Optimal. Leaf size=59 \[ \frac{2 i (a+i a \tan (c+d x))^{13/2}}{13 a^3 d}-\frac{4 i (a+i a \tan (c+d x))^{11/2}}{11 a^2 d} \]
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Rubi [A] time = 0.0679677, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {3487, 43} \[ \frac{2 i (a+i a \tan (c+d x))^{13/2}}{13 a^3 d}-\frac{4 i (a+i a \tan (c+d x))^{11/2}}{11 a^2 d} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 43
Rubi steps
\begin{align*} \int \sec ^4(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx &=-\frac{i \operatorname{Subst}\left (\int (a-x) (a+x)^{9/2} \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (2 a (a+x)^{9/2}-(a+x)^{11/2}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac{4 i (a+i a \tan (c+d x))^{11/2}}{11 a^2 d}+\frac{2 i (a+i a \tan (c+d x))^{13/2}}{13 a^3 d}\\ \end{align*}
Mathematica [A] time = 0.590619, size = 85, normalized size = 1.44 \[ -\frac{2 a^3 (11 \tan (c+d x)+15 i) \sec ^5(c+d x) \sqrt{a+i a \tan (c+d x)} (\cos (5 c+8 d x)+i \sin (5 c+8 d x))}{143 d (\cos (d x)+i \sin (d x))^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.365, size = 127, normalized size = 2.2 \begin{align*}{\frac{2\,{a}^{3} \left ( -64\,i \left ( \cos \left ( dx+c \right ) \right ) ^{6}+64\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}\sin \left ( dx+c \right ) -8\,i \left ( \cos \left ( dx+c \right ) \right ) ^{4}+40\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) +68\,i \left ( \cos \left ( dx+c \right ) \right ) ^{2}-40\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) -11\,i \right ) }{143\,d \left ( \cos \left ( dx+c \right ) \right ) ^{6}}\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 [A] time = 1.09262, size = 54, normalized size = 0.92 \begin{align*} \frac{2 i \,{\left (11 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{13}{2}} - 26 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{11}{2}} a\right )}}{143 \, a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.98042, size = 405, normalized size = 6.86 \begin{align*} \frac{\sqrt{2}{\left (-256 i \, a^{3} e^{\left (12 i \, d x + 12 i \, c\right )} - 1664 i \, a^{3} e^{\left (10 i \, d x + 10 i \, c\right )}\right )} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (i \, d x + i \, c\right )}}{143 \,{\left (d e^{\left (12 i \, d x + 12 i \, c\right )} + 6 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 20 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \, 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 )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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