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

Optimal result

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

[Out]

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*t
an(d*x+c))^(5/2)/d-1/11*I*cos(d*x+c)^11*(a+I*a*tan(d*x+c))^(7/2)/d

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {3578, 3583, 3571, 3570, 212} \[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\frac {195 i a^{7/2} \text {arctanh}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{1024 \sqrt {2} d}+\frac {39 i a^4 \cos ^3(c+d x)}{448 d \sqrt {a+i a \tan (c+d x)}}+\frac {65 i a^4 \cos (c+d x)}{512 d \sqrt {a+i a \tan (c+d x)}}-\frac {13 i a^3 \cos ^5(c+d x) \sqrt {a+i a \tan (c+d x)}}{168 d}-\frac {13 i a^3 \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{128 d}-\frac {195 i a^3 \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{1024 d}-\frac {65 i a^2 \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2}}{924 d}-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}-\frac {5 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{66 d} \]

[In]

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

[Out]

(((195*I)/1024)*a^(7/2)*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(Sqrt[2]*d) + ((
(65*I)/512)*a^4*Cos[c + d*x])/(d*Sqrt[a + I*a*Tan[c + d*x]]) + (((39*I)/448)*a^4*Cos[c + d*x]^3)/(d*Sqrt[a + I
*a*Tan[c + d*x]]) - (((195*I)/1024)*a^3*Cos[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/d - (((13*I)/128)*a^3*Cos[c +
d*x]^3*Sqrt[a + I*a*Tan[c + d*x]])/d - (((13*I)/168)*a^3*Cos[c + d*x]^5*Sqrt[a + I*a*Tan[c + d*x]])/d - (((65
*I)/924)*a^2*Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^(3/2))/d - (((5*I)/66)*a*Cos[c + d*x]^9*(a + I*a*Tan[c + d*
x])^(5/2))/d - ((I/11)*Cos[c + d*x]^11*(a + I*a*Tan[c + d*x])^(7/2))/d

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 3570

Int[sec[(e_.) + (f_.)*(x_)]/Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[-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 3571

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] + Dist[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 3578

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(d*S
ec[e + f*x])^m*((a + b*Tan[e + f*x])^n/(a*f*m)), x] + Dist[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 3583

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] + Dist[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]

Rubi steps \begin{align*} \text {integral}& = -\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 ) \text {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} \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} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.02 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.57 \[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=-\frac {i a^3 e^{-5 i (c+d x)} \left (-462-7161 e^{2 i (c+d x)}+47413 e^{4 i (c+d x)}+78800 e^{6 i (c+d x)}+38512 e^{8 i (c+d x)}+19552 e^{10 i (c+d x)}+7184 e^{12 i (c+d x)}+1624 e^{14 i (c+d x)}+168 e^{16 i (c+d x)}-45045 e^{4 i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \text {arctanh}\left (\sqrt {1+e^{2 i (c+d x)}}\right )\right ) \sqrt {a+i a \tan (c+d x)}}{473088 d} \]

[In]

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

[Out]

((-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)))

Maple [B] (verified)

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

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

[In]

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

[Out]

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))*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)*sin(d*x+c)+45045*(-cos(d*x+c
)/(cos(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)+13513
5*(-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)-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))-360360*cos(d*x+c)^4*(-cos(d*x+c
)/(cos(d*x+c)+1))^(1/2)*arctan((-cos(d*x+c)/(cos(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((-co
s(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((-c
os(d*x+c)/(cos(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)*sin(d*x+c)-45045*I*(-cos(d*x+c)/(cos(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))-80640*I*cos(d*x+c)^7*sin(d*x+c)-160160*I*cos(d*x+c)
^5*sin(d*x+c)+552552*I*cos(d*x+c)^3*sin(d*x+c)-90090*I*sin(d*x+c)*cos(d*x+c)-180180*I*(-cos(d*x+c)/(cos(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))*cos(d*x+c)^3+360360*I*(-cos(d
*x+c)/(cos(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))*cos(d*x+c)^2
+135135*I*(-cos(d*x+c)/(cos(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))*cos(d*x+c)-45045*I*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan((-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*sin(d*x+
c)-360360*I*(-cos(d*x+c)/(cos(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))*cos(d*x+c)^4)

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.00 \[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=-\frac {{\left (45045 \, \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{7}}{d^{2}}} d e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (-\frac {195 \, {\left (-i \, a^{4} + \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{7}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{512 \, d}\right ) - 45045 \, \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{7}}{d^{2}}} d e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (-\frac {195 \, {\left (-i \, a^{4} - \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{7}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{512 \, d}\right ) - \sqrt {2} {\left (-168 i \, a^{3} e^{\left (16 i \, d x + 16 i \, c\right )} - 1624 i \, a^{3} e^{\left (14 i \, d x + 14 i \, c\right )} - 7184 i \, a^{3} e^{\left (12 i \, d x + 12 i \, c\right )} - 19552 i \, a^{3} e^{\left (10 i \, d x + 10 i \, c\right )} - 38512 i \, a^{3} e^{\left (8 i \, d x + 8 i \, c\right )} - 78800 i \, a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} - 47413 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 7161 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 462 i \, a^{3}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-4 i \, d x - 4 i \, c\right )}}{473088 \, d} \]

[In]

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

[Out]

-1/473088*(45045*sqrt(1/2)*sqrt(-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) - 45045*sqrt(1/2)
*sqrt(-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)
- 1624*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) - 3
8512*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. \[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-1)]

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

[In]

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

[Out]

Timed out

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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