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 = 0.02, antiderivative size = 56, normalized size of antiderivative = 1.00, 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 2639
Rule 3771
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*}
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Mathematica [A] time = 0.05, 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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fricas [F] time = 1.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \operatorname {csch}\left (d x + c\right )}}{b \operatorname {csch}\left (d x + c\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b \operatorname {csch}\left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.33, size = 227, normalized size = 4.05 \[ \frac {\sqrt {2}}{d \sqrt {\frac {b \,{\mathrm e}^{d x +c}}{{\mathrm e}^{2 d x +2 c}-1}}}-\frac {\left (\frac {2 b \,{\mathrm e}^{2 d x +2 c}-2 b}{b \sqrt {{\mathrm e}^{d x +c} \left (b \,{\mathrm e}^{2 d x +2 c}-b \right )}}-\frac {\sqrt {{\mathrm e}^{d x +c}+1}\, \sqrt {2-2 \,{\mathrm e}^{d x +c}}\, \sqrt {-{\mathrm e}^{d x +c}}\, \left (-2 \EllipticE \left (\sqrt {{\mathrm e}^{d x +c}+1}, \frac {\sqrt {2}}{2}\right )+\EllipticF \left (\sqrt {{\mathrm e}^{d x +c}+1}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {{\mathrm e}^{3 d x +3 c} b -b \,{\mathrm e}^{d x +c}}}\right ) \sqrt {2}\, \sqrt {b \,{\mathrm e}^{d x +c} \left ({\mathrm e}^{2 d x +2 c}-1\right )}}{d \sqrt {\frac {b \,{\mathrm e}^{d x +c}}{{\mathrm e}^{2 d x +2 c}-1}}\, \left ({\mathrm e}^{2 d x +2 c}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b \operatorname {csch}\left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {\frac {b}{\mathrm {sinh}\left (c+d\,x\right )}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b \operatorname {csch}{\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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