Optimal. Leaf size=60 \[ \frac{2 a \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d \sqrt{e \cos (c+d x)}}+\frac{2 i a}{d \sqrt{e \cos (c+d x)}} \]
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Rubi [A] time = 0.0793143, 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, 2641} \[ \frac{2 a \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d \sqrt{e \cos (c+d x)}}+\frac{2 i a}{d \sqrt{e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3515
Rule 3486
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \frac{a+i a \tan (c+d x)}{\sqrt{e \cos (c+d x)}} \, dx &=\frac{\int \sqrt{e \sec (c+d x)} (a+i a \tan (c+d x)) \, dx}{\sqrt{e \cos (c+d x)} \sqrt{e \sec (c+d x)}}\\ &=\frac{2 i a}{d \sqrt{e \cos (c+d x)}}+\frac{a \int \sqrt{e \sec (c+d x)} \, dx}{\sqrt{e \cos (c+d x)} \sqrt{e \sec (c+d x)}}\\ &=\frac{2 i a}{d \sqrt{e \cos (c+d x)}}+\frac{\left (a \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{\sqrt{e \cos (c+d x)}}\\ &=\frac{2 i a}{d \sqrt{e \cos (c+d x)}}+\frac{2 a \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d \sqrt{e \cos (c+d x)}}\\ \end{align*}
Mathematica [C] time = 1.11397, size = 143, normalized size = 2.38 \[ -\frac{\sqrt{2} a \sin (c) (\cot (c)-i) (\tan (c+d x)-i) (\cos (d x)-i \sin (d x)) \sqrt{e \cos (c+d x)} \left (\sqrt{2} \sqrt{\csc ^2(c)}+i \csc (c) \cos (c+d x) \sqrt{\cos \left (2 d x-2 \tan ^{-1}(\cot (c))\right )+1} \sec \left (d x-\tan ^{-1}(\cot (c))\right ) \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left (d x-\tan ^{-1}(\cot (c))\right )\right )\right )}{d e \sqrt{\csc ^2(c)}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 2.592, size = 94, normalized size = 1.6 \begin{align*} 2\,{\frac{ \left ( -\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +i\sin \left ( 1/2\,dx+c/2 \right ) \right ) a}{\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}e+e}\sin \left ( 1/2\,dx+c/2 \right ) 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 \frac{i \, a \tan \left (d x + c\right ) + a}{\sqrt{e \cos \left (d x + c\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 e^{\left (2 i \, d x + 2 i \, c\right )} + d e\right )}{\rm integral}\left (-\frac{2 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 )}}{d e e^{\left (2 i \, d x + 2 i \, c\right )} + d e}, x\right )}{d e e^{\left (2 i \, d x + 2 i \, c\right )} + d e} \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 \frac{i \, a \tan \left (d x + c\right ) + a}{\sqrt{e \cos \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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