Integrand size = 10, antiderivative size = 60 \[ \int \sqrt {a+b \sinh (x)} \, dx=\frac {2 i E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right ) \sqrt {a+b \sinh (x)}}{\sqrt {\frac {a+b \sinh (x)}{a-i b}}} \]
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Time = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2734, 2732} \[ \int \sqrt {a+b \sinh (x)} \, dx=\frac {2 i \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{\sqrt {\frac {a+b \sinh (x)}{a-i b}}} \]
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Rule 2732
Rule 2734
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a+b \sinh (x)} \int \sqrt {\frac {a}{a-i b}+\frac {b \sinh (x)}{a-i b}} \, dx}{\sqrt {\frac {a+b \sinh (x)}{a-i b}}} \\ & = \frac {2 i E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right ) \sqrt {a+b \sinh (x)}}{\sqrt {\frac {a+b \sinh (x)}{a-i b}}} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.08 \[ \int \sqrt {a+b \sinh (x)} \, dx=\frac {2 (i a+b) E\left (\frac {1}{4} (\pi -2 i x)|-\frac {2 i b}{a-i b}\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}{\sqrt {a+b \sinh (x)}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 261 vs. \(2 (78 ) = 156\).
Time = 1.92 (sec) , antiderivative size = 262, normalized size of antiderivative = 4.37
method | result | size |
default | \(-\frac {2 \left (i b -a \right ) \sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \left (x \right )\right ) b}{i b +a}}\, \sqrt {\frac {\left (i+\sinh \left (x \right )\right ) b}{i b -a}}\, \left (i \operatorname {EllipticF}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) b -i \operatorname {EllipticE}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) b +\operatorname {EllipticF}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) a -\operatorname {EllipticE}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) a \right )}{b \cosh \left (x \right ) \sqrt {a +b \sinh \left (x \right )}}\) | \(262\) |
risch | \(\sqrt {2}\, \sqrt {\left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right ) {\mathrm e}^{-x}}+\frac {\left (\frac {4 a \left (a +\sqrt {a^{2}+b^{2}}\right ) \sqrt {\frac {\left ({\mathrm e}^{x}+\frac {a +\sqrt {a^{2}+b^{2}}}{b}\right ) b}{a +\sqrt {a^{2}+b^{2}}}}\, \sqrt {\frac {{\mathrm e}^{x}-\frac {-a +\sqrt {a^{2}+b^{2}}}{b}}{-\frac {a +\sqrt {a^{2}+b^{2}}}{b}-\frac {-a +\sqrt {a^{2}+b^{2}}}{b}}}\, \sqrt {-\frac {{\mathrm e}^{x} b}{a +\sqrt {a^{2}+b^{2}}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left ({\mathrm e}^{x}+\frac {a +\sqrt {a^{2}+b^{2}}}{b}\right ) b}{a +\sqrt {a^{2}+b^{2}}}}, \sqrt {-\frac {a +\sqrt {a^{2}+b^{2}}}{b \left (-\frac {a +\sqrt {a^{2}+b^{2}}}{b}-\frac {-a +\sqrt {a^{2}+b^{2}}}{b}\right )}}\right )}{b \sqrt {{\mathrm e}^{3 x} b +2 \,{\mathrm e}^{2 x} a -{\mathrm e}^{x} b}}-2 b \left (\frac {2 b \,{\mathrm e}^{2 x}+4 \,{\mathrm e}^{x} a -2 b}{b \sqrt {\left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right ) {\mathrm e}^{x}}}-\frac {2 \left (a +\sqrt {a^{2}+b^{2}}\right ) \sqrt {\frac {\left ({\mathrm e}^{x}+\frac {a +\sqrt {a^{2}+b^{2}}}{b}\right ) b}{a +\sqrt {a^{2}+b^{2}}}}\, \sqrt {\frac {{\mathrm e}^{x}-\frac {-a +\sqrt {a^{2}+b^{2}}}{b}}{-\frac {a +\sqrt {a^{2}+b^{2}}}{b}-\frac {-a +\sqrt {a^{2}+b^{2}}}{b}}}\, \sqrt {-\frac {{\mathrm e}^{x} b}{a +\sqrt {a^{2}+b^{2}}}}\, \left (\left (-\frac {a +\sqrt {a^{2}+b^{2}}}{b}-\frac {-a +\sqrt {a^{2}+b^{2}}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left ({\mathrm e}^{x}+\frac {a +\sqrt {a^{2}+b^{2}}}{b}\right ) b}{a +\sqrt {a^{2}+b^{2}}}}, \sqrt {-\frac {a +\sqrt {a^{2}+b^{2}}}{b \left (-\frac {a +\sqrt {a^{2}+b^{2}}}{b}-\frac {-a +\sqrt {a^{2}+b^{2}}}{b}\right )}}\right )+\frac {\left (-a +\sqrt {a^{2}+b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {\left ({\mathrm e}^{x}+\frac {a +\sqrt {a^{2}+b^{2}}}{b}\right ) b}{a +\sqrt {a^{2}+b^{2}}}}, \sqrt {-\frac {a +\sqrt {a^{2}+b^{2}}}{b \left (-\frac {a +\sqrt {a^{2}+b^{2}}}{b}-\frac {-a +\sqrt {a^{2}+b^{2}}}{b}\right )}}\right )}{b}\right )}{b \sqrt {{\mathrm e}^{3 x} b +2 \,{\mathrm e}^{2 x} a -{\mathrm e}^{x} b}}\right )\right ) \sqrt {2}\, \sqrt {\left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right ) {\mathrm e}^{-x}}\, \sqrt {\left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right ) {\mathrm e}^{x}}}{2 b \,{\mathrm e}^{2 x}+4 \,{\mathrm e}^{x} a -2 b}\) | \(771\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 173, normalized size of antiderivative = 2.88 \[ \int \sqrt {a+b \sinh (x)} \, dx=\frac {2 \, {\left (\sqrt {2} a \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right ) - 3 \, \sqrt {2} b^{\frac {3}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right )\right ) - 3 \, \sqrt {b \sinh \left (x\right ) + a} b\right )}}{3 \, b} \]
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\[ \int \sqrt {a+b \sinh (x)} \, dx=\int \sqrt {a + b \sinh {\left (x \right )}}\, dx \]
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\[ \int \sqrt {a+b \sinh (x)} \, dx=\int { \sqrt {b \sinh \left (x\right ) + a} \,d x } \]
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\[ \int \sqrt {a+b \sinh (x)} \, dx=\int { \sqrt {b \sinh \left (x\right ) + a} \,d x } \]
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Timed out. \[ \int \sqrt {a+b \sinh (x)} \, dx=\int \sqrt {a+b\,\mathrm {sinh}\left (x\right )} \,d x \]
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