Integrand size = 18, antiderivative size = 57 \[ \int \frac {A+B \cosh (x)}{\sqrt {a-a \cosh (x)}} \, dx=-\frac {\sqrt {2} (A+B) \arctan \left (\frac {\sqrt {a} \sinh (x)}{\sqrt {2} \sqrt {a-a \cosh (x)}}\right )}{\sqrt {a}}+\frac {2 B \sinh (x)}{\sqrt {a-a \cosh (x)}} \]
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Time = 0.05 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2830, 2728, 212} \[ \int \frac {A+B \cosh (x)}{\sqrt {a-a \cosh (x)}} \, dx=\frac {2 B \sinh (x)}{\sqrt {a-a \cosh (x)}}-\frac {\sqrt {2} (A+B) \arctan \left (\frac {\sqrt {a} \sinh (x)}{\sqrt {2} \sqrt {a-a \cosh (x)}}\right )}{\sqrt {a}} \]
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Rule 212
Rule 2728
Rule 2830
Rubi steps \begin{align*} \text {integral}& = \frac {2 B \sinh (x)}{\sqrt {a-a \cosh (x)}}+(A+B) \int \frac {1}{\sqrt {a-a \cosh (x)}} \, dx \\ & = \frac {2 B \sinh (x)}{\sqrt {a-a \cosh (x)}}+(2 i (A+B)) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\frac {i a \sinh (x)}{\sqrt {a-a \cosh (x)}}\right ) \\ & = -\frac {\sqrt {2} (A+B) \arctan \left (\frac {\sqrt {a} \sinh (x)}{\sqrt {2} \sqrt {a-a \cosh (x)}}\right )}{\sqrt {a}}+\frac {2 B \sinh (x)}{\sqrt {a-a \cosh (x)}} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.89 \[ \int \frac {A+B \cosh (x)}{\sqrt {a-a \cosh (x)}} \, dx=\frac {2 \left (2 B \cosh \left (\frac {x}{2}\right )-(A+B) \left (\log \left (\cosh \left (\frac {x}{4}\right )\right )-\log \left (\sinh \left (\frac {x}{4}\right )\right )\right )\right ) \sinh \left (\frac {x}{2}\right )}{\sqrt {a-a \cosh (x)}} \]
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Time = 0.34 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.11
method | result | size |
default | \(\frac {\sinh \left (\frac {x}{2}\right ) \left (\ln \left (\cosh \left (\frac {x}{2}\right )-1\right ) A -\ln \left (\cosh \left (\frac {x}{2}\right )+1\right ) A +\ln \left (\cosh \left (\frac {x}{2}\right )-1\right ) B -\ln \left (\cosh \left (\frac {x}{2}\right )+1\right ) B +4 B \cosh \left (\frac {x}{2}\right )\right )}{\sqrt {-2 \sinh \left (\frac {x}{2}\right )^{2} a}}\) | \(63\) |
parts | \(-\frac {2 A \sinh \left (\frac {x}{2}\right ) \operatorname {arctanh}\left (\cosh \left (\frac {x}{2}\right )\right )}{\sqrt {-2 \sinh \left (\frac {x}{2}\right )^{2} a}}+\frac {B \sinh \left (\frac {x}{2}\right ) \left (4 \cosh \left (\frac {x}{2}\right )+\ln \left (\cosh \left (\frac {x}{2}\right )-1\right )-\ln \left (\cosh \left (\frac {x}{2}\right )+1\right )\right )}{\sqrt {-2 \sinh \left (\frac {x}{2}\right )^{2} a}}\) | \(65\) |
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Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (46) = 92\).
Time = 0.26 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.74 \[ \int \frac {A+B \cosh (x)}{\sqrt {a-a \cosh (x)}} \, dx=\frac {\sqrt {2} {\left (A + B\right )} a \sqrt {-\frac {1}{a}} \log \left (\frac {2 \, \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {-\frac {a}{\cosh \left (x\right ) + \sinh \left (x\right )}} \sqrt {-\frac {1}{a}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} - \cosh \left (x\right ) - \sinh \left (x\right ) - 1}{\cosh \left (x\right ) + \sinh \left (x\right ) - 1}\right ) - 2 \, \sqrt {\frac {1}{2}} {\left (B \cosh \left (x\right ) + B \sinh \left (x\right ) + B\right )} \sqrt {-\frac {a}{\cosh \left (x\right ) + \sinh \left (x\right )}}}{a} \]
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\[ \int \frac {A+B \cosh (x)}{\sqrt {a-a \cosh (x)}} \, dx=\int \frac {A + B \cosh {\left (x \right )}}{\sqrt {- a \left (\cosh {\left (x \right )} - 1\right )}}\, dx \]
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\[ \int \frac {A+B \cosh (x)}{\sqrt {a-a \cosh (x)}} \, dx=\int { \frac {B \cosh \left (x\right ) + A}{\sqrt {-a \cosh \left (x\right ) + a}} \,d x } \]
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none
Time = 0.26 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.47 \[ \int \frac {A+B \cosh (x)}{\sqrt {a-a \cosh (x)}} \, dx=-\frac {2 \, {\left (\sqrt {2} A + \sqrt {2} B\right )} \arctan \left (\frac {\sqrt {-a e^{x}}}{\sqrt {a}}\right )}{\sqrt {a} \mathrm {sgn}\left (-e^{x} + 1\right )} - \frac {\sqrt {2} B}{\sqrt {-a e^{x}} \mathrm {sgn}\left (-e^{x} + 1\right )} + \frac {\sqrt {2} \sqrt {-a e^{x}} B}{a \mathrm {sgn}\left (-e^{x} + 1\right )} \]
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Timed out. \[ \int \frac {A+B \cosh (x)}{\sqrt {a-a \cosh (x)}} \, dx=\int \frac {A+B\,\mathrm {cosh}\left (x\right )}{\sqrt {a-a\,\mathrm {cosh}\left (x\right )}} \,d x \]
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