Optimal. Leaf size=27 \[ -\frac{i (a+i a \tan (c+d x))^6}{6 a d} \]
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Rubi [A] time = 0.0361545, antiderivative size = 27, normalized size of antiderivative = 1., 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} \[ -\frac{i (a+i a \tan (c+d x))^6}{6 a d} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 32
Rubi steps
\begin{align*} \int \sec ^2(c+d x) (a+i a \tan (c+d x))^5 \, dx &=-\frac{i \operatorname{Subst}\left (\int (a+x)^5 \, dx,x,i a \tan (c+d x)\right )}{a d}\\ &=-\frac{i (a+i a \tan (c+d x))^6}{6 a d}\\ \end{align*}
Mathematica [B] time = 1.45354, size = 134, normalized size = 4.96 \[ \frac{a^5 \sec (c) \sec ^6(c+d x) (15 \sin (c+2 d x)-15 \sin (3 c+2 d x)+6 \sin (3 c+4 d x)-6 \sin (5 c+4 d x)+2 \sin (5 c+6 d x)+15 i \cos (c+2 d x)+15 i \cos (3 c+2 d x)+6 i \cos (3 c+4 d x)+6 i \cos (5 c+4 d x)-20 \sin (c)+20 i \cos (c))}{12 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.076, size = 115, normalized size = 4.3 \begin{align*}{\frac{1}{d} \left ({\frac{{\frac{i}{6}}{a}^{5} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}}+{\frac{{a}^{5} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}-{\frac{{\frac{5\,i}{2}}{a}^{5} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}-{\frac{10\,{a}^{5} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{{\frac{5\,i}{2}}{a}^{5}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{a}^{5}\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 = 1.1303, size = 28, normalized size = 1.04 \begin{align*} -\frac{i \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{6}}{6 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.10871, size = 463, normalized size = 17.15 \begin{align*} \frac{192 i \, a^{5} e^{\left (10 i \, d x + 10 i \, c\right )} + 480 i \, a^{5} e^{\left (8 i \, d x + 8 i \, c\right )} + 640 i \, a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} + 480 i \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} + 192 i \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} + 32 i \, a^{5}}{3 \,{\left (d e^{\left (12 i \, d x + 12 i \, c\right )} + 6 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 20 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \, 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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} a^{5} \left (\int - 10 \tan ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 5 \tan ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 5 i \tan{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int - 10 i \tan ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int i \tan ^{5}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \sec ^{2}{\left (c + d x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.58311, size = 111, normalized size = 4.11 \begin{align*} -\frac{-i \, a^{5} \tan \left (d x + c\right )^{6} - 6 \, a^{5} \tan \left (d x + c\right )^{5} + 15 i \, a^{5} \tan \left (d x + c\right )^{4} + 20 \, a^{5} \tan \left (d x + c\right )^{3} - 15 i \, a^{5} \tan \left (d x + c\right )^{2} - 6 \, a^{5} \tan \left (d x + c\right )}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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