Optimal. Leaf size=46 \[ \frac{a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{i a \sec ^4(c+d x)}{4 d} \]
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Rubi [A] time = 0.0342313, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {3486, 3767} \[ \frac{a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{i a \sec ^4(c+d x)}{4 d} \]
Antiderivative was successfully verified.
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Rule 3486
Rule 3767
Rubi steps
\begin{align*} \int \sec ^4(c+d x) (a+i a \tan (c+d x)) \, dx &=\frac{i a \sec ^4(c+d x)}{4 d}+a \int \sec ^4(c+d x) \, dx\\ &=\frac{i a \sec ^4(c+d x)}{4 d}-\frac{a \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d}\\ &=\frac{i a \sec ^4(c+d x)}{4 d}+\frac{a \tan (c+d x)}{d}+\frac{a \tan ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.0422621, size = 43, normalized size = 0.93 \[ \frac{a \left (\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right )}{d}+\frac{i a \sec ^4(c+d x)}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.08, size = 39, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({\frac{{\frac{i}{4}}a}{ \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}-a \left ( -{\frac{2}{3}}-{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \tan \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06162, size = 65, normalized size = 1.41 \begin{align*} \frac{3 i \, a \tan \left (d x + c\right )^{4} + 4 \, a \tan \left (d x + c\right )^{3} + 6 i \, a \tan \left (d x + c\right )^{2} + 12 \, a \tan \left (d x + c\right )}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.10474, size = 239, normalized size = 5.2 \begin{align*} \frac{24 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 16 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} + 4 i \, a}{3 \,{\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.66531, size = 48, normalized size = 1.04 \begin{align*} \begin{cases} \frac{a \left (\frac{\tan ^{3}{\left (c + d x \right )}}{3} + \tan{\left (c + d x \right )}\right ) + \frac{i a \sec ^{4}{\left (c + d x \right )}}{4}}{d} & \text{for}\: d \neq 0 \\x \left (i a \tan{\left (c \right )} + a\right ) \sec ^{4}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12926, size = 65, normalized size = 1.41 \begin{align*} -\frac{-3 i \, a \tan \left (d x + c\right )^{4} - 4 \, a \tan \left (d x + c\right )^{3} - 6 i \, a \tan \left (d x + c\right )^{2} - 12 \, a \tan \left (d x + c\right )}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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