\(\int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx\) [332]

Optimal result

Integrand size = 26, antiderivative size = 342 \[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\frac {195 i a^{7/2} \text {arctanh}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{1024 \sqrt {2} d}+\frac {65 i a^4 \cos (c+d x)}{512 d \sqrt {a+i a \tan (c+d x)}}+\frac {39 i a^4 \cos ^3(c+d x)}{448 d \sqrt {a+i a \tan (c+d x)}}-\frac {195 i a^3 \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{1024 d}-\frac {13 i a^3 \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{128 d}-\frac {13 i a^3 \cos ^5(c+d x) \sqrt {a+i a \tan (c+d x)}}{168 d}-\frac {65 i a^2 \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2}}{924 d}-\frac {5 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{66 d}-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d} \] Output:

195/2048*I*a^(7/2)*arctanh(1/2*a^(1/2)*sec(d*x+c)*2^(1/2)/(a+I*a*tan(d*x+c 
))^(1/2))*2^(1/2)/d+65/512*I*a^4*cos(d*x+c)/d/(a+I*a*tan(d*x+c))^(1/2)+39/ 
448*I*a^4*cos(d*x+c)^3/d/(a+I*a*tan(d*x+c))^(1/2)-195/1024*I*a^3*cos(d*x+c 
)*(a+I*a*tan(d*x+c))^(1/2)/d-13/128*I*a^3*cos(d*x+c)^3*(a+I*a*tan(d*x+c))^ 
(1/2)/d-13/168*I*a^3*cos(d*x+c)^5*(a+I*a*tan(d*x+c))^(1/2)/d-65/924*I*a^2* 
cos(d*x+c)^7*(a+I*a*tan(d*x+c))^(3/2)/d-5/66*I*a*cos(d*x+c)^9*(a+I*a*tan(d 
*x+c))^(5/2)/d-1/11*I*cos(d*x+c)^11*(a+I*a*tan(d*x+c))^(7/2)/d
 

Mathematica [A] (verified)

Time = 5.75 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.57 \[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=-\frac {i a^3 e^{-5 i (c+d x)} \left (-462-7161 e^{2 i (c+d x)}+47413 e^{4 i (c+d x)}+78800 e^{6 i (c+d x)}+38512 e^{8 i (c+d x)}+19552 e^{10 i (c+d x)}+7184 e^{12 i (c+d x)}+1624 e^{14 i (c+d x)}+168 e^{16 i (c+d x)}-45045 e^{4 i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \text {arctanh}\left (\sqrt {1+e^{2 i (c+d x)}}\right )\right ) \sqrt {a+i a \tan (c+d x)}}{473088 d} \] Input:

Integrate[Cos[c + d*x]^11*(a + I*a*Tan[c + d*x])^(7/2),x]
 

Output:

((-1/473088*I)*a^3*(-462 - 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))]*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])*Sqrt[a + I*a*Tan[c + d*x 
]])/(d*E^((5*I)*(c + d*x)))
 

Rubi [A] (verified)

Time = 1.75 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.07, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.731, Rules used = {3042, 3978, 3042, 3978, 3042, 3978, 3042, 3978, 3042, 3983, 3042, 3978, 3042, 3983, 3042, 3971, 3042, 3970, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a+i a \tan (c+d x))^{7/2}}{\sec (c+d x)^{11}}dx\)

\(\Big \downarrow \) 3978

\(\displaystyle \frac {15}{22} a \int \cos ^9(c+d x) (i \tan (c+d x) a+a)^{5/2}dx-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {15}{22} a \int \frac {(i \tan (c+d x) a+a)^{5/2}}{\sec (c+d x)^9}dx-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}\)

\(\Big \downarrow \) 3978

\(\displaystyle \frac {15}{22} a \left (\frac {13}{18} a \int \cos ^7(c+d x) (i \tan (c+d x) a+a)^{3/2}dx-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{9 d}\right )-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {15}{22} a \left (\frac {13}{18} a \int \frac {(i \tan (c+d x) a+a)^{3/2}}{\sec (c+d x)^7}dx-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{9 d}\right )-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}\)

