Optimal. Leaf size=90 \[ -\frac{i (a-i a \tan (c+d x))^3}{3 a^7 d}-\frac{i (a-i a \tan (c+d x))^2}{a^6 d}-\frac{4 \tan (c+d x)}{a^4 d}+\frac{8 i \log (\cos (c+d x))}{a^4 d}+\frac{8 x}{a^4} \]
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Rubi [A] time = 0.0560963, antiderivative size = 90, 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 = {3487, 43} \[ -\frac{i (a-i a \tan (c+d x))^3}{3 a^7 d}-\frac{i (a-i a \tan (c+d x))^2}{a^6 d}-\frac{4 \tan (c+d x)}{a^4 d}+\frac{8 i \log (\cos (c+d x))}{a^4 d}+\frac{8 x}{a^4} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 43
Rubi steps
\begin{align*} \int \frac{\sec ^8(c+d x)}{(a+i a \tan (c+d x))^4} \, dx &=-\frac{i \operatorname{Subst}\left (\int \frac{(a-x)^3}{a+x} \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (-4 a^2-2 a (a-x)-(a-x)^2+\frac{8 a^3}{a+x}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=\frac{8 x}{a^4}+\frac{8 i \log (\cos (c+d x))}{a^4 d}-\frac{4 \tan (c+d x)}{a^4 d}-\frac{i (a-i a \tan (c+d x))^2}{a^6 d}-\frac{i (a-i a \tan (c+d x))^3}{3 a^7 d}\\ \end{align*}
Mathematica [A] time = 0.669694, size = 168, normalized size = 1.87 \[ \frac{\sec (c) \sec ^3(c+d x) (12 \sin (2 c+d x)-11 \sin (2 c+3 d x)+6 d x \cos (2 c+3 d x)+6 d x \cos (4 c+3 d x)+6 i \cos (2 c+3 d x) \log (\cos (c+d x))+6 \cos (d x) (3 i \log (\cos (c+d x))+3 d x+i)+6 \cos (2 c+d x) (3 i \log (\cos (c+d x))+3 d x+i)+6 i \cos (4 c+3 d x) \log (\cos (c+d x))-21 \sin (d x))}{6 a^4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.085, size = 68, normalized size = 0.8 \begin{align*} -7\,{\frac{\tan \left ( dx+c \right ) }{{a}^{4}d}}+{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{3\,{a}^{4}d}}+{\frac{2\,i \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{{a}^{4}d}}-{\frac{8\,i\ln \left ( \tan \left ( dx+c \right ) -i \right ) }{{a}^{4}d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.995178, size = 72, normalized size = 0.8 \begin{align*} \frac{\frac{\tan \left (d x + c\right )^{3} + 6 i \, \tan \left (d x + c\right )^{2} - 21 \, \tan \left (d x + c\right )}{a^{4}} - \frac{24 i \, \log \left (i \, \tan \left (d x + c\right ) + 1\right )}{a^{4}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.48496, size = 463, normalized size = 5.14 \begin{align*} \frac{48 \, d x e^{\left (6 i \, d x + 6 i \, c\right )} + 48 \, d x +{\left (144 \, d x - 24 i\right )} e^{\left (4 i \, d x + 4 i \, c\right )} +{\left (144 \, d x - 60 i\right )} e^{\left (2 i \, d x + 2 i \, c\right )} +{\left (24 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 72 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 72 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 24 i\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 44 i}{3 \,{\left (a^{4} d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, a^{4} d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, a^{4} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{4} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.20122, size = 211, normalized size = 2.34 \begin{align*} -\frac{2 \,{\left (\frac{24 i \, \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}{a^{4}} - \frac{12 i \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac{12 i \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a^{4}} + \frac{22 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 21 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 78 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 46 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 78 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 21 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 22 i}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{3} a^{4}}\right )}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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