Optimal. Leaf size=40 \[ \frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \coth (c+d x)}{\sqrt{a-i a \text{csch}(c+d x)}}\right )}{d} \]
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Rubi [A] time = 0.0226346, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {3774, 203} \[ \frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \coth (c+d x)}{\sqrt{a-i a \text{csch}(c+d x)}}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 3774
Rule 203
Rubi steps
\begin{align*} \int \sqrt{a-i a \text{csch}(c+d x)} \, dx &=\frac{(2 i a) \operatorname{Subst}\left (\int \frac{1}{a+x^2} \, dx,x,-\frac{i a \coth (c+d x)}{\sqrt{a-i a \text{csch}(c+d x)}}\right )}{d}\\ &=\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \coth (c+d x)}{\sqrt{a-i a \text{csch}(c+d x)}}\right )}{d}\\ \end{align*}
Mathematica [A] time = 0.922423, size = 80, normalized size = 2. \[ -\frac{2 (-1)^{3/4} \coth (c+d x) \sqrt{a-i a \text{csch}(c+d x)} \tan ^{-1}\left ((-1)^{3/4} \sqrt{\text{csch}(c+d x)-i}\right )}{d \sqrt{\text{csch}(c+d x)-i} (\text{csch}(c+d x)+i)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.578, size = 0, normalized size = 0. \begin{align*} \int \sqrt{a-ia{\rm csch} \left (dx+c\right )}\, dx \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 \sqrt{-i \, a \operatorname{csch}\left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.38993, size = 815, normalized size = 20.38 \begin{align*} \frac{1}{2} \, \sqrt{\frac{a}{d^{2}}} \log \left (\frac{2 \,{\left (-\left (4 i - 1\right ) \, d e^{\left (3 \, d x + 3 \, c\right )} + \left (i + 4\right ) \, d\right )} \sqrt{\frac{a}{d^{2}}} + \sqrt{\frac{a e^{\left (2 \, d x + 2 \, c\right )} - 2 i \, a e^{\left (d x + c\right )} - a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}}{\left (-\left (8 i - 2\right ) \, e^{\left (3 \, d x + 3 \, c\right )} + \left (2 i + 8\right ) \, e^{\left (2 \, d x + 2 \, c\right )} + \left (8 i - 2\right ) \, e^{\left (d x + c\right )} - 2 i - 8\right )}}{\left (48 i + 20\right ) \, e^{\left (2 \, d x + 2 \, c\right )} - \left (20 i - 48\right ) \, e^{\left (d x + c\right )}}\right ) - \frac{1}{2} \, \sqrt{\frac{a}{d^{2}}} \log \left (\frac{2 \,{\left (\left (4 i - 1\right ) \, d e^{\left (3 \, d x + 3 \, c\right )} - \left (i + 4\right ) \, d\right )} \sqrt{\frac{a}{d^{2}}} + \sqrt{\frac{a e^{\left (2 \, d x + 2 \, c\right )} - 2 i \, a e^{\left (d x + c\right )} - a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}}{\left (-\left (8 i - 2\right ) \, e^{\left (3 \, d x + 3 \, c\right )} + \left (2 i + 8\right ) \, e^{\left (2 \, d x + 2 \, c\right )} + \left (8 i - 2\right ) \, e^{\left (d x + c\right )} - 2 i - 8\right )}}{\left (48 i + 20\right ) \, e^{\left (2 \, d x + 2 \, c\right )} - \left (20 i - 48\right ) \, e^{\left (d x + c\right )}}\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 \sqrt{- i a \operatorname{csch}{\left (c + d x \right )} + a}\, 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 \sqrt{-i \, a \operatorname{csch}\left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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