\(\Big \downarrow \) 3978

\(\displaystyle \frac {15}{22} a \left (\frac {13}{18} a \left (\frac {11}{14} a \int \cos ^5(c+d x) \sqrt {i \tan (c+d x) a+a}dx-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2}}{7 d}\right )-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{9 d}\right )-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {15}{22} a \left (\frac {13}{18} a \left (\frac {11}{14} a \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sec (c+d x)^5}dx-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2}}{7 d}\right )-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{9 d}\right )-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}\)

\(\Big \downarrow \) 3978

\(\displaystyle \frac {15}{22} a \left (\frac {13}{18} a \left (\frac {11}{14} a \left (\frac {9}{10} a \int \frac {\cos ^3(c+d x)}{\sqrt {i \tan (c+d x) a+a}}dx-\frac {i \cos ^5(c+d x) \sqrt {a+i a \tan (c+d x)}}{5 d}\right )-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2}}{7 d}\right )-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{9 d}\right )-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {15}{22} a \left (\frac {13}{18} a \left (\frac {11}{14} a \left (\frac {9}{10} a \int \frac {1}{\sec (c+d x)^3 \sqrt {i \tan (c+d x) a+a}}dx-\frac {i \cos ^5(c+d x) \sqrt {a+i a \tan (c+d x)}}{5 d}\right )-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2}}{7 d}\right )-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{9 d}\right )-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}\)

\(\Big \downarrow \) 3983

\(\displaystyle \frac {15}{22} a \left (\frac {13}{18} a \left (\frac {11}{14} a \left (\frac {9}{10} a \left (\frac {7 \int \cos ^3(c+d x) \sqrt {i \tan (c+d x) a+a}dx}{8 a}+\frac {i \cos ^3(c+d x)}{4 d \sqrt {a+i a \tan (c+d x)}}\right )-\frac {i \cos ^5(c+d x) \sqrt {a+i a \tan (c+d x)}}{5 d}\right )-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2}}{7 d}\right )-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{9 d}\right )-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {15}{22} a \left (\frac {13}{18} a \left (\frac {11}{14} a \left (\frac {9}{10} a \left (\frac {7 \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sec (c+d x)^3}dx}{8 a}+\frac {i \cos ^3(c+d x)}{4 d \sqrt {a+i a \tan (c+d x)}}\right )-\frac {i \cos ^5(c+d x) \sqrt {a+i a \tan (c+d x)}}{5 d}\right )-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2}}{7 d}\right )-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{9 d}\right )-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}\)

\(\Big \downarrow \) 3978

\(\displaystyle \frac {15}{22} a \left (\frac {13}{18} a \left (\frac {11}{14} a \left (\frac {9}{10} a \left (\frac {7 \left (\frac {5}{6} a \int \frac {\cos (c+d x)}{\sqrt {i \tan (c+d x) a+a}}dx-\frac {i \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}\right )}{8 a}+\frac {i \cos ^3(c+d x)}{4 d \sqrt {a+i a \tan (c+d x)}}\right )-\frac {i \cos ^5(c+d x) \sqrt {a+i a \tan (c+d x)}}{5 d}\right )-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2}}{7 d}\right )-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{9 d}\right )-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {15}{22} a \left (\frac {13}{18} a \left (\frac {11}{14} a \left (\frac {9}{10} a \left (\frac {7 \left (\frac {5}{6} a \int \frac {1}{\sec (c+d x) \sqrt {i \tan (c+d x) a+a}}dx-\frac {i \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}\right )}{8 a}+\frac {i \cos ^3(c+d x)}{4 d \sqrt {a+i a \tan (c+d x)}}\right )-\frac {i \cos ^5(c+d x) \sqrt {a+i a \tan (c+d x)}}{5 d}\right )-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2}}{7 d}\right )-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{9 d}\right )-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}\)

