Optimal. Leaf size=104 \[ -\frac{64 i a^3 \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{3 d}+\frac{16 i a^2 \cos (c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d}+\frac{2 i a \cos (c+d x) (a+i a \tan (c+d x))^{5/2}}{3 d} \]
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Rubi [A] time = 0.153474, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3494, 3493} \[ -\frac{64 i a^3 \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{3 d}+\frac{16 i a^2 \cos (c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d}+\frac{2 i a \cos (c+d x) (a+i a \tan (c+d x))^{5/2}}{3 d} \]
Antiderivative was successfully verified.
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Rule 3494
Rule 3493
Rubi steps
\begin{align*} \int \cos (c+d x) (a+i a \tan (c+d x))^{7/2} \, dx &=\frac{2 i a \cos (c+d x) (a+i a \tan (c+d x))^{5/2}}{3 d}+\frac{1}{3} (8 a) \int \cos (c+d x) (a+i a \tan (c+d x))^{5/2} \, dx\\ &=\frac{16 i a^2 \cos (c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d}+\frac{2 i a \cos (c+d x) (a+i a \tan (c+d x))^{5/2}}{3 d}+\frac{1}{3} \left (32 a^2\right ) \int \cos (c+d x) (a+i a \tan (c+d x))^{3/2} \, dx\\ &=-\frac{64 i a^3 \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{3 d}+\frac{16 i a^2 \cos (c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d}+\frac{2 i a \cos (c+d x) (a+i a \tan (c+d x))^{5/2}}{3 d}\\ \end{align*}
Mathematica [A] time = 0.357459, size = 59, normalized size = 0.57 \[ -\frac{2 i a^3 \sec (c+d x) \sqrt{a+i a \tan (c+d x)} (-5 i \sin (2 (c+d x))+11 \cos (2 (c+d x))+12)}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.283, size = 73, normalized size = 0.7 \begin{align*} -{\frac{2\,{a}^{3} \left ( 22\,i \left ( \cos \left ( dx+c \right ) \right ) ^{2}+10\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +i \right ) }{3\,d\cos \left ( dx+c \right ) }\sqrt{{\frac{a \left ( i\sin \left ( dx+c \right ) +\cos \left ( dx+c \right ) \right ) }{\cos \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.0588, size = 564, normalized size = 5.42 \begin{align*} \frac{2 \,{\left (23 i \, a^{\frac{7}{2}} + \frac{20 \, a^{\frac{7}{2}} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{88 i \, a^{\frac{7}{2}} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{60 \, a^{\frac{7}{2}} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{130 i \, a^{\frac{7}{2}} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{60 \, a^{\frac{7}{2}} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{88 i \, a^{\frac{7}{2}} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{20 \, a^{\frac{7}{2}} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac{23 i \, a^{\frac{7}{2}} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )}{\left (-\frac{2 i \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )}^{\frac{7}{2}}}{d{\left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac{7}{2}}{\left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}^{\frac{7}{2}}{\left (-\frac{18 i \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{42 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{42 i \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{42 i \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{42 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{18 i \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac{3 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.97789, size = 244, normalized size = 2.35 \begin{align*} \frac{\sqrt{2}{\left (-12 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} - 48 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - 32 i \, a^{3}\right )} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (i \, d x + i \, c\right )}}{3 \,{\left (d e^{\left (3 i \, d x + 3 i \, c\right )} + d e^{\left (i \, d x + i \, c\right )}\right )}} \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 (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{7}{2}} \cos \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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