Optimal. Leaf size=29 \[ -\frac{2 i (a+i a \tan (c+d x))^{9/2}}{9 a d} \]
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Rubi [A] time = 0.0615514, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {3487, 32} \[ -\frac{2 i (a+i a \tan (c+d x))^{9/2}}{9 a d} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 32
Rubi steps
\begin{align*} \int \sec ^2(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx &=-\frac{i \operatorname{Subst}\left (\int (a+x)^{7/2} \, dx,x,i a \tan (c+d x)\right )}{a d}\\ &=-\frac{2 i (a+i a \tan (c+d x))^{9/2}}{9 a d}\\ \end{align*}
Mathematica [B] time = 0.339362, size = 73, normalized size = 2.52 \[ \frac{2 a^3 \sec ^4(c+d x) \sqrt{a+i a \tan (c+d x)} (\sin (4 c+7 d x)-i \cos (4 c+7 d x))}{9 d (\cos (d x)+i \sin (d x))^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 24, normalized size = 0.8 \begin{align*}{\frac{-{\frac{2\,i}{9}}}{ad} \left ( a+ia\tan \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 [A] time = 1.18487, size = 28, normalized size = 0.97 \begin{align*} -\frac{2 i \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{9}{2}}}{9 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.14733, size = 244, normalized size = 8.41 \begin{align*} -\frac{32 i \, \sqrt{2} a^{3} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (9 i \, d x + 9 i \, c\right )}}{9 \,{\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\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}} \sec \left (d x + c\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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