Optimal. Leaf size=29 \[ \frac {\tan (c+d x)}{d \left (a^2+i a^2 \tan (c+d x)\right )^2} \]
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Rubi [A] time = 0.04, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3487, 34} \[ \frac {\tan (c+d x)}{d \left (a^2+i a^2 \tan (c+d x)\right )^2} \]
Antiderivative was successfully verified.
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Rule 34
Rule 3487
Rubi steps
\begin {align*} \int \frac {\sec ^4(c+d x)}{(a+i a \tan (c+d x))^4} \, dx &=-\frac {i \operatorname {Subst}\left (\int \frac {a-x}{(a+x)^3} \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=\frac {\tan (c+d x)}{d \left (a^2+i a^2 \tan (c+d x)\right )^2}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 32, normalized size = 1.10 \[ \frac {i \sec ^4(c+d x)}{4 d (a+i a \tan (c+d x))^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 17, normalized size = 0.59 \[ \frac {i \, e^{\left (-4 i \, d x - 4 i \, c\right )}}{4 \, a^{4} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.69, size = 44, normalized size = 1.52 \[ -\frac {2 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a^{4} d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.45, size = 36, normalized size = 1.24 \[ \frac {-\frac {1}{\tan \left (d x +c \right )-i}-\frac {i}{\left (\tan \left (d x +c \right )-i\right )^{2}}}{d \,a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.68, size = 67, normalized size = 2.31 \[ -\frac {3 \, {\left (\tan \left (d x + c\right )^{2} - i \, \tan \left (d x + c\right )\right )}}{{\left (3 \, a^{4} \tan \left (d x + c\right )^{3} - 9 i \, a^{4} \tan \left (d x + c\right )^{2} - 9 \, a^{4} \tan \left (d x + c\right ) + 3 i \, a^{4}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.39, size = 25, normalized size = 0.86 \[ -\frac {\mathrm {tan}\left (c+d\,x\right )}{a^4\,d\,{\left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.79, size = 95, normalized size = 3.28 \[ \begin {cases} \frac {i \sec ^{4}{\left (c + d x \right )}}{4 a^{4} d \tan ^{4}{\left (c + d x \right )} - 16 i a^{4} d \tan ^{3}{\left (c + d x \right )} - 24 a^{4} d \tan ^{2}{\left (c + d x \right )} + 16 i a^{4} d \tan {\left (c + d x \right )} + 4 a^{4} d} & \text {for}\: d \neq 0 \\\frac {x \sec ^{4}{\relax (c )}}{\left (i a \tan {\relax (c )} + a\right )^{4}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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