\(\Big \downarrow \) 3983

\(\displaystyle \frac {15}{22} a \left (\frac {13}{18} a \left (\frac {11}{14} a \left (\frac {9}{10} a \left (\frac {7 \left (\frac {5}{6} a \left (\frac {3 \int \cos (c+d x) \sqrt {i \tan (c+d x) a+a}dx}{4 a}+\frac {i \cos (c+d x)}{2 d \sqrt {a+i a \tan (c+d x)}}\right )-\frac {i \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}\right )}{8 a}+\frac {i \cos ^3(c+d x)}{4 d \sqrt {a+i a \tan (c+d x)}}\right )-\frac {i \cos ^5(c+d x) \sqrt {a+i a \tan (c+d x)}}{5 d}\right )-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2}}{7 d}\right )-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{9 d}\right )-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {15}{22} a \left (\frac {13}{18} a \left (\frac {11}{14} a \left (\frac {9}{10} a \left (\frac {7 \left (\frac {5}{6} a \left (\frac {3 \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sec (c+d x)}dx}{4 a}+\frac {i \cos (c+d x)}{2 d \sqrt {a+i a \tan (c+d x)}}\right )-\frac {i \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}\right )}{8 a}+\frac {i \cos ^3(c+d x)}{4 d \sqrt {a+i a \tan (c+d x)}}\right )-\frac {i \cos ^5(c+d x) \sqrt {a+i a \tan (c+d x)}}{5 d}\right )-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2}}{7 d}\right )-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{9 d}\right )-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}\)

\(\Big \downarrow \) 3971

\(\displaystyle \frac {15}{22} a \left (\frac {13}{18} a \left (\frac {11}{14} a \left (\frac {9}{10} a \left (\frac {7 \left (\frac {5}{6} a \left (\frac {3 \left (\frac {1}{2} a \int \frac {\sec (c+d x)}{\sqrt {i \tan (c+d x) a+a}}dx-\frac {i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}\right )}{4 a}+\frac {i \cos (c+d x)}{2 d \sqrt {a+i a \tan (c+d x)}}\right )-\frac {i \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}\right )}{8 a}+\frac {i \cos ^3(c+d x)}{4 d \sqrt {a+i a \tan (c+d x)}}\right )-\frac {i \cos ^5(c+d x) \sqrt {a+i a \tan (c+d x)}}{5 d}\right )-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2}}{7 d}\right )-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{9 d}\right )-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {15}{22} a \left (\frac {13}{18} a \left (\frac {11}{14} a \left (\frac {9}{10} a \left (\frac {7 \left (\frac {5}{6} a \left (\frac {3 \left (\frac {1}{2} a \int \frac {\sec (c+d x)}{\sqrt {i \tan (c+d x) a+a}}dx-\frac {i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}\right )}{4 a}+\frac {i \cos (c+d x)}{2 d \sqrt {a+i a \tan (c+d x)}}\right )-\frac {i \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}\right )}{8 a}+\frac {i \cos ^3(c+d x)}{4 d \sqrt {a+i a \tan (c+d x)}}\right )-\frac {i \cos ^5(c+d x) \sqrt {a+i a \tan (c+d x)}}{5 d}\right )-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2}}{7 d}\right )-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{9 d}\right )-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}\)

\(\Big \downarrow \) 3970

\(\displaystyle \frac {15}{22} a \left (\frac {13}{18} a \left (\frac {11}{14} a \left (\frac {9}{10} a \left (\frac {7 \left (\frac {5}{6} a \left (\frac {3 \left (\frac {i a \int \frac {1}{2-\frac {a \sec ^2(c+d x)}{i \tan (c+d x) a+a}}d\frac {\sec (c+d x)}{\sqrt {i \tan (c+d x) a+a}}}{d}-\frac {i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}\right )}{4 a}+\frac {i \cos (c+d x)}{2 d \sqrt {a+i a \tan (c+d x)}}\right )-\frac {i \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}\right )}{8 a}+\frac {i \cos ^3(c+d x)}{4 d \sqrt {a+i a \tan (c+d x)}}\right )-\frac {i \cos ^5(c+d x) \sqrt {a+i a \tan (c+d x)}}{5 d}\right )-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2}}{7 d}\right )-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{9 d}\right )-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {15}{22} a \left (\frac {13}{18} a \left (\frac {11}{14} a \left (\frac {9}{10} a \left (\frac {7 \left (\frac {5}{6} a \left (\frac {3 \left (\frac {i \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{\sqrt {2} d}-\frac {i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}\right )}{4 a}+\frac {i \cos (c+d x)}{2 d \sqrt {a+i a \tan (c+d x)}}\right )-\frac {i \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}\right )}{8 a}+\frac {i \cos ^3(c+d x)}{4 d \sqrt {a+i a \tan (c+d x)}}\right )-\frac {i \cos ^5(c+d x) \sqrt {a+i a \tan (c+d x)}}{5 d}\right )-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2}}{7 d}\right )-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{9 d}\right )-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}\)

