Integrand size = 14, antiderivative size = 156 \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{3/2}} \, dx=-\frac {2 (c \cosh (x)+b \sinh (x))}{\left (a^2-b^2+c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}-\frac {2 i E\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}{\left (a^2-b^2+c^2\right ) \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}} \]
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Time = 0.08 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3207, 3198, 2732} \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{3/2}} \, dx=-\frac {2 (b \sinh (x)+c \cosh (x))}{\left (a^2-b^2+c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}-\frac {2 i \sqrt {a+b \cosh (x)+c \sinh (x)} E\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right )}{\left (a^2-b^2+c^2\right ) \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}} \]
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Rule 2732
Rule 3198
Rule 3207
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (c \cosh (x)+b \sinh (x))}{\left (a^2-b^2+c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}+\frac {\int \sqrt {a+b \cosh (x)+c \sinh (x)} \, dx}{a^2-b^2+c^2} \\ & = -\frac {2 (c \cosh (x)+b \sinh (x))}{\left (a^2-b^2+c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}+\frac {\sqrt {a+b \cosh (x)+c \sinh (x)} \int \sqrt {\frac {a}{a+\sqrt {b^2-c^2}}+\frac {\sqrt {b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}{a+\sqrt {b^2-c^2}}} \, dx}{\left (a^2-b^2+c^2\right ) \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}} \\ & = -\frac {2 (c \cosh (x)+b \sinh (x))}{\left (a^2-b^2+c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}-\frac {2 i E\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}{\left (a^2-b^2+c^2\right ) \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 6.02 (sec) , antiderivative size = 806, normalized size of antiderivative = 5.17 \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{3/2}} \, dx=\frac {2 \left (a b+\left (b^2-c^2\right ) \cosh (x)\right )-\frac {2 b^3 (a+b \cosh (x)+c \sinh (x))}{b^2-c^2}+\frac {2 a \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},\frac {3}{2},\frac {a+b \cosh (x)+c \sinh (x)}{a+i \sqrt {1-\frac {b^2}{c^2}} c},\frac {a+b \cosh (x)+c \sinh (x)}{a-i \sqrt {1-\frac {b^2}{c^2}} c}\right ) \text {sech}\left (x+\text {arctanh}\left (\frac {b}{c}\right )\right ) (a+b \cosh (x)+c \sinh (x)) \sqrt {-\frac {-i \sqrt {1-\frac {b^2}{c^2}} c+b \cosh (x)+c \sinh (x)}{a+i \sqrt {1-\frac {b^2}{c^2}} c}} \sqrt {-\frac {i \sqrt {1-\frac {b^2}{c^2}} c+b \cosh (x)+c \sinh (x)}{a-i \sqrt {1-\frac {b^2}{c^2}} c}}}{\sqrt {1-\frac {b^2}{c^2}}}+\frac {b c \sinh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )}{\sqrt {1-\frac {c^2}{b^2}}}-\frac {b c \operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2},\frac {1}{2},\frac {a+b \cosh (x)+c \sinh (x)}{a+b \sqrt {1-\frac {c^2}{b^2}}},\frac {a+b \cosh (x)+c \sinh (x)}{a-b \sqrt {1-\frac {c^2}{b^2}}}\right ) \sinh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )}{\sqrt {1-\frac {c^2}{b^2}} \sqrt {\frac {b \sqrt {1-\frac {c^2}{b^2}}-b \cosh (x)-c \sinh (x)}{a+b \sqrt {1-\frac {c^2}{b^2}}}} \sqrt {\frac {b \sqrt {1-\frac {c^2}{b^2}}+b \cosh (x)+c \sinh (x)}{-a+b \sqrt {1-\frac {c^2}{b^2}}}}}+\frac {c^2 \left (\frac {2 b^2 (a+b \cosh (x)+c \sinh (x))}{b^2-c^2}-\frac {c \sinh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )}{\sqrt {1-\frac {c^2}{b^2}}}+\frac {c \operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2},\frac {1}{2},\frac {a+b \cosh (x)+c \sinh (x)}{a+b \sqrt {1-\frac {c^2}{b^2}}},\frac {a+b \cosh (x)+c \sinh (x)}{a-b \sqrt {1-\frac {c^2}{b^2}}}\right ) \sinh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )}{\sqrt {1-\frac {c^2}{b^2}} \sqrt {\frac {b \sqrt {1-\frac {c^2}{b^2}}-b \cosh (x)-c \sinh (x)}{a+b \sqrt {1-\frac {c^2}{b^2}}}} \sqrt {\frac {b \sqrt {1-\frac {c^2}{b^2}}+b \cosh (x)+c \sinh (x)}{-a+b \sqrt {1-\frac {c^2}{b^2}}}}}\right )}{b}}{c \left (a^2-b^2+c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(887\) vs. \(2(177)=354\).
