Optimal. Leaf size=80 \[ -\frac{4 i a^{13}}{5 d (a-i a \tan (c+d x))^5}+\frac{i a^{12}}{d (a-i a \tan (c+d x))^4}-\frac{i a^{11}}{3 d (a-i a \tan (c+d x))^3} \]
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Rubi [A] time = 0.0562357, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3487, 43} \[ -\frac{4 i a^{13}}{5 d (a-i a \tan (c+d x))^5}+\frac{i a^{12}}{d (a-i a \tan (c+d x))^4}-\frac{i a^{11}}{3 d (a-i a \tan (c+d x))^3} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 43
Rubi steps
\begin{align*} \int \cos ^{10}(c+d x) (a+i a \tan (c+d x))^8 \, dx &=-\frac{\left (i a^{11}\right ) \operatorname{Subst}\left (\int \frac{(a+x)^2}{(a-x)^6} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac{\left (i a^{11}\right ) \operatorname{Subst}\left (\int \left (\frac{4 a^2}{(a-x)^6}-\frac{4 a}{(a-x)^5}+\frac{1}{(a-x)^4}\right ) \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac{4 i a^{13}}{5 d (a-i a \tan (c+d x))^5}+\frac{i a^{12}}{d (a-i a \tan (c+d x))^4}-\frac{i a^{11}}{3 d (a-i a \tan (c+d x))^3}\\ \end{align*}
Mathematica [A] time = 1.15056, size = 55, normalized size = 0.69 \[ \frac{a^8 (-4 i \sin (2 (c+d x))+16 \cos (2 (c+d x))+15) (\sin (8 (c+d x))-i \cos (8 (c+d x)))}{240 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.092, size = 588, normalized size = 7.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.59788, size = 205, normalized size = 2.56 \begin{align*} -\frac{1280 \, a^{8} \tan \left (d x + c\right )^{7} - 7680 i \, a^{8} \tan \left (d x + c\right )^{6} - 19712 \, a^{8} \tan \left (d x + c\right )^{5} + 28160 i \, a^{8} \tan \left (d x + c\right )^{4} + 24320 \, a^{8} \tan \left (d x + c\right )^{3} - 12800 i \, a^{8} \tan \left (d x + c\right )^{2} - 3840 \, a^{8} \tan \left (d x + c\right ) + 512 i \, a^{8}}{3840 \,{\left (\tan \left (d x + c\right )^{10} + 5 \, \tan \left (d x + c\right )^{8} + 10 \, \tan \left (d x + c\right )^{6} + 10 \, \tan \left (d x + c\right )^{4} + 5 \, \tan \left (d x + c\right )^{2} + 1\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65638, size = 140, normalized size = 1.75 \begin{align*} \frac{-6 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} - 15 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} - 10 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )}}{240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.32546, size = 122, normalized size = 1.52 \begin{align*} \begin{cases} \frac{- 384 i a^{8} d^{2} e^{10 i c} e^{10 i d x} - 960 i a^{8} d^{2} e^{8 i c} e^{8 i d x} - 640 i a^{8} d^{2} e^{6 i c} e^{6 i d x}}{15360 d^{3}} & \text{for}\: 15360 d^{3} \neq 0 \\x \left (\frac{a^{8} e^{10 i c}}{4} + \frac{a^{8} e^{8 i c}}{2} + \frac{a^{8} e^{6 i c}}{4}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.42252, size = 552, normalized size = 6.9 \begin{align*} \frac{-10752 i \, a^{8} e^{\left (38 i \, d x + 24 i \, c\right )} - 177408 i \, a^{8} e^{\left (36 i \, d x + 22 i \, c\right )} - 1372672 i \, a^{8} e^{\left (34 i \, d x + 20 i \, c\right )} - 6610688 i \, a^{8} e^{\left (32 i \, d x + 18 i \, c\right )} - 22177792 i \, a^{8} e^{\left (30 i \, d x + 16 i \, c\right )} - 54955264 i \, a^{8} e^{\left (28 i \, d x + 14 i \, c\right )} - 104039936 i \, a^{8} e^{\left (26 i \, d x + 12 i \, c\right )} - 153497344 i \, a^{8} e^{\left (24 i \, d x + 10 i \, c\right )} - 178354176 i \, a^{8} e^{\left (22 i \, d x + 8 i \, c\right )} - 163747584 i \, a^{8} e^{\left (20 i \, d x + 6 i \, c\right )} - 118390272 i \, a^{8} e^{\left (18 i \, d x + 4 i \, c\right )} - 66696448 i \, a^{8} e^{\left (16 i \, d x + 2 i \, c\right )} - 9119488 i \, a^{8} e^{\left (12 i \, d x - 2 i \, c\right )} - 2017792 i \, a^{8} e^{\left (10 i \, d x - 4 i \, c\right )} - 277760 i \, a^{8} e^{\left (8 i \, d x - 6 i \, c\right )} - 17920 i \, a^{8} e^{\left (6 i \, d x - 8 i \, c\right )} - 28700672 i \, a^{8} e^{\left (14 i \, d x\right )}}{430080 \,{\left (d e^{\left (28 i \, d x + 14 i \, c\right )} + 14 \, d e^{\left (26 i \, d x + 12 i \, c\right )} + 91 \, d e^{\left (24 i \, d x + 10 i \, c\right )} + 364 \, d e^{\left (22 i \, d x + 8 i \, c\right )} + 1001 \, d e^{\left (20 i \, d x + 6 i \, c\right )} + 2002 \, d e^{\left (18 i \, d x + 4 i \, c\right )} + 3003 \, d e^{\left (16 i \, d x + 2 i \, c\right )} + 3003 \, d e^{\left (12 i \, d x - 2 i \, c\right )} + 2002 \, d e^{\left (10 i \, d x - 4 i \, c\right )} + 1001 \, d e^{\left (8 i \, d x - 6 i \, c\right )} + 364 \, d e^{\left (6 i \, d x - 8 i \, c\right )} + 91 \, d e^{\left (4 i \, d x - 10 i \, c\right )} + 14 \, d e^{\left (2 i \, d x - 12 i \, c\right )} + 3432 \, d e^{\left (14 i \, d x\right )} + d e^{\left (-14 i \, c\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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