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 = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, 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*}
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Mathematica [B] time = 0.04, 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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fricas [A] time = 0.49, size = 28, normalized size = 0.90 \[ \frac {\sqrt {\frac {1}{2} i \, a e^{\left (-d x - c\right )}} {\left (2 \, e^{\left (d x + c\right )} + 2 i\right )}}{d} \]
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 \[ \int \sqrt {i \, a \sinh \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.13, size = 89, normalized size = 2.87 \[ \frac {i \sqrt {2}\, \sqrt {a \left (i {\mathrm e}^{2 d x +2 c}-i+2 \,{\mathrm e}^{d x +c}\right ) {\mathrm e}^{-d x -c}}\, \left ({\mathrm e}^{d x +c}+i\right ) \left ({\mathrm e}^{d x +c}-i\right )}{\left (i {\mathrm e}^{2 d x +2 c}-i+2 \,{\mathrm e}^{d x +c}\right ) d} \]
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 \[ \int \sqrt {i \, a \sinh \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.75, size = 53, normalized size = 1.71 \[ \frac {\sqrt {2}\,\left ({\mathrm {e}}^{c+d\,x}+1{}\mathrm {i}\right )\,\sqrt {a\,{\mathrm {e}}^{-c-d\,x}\,{\left ({\mathrm {e}}^{c+d\,x}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}}{d\,\left ({\mathrm {e}}^{c+d\,x}-\mathrm {i}\right )} \]
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 \[ \int \sqrt {i a \sinh {\left (c + d x \right )} + a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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