\(\int \frac {\cos ^3(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx\) [393]

Optimal result

Integrand size = 26, antiderivative size = 307 \[ \int \frac {\cos ^3(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\frac {3003 i \text {arctanh}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{16384 \sqrt {2} a^{7/2} d}+\frac {i \cos ^3(c+d x)}{10 d (a+i a \tan (c+d x))^{7/2}}+\frac {13 i \cos ^3(c+d x)}{160 a d (a+i a \tan (c+d x))^{5/2}}+\frac {143 i \cos ^3(c+d x)}{1920 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac {1001 i \cos (c+d x)}{8192 a^3 d \sqrt {a+i a \tan (c+d x)}}+\frac {429 i \cos ^3(c+d x)}{5120 a^3 d \sqrt {a+i a \tan (c+d x)}}-\frac {3003 i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{16384 a^4 d}-\frac {1001 i \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{10240 a^4 d} \]

[Out]

3003/32768*I*arctanh(1/2*sec(d*x+c)*a^(1/2)*2^(1/2)/(a+I*a*tan(d*x+c))^(1/2))/a^(7/2)/d*2^(1/2)+1001/8192*I*co
s(d*x+c)/a^3/d/(a+I*a*tan(d*x+c))^(1/2)+429/5120*I*cos(d*x+c)^3/a^3/d/(a+I*a*tan(d*x+c))^(1/2)-3003/16384*I*co
s(d*x+c)*(a+I*a*tan(d*x+c))^(1/2)/a^4/d-1001/10240*I*cos(d*x+c)^3*(a+I*a*tan(d*x+c))^(1/2)/a^4/d+1/10*I*cos(d*
x+c)^3/d/(a+I*a*tan(d*x+c))^(7/2)+13/160*I*cos(d*x+c)^3/a/d/(a+I*a*tan(d*x+c))^(5/2)+143/1920*I*cos(d*x+c)^3/a
^2/d/(a+I*a*tan(d*x+c))^(3/2)

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {3583, 3578, 3571, 3570, 212} \[ \int \frac {\cos ^3(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\frac {3003 i \text {arctanh}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{16384 \sqrt {2} a^{7/2} d}-\frac {1001 i \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{10240 a^4 d}-\frac {3003 i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{16384 a^4 d}+\frac {429 i \cos ^3(c+d x)}{5120 a^3 d \sqrt {a+i a \tan (c+d x)}}+\frac {1001 i \cos (c+d x)}{8192 a^3 d \sqrt {a+i a \tan (c+d x)}}+\frac {143 i \cos ^3(c+d x)}{1920 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac {13 i \cos ^3(c+d x)}{160 a d (a+i a \tan (c+d x))^{5/2}}+\frac {i \cos ^3(c+d x)}{10 d (a+i a \tan (c+d x))^{7/2}} \]

[In]

Int[Cos[c + d*x]^3/(a + I*a*Tan[c + d*x])^(7/2),x]

[Out]

