Integrand size = 14, antiderivative size = 102 \[ \int \sqrt {a+b \cosh (x)+c \sinh (x)} \, dx=-\frac {2 i E\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}{\sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}} \]
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Time = 0.06 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3198, 2732} \[ \int \sqrt {a+b \cosh (x)+c \sinh (x)} \, dx=-\frac {2 i \sqrt {a+b \cosh (x)+c \sinh (x)} E\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right )}{\sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}} \]
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Rule 2732
Rule 3198
Rubi steps \begin{align*} \text {integral}& = \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}{\sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}} \\ & = -\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)}}{\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 = 6.09 (sec) , antiderivative size = 1401, normalized size of antiderivative = 13.74 \[ \int \sqrt {a+b \cosh (x)+c \sinh (x)} \, dx=\frac {2 b \sqrt {a+b \cosh (x)+c \sinh (x)}}{c}+\frac {2 a \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},\frac {3}{2},-\frac {i \left (a+\sqrt {1-\frac {b^2}{c^2}} c \sinh \left (x+\text {arctanh}\left (\frac {b}{c}\right )\right )\right )}{\sqrt {1-\frac {b^2}{c^2}} \left (1-\frac {i a}{\sqrt {1-\frac {b^2}{c^2}} c}\right ) c},-\frac {i \left (a+\sqrt {1-\frac {b^2}{c^2}} c \sinh \left (x+\text {arctanh}\left (\frac {b}{c}\right )\right )\right )}{\sqrt {1-\frac {b^2}{c^2}} \left (-1-\frac {i a}{\sqrt {1-\frac {b^2}{c^2}} c}\right ) c}\right ) \text {sech}\left (x+\text {arctanh}\left (\frac {b}{c}\right )\right ) \sqrt {-1+i \sinh \left (x+\text {arctanh}\left (\frac {b}{c}\right )\right )} \sqrt {\frac {c \sqrt {\frac {-b^2+c^2}{c^2}}-i c \sqrt {\frac {-b^2+c^2}{c^2}} \sinh \left (x+\text {arctanh}\left (\frac {b}{c}\right )\right )}{i a+c \sqrt {\frac {-b^2+c^2}{c^2}}}} \sqrt {\frac {c \sqrt {\frac {-b^2+c^2}{c^2}}+i c \sqrt {\frac {-b^2+c^2}{c^2}} \sinh \left (x+\text {arctanh}\left (\frac {b}{c}\right )\right )}{-i a+c \sqrt {\frac {-b^2+c^2}{c^2}}}} \sqrt {a+c \sqrt {\frac {-b^2+c^2}{c^2}} \sinh \left (x+\text {arctanh}\left (\frac {b}{c}\right )\right )}}{\sqrt {1-\frac {b^2}{c^2}} c \sqrt {i \left (i+\sinh \left (x+\text {arctanh}\left (\frac {b}{c}\right )\right )\right )}}-\frac {b^2 \left (\frac {c \operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2},\frac {1}{2},\frac {a+b \sqrt {1-\frac {c^2}{b^2}} \cosh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )}{b \sqrt {1-\frac {c^2}{b^2}} \left (1+\frac {a}{b \sqrt {1-\frac {c^2}{b^2}}}\right )},\frac {a+b \sqrt {1-\frac {c^2}{b^2}} \cosh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )}{b \sqrt {1-\frac {c^2}{b^2}} \left (-1+\frac {a}{b \sqrt {1-\frac {c^2}{b^2}}}\right )}\right ) \sinh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )}{b \sqrt {1-\frac {c^2}{b^2}} \sqrt {\frac {b \sqrt {\frac {b^2-c^2}{b^2}}-b \sqrt {\frac {b^2-c^2}{b^2}} \cosh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )}{a+b \sqrt {\frac {b^2-c^2}{b^2}}}} \sqrt {a+b \sqrt {\frac {b^2-c^2}{b^2}} \cosh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )} \sqrt {\frac {b \sqrt {\frac {b^2-c^2}{b^2}}+b \sqrt {\frac {b^2-c^2}{b^2}} \cosh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )}{-a+b \sqrt {\frac {b^2-c^2}{b^2}}}}}-\frac {-\frac {2 b \left (a+b \sqrt {1-\frac {c^2}{b^2}} \cosh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )\right )}{b^2-c^2}+\frac {c \sinh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )}{b \sqrt {1-\frac {c^2}{b^2}}}}{\sqrt {a+b \sqrt {1-\frac {c^2}{b^2}} \cosh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )}}\right )}{c}+c \left (\frac {c \operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2},\frac {1}{2},\frac {a+b \sqrt {1-\frac {c^2}{b^2}} \cosh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )}{b \sqrt {1-\frac {c^2}{b^2}} \left (1+\frac {a}{b \sqrt {1-\frac {c^2}{b^2}}}\right )},\frac {a+b \sqrt {1-\frac {c^2}{b^2}} \cosh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )}{b \sqrt {1-\frac {c^2}{b^2}} \left (-1+\frac {a}{b \sqrt {1-\frac {c^2}{b^2}}}\right )}\right ) \sinh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )}{b \sqrt {1-\frac {c^2}{b^2}} \sqrt {\frac {b \sqrt {\frac {b^2-c^2}{b^2}}-b \sqrt {\frac {b^2-c^2}{b^2}} \cosh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )}{a+b \sqrt {\frac {b^2-c^2}{b^2}}}} \sqrt {a+b \sqrt {\frac {b^2-c^2}{b^2}} \cosh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )} \sqrt {\frac {b \sqrt {\frac {b^2-c^2}{b^2}}+b \sqrt {\frac {b^2-c^2}{b^2}} \cosh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )}{-a+b \sqrt {\frac {b^2-c^2}{b^2}}}}}-\frac {-\frac {2 b \left (a+b \sqrt {1-\frac {c^2}{b^2}} \cosh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )\right )}{b^2-c^2}+\frac {c \sinh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )}{b \sqrt {1-\frac {c^2}{b^2}}}}{\sqrt {a+b \sqrt {1-\frac {c^2}{b^2}} \cosh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(316\) vs. \(2(125)=250\).
