Optimal. Leaf size=60 \[ \frac{2 a E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}-\frac{2 i a}{d \sqrt{e \sec (c+d x)}} \]
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Rubi [A] time = 0.0467685, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {3486, 3771, 2639} \[ \frac{2 a E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}-\frac{2 i a}{d \sqrt{e \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3486
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \frac{a+i a \tan (c+d x)}{\sqrt{e \sec (c+d x)}} \, dx &=-\frac{2 i a}{d \sqrt{e \sec (c+d x)}}+a \int \frac{1}{\sqrt{e \sec (c+d x)}} \, dx\\ &=-\frac{2 i a}{d \sqrt{e \sec (c+d x)}}+\frac{a \int \sqrt{\cos (c+d x)} \, dx}{\sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}\\ &=-\frac{2 i a}{d \sqrt{e \sec (c+d x)}}+\frac{2 a E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}\\ \end{align*}
Mathematica [C] time = 0.343589, size = 73, normalized size = 1.22 \[ -\frac{4 i a e^{2 i (c+d x)} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )}{3 d \sqrt{1+e^{2 i (c+d x)}} \sqrt{e \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.243, size = 910, normalized size = 15.2 \begin{align*} \text{result too large to display} \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 \sec \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{\sqrt{2}{\left (-2 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} - 2 i \, a\right )} \sqrt{\frac{e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac{1}{2} i \, d x + \frac{1}{2} i \, c\right )} +{\left (d e e^{\left (i \, d x + i \, c\right )} - d e\right )}{\rm integral}\left (\frac{\sqrt{2}{\left (-i \, a e^{\left (2 i \, d x + 2 i \, c\right )} - 2 i \, a e^{\left (i \, d x + i \, c\right )} - i \, a\right )} \sqrt{\frac{e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac{1}{2} i \, d x + \frac{1}{2} i \, c\right )}}{d e e^{\left (3 i \, d x + 3 i \, c\right )} - 2 \, d e e^{\left (2 i \, d x + 2 i \, c\right )} + d e e^{\left (i \, d x + i \, c\right )}}, x\right )}{d e e^{\left (i \, d x + i \, c\right )} - d e} \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*} a \left (\int \frac{1}{\sqrt{e \sec{\left (c + d x \right )}}}\, dx + \int \frac{i \tan{\left (c + d x \right )}}{\sqrt{e \sec{\left (c + d x \right )}}}\, dx\right ) \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 \sec \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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