Optimal. Leaf size=22 \[ \frac {\tan ^{-1}\left (\frac {1}{2} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2659, 206} \[ \frac {\tan ^{-1}\left (\frac {1}{2} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 2659
Rubi steps
\begin {align*} \int \frac {1}{3+5 \cosh (c+d x)} \, dx &=-\frac {(2 i) \operatorname {Subst}\left (\int \frac {1}{8-2 x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{d}\\ &=\frac {\tan ^{-1}\left (\frac {1}{2} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 23, normalized size = 1.05 \[ -\frac {\tan ^{-1}\left (2 \coth \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.53, size = 24, normalized size = 1.09 \[ \frac {\arctan \left (\frac {5}{4} \, \cosh \left (d x + c\right ) + \frac {5}{4} \, \sinh \left (d x + c\right ) + \frac {3}{4}\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.14, size = 16, normalized size = 0.73 \[ \frac {\arctan \left (\frac {5}{4} \, e^{\left (d x + c\right )} + \frac {3}{4}\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 18, normalized size = 0.82 \[ \frac {\arctan \left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.41, size = 19, normalized size = 0.86 \[ -\frac {\arctan \left (\frac {5}{4} \, e^{\left (-d x - c\right )} + \frac {3}{4}\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.13, size = 34, normalized size = 1.55 \[ \frac {\mathrm {atan}\left (\frac {3\,\sqrt {d^2}+5\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {d^2}}{4\,d}\right )}{2\,\sqrt {d^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.78, size = 24, normalized size = 1.09 \[ \begin {cases} \frac {\operatorname {atan}{\left (\frac {\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2} \right )}}{2 d} & \text {for}\: d \neq 0 \\\frac {x}{5 \cosh {\relax (c )} + 3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________