Optimal. Leaf size=31 \[ \frac{2 i a \cosh (c+d x)}{d \sqrt{a+i a \sinh (c+d x)}} \]
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Rubi [A] time = 0.0139036, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {2646} \[ \frac{2 i a \cosh (c+d x)}{d \sqrt{a+i a \sinh (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2646
Rubi steps
\begin{align*} \int \sqrt{a+i a \sinh (c+d x)} \, dx &=\frac{2 i a \cosh (c+d x)}{d \sqrt{a+i a \sinh (c+d x)}}\\ \end{align*}
Mathematica [B] time = 0.041178, size = 74, normalized size = 2.39 \[ \frac{2 \sqrt{a+i a \sinh (c+d x)} \left (\sinh \left (\frac{1}{2} (c+d x)\right )+i \cosh \left (\frac{1}{2} (c+d x)\right )\right )}{d \left (\cosh \left (\frac{1}{2} (c+d x)\right )+i \sinh \left (\frac{1}{2} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.125, size = 89, normalized size = 2.9 \begin{align*}{\frac{i\sqrt{2} \left ({{\rm e}^{dx+c}}+i \right ) \left ({{\rm e}^{dx+c}}-i \right ) }{ \left ( i{{\rm e}^{2\,dx+2\,c}}-i+2\,{{\rm e}^{dx+c}} \right ) d}\sqrt{a \left ( i{{\rm e}^{2\,dx+2\,c}}-i+2\,{{\rm e}^{dx+c}} \right ){{\rm e}^{-dx-c}}}} \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 \sinh \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 = 2.00736, size = 170, normalized size = 5.48 \begin{align*} \frac{\sqrt{\frac{1}{2}} \sqrt{i \, a e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a e^{\left (d x + c\right )} - i \, a}{\left (2 \, e^{\left (d x + c\right )} + 2 i\right )} e^{\left (-\frac{1}{2} \, d x - \frac{1}{2} \, c\right )}}{d e^{\left (d x + c\right )} - i \, d} \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 \sinh{\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 \sinh \left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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