Optimal. Leaf size=342 \[ -\frac{65 i a^2 \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2}}{924 d}-\frac{13 i a^3 \cos ^5(c+d x) \sqrt{a+i a \tan (c+d x)}}{168 d}-\frac{13 i a^3 \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{128 d}+\frac{39 i a^4 \cos ^3(c+d x)}{448 d \sqrt{a+i a \tan (c+d x)}}-\frac{195 i a^3 \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{1024 d}+\frac{65 i a^4 \cos (c+d x)}{512 d \sqrt{a+i a \tan (c+d x)}}+\frac{195 i a^{7/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right )}{1024 \sqrt{2} d}-\frac{i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}-\frac{5 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{66 d} \]
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Rubi [A] time = 0.56461, antiderivative size = 342, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ -\frac{65 i a^2 \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2}}{924 d}-\frac{13 i a^3 \cos ^5(c+d x) \sqrt{a+i a \tan (c+d x)}}{168 d}-\frac{13 i a^3 \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{128 d}+\frac{39 i a^4 \cos ^3(c+d x)}{448 d \sqrt{a+i a \tan (c+d x)}}-\frac{195 i a^3 \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{1024 d}+\frac{65 i a^4 \cos (c+d x)}{512 d \sqrt{a+i a \tan (c+d x)}}+\frac{195 i a^{7/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right )}{1024 \sqrt{2} d}-\frac{i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}-\frac{5 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{66 d} \]
Antiderivative was successfully verified.
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Rule 3497
Rule 3502
Rule 3490
Rule 3489
Rule 206
Rubi steps
\begin{align*} \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx &=-\frac{i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}+\frac{1}{22} (15 a) \int \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx\\ &=-\frac{5 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{66 d}-\frac{i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}+\frac{1}{132} \left (65 a^2\right ) \int \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx\\ &=-\frac{65 i a^2 \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2}}{924 d}-\frac{5 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{66 d}-\frac{i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}+\frac{1}{168} \left (65 a^3\right ) \int \cos ^5(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx\\ &=-\frac{13 i a^3 \cos ^5(c+d x) \sqrt{a+i a \tan (c+d x)}}{168 d}-\frac{65 i a^2 \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2}}{924 d}-\frac{5 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{66 d}-\frac{i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}+\frac{1}{112} \left (39 a^4\right ) \int \frac{\cos ^3(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx\\ &=\frac{39 i a^4 \cos ^3(c+d x)}{448 d \sqrt{a+i a \tan (c+d x)}}-\frac{13 i a^3 \cos ^5(c+d x) \sqrt{a+i a \tan (c+d x)}}{168 d}-\frac{65 i a^2 \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2}}{924 d}-\frac{5 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{66 d}-\frac{i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}+\frac{1}{128} \left (39 a^3\right ) \int \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx\\ &=\frac{39 i a^4 \cos ^3(c+d x)}{448 d \sqrt{a+i a \tan (c+d x)}}-\frac{13 i a^3 \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{128 d}-\frac{13 i a^3 \cos ^5(c+d x) \sqrt{a+i a \tan (c+d x)}}{168 d}-\frac{65 i a^2 \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2}}{924 d}-\frac{5 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{66 d}-\frac{i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}+\frac{1}{256} \left (65 a^4\right ) \int \frac{\cos (c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx\\ &=\frac{65 i a^4 \cos (c+d x)}{512 d \sqrt{a+i a \tan (c+d x)}}+\frac{39 i a^4 \cos ^3(c+d x)}{448 d \sqrt{a+i a \tan (c+d x)}}-\frac{13 i a^3 \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{128 d}-\frac{13 i a^3 \cos ^5(c+d x) \sqrt{a+i a \tan (c+d x)}}{168 d}-\frac{65 i a^2 \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2}}{924 d}-\frac{5 