Optimal. Leaf size=157 \[ -\frac{10 e^5 \sin (c+d x) (e \sec (c+d x))^{3/2}}{a^4 d}+\frac{12 i e^4 (e \sec (c+d x))^{5/2}}{d \left (a^4+i a^4 \tan (c+d x)\right )}-\frac{10 e^6 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \sec (c+d x)}}{a^4 d}+\frac{4 i e^2 (e \sec (c+d x))^{9/2}}{3 a d (a+i a \tan (c+d x))^3} \]
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Rubi [A] time = 0.154751, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3500, 3768, 3771, 2641} \[ -\frac{10 e^5 \sin (c+d x) (e \sec (c+d x))^{3/2}}{a^4 d}+\frac{12 i e^4 (e \sec (c+d x))^{5/2}}{d \left (a^4+i a^4 \tan (c+d x)\right )}-\frac{10 e^6 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \sec (c+d x)}}{a^4 d}+\frac{4 i e^2 (e \sec (c+d x))^{9/2}}{3 a d (a+i a \tan (c+d x))^3} \]
Antiderivative was successfully verified.
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Rule 3500
Rule 3768
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \frac{(e \sec (c+d x))^{13/2}}{(a+i a \tan (c+d x))^4} \, dx &=\frac{4 i e^2 (e \sec (c+d x))^{9/2}}{3 a d (a+i a \tan (c+d x))^3}-\frac{\left (3 e^2\right ) \int \frac{(e \sec (c+d x))^{9/2}}{(a+i a \tan (c+d x))^2} \, dx}{a^2}\\ &=\frac{4 i e^2 (e \sec (c+d x))^{9/2}}{3 a d (a+i a \tan (c+d x))^3}+\frac{12 i e^4 (e \sec (c+d x))^{5/2}}{d \left (a^4+i a^4 \tan (c+d x)\right )}-\frac{\left (15 e^4\right ) \int (e \sec (c+d x))^{5/2} \, dx}{a^4}\\ &=-\frac{10 e^5 (e \sec (c+d x))^{3/2} \sin (c+d x)}{a^4 d}+\frac{4 i e^2 (e \sec (c+d x))^{9/2}}{3 a d (a+i a \tan (c+d x))^3}+\frac{12 i e^4 (e \sec (c+d x))^{5/2}}{d \left (a^4+i a^4 \tan (c+d x)\right )}-\frac{\left (5 e^6\right ) \int \sqrt{e \sec (c+d x)} \, dx}{a^4}\\ &=-\frac{10 e^5 (e \sec (c+d x))^{3/2} \sin (c+d x)}{a^4 d}+\frac{4 i e^2 (e \sec (c+d x))^{9/2}}{3 a d (a+i a \tan (c+d x))^3}+\frac{12 i e^4 (e \sec (c+d x))^{5/2}}{d \left (a^4+i a^4 \tan (c+d x)\right )}-\frac{\left (5 e^6 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{a^4}\\ &=-\frac{10 e^6 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \sec (c+d x)}}{a^4 d}-\frac{10 e^5 (e \sec (c+d x))^{3/2} \sin (c+d x)}{a^4 d}+\frac{4 i e^2 (e \sec (c+d x))^{9/2}}{3 a d (a+i a \tan (c+d x))^3}+\frac{12 i e^4 (e \sec (c+d x))^{5/2}}{d \left (a^4+i a^4 \tan (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.500015, size = 134, normalized size = 0.85 \[ \frac{i e^6 \sec ^5(c+d x) \sqrt{e \sec (c+d x)} (\cos (3 (c+d x))+i \sin (3 (c+d x))) \left (11 i \sin (2 (c+d x))+19 \cos (2 (c+d x))+30 i \cos ^{\frac{3}{2}}(c+d x) F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (\cos (c+d x)+i \sin (c+d x))+21\right )}{3 a^4 d (\tan (c+d x)-i)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.3, size = 198, normalized size = 1.3 \begin{align*}{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{3\,{a}^{4}d} \left ({\frac{e}{\cos \left ( dx+c \right ) }} \right ) ^{{\frac{13}{2}}} \left ( -15\,i \left ( \cos \left ( dx+c \right ) \right ) ^{2}\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 ) -15\,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 ) +8\,i \left ( \cos \left ( dx+c \right ) \right ) ^{3}+8\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) +12\,i\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{\sqrt{2}{\left (30 i \, e^{6} e^{\left (4 i \, d x + 4 i \, c\right )} + 42 i \, e^{6} e^{\left (2 i \, d x + 2 i \, c\right )} + 8 i \, e^{6}\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 )} + 3 \,{\left (a^{4} d e^{\left (4 i \, d x + 4 i \, c\right )} + a^{4} d e^{\left (2 i \, d x + 2 i \, c\right )}\right )}{\rm integral}\left (\frac{5 i \, \sqrt{2} e^{6} \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 )}}{a^{4} d}, x\right )}{3 \,{\left (a^{4} d e^{\left (4 i \, d x + 4 i \, c\right )} + a^{4} d e^{\left (2 i \, d x + 2 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{\left (e \sec \left (d x + c\right )\right )^{\frac{13}{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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