Optimal. Leaf size=25 \[ -\frac {i a^3}{d (a-i a \tan (c+d x))} \]
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Rubi [A] time = 0.04, antiderivative size = 25, 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} \[ -\frac {i a^3}{d (a-i a \tan (c+d x))} \]
Antiderivative was successfully verified.
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Rule 32
Rule 3487
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+i a \tan (c+d x))^2 \, dx &=-\frac {\left (i a^3\right ) \operatorname {Subst}\left (\int \frac {1}{(a-x)^2} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac {i a^3}{d (a-i a \tan (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 31, normalized size = 1.24 \[ -\frac {i a^2 (\cos (c+d x)+i \sin (c+d x))^2}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 17, normalized size = 0.68 \[ -\frac {i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.96, size = 17, normalized size = 0.68 \[ -\frac {i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.31, size = 73, normalized size = 2.92 \[ \frac {-a^{2} \left (-\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-i a^{2} \left (\cos ^{2}\left (d x +c \right )\right )+a^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 32, normalized size = 1.28 \[ \frac {a^{2} \tan \left (d x + c\right ) - i \, a^{2}}{{\left (\tan \left (d x + c\right )^{2} + 1\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.28, size = 18, normalized size = 0.72 \[ \frac {a^2}{d\,\left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 37, normalized size = 1.48 \[ \begin {cases} - \frac {i a^{2} e^{2 i c} e^{2 i d x}}{2 d} & \text {for}\: 2 d \neq 0 \\a^{2} x e^{2 i c} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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