Integrand size = 26, antiderivative size = 342 \[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\frac {195 i a^{7/2} \text {arctanh}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{1024 \sqrt {2} d}+\frac {65 i a^4 \cos (c+d x)}{512 d \sqrt {a+i a \tan (c+d x)}}+\frac {39 i a^4 \cos ^3(c+d x)}{448 d \sqrt {a+i a \tan (c+d x)}}-\frac {195 i a^3 \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{1024 d}-\frac {13 i a^3 \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{128 d}-\frac {13 i a^3 \cos ^5(c+d x) \sqrt {a+i a \tan (c+d x)}}{168 d}-\frac {65 i a^2 \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2}}{924 d}-\frac {5 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{66 d}-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d} \]
195/2048*I*a^(7/2)*arctanh(1/2*sec(d*x+c)*a^(1/2)*2^(1/2)/(a+I*a*tan(d*x+c ))^(1/2))/d*2^(1/2)+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
Time = 6.02 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.57 \[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=-\frac {i a^3 e^{-5 i (c+d x)} \left (-462-7161 e^{2 i (c+d x)}+47413 e^{4 i (c+d x)}+78800 e^{6 i (c+d x)}+38512 e^{8 i (c+d x)}+19552 e^{10 i (c+d x)}+7184 e^{12 i (c+d x)}+1624 e^{14 i (c+d x)}+168 e^{16 i (c+d x)}-45045 e^{4 i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \text {arctanh}\left (\sqrt {1+e^{2 i (c+d x)}}\right )\right ) \sqrt {a+i a \tan (c+d x)}}{473088 d} \]
((-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)))
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 |
\(\displaystyle \frac {15}{22} a \left (\frac {13}{18} a \left (\frac {11}{14} a \left (\frac {9}{10} a \left (\frac {7 \left (\frac {5}{6} a \left (\frac {3 \left (\frac {i \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{\sqrt {2} d}-\frac {i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}\right )}{4 a}+\frac {i \cos (c+d x)}{2 d \sqrt {a+i a \tan (c+d x)}}\right )-\frac {i \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}\right )}{8 a}+\frac {i \cos ^3(c+d x)}{4 d \sqrt {a+i a \tan (c+d x)}}\right )-\frac {i \cos ^5(c+d x) \sqrt {a+i a \tan (c+d x)}}{5 d}\right )-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2}}{7 d}\right )-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{9 d}\right )-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}\) |
((-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
3.4.32.3.1 Defintions of rubi rules used
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])
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 ]
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]
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]
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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1237 vs. \(2 (281 ) = 562\).
Time = 6.02 (sec) , antiderivative size = 1238, normalized size of antiderivative = 3.62
\[\text {Expression too large to display}\]
1/473088/d*(tan(d*x+c)-I)^3*(a*(1+I*tan(d*x+c)))^(1/2)*a^3*cos(d*x+c)^3*(- 360360*cos(d*x+c)^3*arctanh(sin(d*x+c)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d* x+c)+1))^(1/2))*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)+37632*cos(d* x+c)^8+101920*cos(d*x+c)^6+330330*cos(d*x+c)^2-655512*cos(d*x+c)^4+180180* arctanh(sin(d*x+c)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*(-co s(d*x+c)/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)*sin(d*x+c)+45045*(-cos(d*x+c)/(c os(d*x+c)+1))^(1/2)*arctanh(sin(d*x+c)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d* x+c)+1))^(1/2))*sin(d*x+c)+135135*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arcta n((-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)-180180*arctanh(sin(d*x+c) /(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)^2*sin(d*x+c )*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+360360*(-cos(d*x+c)/(cos(d*x+c)+1))^( 1/2)*arctan((-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)^2-45045*(-cos(d *x+c)/(cos(d*x+c)+1))^(1/2)*arctan((-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))-360 360*cos(d*x+c)^4*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan((-cos(d*x+c)/(c os(d*x+c)+1))^(1/2))-180180*cos(d*x+c)^3*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2 )*arctan((-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+360360*I*(-cos(d*x+c)/(cos(d* x+c)+1))^(1/2)*arctan((-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)^3*sin (d*x+c)+180180*I*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan((-cos(d*x+c)/(c os(d*x+c)+1))^(1/2))*cos(d*x+c)^2*sin(d*x+c)-180180*I*(-cos(d*x+c)/(cos(d* x+c)+1))^(1/2)*arctan((-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)*si...
Time = 0.31 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.00 \[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=-\frac {{\left (45045 \, \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{7}}{d^{2}}} d e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (-\frac {195 \, {\left (-i \, a^{4} + \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{7}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{512 \, d}\right ) - 45045 \, \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{7}}{d^{2}}} d e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (-\frac {195 \, {\left (-i \, a^{4} - \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{7}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{512 \, d}\right ) - \sqrt {2} {\left (-168 i \, a^{3} e^{\left (16 i \, d x + 16 i \, c\right )} - 1624 i \, a^{3} e^{\left (14 i \, d x + 14 i \, c\right )} - 7184 i \, a^{3} e^{\left (12 i \, d x + 12 i \, c\right )} - 19552 i \, a^{3} e^{\left (10 i \, d x + 10 i \, c\right )} - 38512 i \, a^{3} e^{\left (8 i \, d x + 8 i \, c\right )} - 78800 i \, a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} - 47413 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 7161 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 462 i \, a^{3}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-4 i \, d x - 4 i \, c\right )}}{473088 \, d} \]
-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
Timed out. \[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\text {Timed out} \]
Timed out. \[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\text {Timed out} \]
\[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\int { {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} \cos \left (d x + c\right )^{11} \,d x } \]
Timed out. \[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\int {\cos \left (c+d\,x\right )}^{11}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{7/2} \,d x \]