Optimal. Leaf size=268 \[ -\frac{11 i a^2 \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{140 d}-\frac{11 i a^3 \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{120 d}-\frac{11 i a^3 \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{64 d}+\frac{11 i a^4 \cos (c+d x)}{96 d \sqrt{a+i a \tan (c+d x)}}+\frac{11 i a^{7/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right )}{64 \sqrt{2} d}-\frac{i \cos ^9(c+d x) (a+i a \tan (c+d x))^{7/2}}{9 d}-\frac{11 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{126 d} \]
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Rubi [A] time = 0.411437, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {3497, 3502, 3490, 3489, 206} \[ -\frac{11 i a^2 \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{140 d}-\frac{11 i a^3 \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{120 d}-\frac{11 i a^3 \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{64 d}+\frac{11 i a^4 \cos (c+d x)}{96 d \sqrt{a+i a \tan (c+d x)}}+\frac{11 i a^{7/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right )}{64 \sqrt{2} d}-\frac{i \cos ^9(c+d x) (a+i a \tan (c+d x))^{7/2}}{9 d}-\frac{11 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{126 d} \]
Antiderivative was successfully verified.
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Rule 3497
Rule 3502
Rule 3490
Rule 3489
Rule 206
Rubi steps
\begin{align*} \int \cos ^9(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx &=-\frac{i \cos ^9(c+d x) (a+i a \tan (c+d x))^{7/2}}{9 d}+\frac{1}{18} (11 a) \int \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx\\ &=-\frac{11 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{126 d}-\frac{i \cos ^9(c+d x) (a+i a \tan (c+d x))^{7/2}}{9 d}+\frac{1}{28} \left (11 a^2\right ) \int \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx\\ &=-\frac{11 i a^2 \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{140 d}-\frac{11 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{126 d}-\frac{i \cos ^9(c+d x) (a+i a \tan (c+d x))^{7/2}}{9 d}+\frac{1}{40} \left (11 a^3\right ) \int \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx\\ &=-\frac{11 i a^3 \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{120 d}-\frac{11 i a^2 \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{140 d}-\frac{11 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{126 d}-\frac{i \cos ^9(c+d x) (a+i a \tan (c+d x))^{7/2}}{9 d}+\frac{1}{48} \left (11 a^4\right ) \int \frac{\cos (c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx\\ &=\frac{11 i a^4 \cos (c+d x)}{96 d \sqrt{a+i a \tan (c+d x)}}-\frac{11 i a^3 \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{120 d}-\frac{11 i a^2 \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{140 d}-\frac{11 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{126 d}-\frac{i \cos ^9(c+d x) (a+i a \tan (c+d x))^{7/2}}{9 d}+\frac{1}{64} \left (11 a^3\right ) \int \cos (c+d x) \sqrt{a+i a \tan (c+d x)} \, dx\\ &=\frac{11 i a^4 \cos (c+d x)}{96 d \sqrt{a+i a \tan (c+d x)}}-\frac{11 i a^3 \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{64 d}-\frac{11 i a^3 \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{120 d}-\frac{11 i a^2 \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{140 d}-\frac{11 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{126 d}-\frac{i \cos ^9(c+d x) (a+i a \tan (c+d x))^{7/2}}{9 d}+\frac{1}{128} \left (11 a^4\right ) \int \frac{\sec (c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx\\ &=\frac{11 i a^4 \cos (c+d x)}{96 d \sqrt{a+i a \tan (c+d x)}}-\frac{11 i a^3 \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{64 d}-\frac{11 i a^3 \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{120 d}-\frac{11 i a^2 \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{140 d}-\frac{11 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{126 d}-\frac{i \cos ^9(c+d x) (a+i a \tan (c+d x))^{7/2}}{9 d}+\frac{\left (11 i a^4\right ) \operatorname{Subst}\left (\int \frac{1}{2-a x^2} \, dx,x,\frac{\sec (c+d x)}{\sqrt{a+i a \tan (c+d x)}}\right )}{64 d}\\ &=\frac{11 i a^{7/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right )}{64 \sqrt{2} d}+\frac{11 i a^4 \cos (c+d x)}{96 d \sqrt{a+i a \tan (c+d x)}}-\frac{11 i a^3 \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{64 d}-\frac{11 i a^3 \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{120 d}-\frac{11 i a^2 \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{140 d}-\frac{11 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{126 d}-\frac{i \cos ^9(c+d x) (a+i a \tan (c+d x))^{7/2}}{9 d}\\ \end{align*}
Mathematica [A] time = 3.20505, size = 188, normalized size = 0.7 \[ -\frac{i a^3 e^{-3 i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \left (4303 e^{2 i (c+d x)}+7034 e^{4 i (c+d x)}+3754 e^{6 i (c+d x)}+1798 e^{8 i (c+d x)}+530 e^{10 i (c+d x)}+70 e^{12 i (c+d x)}-3465 e^{2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left (\sqrt{1+e^{2 i (c+d x)}}\right )-315\right )}{20160 \sqrt{2} d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.615, size = 1604, normalized size = 6. \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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.25117, size = 1034, normalized size = 3.86 \begin{align*} -\frac{{\left (3465 \, \sqrt{\frac{1}{2}} \sqrt{-\frac{a^{7}}{d^{2}}} d e^{\left (3 i \, d x + 3 i \, c\right )} \log \left (\frac{{\left (22 i \, \sqrt{\frac{1}{2}} \sqrt{-\frac{a^{7}}{d^{2}}} d e^{\left (i \, d x + i \, c\right )} + 11 \, \sqrt{2}{\left (a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{3}\right )} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{11 \, a^{3}}\right ) - 3465 \, \sqrt{\frac{1}{2}} \sqrt{-\frac{a^{7}}{d^{2}}} d e^{\left (3 i \, d x + 3 i \, c\right )} \log \left (\frac{{\left (-22 i \, \sqrt{\frac{1}{2}} \sqrt{-\frac{a^{7}}{d^{2}}} d e^{\left (i \, d x + i \, c\right )} + 11 \, \sqrt{2}{\left (a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{3}\right )} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{11 \, a^{3}}\right ) - \sqrt{2}{\left (-70 i \, a^{3} e^{\left (12 i \, d x + 12 i \, c\right )} - 530 i \, a^{3} e^{\left (10 i \, d x + 10 i \, c\right )} - 1798 i \, a^{3} e^{\left (8 i \, d x + 8 i \, c\right )} - 3754 i \, a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} - 7034 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} - 4303 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 315 i \, a^{3}\right )} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-3 i \, d x - 3 i \, c\right )}}{40320 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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