Input:

Int[Cos[c + d*x]^11*(a + I*a*Tan[c + d*x])^(7/2),x]
 

Output:

((-1/11*I)*Cos[c + d*x]^11*(a + I*a*Tan[c + d*x])^(7/2))/d + (15*a*(((-1/9 
*I)*Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^(5/2))/d + (13*a*(((-1/7*I)*Cos[ 
c + d*x]^7*(a + I*a*Tan[c + d*x])^(3/2))/d + (11*a*(((-1/5*I)*Cos[c + d*x] 
^5*Sqrt[a + I*a*Tan[c + d*x]])/d + (9*a*(((I/4)*Cos[c + d*x]^3)/(d*Sqrt[a 
+ I*a*Tan[c + d*x]]) + (7*(((-1/3*I)*Cos[c + d*x]^3*Sqrt[a + I*a*Tan[c + d 
*x]])/d + (5*a*(((I/2)*Cos[c + d*x])/(d*Sqrt[a + I*a*Tan[c + d*x]]) + (3*( 
(I*Sqrt[a]*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d* 
x]])])/(Sqrt[2]*d) - (I*Cos[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/d))/(4*a) 
))/6))/(8*a)))/10))/14))/18))/22
 

Defintions of rubi rules used

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3970

Int[sec[(e_.) + (f_.)*(x_)]/Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]], x_S 
ymbol] :> Simp[-2*(a/(b*f))   Subst[Int[1/(2 - a*x^2), x], x, Sec[e + f*x]/ 
Sqrt[a + b*Tan[e + f*x]]], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 + b^2, 0 
]
 

rule 3971

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*( 
x_)])^(n_), x_Symbol] :> Simp[b*(d*Sec[e + f*x])^m*((a + b*Tan[e + f*x])^n/ 
(a*f*m)), x] + Simp[a/(2*d^2)   Int[(d*Sec[e + f*x])^(m + 2)*(a + b*Tan[e + 
 f*x])^(n - 1), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0] && 
 EqQ[m/2 + n, 0] && GtQ[n, 0]
 

rule 3978

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x 
_)])^(n_), x_Symbol] :> Simp[b*(d*Sec[e + f*x])^m*((a + b*Tan[e + f*x])^n/( 
a*f*m)), x] + Simp[a*((m + n)/(m*d^2))   Int[(d*Sec[e + f*x])^(m + 2)*(a + 
b*Tan[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b 
^2, 0] && GtQ[n, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*n]
 

rule 3983

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*( 
x_)])^(n_), x_Symbol] :> Simp[a*(d*Sec[e + f*x])^m*((a + b*Tan[e + f*x])^n/ 
(b*f*(m + 2*n))), x] + Simp[Simplify[m + n]/(a*(m + 2*n))   Int[(d*Sec[e + 
f*x])^m*(a + b*Tan[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, m}, x 
] && EqQ[a^2 + b^2, 0] && LtQ[n, 0] && NeQ[m + 2*n, 0] && IntegersQ[2*m, 2* 
n]
 

Maple [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2084 vs. \(2 (281 ) = 562\).

