Optimal. Leaf size=60 \[ \frac{2 a E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{2 i a \sqrt{e \cos (c+d x)}}{d} \]
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Rubi [A] time = 0.0782001, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3515, 3486, 3771, 2639} \[ \frac{2 a E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{2 i a \sqrt{e \cos (c+d x)}}{d} \]
Antiderivative was successfully verified.
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Rule 3515
Rule 3486
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \sqrt{e \cos (c+d x)} (a+i a \tan (c+d x)) \, dx &=\left (\sqrt{e \cos (c+d x)} \sqrt{e \sec (c+d x)}\right ) \int \frac{a+i a \tan (c+d x)}{\sqrt{e \sec (c+d x)}} \, dx\\ &=-\frac{2 i a \sqrt{e \cos (c+d x)}}{d}+\left (a \sqrt{e \cos (c+d x)} \sqrt{e \sec (c+d x)}\right ) \int \frac{1}{\sqrt{e \sec (c+d x)}} \, dx\\ &=-\frac{2 i a \sqrt{e \cos (c+d x)}}{d}+\frac{\left (a \sqrt{e \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{\sqrt{\cos (c+d x)}}\\ &=-\frac{2 i a \sqrt{e \cos (c+d x)}}{d}+\frac{2 a \sqrt{e \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d \sqrt{\cos (c+d x)}}\\ \end{align*}
Mathematica [C] time = 0.690825, size = 244, normalized size = 4.07 \[ \frac{a e (\cot (c)+i) e^{-i (c+d x)} \left (e^{2 i d x} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right )-3 \sqrt{1-i e^{i (c+d x)}} \sqrt{e^{i (c+d x)} \left (e^{i (c+d x)}-i\right )} F\left (\left .\sin ^{-1}\left (\sqrt{\sin (c+d x)-i \cos (c+d x)}\right )\right |-1\right )+3 \sqrt{1-i e^{i (c+d x)}} \sqrt{e^{i (c+d x)} \left (e^{i (c+d x)}-i\right )} E\left (\left .\sin ^{-1}\left (\sqrt{\sin (c+d x)-i \cos (c+d x)}\right )\right |-1\right )\right )}{3 d \sqrt{e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.352, size = 108, normalized size = 1.8 \begin{align*} 2\,{\frac{ae \left ( 2\,i \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}+{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}-i\sin \left ( 1/2\,dx+c/2 \right ) \right ) }{\sin \left ( 1/2\,dx+c/2 \right ) \sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}e+e}d}} \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{e \cos \left (d x + c\right )}{\left (i \, a \tan \left (d x + c\right ) + a\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*} \frac{-4 i \, \sqrt{\frac{1}{2}} \sqrt{e e^{\left (2 i \, d x + 2 i \, c\right )} + e} a e^{\left (\frac{1}{2} i \, d x + \frac{1}{2} i \, c\right )} +{\left (d e^{\left (i \, d x + i \, c\right )} - d\right )}{\rm integral}\left (\frac{\sqrt{\frac{1}{2}}{\left (-2 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} - 4 i \, a e^{\left (i \, d x + i \, c\right )} - 2 i \, a\right )} \sqrt{e e^{\left (2 i \, d x + 2 i \, c\right )} + e} e^{\left (-\frac{1}{2} i \, d x - \frac{1}{2} i \, c\right )}}{d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (3 i \, d x + 3 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} - 2 \, d e^{\left (i \, d x + i \, c\right )} + d}, x\right )}{d e^{\left (i \, d x + i \, c\right )} - d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{e \cos \left (d x + c\right )}{\left (i \, a \tan \left (d x + c\right ) + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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