Optimal. Leaf size=57 \[ \frac{2 x}{b \sqrt{\text{sech}(a+b x)}}+\frac{4 i \sqrt{\cosh (a+b x)} \sqrt{\text{sech}(a+b x)} E\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{b^2} \]
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Rubi [A] time = 0.039545, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5444, 3771, 2639} \[ \frac{2 x}{b \sqrt{\text{sech}(a+b x)}}+\frac{4 i \sqrt{\cosh (a+b x)} \sqrt{\text{sech}(a+b x)} E\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{b^2} \]
Antiderivative was successfully verified.
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Rule 5444
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int x \sqrt{\text{sech}(a+b x)} \sinh (a+b x) \, dx &=\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*}
Mathematica [C] time = 1.07534, size = 100, normalized size = 1.75 \[ \frac{\sqrt{2} e^{-a-b x} \sqrt{\frac{e^{a+b x}}{e^{2 (a+b x)}+1}} \left (4 \sqrt{e^{2 (a+b x)}+1} \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 (a+b x)}\right )+(b x-2) \left (e^{2 (a+b x)}+1\right )\right )}{b^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.053, size = 250, normalized size = 4.4 \begin{align*}{\frac{ \left ( bx-2 \right ) \left ( \left ({{\rm e}^{bx+a}} \right ) ^{2}+1 \right ) \sqrt{2}}{{b}^{2}{{\rm e}^{bx+a}}}\sqrt{{\frac{{{\rm e}^{bx+a}}}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}}}}-2\,{\frac{\sqrt{2}\sqrt{ \left ( \left ({{\rm e}^{bx+a}} \right ) ^{2}+1 \right ){{\rm e}^{bx+a}}}}{{b}^{2}{{\rm e}^{bx+a}}} \left ( -2\,{\frac{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}{\sqrt{ \left ( \left ({{\rm e}^{bx+a}} \right ) ^{2}+1 \right ){{\rm e}^{bx+a}}}}}+{\frac{i\sqrt{-i \left ({{\rm e}^{bx+a}}+i \right ) }\sqrt{2}\sqrt{i \left ({{\rm e}^{bx+a}}-i \right ) }\sqrt{i{{\rm e}^{bx+a}}} \left ( -2\,i{\it EllipticE} \left ( \sqrt{-i \left ({{\rm e}^{bx+a}}+i \right ) },1/2\,\sqrt{2} \right ) +i{\it EllipticF} \left ( \sqrt{-i \left ({{\rm e}^{bx+a}}+i \right ) },1/2\,\sqrt{2} \right ) \right ) }{\sqrt{ \left ({{\rm e}^{bx+a}} \right ) ^{3}+{{\rm e}^{bx+a}}}}} \right ) \sqrt{{\frac{{{\rm e}^{bx+a}}}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sqrt{\operatorname{sech}\left (b x + a\right )} \sinh \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sinh{\left (a + b x \right )} \sqrt{\operatorname{sech}{\left (a + b x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sqrt{\operatorname{sech}\left (b x + a\right )} \sinh \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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