Optimal. Leaf size=94 \[ \frac{a \tan ^9(c+d x)}{9 d}+\frac{4 a \tan ^7(c+d x)}{7 d}+\frac{6 a \tan ^5(c+d x)}{5 d}+\frac{4 a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{i a \sec ^{10}(c+d x)}{10 d} \]
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Rubi [A] time = 0.188289, antiderivative size = 94, 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} \[ \frac{a \tan ^9(c+d x)}{9 d}+\frac{4 a \tan ^7(c+d x)}{7 d}+\frac{6 a \tan ^5(c+d x)}{5 d}+\frac{4 a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{i a \sec ^{10}(c+d x)}{10 d} \]
Antiderivative was successfully verified.
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Rule 3486
Rule 3767
Rubi steps
\begin{align*} \int \sec ^{10}(c+d x) (a+i a \tan (c+d x)) \, dx &=\frac{i a \sec ^{10}(c+d x)}{10 d}+a \int \sec ^{10}(c+d x) \, dx\\ &=\frac{i a \sec ^{10}(c+d x)}{10 d}-\frac{a \operatorname{Subst}\left (\int \left (1+4 x^2+6 x^4+4 x^6+x^8\right ) \, dx,x,-\tan (c+d x)\right )}{d}\\ &=\frac{i a \sec ^{10}(c+d x)}{10 d}+\frac{a \tan (c+d x)}{d}+\frac{4 a \tan ^3(c+d x)}{3 d}+\frac{6 a \tan ^5(c+d x)}{5 d}+\frac{4 a \tan ^7(c+d x)}{7 d}+\frac{a \tan ^9(c+d x)}{9 d}\\ \end{align*}
Mathematica [A] time = 0.361365, size = 79, normalized size = 0.84 \[ \frac{a \left (\frac{1}{9} \tan ^9(c+d x)+\frac{4}{7} \tan ^7(c+d x)+\frac{6}{5} \tan ^5(c+d x)+\frac{4}{3} \tan ^3(c+d x)+\tan (c+d x)\right )}{d}+\frac{i a \sec ^{10}(c+d x)}{10 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.093, size = 69, normalized size = 0.7 \begin{align*}{\frac{1}{d} \left ({\frac{{\frac{i}{10}}a}{ \left ( \cos \left ( dx+c \right ) \right ) ^{10}}}-a \left ( -{\frac{128}{315}}-{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{8}}{9}}-{\frac{8\, \left ( \sec \left ( dx+c \right ) \right ) ^{6}}{63}}-{\frac{16\, \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{105}}-{\frac{64\, \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{315}} \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 = 1.10113, size = 154, normalized size = 1.64 \begin{align*} \frac{63 i \, a \tan \left (d x + c\right )^{10} + 70 \, a \tan \left (d x + c\right )^{9} + 315 i \, a \tan \left (d x + c\right )^{8} + 360 \, a \tan \left (d x + c\right )^{7} + 630 i \, a \tan \left (d x + c\right )^{6} + 756 \, a \tan \left (d x + c\right )^{5} + 630 i \, a \tan \left (d x + c\right )^{4} + 840 \, a \tan \left (d x + c\right )^{3} + 315 i \, a \tan \left (d x + c\right )^{2} + 630 \, a \tan \left (d x + c\right )}{630 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.301, size = 629, normalized size = 6.69 \begin{align*} \frac{64512 i \, a e^{\left (10 i \, d x + 10 i \, c\right )} + 53760 i \, a e^{\left (8 i \, d x + 8 i \, c\right )} + 30720 i \, a e^{\left (6 i \, d x + 6 i \, c\right )} + 11520 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 2560 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} + 256 i \, a}{315 \,{\left (d e^{\left (20 i \, d x + 20 i \, c\right )} + 10 \, d e^{\left (18 i \, d x + 18 i \, c\right )} + 45 \, d e^{\left (16 i \, d x + 16 i \, c\right )} + 120 \, d e^{\left (14 i \, d x + 14 i \, c\right )} + 210 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 252 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 210 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 120 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 45 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 10 \, 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 = 123.932, size = 83, normalized size = 0.88 \begin{align*} \begin{cases} \frac{a \left (\frac{\tan ^{9}{\left (c + d x \right )}}{9} + \frac{4 \tan ^{7}{\left (c + d x \right )}}{7} + \frac{6 \tan ^{5}{\left (c + d x \right )}}{5} + \frac{4 \tan ^{3}{\left (c + d x \right )}}{3} + \tan{\left (c + d x \right )}\right ) + \frac{i a \sec ^{10}{\left (c + d x \right )}}{10}}{d} & \text{for}\: d \neq 0 \\x \left (i a \tan{\left (c \right )} + a\right ) \sec ^{10}{\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 = 1.17452, size = 154, normalized size = 1.64 \begin{align*} -\frac{-63 i \, a \tan \left (d x + c\right )^{10} - 70 \, a \tan \left (d x + c\right )^{9} - 315 i \, a \tan \left (d x + c\right )^{8} - 360 \, a \tan \left (d x + c\right )^{7} - 630 i \, a \tan \left (d x + c\right )^{6} - 756 \, a \tan \left (d x + c\right )^{5} - 630 i \, a \tan \left (d x + c\right )^{4} - 840 \, a \tan \left (d x + c\right )^{3} - 315 i \, a \tan \left (d x + c\right )^{2} - 630 \, a \tan \left (d x + c\right )}{630 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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