Optimal. Leaf size=124 \[ -\frac{2 i a (e \cos (c+d x))^{7/2}}{7 d}+\frac{10 a F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (e \cos (c+d x))^{7/2}}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 a \tan (c+d x) (e \cos (c+d x))^{7/2}}{7 d}+\frac{10 a \tan (c+d x) \sec ^2(c+d x) (e \cos (c+d x))^{7/2}}{21 d} \]
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Rubi [A] time = 0.13542, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {3515, 3486, 3769, 3771, 2641} \[ -\frac{2 i a (e \cos (c+d x))^{7/2}}{7 d}+\frac{10 a F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (e \cos (c+d x))^{7/2}}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 a \tan (c+d x) (e \cos (c+d x))^{7/2}}{7 d}+\frac{10 a \tan (c+d x) \sec ^2(c+d x) (e \cos (c+d x))^{7/2}}{21 d} \]
Antiderivative was successfully verified.
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Rule 3515
Rule 3486
Rule 3769
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int (e \cos (c+d x))^{7/2} (a+i a \tan (c+d x)) \, dx &=\left ((e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}\right ) \int \frac{a+i a \tan (c+d x)}{(e \sec (c+d x))^{7/2}} \, dx\\ &=-\frac{2 i a (e \cos (c+d x))^{7/2}}{7 d}+\left (a (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}\right ) \int \frac{1}{(e \sec (c+d x))^{7/2}} \, dx\\ &=-\frac{2 i a (e \cos (c+d x))^{7/2}}{7 d}+\frac{2 a (e \cos (c+d x))^{7/2} \tan (c+d x)}{7 d}+\frac{\left (5 a (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}\right ) \int \frac{1}{(e \sec (c+d x))^{3/2}} \, dx}{7 e^2}\\ &=-\frac{2 i a (e \cos (c+d x))^{7/2}}{7 d}+\frac{2 a (e \cos (c+d x))^{7/2} \tan (c+d x)}{7 d}+\frac{10 a (e \cos (c+d x))^{7/2} \sec ^2(c+d x) \tan (c+d x)}{21 d}+\frac{\left (5 a (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}\right ) \int \sqrt{e \sec (c+d x)} \, dx}{21 e^4}\\ &=-\frac{2 i a (e \cos (c+d x))^{7/2}}{7 d}+\frac{2 a (e \cos (c+d x))^{7/2} \tan (c+d x)}{7 d}+\frac{10 a (e \cos (c+d x))^{7/2} \sec ^2(c+d x) \tan (c+d x)}{21 d}+\frac{\left (5 a (e \cos (c+d x))^{7/2}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{21 \cos ^{\frac{7}{2}}(c+d x)}\\ &=-\frac{2 i a (e \cos (c+d x))^{7/2}}{7 d}+\frac{10 a (e \cos (c+d x))^{7/2} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 a (e \cos (c+d x))^{7/2} \tan (c+d x)}{7 d}+\frac{10 a (e \cos (c+d x))^{7/2} \sec ^2(c+d x) \tan (c+d x)}{21 d}\\ \end{align*}
Mathematica [A] time = 0.689215, size = 133, normalized size = 1.07 \[ \frac{a e^3 (\cos (d x)-i \sin (d x)) \sqrt{e \cos (c+d x)} (\cos (c+2 d x)+i \sin (c+2 d x)) \left (\sqrt{\cos (c+d x)} (5 \sin (2 (c+d x))+2 i \cos (2 (c+d x))-8 i)+10 F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (\cos (c+d x)-i \sin (c+d x))\right )}{21 d \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 2.749, size = 241, normalized size = 1.9 \begin{align*} -{\frac{2\,a{e}^{4}}{21\,d} \left ( 48\,i \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{9}+48\,\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}-96\,i \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}-72\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}\cos \left ( 1/2\,dx+c/2 \right ) +72\,i \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}+56\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}\cos \left ( 1/2\,dx+c/2 \right ) -24\,i \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}+5\,\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -16\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) +3\,i\sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}e+e}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e \cos \left (d x + c\right )\right )^{\frac{7}{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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (\sqrt{\frac{1}{2}}{\left (-3 i \, a e^{3} e^{\left (4 i \, d x + 4 i \, c\right )} - 16 i \, a e^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 7 i \, a e^{3}\right )} \sqrt{e e^{\left (2 i \, d x + 2 i \, c\right )} + e} e^{\left (-\frac{1}{2} i \, d x - \frac{1}{2} i \, c\right )} + 42 \, d e^{\left (i \, d x + i \, c\right )}{\rm integral}\left (-\frac{10 i \, \sqrt{\frac{1}{2}} \sqrt{e e^{\left (2 i \, d x + 2 i \, c\right )} + e} a e^{3} e^{\left (-\frac{1}{2} i \, d x - \frac{1}{2} i \, c\right )}}{21 \,{\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}}, x\right )\right )} e^{\left (-i \, d x - i \, c\right )}}{42 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 \left (e \cos \left (d x + c\right )\right )^{\frac{7}{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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