Time = 7.72 (sec) , antiderivative size = 2085, normalized size of antiderivative = 6.10

\[\text {Expression too large to display}\]

Input:

int(cos(d*x+c)^11*(a+I*a*tan(d*x+c))^(7/2),x)
 

Output:

-1/473088/d*cos(d*x+c)^3*((360360*cos(d*x+c)^3+180180*cos(d*x+c)^2-180180* 
cos(d*x+c)-45045)*sin(d*x+c)*arctanh(1/(cot(d*x+c)^2-2*cot(d*x+c)*csc(d*x+ 
c)+csc(d*x+c)^2-1)^(1/2)*(csc(d*x+c)-cot(d*x+c))*2^(1/2))*(-cos(d*x+c)/(co 
s(d*x+c)+1))^(1/2)*tan(d*x+c)^3+I*(360360*cos(d*x+c)^4+180180*cos(d*x+c)^3 
-360360*cos(d*x+c)^2-135135*cos(d*x+c)+45045)*arctanh(1/(cot(d*x+c)^2-2*co 
t(d*x+c)*csc(d*x+c)+csc(d*x+c)^2-1)^(1/2)*(csc(d*x+c)-cot(d*x+c))*2^(1/2)) 
*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*tan(d*x+c)^3+I*sin(d*x+c)^2*tan(d*x+c) 
^2*(80640*cos(d*x+c)^6+160160*cos(d*x+c)^4-552552*cos(d*x+c)^2+90090)+(108 
1080*cos(d*x+c)^4+540540*cos(d*x+c)^3-1081080*cos(d*x+c)^2-405405*cos(d*x+ 
c)+135135)*arctanh(1/(cot(d*x+c)^2-2*cot(d*x+c)*csc(d*x+c)+csc(d*x+c)^2-1) 
^(1/2)*(csc(d*x+c)-cot(d*x+c))*2^(1/2))*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2) 
*tan(d*x+c)^2+(-1081080*cos(d*x+c)^3-540540*cos(d*x+c)^2+540540*cos(d*x+c) 
+135135)*sin(d*x+c)*arctanh(1/(cot(d*x+c)^2-2*cot(d*x+c)*csc(d*x+c)+csc(d* 
x+c)^2-1)^(1/2)*(csc(d*x+c)-cot(d*x+c))*2^(1/2))*(-cos(d*x+c)/(cos(d*x+c)+ 
1))^(1/2)*tan(d*x+c)+I*sin(d*x+c)^2*(-241920*cos(d*x+c)^6-480480*cos(d*x+c 
)^4+1657656*cos(d*x+c)^2-270270)+I*(-1081080*cos(d*x+c)^3-540540*cos(d*x+c 
)^2+540540*cos(d*x+c)+135135)*sin(d*x+c)*arctanh(1/(cot(d*x+c)^2-2*cot(d*x 
+c)*csc(d*x+c)+csc(d*x+c)^2-1)^(1/2)*(csc(d*x+c)-cot(d*x+c))*2^(1/2))*(-co 
s(d*x+c)/(cos(d*x+c)+1))^(1/2)*tan(d*x+c)^2+(-360360*cos(d*x+c)^4-180180*c 
os(d*x+c)^3+360360*cos(d*x+c)^2+135135*cos(d*x+c)-45045)*arctanh(1/(cot...
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.00 \[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=-\frac {{\left (45045 \, \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{7}}{d^{2}}} d e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (-\frac {195 \, {\left (-i \, a^{4} + \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{7}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{512 \, d}\right ) - 45045 \, \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{7}}{d^{2}}} d e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (-\frac {195 \, {\left (-i \, a^{4} - \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{7}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{512 \, d}\right ) - \sqrt {2} {\left (-168 i \, a^{3} e^{\left (16 i \, d x + 16 i \, c\right )} - 1624 i \, a^{3} e^{\left (14 i \, d x + 14 i \, c\right )} - 7184 i \, a^{3} e^{\left (12 i \, d x + 12 i \, c\right )} - 19552 i \, a^{3} e^{\left (10 i \, d x + 10 i \, c\right )} - 38512 i \, a^{3} e^{\left (8 i \, d x + 8 i \, c\right )} - 78800 i \, a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} - 47413 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 7161 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 462 i \, a^{3}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-4 i \, d x - 4 i \, c\right )}}{473088 \, d} \] Input:

integrate(cos(d*x+c)^11*(a+I*a*tan(d*x+c))^(7/2),x, algorithm="fricas")
 