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{66 d}-\frac{i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}+\frac{\left (195 a^3\right ) \int \cos (c+d x) \sqrt{a+i a \tan (c+d x)} \, dx}{1024}\\ &=\frac{65 i a^4 \cos (c+d x)}{512 d \sqrt{a+i a \tan (c+d x)}}+\frac{39 i a^4 \cos ^3(c+d x)}{448 d \sqrt{a+i a \tan (c+d x)}}-\frac{195 i a^3 \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{1024 d}-\frac{13 i a^3 \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{128 d}-\frac{13 i a^3 \cos ^5(c+d x) \sqrt{a+i a \tan (c+d x)}}{168 d}-\frac{65 i a^2 \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2}}{924 d}-\frac{5 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{66 d}-\frac{i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}+\frac{\left (195 a^4\right ) \int \frac{\sec (c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx}{2048}\\ &=\frac{65 i a^4 \cos (c+d x)}{512 d \sqrt{a+i a \tan (c+d x)}}+\frac{39 i a^4 \cos ^3(c+d x)}{448 d \sqrt{a+i a \tan (c+d x)}}-\frac{195 i a^3 \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{1024 d}-\frac{13 i a^3 \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{128 d}-\frac{13 i a^3 \cos ^5(c+d x) \sqrt{a+i a \tan (c+d x)}}{168 d}-\frac{65 i a^2 \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2}}{924 d}-\frac{5 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{66 d}-\frac{i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}+\frac{\left (195 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 )}{1024 d}\\ &=\frac{195 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 )}{1024 \sqrt{2} d}+\frac{65 i a^4 \cos (c+d x)}{512 d \sqrt{a+i a \tan (c+d x)}}+\frac{39 i a^4 \cos ^3(c+d x)}{448 d \sqrt{a+i a \tan (c+d x)}}-\frac{195 i a^3 \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{1024 d}-\frac{13 i a^3 \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{128 d}-\frac{13 i a^3 \cos ^5(c+d x) \sqrt{a+i a \tan (c+d x)}}{168 d}-\frac{65 i a^2 \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2}}{924 d}-\frac{5 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{66 d}-\frac{i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}\\ \end{align*}
Mathematica [A] time = 6.32674, size = 194, normalized size = 0.57 \[ -\frac{i a^3 e^{-5 i (c+d x)} \left (-7161 e^{2 i (c+d x)}+47413 e^{4 i (c+d x)}+78800 e^{6 i (c+d x)}+38512 e^{8 i (c+d x)}+19552 e^{10 i (c+d x)}+7184 e^{12 i (c+d x)}+1624 e^{14 i (c+d x)}+168 e^{16 i (c+d x)}-45045 e^{4 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left (\sqrt{1+e^{2 i (c+d x)}}\right )-462\right ) \sqrt{a+i a \tan (c+d x)}}{473088 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.816, size = 1948, normalized size = 5.7 \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 = 2.26877, size = 1149, normalized size = 3.36 \begin{align*} -\frac{{\left (45045 \, \sqrt{\frac{1}{2}} \sqrt{-\frac{a^{7}}{d^{2}}} d e^{\left (5 i \, d x + 5 i \, c\right )} \log \left (\frac{{\left (390 i \, \sqrt{\frac{1}{2}} \sqrt{-\frac{a^{7}}{d^{2}}} d e^{\left (i \, d x + i \, c\right )} + 195 \, \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 )}}{195 \, a^{3}}\right ) - 45045 \, \sqrt{\frac{1}{2}} \sqrt{-\frac{a^{7}}{d^{2}}} d e^{\left (5 i \, d x + 5 i \, c\right )} \log \left (\frac{{\left (-390 i \, \sqrt{\frac{1}{2}} \sqrt{-\frac{a^{7}}{d^{2}}} d e^{\left (i \, d x + i \, c\right )} + 195 \, \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 )}}{195 \, a^{3}}\right ) - \sqrt{2}{\left (-168 i \, a^{3} e^{\left (16 i \, d x + 16 i \, c\right )} - 1624 i \, a^{3} e^{\left (14 i \, d x + 14 i \, c\right )} - 7184 i \, a^{3} e^{\left (12 i \, d x + 12 i \, c\right )} - 19552 i \, a^{3} e^{\left (10 i \, d x + 10 i \, c\right )} - 38512 i \, a^{3} e^{\left (8 i \, d x + 8 i \, c\right )} - 78800 i \, a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} - 47413 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 7161 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 462 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 (-5 i \, d x - 5 i \, c\right )}}{473088 \, 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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