Integrand size = 11, antiderivative size = 38 \[ \int \frac {1}{a \cosh (x)+b \sinh (x)} \, dx=\frac {\arctan \left (\frac {b \cosh (x)+a \sinh (x)}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3153, 212} \[ \int \frac {1}{a \cosh (x)+b \sinh (x)} \, dx=\frac {\arctan \left (\frac {a \sinh (x)+b \cosh (x)}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}} \]
[In]
[Out]
Rule 212
Rule 3153
Rubi steps \begin{align*} \text {integral}& = i \text {Subst}\left (\int \frac {1}{a^2-b^2-x^2} \, dx,x,-i b \cosh (x)-i a \sinh (x)\right ) \\ & = \frac {\arctan \left (\frac {b \cosh (x)+a \sinh (x)}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.21 \[ \int \frac {1}{a \cosh (x)+b \sinh (x)} \, dx=\frac {2 \arctan \left (\frac {b+a \tanh \left (\frac {x}{2}\right )}{\sqrt {a-b} \sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b}} \]
[In]
[Out]
Time = 0.66 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.03
method | result | size |
default | \(\frac {2 \arctan \left (\frac {2 a \tanh \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}\) | \(39\) |
risch | \(-\frac {\ln \left ({\mathrm e}^{x}-\frac {a -b}{\sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {a -b}{\sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}}\) | \(70\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 148, normalized size of antiderivative = 3.89 \[ \int \frac {1}{a \cosh (x)+b \sinh (x)} \, dx=\left [-\frac {\sqrt {-a^{2} + b^{2}} \log \left (\frac {{\left (a + b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )^{2} - 2 \, \sqrt {-a^{2} + b^{2}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} - a + b}{{\left (a + b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )^{2} + a - b}\right )}{a^{2} - b^{2}}, -\frac {2 \, \arctan \left (\frac {\sqrt {a^{2} - b^{2}}}{{\left (a + b\right )} \cosh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )}\right )}{\sqrt {a^{2} - b^{2}}}\right ] \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (31) = 62\).
Time = 2.11 (sec) , antiderivative size = 105, normalized size of antiderivative = 2.76 \[ \int \frac {1}{a \cosh (x)+b \sinh (x)} \, dx=\begin {cases} \tilde {\infty } \log {\left (\tanh {\left (\frac {x}{2} \right )} \right )} & \text {for}\: a = 0 \wedge b = 0 \\\frac {\log {\left (\tanh {\left (\frac {x}{2} \right )} \right )}}{b} & \text {for}\: a = 0 \\- \frac {1}{- b \sinh {\left (x \right )} + b \cosh {\left (x \right )}} & \text {for}\: a = - b \\- \frac {1}{b \sinh {\left (x \right )} + b \cosh {\left (x \right )}} & \text {for}\: a = b \\\frac {\log {\left (\tanh {\left (\frac {x}{2} \right )} + \frac {b}{a} - \frac {\sqrt {- a^{2} + b^{2}}}{a} \right )}}{\sqrt {- a^{2} + b^{2}}} - \frac {\log {\left (\tanh {\left (\frac {x}{2} \right )} + \frac {b}{a} + \frac {\sqrt {- a^{2} + b^{2}}}{a} \right )}}{\sqrt {- a^{2} + b^{2}}} & \text {otherwise} \end {cases} \]
[In]
[Out]
Exception generated. \[ \int \frac {1}{a \cosh (x)+b \sinh (x)} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.92 \[ \int \frac {1}{a \cosh (x)+b \sinh (x)} \, dx=\frac {2 \, \arctan \left (\frac {a e^{x} + b e^{x}}{\sqrt {a^{2} - b^{2}}}\right )}{\sqrt {a^{2} - b^{2}}} \]
[In]
[Out]
Time = 0.11 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.92 \[ \int \frac {1}{a \cosh (x)+b \sinh (x)} \, dx=\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {a^2-b^2}}{a-b}\right )}{\sqrt {a^2-b^2}} \]
[In]
[Out]