(((3003*I)/16384)*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(Sqrt[2]*a^(7/2)*d) +
((I/10)*Cos[c + d*x]^3)/(d*(a + I*a*Tan[c + d*x])^(7/2)) + (((13*I)/160)*Cos[c + d*x]^3)/(a*d*(a + I*a*Tan[c +
d*x])^(5/2)) + (((143*I)/1920)*Cos[c + d*x]^3)/(a^2*d*(a + I*a*Tan[c + d*x])^(3/2)) + (((1001*I)/8192)*Cos[c
+ d*x])/(a^3*d*Sqrt[a + I*a*Tan[c + d*x]]) + (((429*I)/5120)*Cos[c + d*x]^3)/(a^3*d*Sqrt[a + I*a*Tan[c + d*x]]
) - (((3003*I)/16384)*Cos[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(a^4*d) - (((1001*I)/10240)*Cos[c + d*x]^3*Sqrt
[a + I*a*Tan[c + d*x]])/(a^4*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 ^3(c+d x)}{10 d (a+i a \tan (c+d x))^{7/2}}+\frac {13 \int \frac {\cos ^3(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx}{20 a} \\ & = \frac {i \cos ^3(c+d x)}{10 d (a+i a \tan (c+d x))^{7/2}}+\frac {13 i \cos ^3(c+d x)}{160 a d (a+i a \tan (c+d x))^{5/2}}+\frac {143 \int \frac {\cos ^3(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx}{320 a^2} \\ & = \frac {i \cos ^3(c+d x)}{10 d (a+i a \tan (c+d x))^{7/2}}+\frac {13 i \cos ^3(c+d x)}{160 a d (a+i a \tan (c+d x))^{5/2}}+\frac {143 i \cos ^3(c+d x)}{1920 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac {429 \int \frac {\cos ^3(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx}{1280 a^3} \\ & = \frac {i \cos ^3(c+d x)}{10 d (a+i a \tan (c+d x))^{7/2}}+\frac {13 i \cos ^3(c+d x)}{160 a d (a+i a \tan (c+d x))^{5/2}}+\frac {143 i \cos ^3(c+d x)}{1920 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac {429 i \cos ^3(c+d x)}{5120 a^3 d \sqrt {a+i a \tan (c+d x)}}+\frac {3003 \int \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx}{10240 a^4} \\ & = \frac {i \cos ^3(c+d x)}{10 d (a+i a \tan (c+d x))^{7/2}}+\frac {13 i \cos ^3(c+d x)}{160 a d (a+i a \tan (c+d x))^{5/2}}+\frac {143 i \cos ^3(c+d x)}{1920 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac {429 i \cos ^3(c+d x)}{5120 a^3 d \sqrt {a+i a \tan (c+d x)}}-\frac {1001 i \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{10240 a^4 d}+\frac {1001 \int \frac {\cos (c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx}{4096 a^3} \\ & = \frac {i \cos ^3(c+d x)}{10 d (a+i a \tan (c+d x))^{7/2}}+\frac {13 i \cos ^3(c+d x)}{160 a d (a+i a \tan (c+d x))^{5/2}}+\frac {143 i \cos ^3(c+d x)}{1920 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac {1001 i \cos (c+d x)}{8192 a^3 d \sqrt {a+i a \tan (c+d x)}}+\frac {429 i \cos ^3(c+d x)}{5120 a^3 d \sqrt {a+i a \tan (c+d x)}}-\frac {1001 i \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{10240 a^4 d}+\frac {3003 \int \cos (c+d x) \sqrt {a+i a \tan (c+d x)} \, dx}{16384 a^4} \\ & = \frac {i \cos ^3(c+d x)}{10 d (a+i a \tan (c+d x))^{7/2}}+\frac {13 i \cos ^3(c+d x)}{160 a d (a+i a \tan (c+d x))^{5/2}}+\frac {143 i \cos ^3(c+d x)}{1920 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac {1001 i \cos (c+d x)}{8192 a^3 d \sqrt {a+i a \tan (c+d x)}}+\frac {429 i \cos ^3(c+d x)}{5120 a^3 d \sqrt {a+i a \tan (c+d x)}}-\frac {3003 i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{16384 a^4 d}-\frac {1001 i \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{10240 a^4 d}+\frac {3003 \int \frac {\sec (c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx}{32768 a^3} \\ & = \frac {i \cos ^3(c+d x)}{10 d (a+i a \tan (c+d x))^{7/2}}+\frac {13 i \cos ^3(c+d x)}{160 a d (a+i a \tan (c+d x))^{5/2}}+\frac {143 i \cos ^3(c+d x)}{1920 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac {1001 i \cos (c+d x)}{8192 a^3 d \sqrt {a+i a \tan (c+d x)}}+\frac {429 i \cos ^3(c+d x)}{5120 a^3 d \sqrt {a+i a \tan (c+d x)}}-\frac {3003 i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{16384 a^4 d}-\frac {1001 i \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{10240 a^4 d}+\frac {(3003 i) \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 )}{16384 a^3 d} \\ & = \frac {3003 i \text {arctanh}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{16384 \sqrt {2} a^{7/2} d}+\frac {i \cos ^3(c+d x)}{10 d (a+i a \tan (c+d x))^{7/2}}+\frac {13 i \cos ^3(c+d x)}{160 a d (a+i a \tan (c+d x))^{5/2}}+\frac {143 i \cos ^3(c+d x)}{1920 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac {1001 i \cos (c+d x)}{8192 a^3 d \sqrt {a+i a \tan (c+d x)}}+\frac {429 i \cos ^3(c+d x)}{5120 a^3 d \sqrt {a+i a \tan (c+d x)}}-\frac {3003 i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{16384 a^4 d}-\frac {1001 i \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{10240 a^4 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.97 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.57 \[ \int \frac {\cos ^3(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=-\frac {\left (42140+20048 e^{-2 i (c+d x)}+71190 e^{2 i (c+d x)}+5856 e^{-4 i (c+d x)}-48640 e^{4 i (c+d x)}+768 e^{-6 i (c+d x)}-2560 e^{6 i (c+d x)}+\frac {90090 e^{4 i (c+d x)} \text {arctanh}\left (\sqrt {1+e^{2 i (c+d x)}}\right )}{\sqrt {1+e^{2 i (c+d x)}}}\right ) \sec ^3(c+d x)}{491520 a^3 d (-i+\tan (c+d x))^3 \sqrt {a+i a \tan (c+d x)}} \]

