Optimal. Leaf size=80 \[ \frac{6 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{2 i}{5 d (a+i a \tan (c+d x)) \sqrt{e \sec (c+d x)}} \]
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Rubi [A] time = 0.0696769, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {3502, 3771, 2639} \[ \frac{6 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{2 i}{5 d (a+i a \tan (c+d x)) \sqrt{e \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3502
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{e \sec (c+d x)} (a+i a \tan (c+d x))} \, dx &=\frac{2 i}{5 d \sqrt{e \sec (c+d x)} (a+i a \tan (c+d x))}+\frac{3 \int \frac{1}{\sqrt{e \sec (c+d x)}} \, dx}{5 a}\\ &=\frac{2 i}{5 d \sqrt{e \sec (c+d x)} (a+i a \tan (c+d x))}+\frac{3 \int \sqrt{\cos (c+d x)} \, dx}{5 a \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}\\ &=\frac{6 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{2 i}{5 d \sqrt{e \sec (c+d x)} (a+i a \tan (c+d x))}\\ \end{align*}
Mathematica [C] time = 0.799048, size = 109, normalized size = 1.36 \[ \frac{(\tan (c+d x)+i) \left (-2 e^{2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )+3 i \sin (2 (c+d x))+4 \cos (2 (c+d x))+4\right )}{5 a d \sqrt{e \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.444, size = 358, normalized size = 4.5 \begin{align*}{\frac{2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2} \left ( \cos \left ( dx+c \right ) -1 \right ) ^{2}}{5\,ad \left ( \sin \left ( dx+c \right ) \right ) ^{5}e} \left ( 3\,i{\it EllipticF} \left ({\frac{i \left ( \cos \left ( dx+c \right ) -1 \right ) }{\sin \left ( dx+c \right ) }},i \right ) \sin \left ( dx+c \right ) \cos \left ( dx+c \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}-3\,i\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}{\it EllipticE} \left ({\frac{i \left ( \cos \left ( dx+c \right ) -1 \right ) }{\sin \left ( dx+c \right ) }},i \right ) \cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +i\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3\,i{\it EllipticF} \left ({\frac{i \left ( \cos \left ( dx+c \right ) -1 \right ) }{\sin \left ( dx+c \right ) }},i \right ) \sin \left ( dx+c \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}-3\,i\sin \left ( dx+c \right ){\it EllipticE} \left ({\frac{i \left ( \cos \left ( dx+c \right ) -1 \right ) }{\sin \left ( dx+c \right ) }},i \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}- \left ( \cos \left ( dx+c \right ) \right ) ^{4}-2\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+3\,\cos \left ( dx+c \right ) \right ) \sqrt{{\frac{e}{\cos \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \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} \sqrt{\frac{e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (-5 i \, e^{\left (5 i \, d x + 5 i \, c\right )} - 7 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 4 i \, e^{\left (3 i \, d x + 3 i \, c\right )} - 8 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i \, e^{\left (i \, d x + i \, c\right )} - i\right )} e^{\left (\frac{1}{2} i \, d x + \frac{1}{2} i \, c\right )} + 10 \,{\left (a d e e^{\left (4 i \, d x + 4 i \, c\right )} - a d e e^{\left (3 i \, d x + 3 i \, c\right )}\right )}{\rm integral}\left (\frac{\sqrt{2} \sqrt{\frac{e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (-3 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 6 i \, e^{\left (i \, d x + i \, c\right )} - 3 i\right )} e^{\left (\frac{1}{2} i \, d x + \frac{1}{2} i \, c\right )}}{5 \,{\left (a d e e^{\left (3 i \, d x + 3 i \, c\right )} - 2 \, a d e e^{\left (2 i \, d x + 2 i \, c\right )} + a d e e^{\left (i \, d x + i \, c\right )}\right )}}, x\right )}{10 \,{\left (a d e e^{\left (4 i \, d x + 4 i \, c\right )} - a d e e^{\left (3 i \, d x + 3 i \, c\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \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{1}{\sqrt{e \sec \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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