Optimal. Leaf size=84 \[ -\frac{2 i (a+i a \tan (c+d x))^{3/2}}{3 a^5 d}+\frac{8 i \sqrt{a+i a \tan (c+d x)}}{a^4 d}+\frac{8 i}{a^3 d \sqrt{a+i a \tan (c+d x)}} \]
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Rubi [A] time = 0.0795285, antiderivative size = 84, 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} \[ -\frac{2 i (a+i a \tan (c+d x))^{3/2}}{3 a^5 d}+\frac{8 i \sqrt{a+i a \tan (c+d x)}}{a^4 d}+\frac{8 i}{a^3 d \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 43
Rubi steps
\begin{align*} \int \frac{\sec ^6(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx &=-\frac{i \operatorname{Subst}\left (\int \frac{(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 (\frac{4 a^2}{(a+x)^{3/2}}-\frac{4 a}{\sqrt{a+x}}+\sqrt{a+x}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=\frac{8 i}{a^3 d \sqrt{a+i a \tan (c+d x)}}+\frac{8 i \sqrt{a+i a \tan (c+d x)}}{a^4 d}-\frac{2 i (a+i a \tan (c+d x))^{3/2}}{3 a^5 d}\\ \end{align*}
Mathematica [A] time = 0.295884, size = 61, normalized size = 0.73 \[ \frac{2 i \sec ^2(c+d x) (5 i \sin (2 (c+d x))+11 \cos (2 (c+d x))+12)}{3 a^3 d \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.28, size = 88, normalized size = 1.1 \begin{align*}{\frac{24\,i \left ( \cos \left ( dx+c \right ) \right ) ^{3}+24\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) +22\,i\cos \left ( dx+c \right ) +2\,\sin \left ( dx+c \right ) }{3\,{a}^{4}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 [A] time = 0.978093, size = 84, normalized size = 1. \begin{align*} \frac{2 i \,{\left (\frac{12}{\sqrt{i \, a \tan \left (d x + c\right ) + a} a^{2}} - \frac{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{3}{2}} - 12 \, \sqrt{i \, a \tan \left (d x + c\right ) + a} a}{a^{4}}\right )}}{3 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.16177, size = 243, normalized size = 2.89 \begin{align*} \frac{\sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (32 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 48 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 12 i\right )} e^{\left (i \, d x + i \, c\right )}}{3 \,{\left (a^{4} d e^{\left (4 i \, d x + 4 i \, c\right )} + a^{4} d e^{\left (2 i \, d x + 2 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 \frac{\sec \left (d x + c\right )^{6}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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