Optimal. Leaf size=114 \[ \frac{10 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \sec (c+d x)}}{21 a d e^2}+\frac{10 \sin (c+d x)}{21 a d e \sqrt{e \sec (c+d x)}}+\frac{2 i}{7 d (a+i a \tan (c+d x)) (e \sec (c+d x))^{3/2}} \]
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Rubi [A] time = 0.0930508, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3502, 3769, 3771, 2641} \[ \frac{10 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \sec (c+d x)}}{21 a d e^2}+\frac{10 \sin (c+d x)}{21 a d e \sqrt{e \sec (c+d x)}}+\frac{2 i}{7 d (a+i a \tan (c+d x)) (e \sec (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3502
Rule 3769
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{(e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))} \, dx &=\frac{2 i}{7 d (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))}+\frac{5 \int \frac{1}{(e \sec (c+d x))^{3/2}} \, dx}{7 a}\\ &=\frac{10 \sin (c+d x)}{21 a d e \sqrt{e \sec (c+d x)}}+\frac{2 i}{7 d (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))}+\frac{5 \int \sqrt{e \sec (c+d x)} \, dx}{21 a e^2}\\ &=\frac{10 \sin (c+d x)}{21 a d e \sqrt{e \sec (c+d x)}}+\frac{2 i}{7 d (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))}+\frac{\left (5 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{21 a e^2}\\ &=\frac{10 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \sec (c+d x)}}{21 a d e^2}+\frac{10 \sin (c+d x)}{21 a d e \sqrt{e \sec (c+d x)}}+\frac{2 i}{7 d (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.487852, size = 125, normalized size = 1.1 \[ -\frac{\sec ^3(c+d x) \left (5 i \sin (c+d x)+5 i \sin (3 (c+d x))-14 \cos (c+d x)+2 \cos (3 (c+d x))+20 i \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (\cos (c+d x)+i \sin (c+d x))\right )}{42 a d (\tan (c+d x)-i) (e \sec (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.336, size = 218, normalized size = 1.9 \begin{align*}{\frac{2\, \left ( \cos \left ( dx+c \right ) -1 \right ) ^{2} \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2}\cos \left ( dx+c \right ) }{21\,ad{e}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}} \left ({\frac{e}{\cos \left ( dx+c \right ) }} \right ) ^{{\frac{3}{2}}} \left ( 5\,i\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}}}{\it EllipticF} \left ({\frac{i \left ( \cos \left ( dx+c \right ) -1 \right ) }{\sin \left ( dx+c \right ) }},i \right ) +3\,i \left ( \cos \left ( dx+c \right ) \right ) ^{4}+5\,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 EllipticF} \left ({\frac{i \left ( \cos \left ( dx+c \right ) -1 \right ) }{\sin \left ( dx+c \right ) }},i \right ) +3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) +5\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) \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{{\left (84 \, a d e^{2} e^{\left (4 i \, d x + 4 i \, c\right )}{\rm integral}\left (-\frac{5 i \, \sqrt{2} \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 )}}{21 \, a d e^{2}}, x\right ) + \sqrt{2} \sqrt{\frac{e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (-7 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 9 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 19 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i\right )} e^{\left (\frac{1}{2} i \, d x + \frac{1}{2} i \, c\right )}\right )} e^{\left (-4 i \, d x - 4 i \, c\right )}}{84 \, a d e^{2}} \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}{\left (e \sec \left (d x + c\right )\right )^{\frac{3}{2}}{\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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