Optimal. Leaf size=15 \[ \frac {\log \left (a+c \tanh \left (\frac {x}{2}\right )\right )}{c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 15, 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 = {3124, 31} \[ \frac {\log \left (a+c \tanh \left (\frac {x}{2}\right )\right )}{c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 3124
Rubi steps
\begin {align*} \int \frac {1}{a+a \cosh (x)+c \sinh (x)} \, dx &=2 \operatorname {Subst}\left (\int \frac {1}{2 a+2 c x} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )\\ &=\frac {\log \left (a+c \tanh \left (\frac {x}{2}\right )\right )}{c}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.04, size = 35, normalized size = 2.33 \[ \frac {\log \left (a \cosh \left (\frac {x}{2}\right )+c \sinh \left (\frac {x}{2}\right )\right )}{c}-\frac {\log \left (\cosh \left (\frac {x}{2}\right )\right )}{c} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.40, size = 32, normalized size = 2.13 \[ \frac {\log \left ({\left (a + c\right )} \cosh \relax (x) + {\left (a + c\right )} \sinh \relax (x) + a - c\right ) - \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right )}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.13, size = 39, normalized size = 2.60 \[ \frac {{\left (a + c\right )} \log \left ({\left | a e^{x} + c e^{x} + a - c \right |}\right )}{a c + c^{2}} - \frac {\log \left (e^{x} + 1\right )}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.18, size = 14, normalized size = 0.93 \[ \frac {\ln \left (a +c \tanh \left (\frac {x}{2}\right )\right )}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.31, size = 36, normalized size = 2.40 \[ \frac {\log \left (-{\left (a - c\right )} e^{\left (-x\right )} - a - c\right )}{c} - \frac {\log \left (e^{\left (-x\right )} + 1\right )}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.16, size = 46, normalized size = 3.07 \[ -\frac {2\,\mathrm {atan}\left (\frac {a\,\sqrt {-c^2}+a\,{\mathrm {e}}^x\,\sqrt {-c^2}+c\,{\mathrm {e}}^x\,\sqrt {-c^2}}{c^2}\right )}{\sqrt {-c^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.72, size = 17, normalized size = 1.13 \[ \begin {cases} \frac {\log {\left (\frac {a}{c} + \tanh {\left (\frac {x}{2} \right )} \right )}}{c} & \text {for}\: c \neq 0 \\\frac {\tanh {\left (\frac {x}{2} \right )}}{a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________