Output:

-1/473088*(45045*sqrt(1/2)*sqrt(-a^7/d^2)*d*e^(4*I*d*x + 4*I*c)*log(-195/5 
12*(-I*a^4 + sqrt(2)*sqrt(1/2)*sqrt(-a^7/d^2)*(d*e^(2*I*d*x + 2*I*c) + d)* 
sqrt(a/(e^(2*I*d*x + 2*I*c) + 1)))*e^(-I*d*x - I*c)/d) - 45045*sqrt(1/2)*s 
qrt(-a^7/d^2)*d*e^(4*I*d*x + 4*I*c)*log(-195/512*(-I*a^4 - sqrt(2)*sqrt(1/ 
2)*sqrt(-a^7/d^2)*(d*e^(2*I*d*x + 2*I*c) + d)*sqrt(a/(e^(2*I*d*x + 2*I*c) 
+ 1)))*e^(-I*d*x - I*c)/d) - sqrt(2)*(-168*I*a^3*e^(16*I*d*x + 16*I*c) - 1 
624*I*a^3*e^(14*I*d*x + 14*I*c) - 7184*I*a^3*e^(12*I*d*x + 12*I*c) - 19552 
*I*a^3*e^(10*I*d*x + 10*I*c) - 38512*I*a^3*e^(8*I*d*x + 8*I*c) - 78800*I*a 
^3*e^(6*I*d*x + 6*I*c) - 47413*I*a^3*e^(4*I*d*x + 4*I*c) + 7161*I*a^3*e^(2 
*I*d*x + 2*I*c) + 462*I*a^3)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1)))*e^(-4*I*d* 
x - 4*I*c)/d
 

Sympy [F(-1)]

Timed out. \[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)**11*(a+I*a*tan(d*x+c))**(7/2),x)
 

Output:

Timed out
 

Maxima [F(-1)]

Timed out. \[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)^11*(a+I*a*tan(d*x+c))^(7/2),x, algorithm="maxima")
 

Output:

Timed out
                                                                                    
                                                                                    
 

Giac [F(-2)]

Exception generated. \[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(cos(d*x+c)^11*(a+I*a*tan(d*x+c))^(7/2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument TypeError: Bad 
Argument TypeDone
 

Mupad [F(-1)]

Timed out. \[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\int {\cos \left (c+d\,x\right )}^{11}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{7/2} \,d x \] Input:

int(cos(c + d*x)^11*(a + a*tan(c + d*x)*1i)^(7/2),x)
 

Output:

int(cos(c + d*x)^11*(a + a*tan(c + d*x)*1i)^(7/2), x)
 

Reduce [F]

\[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\sqrt {a}\, a^{3} \left (-\left (\int \sqrt {\tan \left (d x +c \right ) i +1}\, \cos \left (d x +c \right )^{11} \tan \left (d x +c \right )^{3}d x \right ) i -3 \left (\int \sqrt {\tan \left (d x +c \right ) i +1}\, \cos \left (d x +c \right )^{11} \tan \left (d x +c \right )^{2}d x \right )+3 \left (\int \sqrt {\tan \left (d x +c \right ) i +1}\, \cos \left (d x +c \right )^{11} \tan \left (d x +c \right )d x \right ) i +\int \sqrt {\tan \left (d x +c \right ) i +1}\, \cos \left (d x +c \right )^{11}d x \right ) \] Input:

int(cos(d*x+c)^11*(a+I*a*tan(d*x+c))^(7/2),x)
 

Output:

sqrt(a)*a**3*( - int(sqrt(tan(c + d*x)*i + 1)*cos(c + d*x)**11*tan(c + d*x 
)**3,x)*i - 3*int(sqrt(tan(c + d*x)*i + 1)*cos(c + d*x)**11*tan(c + d*x)** 
2,x) + 3*int(sqrt(tan(c + d*x)*i + 1)*cos(c + d*x)**11*tan(c + d*x),x)*i + 
 int(sqrt(tan(c + d*x)*i + 1)*cos(c + d*x)**11,x))