Optimal. Leaf size=132 \[ -\frac{20 i e^4 \sqrt{e \sec (c+d x)}}{21 d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac{10 e^4 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \sec (c+d x)}}{21 a^4 d}+\frac{4 i e^2 (e \sec (c+d x))^{5/2}}{7 a d (a+i a \tan (c+d x))^3} \]
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Rubi [A] time = 0.137437, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {3500, 3771, 2641} \[ -\frac{20 i e^4 \sqrt{e \sec (c+d x)}}{21 d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac{10 e^4 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \sec (c+d x)}}{21 a^4 d}+\frac{4 i e^2 (e \sec (c+d x))^{5/2}}{7 a d (a+i a \tan (c+d x))^3} \]
Antiderivative was successfully verified.
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Rule 3500
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \frac{(e \sec (c+d x))^{9/2}}{(a+i a \tan (c+d x))^4} \, dx &=\frac{4 i e^2 (e \sec (c+d x))^{5/2}}{7 a d (a+i a \tan (c+d x))^3}-\frac{\left (5 e^2\right ) \int \frac{(e \sec (c+d x))^{5/2}}{(a+i a \tan (c+d x))^2} \, dx}{7 a^2}\\ &=\frac{4 i e^2 (e \sec (c+d x))^{5/2}}{7 a d (a+i a \tan (c+d x))^3}-\frac{20 i e^4 \sqrt{e \sec (c+d x)}}{21 d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac{\left (5 e^4\right ) \int \sqrt{e \sec (c+d x)} \, dx}{21 a^4}\\ &=\frac{4 i e^2 (e \sec (c+d x))^{5/2}}{7 a d (a+i a \tan (c+d x))^3}-\frac{20 i e^4 \sqrt{e \sec (c+d x)}}{21 d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac{\left (5 e^4 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{21 a^4}\\ &=\frac{10 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)}}{21 a^4 d}+\frac{4 i e^2 (e \sec (c+d x))^{5/2}}{7 a d (a+i a \tan (c+d x))^3}-\frac{20 i e^4 \sqrt{e \sec (c+d x)}}{21 d \left (a^4+i a^4 \tan (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.494361, size = 137, normalized size = 1.04 \[ \frac{2 e^4 \sec ^4(c+d x) \sqrt{e \sec (c+d x)} (\cos (2 (c+d x))+i \sin (2 (c+d x))) \left (5 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (\cos (2 (c+d x))+i \sin (2 (c+d x)))-2 i (4 i \sin (2 (c+d x))+\cos (2 (c+d x))+1)\right )}{21 a^4 d (\tan (c+d x)-i)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.251, size = 200, normalized size = 1.5 \begin{align*}{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{21\,{a}^{4}d} \left ({\frac{e}{\cos \left ( dx+c \right ) }} \right ) ^{{\frac{9}{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 ) +24\,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 ) +24\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) -28\,i \left ( \cos \left ( dx+c \right ) \right ) ^{2}-16\,\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 (21 \, a^{4} d e^{\left (4 i \, d x + 4 i \, c\right )}{\rm integral}\left (-\frac{5 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 )}}{21 \, a^{4} d}, x\right ) + \sqrt{2}{\left (-10 i \, e^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 4 i \, e^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + 6 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 )}\right )} e^{\left (-4 i \, d x - 4 i \, c\right )}}{21 \, a^{4} d} \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}}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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