Optimal. Leaf size=50 \[ \frac {2 i}{d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac {i \log (\cos (c+d x))}{a^3 d}-\frac {x}{a^3} \]
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Rubi [A] time = 0.05, antiderivative size = 50, 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} \[ \frac {2 i}{d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac {i \log (\cos (c+d x))}{a^3 d}-\frac {x}{a^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 3487
Rubi steps
\begin {align*} \int \frac {\sec ^4(c+d x)}{(a+i a \tan (c+d x))^3} \, dx &=-\frac {i \operatorname {Subst}\left (\int \frac {a-x}{(a+x)^2} \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac {i \operatorname {Subst}\left (\int \left (\frac {1}{-a-x}+\frac {2 a}{(a+x)^2}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac {x}{a^3}-\frac {i \log (\cos (c+d x))}{a^3 d}+\frac {2 i}{d \left (a^3+i a^3 \tan (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 88, normalized size = 1.76 \[ \frac {\sec ^2(c+d x) (\cos (2 (c+d x))+i \sin (2 (c+d x))) (\log (\cos (c+d x))+\tan (c+d x) (i \log (\cos (c+d x))+d x+i)-i d x-1)}{a^3 d (\tan (c+d x)-i)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 55, normalized size = 1.10 \[ -\frac {{\left (2 \, d x e^{\left (2 i \, d x + 2 i \, c\right )} + i \, e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - i\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.45, size = 100, normalized size = 2.00 \[ -\frac {\frac {i \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{3}} - \frac {2 i \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}{a^{3}} + \frac {i \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}{a^{3}} + \frac {3 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 10 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 i}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.43, size = 40, normalized size = 0.80 \[ \frac {2}{a^{3} d \left (\tan \left (d x +c \right )-i\right )}+\frac {i \ln \left (\tan \left (d x +c \right )-i\right )}{a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 66, normalized size = 1.32 \[ -\frac {\frac {4 \, {\left (-i \, \tan \left (d x + c\right ) - 1\right )}}{2 i \, a^{3} \tan \left (d x + c\right )^{2} + 4 \, a^{3} \tan \left (d x + c\right ) - 2 i \, a^{3}} - \frac {i \, \log \left (i \, \tan \left (d x + c\right ) + 1\right )}{a^{3}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.41, size = 42, normalized size = 0.84 \[ \frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{a^3\,d}+\frac {2{}\mathrm {i}}{a^3\,d\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \]
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 \[ \frac {i \int \frac {\sec ^{4}{\left (c + d x \right )}}{\tan ^{3}{\left (c + d x \right )} - 3 i \tan ^{2}{\left (c + d x \right )} - 3 \tan {\left (c + d x \right )} + i}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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