[In]

Integrate[Cos[c + d*x]^3/(a + I*a*Tan[c + d*x])^(7/2),x]

[Out]

-1/491520*((42140 + 20048/E^((2*I)*(c + d*x)) + 71190*E^((2*I)*(c + d*x)) + 5856/E^((4*I)*(c + d*x)) - 48640*E
^((4*I)*(c + d*x)) + 768/E^((6*I)*(c + d*x)) - 2560*E^((6*I)*(c + d*x)) + (90090*E^((4*I)*(c + d*x))*ArcTanh[S
qrt[1 + E^((2*I)*(c + d*x))]])/Sqrt[1 + E^((2*I)*(c + d*x))])*Sec[c + d*x]^3)/(a^3*d*(-I + Tan[c + d*x])^3*Sqr
t[a + I*a*Tan[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. 1056 vs. \(2 (252 ) = 504\).

Time = 11.08 (sec) , antiderivative size = 1057, normalized size of antiderivative = 3.44

method	result	size
default	\(\text {Expression too large to display}\)	\(1057\)

[In]

int(cos(d*x+c)^3/(a+I*a*tan(d*x+c))^(7/2),x,method=_RETURNVERBOSE)

[Out]

1/491520/d/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)/(cos(d*x+c)+1)/(a*(1+I*tan(d*x+c)))^(1/2)/(1+I*tan(d*x+c))^3/a^3
*(-256256*I*cos(d*x+c)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-256256*I*cos(d*x+c)^2*(-cos(d*x+c)/(cos(d*x+c)+1))^(
1/2)+106496*sin(d*x+c)*cos(d*x+c)^3*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+360360*I*cos(d*x+c)*arctan(1/2*(I*sin(d
*x+c)-cos(d*x+c)-1)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+106496*sin(d*x+c)*cos(d*x+c)^2*(-cos(d*
x+c)/(cos(d*x+c)+1))^(1/2)-90090*I*sec(d*x+c)^3*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+45045*I*sec(d*x+c)^3*arctan
(1/2*(I*sin(d*x+c)-cos(d*x+c)-1)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+329472*(-cos(d*x+c)/(cos(d
*x+c)+1))^(1/2)*cos(d*x+c)*sin(d*x+c)+492492*I*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-90090*I*sec(d*x+c)^2*(-cos(d
*x+c)/(cos(d*x+c)+1))^(1/2)-360360*arctan(1/2*(I*sin(d*x+c)-cos(d*x+c)-1)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x
+c)+1))^(1/2))*sin(d*x+c)+329472*sin(d*x+c)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-57344*I*cos(d*x+c)^4*(-cos(d*x+
c)/(cos(d*x+c)+1))^(1/2)+492492*I*sec(d*x+c)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-180180*tan(d*x+c)*arctan(1/2*(
I*sin(d*x+c)-cos(d*x+c)-1)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))-300300*tan(d*x+c)*(-cos(d*x+c)/(
cos(d*x+c)+1))^(1/2)+180180*I*arctan(1/2*(I*sin(d*x+c)-cos(d*x+c)-1)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1
))^(1/2))-57344*I*cos(d*x+c)^3*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+180180*tan(d*x+c)*sec(d*x+c)*arctan(1/2*(I*s
in(d*x+c)-cos(d*x+c)-1)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))-300300*tan(d*x+c)*sec(d*x+c)*(-cos(
d*x+c)/(cos(d*x+c)+1))^(1/2)-135135*I*sec(d*x+c)^2*arctan(1/2*(I*sin(d*x+c)-cos(d*x+c)-1)/(cos(d*x+c)+1)/(-cos
(d*x+c)/(cos(d*x+c)+1))^(1/2))-360360*I*sec(d*x+c)*arctan(1/2*(I*sin(d*x+c)-cos(d*x+c)-1)/(cos(d*x+c)+1)/(-cos
(d*x+c)/(cos(d*x+c)+1))^(1/2))+45045*tan(d*x+c)*sec(d*x+c)^2*arctan(1/2*(I*sin(d*x+c)-cos(d*x+c)-1)/(cos(d*x+c
)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.05 \[ \int \frac {\cos ^3(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\frac {{\left (-45045 i \, \sqrt {\frac {1}{2}} a^{4} d \sqrt {\frac {1}{a^{7} d^{2}}} e^{\left (10 i \, d x + 10 i \, c\right )} \log \left (-\frac {3003 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (i \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a^{3} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{7} d^{2}}} - i\right )} e^{\left (-i \, d x - i \, c\right )}}{8192 \, a^{3} d}\right ) + 45045 i \, \sqrt {\frac {1}{2}} a^{4} d \sqrt {\frac {1}{a^{7} d^{2}}} e^{\left (10 i \, d x + 10 i \, c\right )} \log \left (-\frac {3003 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (-i \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{3} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{7} d^{2}}} - i\right )} e^{\left (-i \, d x - i \, c\right )}}{8192 \, a^{3} d}\right ) + \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-1280 i \, e^{\left (14 i \, d x + 14 i \, c\right )} - 25600 i \, e^{\left (12 i \, d x + 12 i \, c\right )} + 11275 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 56665 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 31094 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 12952 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 3312 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 384 i\right )}\right )} e^{\left (-10 i \, d x - 10 i \, c\right )}}{491520 \, a^{4} d} \]

