Optimal. Leaf size=27 \[ -\frac {i (a+i a \tan (c+d x))^3}{3 a d} \]
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Rubi [A] time = 0.04, 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} \[ -\frac {i (a+i a \tan (c+d x))^3}{3 a d} \]
Antiderivative was successfully verified.
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Rule 32
Rule 3487
Rubi steps
\begin {align*} \int \sec ^2(c+d x) (a+i a \tan (c+d x))^2 \, dx &=-\frac {i \operatorname {Subst}\left (\int (a+x)^2 \, dx,x,i a \tan (c+d x)\right )}{a d}\\ &=-\frac {i (a+i a \tan (c+d x))^3}{3 a d}\\ \end {align*}
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Mathematica [B] time = 0.41, size = 68, normalized size = 2.52 \[ \frac {a^2 \sec (c) \sec ^3(c+d x) (-3 \sin (2 c+d x)+2 \sin (2 c+3 d x)+3 i \cos (2 c+d x)+3 \sin (d x)+3 i \cos (d x))}{6 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 75, normalized size = 2.78 \[ \frac {24 i \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} + 24 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + 8 i \, a^{2}}{3 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.10, size = 42, normalized size = 1.56 \[ -\frac {a^{2} \tan \left (d x + c\right )^{3} - 3 i \, a^{2} \tan \left (d x + c\right )^{2} - 3 \, a^{2} \tan \left (d x + c\right )}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.43, size = 51, normalized size = 1.89 \[ \frac {-\frac {a^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )^{3}}+\frac {i a^{2}}{\cos \left (d x +c \right )^{2}}+a^{2} \tan \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 21, normalized size = 0.78 \[ -\frac {i \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3}}{3 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.22, size = 35, normalized size = 1.30 \[ \frac {a^2\,\mathrm {tan}\left (c+d\,x\right )\,\left (-{\mathrm {tan}\left (c+d\,x\right )}^2+\mathrm {tan}\left (c+d\,x\right )\,3{}\mathrm {i}+3\right )}{3\,d} \]
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 \[ - a^{2} \left (\int \tan ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \left (- 2 i \tan {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\right )\, dx + \int \left (- \sec ^{2}{\left (c + d x \right )}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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