Optimal. Leaf size=27 \[ \frac {i}{2 a d (a+i a \tan (c+d x))^2} \]
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Rubi [A] time = 0.05, antiderivative size = 27, 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, 32} \[ \frac {i}{2 a d (a+i a \tan (c+d x))^2} \]
Antiderivative was successfully verified.
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Rule 32
Rule 3487
Rubi steps
\begin {align*} \int \frac {\sec ^2(c+d x)}{(a+i a \tan (c+d x))^3} \, dx &=-\frac {i \operatorname {Subst}\left (\int \frac {1}{(a+x)^3} \, dx,x,i a \tan (c+d x)\right )}{a d}\\ &=\frac {i}{2 a d (a+i a \tan (c+d x))^2}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 42, normalized size = 1.56 \[ -\frac {i (\tan (c+d x)-3 i) \sec ^2(c+d x)}{8 a^3 d (\tan (c+d x)-i)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 30, normalized size = 1.11 \[ \frac {{\left (2 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (-4 i \, d x - 4 i \, c\right )}}{8 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.55, size = 57, normalized size = 2.11 \[ -\frac {2 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a^{3} d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 24, normalized size = 0.89 \[ \frac {i}{2 a d \left (a +i a \tan \left (d x +c \right )\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 21, normalized size = 0.78 \[ \frac {i}{2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.35, size = 20, normalized size = 0.74 \[ -\frac {1{}\mathrm {i}}{2\,a^3\,d\,{\left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.12, size = 153, normalized size = 5.67 \[ \begin {cases} - \frac {i \tan {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{8 a^{3} d \tan ^{3}{\left (c + d x \right )} - 24 i a^{3} d \tan ^{2}{\left (c + d x \right )} - 24 a^{3} d \tan {\left (c + d x \right )} + 8 i a^{3} d} - \frac {3 \sec ^{2}{\left (c + d x \right )}}{8 a^{3} d \tan ^{3}{\left (c + d x \right )} - 24 i a^{3} d \tan ^{2}{\left (c + d x \right )} - 24 a^{3} d \tan {\left (c + d x \right )} + 8 i a^{3} d} & \text {for}\: d \neq 0 \\\frac {x \sec ^{2}{\relax (c )}}{\left (i a \tan {\relax (c )} + a\right )^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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