Optimal. Leaf size=88 \[ -\frac{2 i (a+i a \tan (c+d x))^{9/2}}{9 a^5 d}+\frac{8 i (a+i a \tan (c+d x))^{7/2}}{7 a^4 d}-\frac{8 i (a+i a \tan (c+d x))^{5/2}}{5 a^3 d} \]
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Rubi [A] time = 0.0720852, antiderivative size = 88, 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} \[ -\frac{2 i (a+i a \tan (c+d x))^{9/2}}{9 a^5 d}+\frac{8 i (a+i a \tan (c+d x))^{7/2}}{7 a^4 d}-\frac{8 i (a+i a \tan (c+d x))^{5/2}}{5 a^3 d} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 43
Rubi steps
\begin{align*} \int \frac{\sec ^6(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx &=-\frac{i \operatorname{Subst}\left (\int (a-x)^2 (a+x)^{3/2} \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (4 a^2 (a+x)^{3/2}-4 a (a+x)^{5/2}+(a+x)^{7/2}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=-\frac{8 i (a+i a \tan (c+d x))^{5/2}}{5 a^3 d}+\frac{8 i (a+i a \tan (c+d x))^{7/2}}{7 a^4 d}-\frac{2 i (a+i a \tan (c+d x))^{9/2}}{9 a^5 d}\\ \end{align*}
Mathematica [A] time = 0.279938, size = 77, normalized size = 0.88 \[ \frac{2 \sec ^5(c+d x) (-55 i \sin (2 (c+d x))+71 \cos (2 (c+d x))+36) (\sin (3 (c+d x))-i \cos (3 (c+d x)))}{315 d \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.375, size = 100, normalized size = 1.1 \begin{align*} -{\frac{128\,i \left ( \cos \left ( dx+c \right ) \right ) ^{4}-128\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) +16\,i \left ( \cos \left ( dx+c \right ) \right ) ^{2}-80\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +70\,i}{315\,ad \left ( \cos \left ( dx+c \right ) \right ) ^{4}}\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 = 1.13443, size = 228, normalized size = 2.59 \begin{align*} -\frac{2 i \,{\left (315 \, \sqrt{i \, a \tan \left (d x + c\right ) + a} - \frac{42 \,{\left (3 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}} - 10 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{3}{2}} a + 15 \, \sqrt{i \, a \tan \left (d x + c\right ) + a} a^{2}\right )}}{a^{2}} + \frac{35 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{9}{2}} - 180 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{7}{2}} a + 378 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}} a^{2} - 420 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{3}{2}} a^{3} + 315 \, \sqrt{i \, a \tan \left (d x + c\right ) + a} a^{4}}{a^{4}}\right )}}{315 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.21809, size = 362, normalized size = 4.11 \begin{align*} \frac{\sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (-256 i \, e^{\left (8 i \, d x + 8 i \, c\right )} - 1152 i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 2016 i \, e^{\left (4 i \, d x + 4 i \, c\right )}\right )} e^{\left (i \, d x + i \, c\right )}}{315 \,{\left (a d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, a d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, a d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, a d e^{\left (2 i \, d x + 2 i \, c\right )} + a 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 \frac{\sec \left (d x + c\right )^{6}}{\sqrt{i \, a \tan \left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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