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 = 0.06, antiderivative size = 90, normalized size of antiderivative = 1.00, 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 43
Rule 3487
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*}
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Mathematica [A] time = 0.78, 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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fricas [A] time = 0.50, size = 153, normalized size = 1.70 \[ \frac {48 \, d x e^{\left (6 i \, d x + 6 i \, c\right )} + 48 \, d x + {\left (144 \, d x - 24 i\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + {\left (144 \, d x - 60 i\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (24 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 72 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 72 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 24 i\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 44 i}{3 \, {\left (a^{4} d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, a^{4} d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, a^{4} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{4} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.99, size = 154, normalized size = 1.71 \[ -\frac {2 \, {\left (-\frac {12 i \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{4}} + \frac {24 i \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}{a^{4}} - \frac {12 i \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}{a^{4}} + \frac {22 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 21 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 78 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 46 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 78 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 21 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 22 i}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3} a^{4}}\right )}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.40, size = 68, normalized size = 0.76 \[ -\frac {7 \tan \left (d x +c \right )}{a^{4} d}+\frac {\tan ^{3}\left (d x +c \right )}{3 a^{4} d}+\frac {2 i \left (\tan ^{2}\left (d x +c \right )\right )}{a^{4} d}-\frac {8 i \ln \left (\tan \left (d x +c \right )-i\right )}{a^{4} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.77, size = 53, normalized size = 0.59 \[ \frac {\frac {\tan \left (d x + c\right )^{3} + 6 i \, \tan \left (d x + c\right )^{2} - 21 \, \tan \left (d x + c\right )}{a^{4}} - \frac {24 i \, \log \left (i \, \tan \left (d x + c\right ) + 1\right )}{a^{4}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.38, size = 60, normalized size = 0.67 \[ -\frac {\frac {7\,\mathrm {tan}\left (c+d\,x\right )}{a^4}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^3}{3\,a^4}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,8{}\mathrm {i}}{a^4}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,2{}\mathrm {i}}{a^4}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\sec ^{8}{\left (c + d x \right )}}{\tan ^{4}{\left (c + d x \right )} - 4 i \tan ^{3}{\left (c + d x \right )} - 6 \tan ^{2}{\left (c + d x \right )} + 4 i \tan {\left (c + d x \right )} + 1}\, dx}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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