Time = 1.82 (sec) , antiderivative size = 317, normalized size of antiderivative = 3.11
method | result | size |
default | \(\frac {\left (-b^{2}+c^{2}\right ) \cosh \left (x \right )}{\sqrt {b^{2}-c^{2}}\, \sqrt {\frac {-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}}{\sqrt {b^{2}-c^{2}}}}}+\frac {\sqrt {\frac {\left (-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}\right ) \sinh \left (x \right )^{2}}{\sqrt {b^{2}-c^{2}}}}\, a \ln \left (\frac {-\sinh \left (x \right ) \cosh \left (x \right ) b^{2}+\sinh \left (x \right ) \cosh \left (x \right ) c^{2}+\cosh \left (x \right ) \sqrt {b^{2}-c^{2}}\, a +\sqrt {\frac {\left (-b^{2}+c^{2}\right ) \sinh \left (x \right )^{3}}{\sqrt {b^{2}-c^{2}}}+a \sinh \left (x \right )^{2}}\, \sqrt {b^{2}-c^{2}}\, \sqrt {\frac {\left (-b^{2}+c^{2}\right ) \sinh \left (x \right )}{\sqrt {b^{2}-c^{2}}}+a}}{\sqrt {b^{2}-c^{2}}\, \sqrt {\frac {-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}}{\sqrt {b^{2}-c^{2}}}}}\right ) \sqrt {b^{2}-c^{2}}}{\left (-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}\right ) \sinh \left (x \right )}\) | \(317\) |
risch | \(\text {Expression too large to display}\) | \(1093\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 314, normalized size of antiderivative = 3.08 \[ \int \sqrt {a+b \cosh (x)+c \sinh (x)} \, dx=\frac {2 \, {\left (\sqrt {2} a \sqrt {b + c} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2} + 3 \, c^{2}\right )}}{3 \, {\left (b^{2} + 2 \, b c + c^{2}\right )}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2} + 9 \, a c^{2}\right )}}{27 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )}}, \frac {3 \, {\left (b + c\right )} \cosh \left (x\right ) + 3 \, {\left (b + c\right )} \sinh \left (x\right ) + 2 \, a}{3 \, {\left (b + c\right )}}\right ) - 3 \, \sqrt {2} {\left (b + c\right )}^{\frac {3}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2} + 3 \, c^{2}\right )}}{3 \, {\left (b^{2} + 2 \, b c + c^{2}\right )}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2} + 9 \, a c^{2}\right )}}{27 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2} + 3 \, c^{2}\right )}}{3 \, {\left (b^{2} + 2 \, b c + c^{2}\right )}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2} + 9 \, a c^{2}\right )}}{27 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )}}, \frac {3 \, {\left (b + c\right )} \cosh \left (x\right ) + 3 \, {\left (b + c\right )} \sinh \left (x\right ) + 2 \, a}{3 \, {\left (b + c\right )}}\right )\right ) - 3 \, \sqrt {b \cosh \left (x\right ) + c \sinh \left (x\right ) + a} {\left (b + c\right )}\right )}}{3 \, {\left (b + c\right )}} \]
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\[ \int \sqrt {a+b \cosh (x)+c \sinh (x)} \, dx=\int \sqrt {a + b \cosh {\left (x \right )} + c \sinh {\left (x \right )}}\, dx \]
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\[ \int \sqrt {a+b \cosh (x)+c \sinh (x)} \, dx=\int { \sqrt {b \cosh \left (x\right ) + c \sinh \left (x\right ) + a} \,d x } \]
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\[ \int \sqrt {a+b \cosh (x)+c \sinh (x)} \, dx=\int { \sqrt {b \cosh \left (x\right ) + c \sinh \left (x\right ) + a} \,d x } \]
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Timed out. \[ \int \sqrt {a+b \cosh (x)+c \sinh (x)} \, dx=\int \sqrt {a+b\,\mathrm {cosh}\left (x\right )+c\,\mathrm {sinh}\left (x\right )} \,d x \]
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