Optimal. Leaf size=56 \[ -\frac{2 i E\left (\left .\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right )\right |2\right )}{d \sqrt{i \sinh (c+d x)} \sqrt{b \text{csch}(c+d x)}} \]
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Rubi [A] time = 0.0226316, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3771, 2639} \[ -\frac{2 i E\left (\left .\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right )\right |2\right )}{d \sqrt{i \sinh (c+d x)} \sqrt{b \text{csch}(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{b \text{csch}(c+d x)}} \, dx &=\frac{\int \sqrt{i \sinh (c+d x)} \, dx}{\sqrt{b \text{csch}(c+d x)} \sqrt{i \sinh (c+d x)}}\\ &=-\frac{2 i E\left (\left .\frac{1}{2} \left (i c-\frac{\pi }{2}+i d x\right )\right |2\right )}{d \sqrt{b \text{csch}(c+d x)} \sqrt{i \sinh (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0445771, size = 52, normalized size = 0.93 \[ \frac{2 i E\left (\left .\frac{1}{4} (-2 i c-2 i d x+\pi )\right |2\right )}{d \sqrt{i \sinh (c+d x)} \sqrt{b \text{csch}(c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.138, size = 227, normalized size = 4.1 \begin{align*}{\frac{\sqrt{2}}{d}{\frac{1}{\sqrt{{\frac{b{{\rm e}^{dx+c}}}{ \left ({{\rm e}^{dx+c}} \right ) ^{2}-1}}}}}}-{\frac{\sqrt{2}}{d \left ( \left ({{\rm e}^{dx+c}} \right ) ^{2}-1 \right ) } \left ( 2\,{\frac{b \left ({{\rm e}^{dx+c}} \right ) ^{2}-b}{b\sqrt{{{\rm e}^{dx+c}} \left ( b \left ({{\rm e}^{dx+c}} \right ) ^{2}-b \right ) }}}-{\sqrt{{{\rm e}^{dx+c}}+1}\sqrt{2-2\,{{\rm e}^{dx+c}}}\sqrt{-{{\rm e}^{dx+c}}} \left ( -2\,{\it EllipticE} \left ( \sqrt{{{\rm e}^{dx+c}}+1},1/2\,\sqrt{2} \right ) +{\it EllipticF} \left ( \sqrt{{{\rm e}^{dx+c}}+1},{\frac{\sqrt{2}}{2}} \right ) \right ){\frac{1}{\sqrt{b \left ({{\rm e}^{dx+c}} \right ) ^{3}-b{{\rm e}^{dx+c}}}}}} \right ) \sqrt{b{{\rm e}^{dx+c}} \left ( \left ({{\rm e}^{dx+c}} \right ) ^{2}-1 \right ) }{\frac{1}{\sqrt{{\frac{b{{\rm e}^{dx+c}}}{ \left ({{\rm e}^{dx+c}} \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 \frac{1}{\sqrt{b \operatorname{csch}\left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b \operatorname{csch}\left (d x + c\right )}}{b \operatorname{csch}\left (d x + c\right )}, x\right ) \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 \frac{1}{\sqrt{b \operatorname{csch}{\left (c + d 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 \frac{1}{\sqrt{b \operatorname{csch}\left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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