Time = 1.02 (sec) , antiderivative size = 888, normalized size of antiderivative = 5.69
method | result | size |
default | \(\frac {\sqrt {b^{2}-c^{2}}\, \operatorname {arctanh}\left (\frac {\left (b^{2}-c^{2}\right ) \cosh \left (x \right )}{\sqrt {\left (a^{2}+b^{2}-c^{2}\right ) \left (b^{2}-c^{2}\right )}}\right )}{\sqrt {\frac {-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}}{\sqrt {b^{2}-c^{2}}}}\, \sqrt {\left (a^{2}+b^{2}-c^{2}\right ) \left (b^{2}-c^{2}\right )}}-\frac {\sqrt {\frac {\left (-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}\right ) \sinh \left (x \right )^{2}}{\sqrt {b^{2}-c^{2}}}}\, a \left (-\frac {\left (-b^{2}+c^{2}\right ) \ln \left (\frac {-\frac {2 \left (\sinh \left (x \right ) \sqrt {b^{2}-c^{2}}-a \right ) a^{2}}{b^{2}-c^{2}}+\frac {2 \left (b^{2} \sinh \left (x \right )-\sinh \left (x \right ) c^{2}-a \sqrt {b^{2}-c^{2}}\right ) \sqrt {\left (a^{2}+b^{2}-c^{2}\right ) \left (b -c \right ) \left (b +c \right )}\, \left (\cosh \left (x \right )+\frac {\sqrt {\left (a^{2}+b^{2}-c^{2}\right ) \left (b -c \right ) \left (b +c \right )}}{b^{2}-c^{2}}\right )}{\left (b^{2}-c^{2}\right )^{\frac {3}{2}}}+2 \sqrt {\frac {\left (-\sinh \left (x \right ) \sqrt {b^{2}-c^{2}}+a \right ) a^{2}}{b^{2}-c^{2}}}\, \sqrt {\frac {\left (-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}\right ) \sinh \left (x \right )^{2}}{\sqrt {b^{2}-c^{2}}}}}{\cosh \left (x \right )+\frac {\sqrt {\left (a^{2}+b^{2}-c^{2}\right ) \left (b -c \right ) \left (b +c \right )}}{b^{2}-c^{2}}}\right )}{2 \sqrt {\left (a^{2}+b^{2}-c^{2}\right ) \left (b -c \right ) \left (b +c \right )}\, \left (b^{2}-c^{2}\right ) \sqrt {\frac {\left (-\sinh \left (x \right ) \sqrt {b^{2}-c^{2}}+a \right ) a^{2}}{b^{2}-c^{2}}}}+\frac {\left (b^{2}-c^{2}\right ) \ln \left (\frac {-\frac {2 \left (\sinh \left (x \right ) \sqrt {b^{2}-c^{2}}-a \right ) a^{2}}{b^{2}-c^{2}}-\frac {2 \left (b^{2} \sinh \left (x \right )-\sinh \left (x \right ) c^{2}-a \sqrt {b^{2}-c^{2}}\right ) \sqrt {\left (a^{2}+b^{2}-c^{2}\right ) \left (b -c \right ) \left (b +c \right )}\, \left (\cosh \left (x \right )+\frac {\sqrt {\left (a^{2}+b^{2}-c^{2}\right ) \left (b -c \right ) \left (b +c \right )}}{-b^{2}+c^{2}}\right )}{\left (b^{2}-c^{2}\right )^{\frac {3}{2}}}+2 \sqrt {\frac {\left (-\sinh \left (x \right ) \sqrt {b^{2}-c^{2}}+a \right ) a^{2}}{b^{2}-c^{2}}}\, \sqrt {\frac {\left (-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}\right ) \sinh \left (x \right )^{2}}{\sqrt {b^{2}-c^{2}}}}}{\cosh \left (x \right )+\frac {\sqrt {\left (a^{2}+b^{2}-c^{2}\right ) \left (b -c \right ) \left (b +c \right )}}{-b^{2}+c^{2}}}\right )}{2 \sqrt {\left (a^{2}+b^{2}-c^{2}\right ) \left (b -c \right ) \left (b +c \right )}\, \left (-b^{2}+c^{2}\right ) \sqrt {\frac {\left (-\sinh \left (x \right ) \sqrt {b^{2}-c^{2}}+a \right ) a^{2}}{b^{2}-c^{2}}}}\right )}{\sinh \left (x \right ) \sqrt {\frac {-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}}{\sqrt {b^{2}-c^{2}}}}}\) | \(888\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 798, normalized size of antiderivative = 5.12 \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{3/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{3/2}} \, dx=\int \frac {1}{\left (a + b \cosh {\left (x \right )} + c \sinh {\left (x \right )}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \cosh \left (x\right ) + c \sinh \left (x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \cosh \left (x\right ) + c \sinh \left (x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{3/2}} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {cosh}\left (x\right )+c\,\mathrm {sinh}\left (x\right )\right )}^{3/2}} \,d x \]
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