Optimal. Leaf size=105 \[ -\frac{2 i e^2 (e \sec (c+d x))^{5/2}}{5 a d}+\frac{2 e^3 \sin (c+d x) (e \sec (c+d x))^{3/2}}{3 a d}+\frac{2 e^4 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \sec (c+d x)}}{3 a d} \]
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Rubi [A] time = 0.0891565, antiderivative size = 105, 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 = {3501, 3768, 3771, 2641} \[ -\frac{2 i e^2 (e \sec (c+d x))^{5/2}}{5 a d}+\frac{2 e^3 \sin (c+d x) (e \sec (c+d x))^{3/2}}{3 a d}+\frac{2 e^4 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \sec (c+d x)}}{3 a d} \]
Antiderivative was successfully verified.
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Rule 3501
Rule 3768
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \frac{(e \sec (c+d x))^{9/2}}{a+i a \tan (c+d x)} \, dx &=-\frac{2 i e^2 (e \sec (c+d x))^{5/2}}{5 a d}+\frac{e^2 \int (e \sec (c+d x))^{5/2} \, dx}{a}\\ &=-\frac{2 i e^2 (e \sec (c+d x))^{5/2}}{5 a d}+\frac{2 e^3 (e \sec (c+d x))^{3/2} \sin (c+d x)}{3 a d}+\frac{e^4 \int \sqrt{e \sec (c+d x)} \, dx}{3 a}\\ &=-\frac{2 i e^2 (e \sec (c+d x))^{5/2}}{5 a d}+\frac{2 e^3 (e \sec (c+d x))^{3/2} \sin (c+d x)}{3 a d}+\frac{\left (e^4 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{3 a}\\ &=\frac{2 e^4 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \sec (c+d x)}}{3 a d}-\frac{2 i e^2 (e \sec (c+d x))^{5/2}}{5 a d}+\frac{2 e^3 (e \sec (c+d x))^{3/2} \sin (c+d x)}{3 a d}\\ \end{align*}
Mathematica [A] time = 0.584932, size = 62, normalized size = 0.59 \[ \frac{e^2 (e \sec (c+d x))^{5/2} \left (5 \sin (2 (c+d x))+10 \cos ^{\frac{5}{2}}(c+d x) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )-6 i\right )}{15 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.239, size = 202, 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} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{15\,ad \left ( \sin \left ( dx+c \right ) \right ) ^{4}} \left ( 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}}} \left ( \cos \left ( dx+c \right ) \right ) ^{3}{\it EllipticF} \left ({\frac{i \left ( \cos \left ( dx+c \right ) -1 \right ) }{\sin \left ( dx+c \right ) }},i \right ) +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}}} \left ( \cos \left ( dx+c \right ) \right ) ^{2}{\it EllipticF} \left ({\frac{i \left ( \cos \left ( dx+c \right ) -1 \right ) }{\sin \left ( dx+c \right ) }},i \right ) +5\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) -3\,i \right ) \left ({\frac{e}{\cos \left ( dx+c \right ) }} \right ) ^{{\frac{9}{2}}}} \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}{\left (-10 i \, e^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 24 i \, e^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + 10 i \, e^{4}\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 )} + 15 \,{\left (a d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, a d e^{\left (2 i \, d x + 2 i \, c\right )} + a d\right )}{\rm integral}\left (-\frac{i \, \sqrt{2} e^{4} \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 )}}{3 \, a d}, x\right )}{15 \,{\left (a d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, a d e^{\left (2 i \, d x + 2 i \, c\right )} + a d\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{\left (e \sec \left (d x + c\right )\right )^{\frac{9}{2}}}{i \, a \tan \left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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