Integrand size = 28, antiderivative size = 102 \[ \int \frac {1}{\sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}} \, dx=-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt [4]{b^2-c^2} \sinh \left (x+i \tan ^{-1}(b,-i c)\right )}{\sqrt {2} \sqrt {-\sqrt {b^2-c^2}+\sqrt {b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}}\right )}{\sqrt [4]{b^2-c^2}} \]
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Time = 0.08 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {3194, 2728, 210} \[ \int \frac {1}{\sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}} \, dx=-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt [4]{b^2-c^2} \sinh \left (x+i \tan ^{-1}(b,-i c)\right )}{\sqrt {2} \sqrt {-\sqrt {b^2-c^2}+\sqrt {b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}}\right )}{\sqrt [4]{b^2-c^2}} \]
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Rule 210
Rule 2728
Rule 3194
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sqrt {-\sqrt {b^2-c^2}+\sqrt {b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}} \, dx \\ & = 2 i \text {Subst}\left (\int \frac {1}{-2 \sqrt {b^2-c^2}-x^2} \, dx,x,-\frac {i \sqrt {b^2-c^2} \sinh \left (x+i \tan ^{-1}(b,-i c)\right )}{\sqrt {-\sqrt {b^2-c^2}+\sqrt {b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}}\right ) \\ & = -\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt [4]{b^2-c^2} \sinh \left (x+i \tan ^{-1}(b,-i c)\right )}{\sqrt {2} \sqrt {-\sqrt {b^2-c^2}+\sqrt {b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}}\right )}{\sqrt [4]{b^2-c^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 54.83 (sec) , antiderivative size = 52609, normalized size of antiderivative = 515.77 \[ \int \frac {1}{\sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}} \, dx=\text {Result too large to show} \]
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Time = 0.40 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.26
method | result | size |
default | \(\frac {\sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )+1\right ) \sinh \left (x \right )^{2}}\, \arctan \left (\frac {\sqrt {\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )+1\right )}\, \cosh \left (x \right )}{\sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )+1\right ) \sinh \left (x \right )^{2}}}\right )}{\sqrt {\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )+1\right )}\, \sinh \left (x \right ) \sqrt {-\frac {\sinh \left (x \right ) b^{2}-\sinh \left (x \right ) c^{2}+b^{2}-c^{2}}{\sqrt {b^{2}-c^{2}}}}}\) | \(129\) |
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Time = 0.32 (sec) , antiderivative size = 680, normalized size of antiderivative = 6.67 \[ \int \frac {1}{\sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}} \, dx=\text {Too large to display} \]
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\[ \int \frac {1}{\sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}} \, dx=\int \frac {1}{\sqrt {b \cosh {\left (x \right )} + c \sinh {\left (x \right )} - \sqrt {b^{2} - c^{2}}}}\, dx \]
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\[ \int \frac {1}{\sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}} \, dx=\int { \frac {1}{\sqrt {b \cosh \left (x\right ) + c \sinh \left (x\right ) - \sqrt {b^{2} - c^{2}}}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 546 vs. \(2 (83) = 166\).
Time = 0.52 (sec) , antiderivative size = 546, normalized size of antiderivative = 5.35 \[ \int \frac {1}{\sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}} \, dx=\frac {2 \, \sqrt {2} {\left (b^{2} - c^{2} - b + c\right )} \sqrt {b + c} \arctan \left (\frac {b^{3} e^{\left (\frac {1}{2} \, x\right )} + b^{2} c e^{\left (\frac {1}{2} \, x\right )} - b c^{2} e^{\left (\frac {1}{2} \, x\right )} - c^{3} e^{\left (\frac {1}{2} \, x\right )} - b^{2} e^{\left (\frac {1}{2} \, x\right )} + c^{2} e^{\left (\frac {1}{2} \, x\right )}}{\sqrt {-\sqrt {b^{2} - c^{2}} b^{5} - \sqrt {b^{2} - c^{2}} b^{4} c + 2 \, \sqrt {b^{2} - c^{2}} b^{3} c^{2} + 2 \, \sqrt {b^{2} - c^{2}} b^{2} c^{3} - \sqrt {b^{2} - c^{2}} b c^{4} - \sqrt {b^{2} - c^{2}} c^{5} + 2 \, \sqrt {b^{2} - c^{2}} b^{4} - 4 \, \sqrt {b^{2} - c^{2}} b^{2} c^{2} + 2 \, \sqrt {b^{2} - c^{2}} c^{4} - \sqrt {b^{2} - c^{2}} b^{3} + \sqrt {b^{2} - c^{2}} b^{2} c + \sqrt {b^{2} - c^{2}} b c^{2} - \sqrt {b^{2} - c^{2}} c^{3}}}\right )}{\sqrt {-\sqrt {b^{2} - c^{2}} b^{5} - \sqrt {b^{2} - c^{2}} b^{4} c + 2 \, \sqrt {b^{2} - c^{2}} b^{3} c^{2} + 2 \, \sqrt {b^{2} - c^{2}} b^{2} c^{3} - \sqrt {b^{2} - c^{2}} b c^{4} - \sqrt {b^{2} - c^{2}} c^{5} + 2 \, \sqrt {b^{2} - c^{2}} b^{4} - 4 \, \sqrt {b^{2} - c^{2}} b^{2} c^{2} + 2 \, \sqrt {b^{2} - c^{2}} c^{4} - \sqrt {b^{2} - c^{2}} b^{3} + \sqrt {b^{2} - c^{2}} b^{2} c + \sqrt {b^{2} - c^{2}} b c^{2} - \sqrt {b^{2} - c^{2}} c^{3}} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} + b - c\right )} \]
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Timed out. \[ \int \frac {1}{\sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}} \, dx=\int \frac {1}{\sqrt {b\,\mathrm {cosh}\left (x\right )-\sqrt {b^2-c^2}+c\,\mathrm {sinh}\left (x\right )}} \,d x \]
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