Integrand size = 18, antiderivative size = 57 \[ \int x \sqrt {\text {sech}(a+b x)} \sinh (a+b x) \, dx=\frac {2 x}{b \sqrt {\text {sech}(a+b x)}}+\frac {4 i \sqrt {\cosh (a+b x)} E\left (\left .\frac {1}{2} i (a+b x)\right |2\right ) \sqrt {\text {sech}(a+b x)}}{b^2} \]
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Time = 0.03 (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 = {5552, 3856, 2719} \[ \int x \sqrt {\text {sech}(a+b x)} \sinh (a+b x) \, dx=\frac {2 x}{b \sqrt {\text {sech}(a+b x)}}+\frac {4 i \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{b^2} \]
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Rule 2719
Rule 3856
Rule 5552
Rubi steps \begin{align*} \text {integral}& = \frac {2 x}{b \sqrt {\text {sech}(a+b x)}}-\frac {2 \int \frac {1}{\sqrt {\text {sech}(a+b x)}} \, dx}{b} \\ & = \frac {2 x}{b \sqrt {\text {sech}(a+b x)}}-\frac {\left (2 \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)}\right ) \int \sqrt {\cosh (a+b x)} \, dx}{b} \\ & = \frac {2 x}{b \sqrt {\text {sech}(a+b x)}}+\frac {4 i \sqrt {\cosh (a+b x)} E\left (\left .\frac {1}{2} i (a+b x)\right |2\right ) \sqrt {\text {sech}(a+b x)}}{b^2} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 1.40 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.75 \[ \int x \sqrt {\text {sech}(a+b x)} \sinh (a+b x) \, dx=\frac {\sqrt {2} e^{-a-b x} \sqrt {\frac {e^{a+b x}}{1+e^{2 (a+b x)}}} \left (\left (1+e^{2 (a+b x)}\right ) (-2+b x)+4 \sqrt {1+e^{2 (a+b x)}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-e^{2 (a+b x)}\right )\right )}{b^2} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 249 vs. \(2 (77 ) = 154\).
Time = 0.29 (sec) , antiderivative size = 250, normalized size of antiderivative = 4.39
method | result | size |
risch | \(\frac {\left (b x -2\right ) \left (1+{\mathrm e}^{2 b x +2 a}\right ) \sqrt {2}\, \sqrt {\frac {{\mathrm e}^{b x +a}}{1+{\mathrm e}^{2 b x +2 a}}}\, {\mathrm e}^{-b x -a}}{b^{2}}-\frac {2 \left (-\frac {2 \left (1+{\mathrm e}^{2 b x +2 a}\right )}{\sqrt {\left (1+{\mathrm e}^{2 b x +2 a}\right ) {\mathrm e}^{b x +a}}}+\frac {i \sqrt {-i \left ({\mathrm e}^{b x +a}+i\right )}\, \sqrt {2}\, \sqrt {i \left ({\mathrm e}^{b x +a}-i\right )}\, \sqrt {i {\mathrm e}^{b x +a}}\, \left (-2 i \operatorname {EllipticE}\left (\sqrt {-i \left ({\mathrm e}^{b x +a}+i\right )}, \frac {\sqrt {2}}{2}\right )+i \operatorname {EllipticF}\left (\sqrt {-i \left ({\mathrm e}^{b x +a}+i\right )}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {{\mathrm e}^{3 b x +3 a}+{\mathrm e}^{b x +a}}}\right ) \sqrt {2}\, \sqrt {\frac {{\mathrm e}^{b x +a}}{1+{\mathrm e}^{2 b x +2 a}}}\, \sqrt {\left (1+{\mathrm e}^{2 b x +2 a}\right ) {\mathrm e}^{b x +a}}\, {\mathrm e}^{-b x -a}}{b^{2}}\) | \(250\) |
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Exception generated. \[ \int x \sqrt {\text {sech}(a+b x)} \sinh (a+b x) \, dx=\text {Exception raised: TypeError} \]
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\[ \int x \sqrt {\text {sech}(a+b x)} \sinh (a+b x) \, dx=\int x \sinh {\left (a + b x \right )} \sqrt {\operatorname {sech}{\left (a + b x \right )}}\, dx \]
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\[ \int x \sqrt {\text {sech}(a+b x)} \sinh (a+b x) \, dx=\int { x \sqrt {\operatorname {sech}\left (b x + a\right )} \sinh \left (b x + a\right ) \,d x } \]
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\[ \int x \sqrt {\text {sech}(a+b x)} \sinh (a+b x) \, dx=\int { x \sqrt {\operatorname {sech}\left (b x + a\right )} \sinh \left (b x + a\right ) \,d x } \]
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Timed out. \[ \int x \sqrt {\text {sech}(a+b x)} \sinh (a+b x) \, dx=\int x\,\mathrm {sinh}\left (a+b\,x\right )\,\sqrt {\frac {1}{\mathrm {cosh}\left (a+b\,x\right )}} \,d x \]
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