 \[ \frac {\frac {2\,A}{3}-\frac {B\,4{}\mathrm {i}}{3}+{\mathrm {e}}^x\,\left (2\,B+A\,2{}\mathrm {i}\right )+B\,{\mathrm {e}}^{2\,x}\,2{}\mathrm {i}}{{\left (1+{\mathrm {e}}^x\,1{}\mathrm {i}\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.22, size = 51, normalized size = 1.04 \[ \frac {- 2 i A + 6 B e^{2 x} - 4 B + \left (6 A - 6 i B\right ) e^{x}}{- 3 e^{3 x} + 9 i e^{2 x} + 9 e^{x} - 3 i} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________