Optimal. Leaf size=34 \[ \frac {\sinh (c+d x)}{a d}-\frac {i \sinh ^2(c+d x)}{2 a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {2667} \[ \frac {\sinh (c+d x)}{a d}-\frac {i \sinh ^2(c+d x)}{2 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2667
Rubi steps
\begin {align*} \int \frac {\cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx &=-\frac {i \operatorname {Subst}(\int (a-x) \, dx,x,i a \sinh (c+d x))}{a^3 d}\\ &=\frac {\sinh (c+d x)}{a d}-\frac {i \sinh ^2(c+d x)}{2 a d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 28, normalized size = 0.82 \[ \frac {(2-i \sinh (c+d x)) \sinh (c+d x)}{2 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.52, size = 49, normalized size = 1.44 \[ \frac {{\left (-i \, e^{\left (4 \, d x + 4 \, c\right )} + 4 \, e^{\left (3 \, d x + 3 \, c\right )} - 4 \, e^{\left (d x + c\right )} - i\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.23, size = 55, normalized size = 1.62 \[ -\frac {\frac {{\left (4 \, e^{\left (d x + c\right )} + i\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{a} + \frac {i \, a e^{\left (2 \, d x + 2 \, c\right )} - 4 \, a e^{\left (d x + c\right )}}{a^{2}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 29, normalized size = 0.85 \[ -\frac {\frac {i \left (\sinh ^{2}\left (d x +c \right )\right )}{2}-\sinh \left (d x +c \right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 60, normalized size = 1.76 \[ -\frac {i \, {\left (4 i \, e^{\left (-d x - c\right )} + 1\right )} e^{\left (2 \, d x + 2 \, c\right )}}{8 \, a d} - \frac {i \, {\left (-4 i \, e^{\left (-d x - c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )}}{8 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.29, size = 29, normalized size = 0.85 \[ \frac {4\,\mathrm {sinh}\left (c+d\,x\right )-\mathrm {cosh}\left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}}{4\,a\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.35, size = 134, normalized size = 3.94 \[ \begin {cases} \frac {\left (- 32 i a^{3} d^{3} e^{5 c} e^{2 d x} + 128 a^{3} d^{3} e^{4 c} e^{d x} - 128 a^{3} d^{3} e^{2 c} e^{- d x} - 32 i a^{3} d^{3} e^{c} e^{- 2 d x}\right ) e^{- 3 c}}{256 a^{4} d^{4}} & \text {for}\: 256 a^{4} d^{4} e^{3 c} \neq 0 \\\frac {x \left (- i e^{4 c} + 2 e^{3 c} + 2 e^{c} + i\right ) e^{- 2 c}}{4 a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________