[In]

integrate(cos(d*x+c)^3/(a+I*a*tan(d*x+c))^(7/2),x, algorithm="fricas")

[Out]

1/491520*(-45045*I*sqrt(1/2)*a^4*d*sqrt(1/(a^7*d^2))*e^(10*I*d*x + 10*I*c)*log(-3003/8192*(sqrt(2)*sqrt(1/2)*(
I*a^3*d*e^(2*I*d*x + 2*I*c) + I*a^3*d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(1/(a^7*d^2)) - I)*e^(-I*d*x - I*
c)/(a^3*d)) + 45045*I*sqrt(1/2)*a^4*d*sqrt(1/(a^7*d^2))*e^(10*I*d*x + 10*I*c)*log(-3003/8192*(sqrt(2)*sqrt(1/2
)*(-I*a^3*d*e^(2*I*d*x + 2*I*c) - I*a^3*d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(1/(a^7*d^2)) - I)*e^(-I*d*x
- I*c)/(a^3*d)) + sqrt(2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*(-1280*I*e^(14*I*d*x + 14*I*c) - 25600*I*e^(12*I*d
*x + 12*I*c) + 11275*I*e^(10*I*d*x + 10*I*c) + 56665*I*e^(8*I*d*x + 8*I*c) + 31094*I*e^(6*I*d*x + 6*I*c) + 129
52*I*e^(4*I*d*x + 4*I*c) + 3312*I*e^(2*I*d*x + 2*I*c) + 384*I))*e^(-10*I*d*x - 10*I*c)/(a^4*d)

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^3(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**3/(a+I*a*tan(d*x+c))**(7/2),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 5821 vs. \(2 (236) = 472\).

Time = 0.62 (sec) , antiderivative size = 5821, normalized size of antiderivative = 18.96 \[ \int \frac {\cos ^3(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\text {Too large to display} \]

[In]

integrate(cos(d*x+c)^3/(a+I*a*tan(d*x+c))^(7/2),x, algorithm="maxima")

[Out]

-1/1966080*(40*(cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + sin(1/5*arctan2(sin(10*d*x + 10*c
), cos(10*d*x + 10*c)))^2 + 2*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1)^(3/4)*((79*(-I*sqr
t(2)*cos(10*d*x + 10*c) - sqrt(2)*sin(10*d*x + 10*c))*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))
^2 + 79*(-I*sqrt(2)*cos(10*d*x + 10*c) - sqrt(2)*sin(10*d*x + 10*c))*sin(1/5*arctan2(sin(10*d*x + 10*c), cos(1
0*d*x + 10*c)))^2 + 837*(-I*sqrt(2)*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 - I*sqrt(2)*sin
(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 - 2*I*sqrt(2)*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(
10*d*x + 10*c))) - I*sqrt(2))*cos(4/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 158*(-I*sqrt(2)*cos(1
0*d*x + 10*c) - sqrt(2)*sin(10*d*x + 10*c))*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) - 837*(sq
rt(2)*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + sqrt(2)*sin(1/5*arctan2(sin(10*d*x + 10*c),
cos(10*d*x + 10*c)))^2 + 2*sqrt(2)*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + sqrt(2))*sin(4/
5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) - 79*I*sqrt(2)*cos(10*d*x + 10*c) - 79*sqrt(2)*sin(10*d*x +
10*c))*cos(7/2*arctan2(sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))), cos(1/5*arctan2(sin(10*d*x +
10*c), cos(10*d*x + 10*c))) + 1)) + (-49*I*sqrt(2)*cos(10*d*x + 10*c) - 1155*I*sqrt(2)*cos(4/5*arctan2(sin(10
*d*x + 10*c), cos(10*d*x + 10*c))) + 3264*I*sqrt(2)*cos(3/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) -
624*I*sqrt(2)*cos(2/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) - 49*sqrt(2)*sin(10*d*x + 10*c) - 1155
*sqrt(2)*sin(4/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 3264*sqrt(2)*sin(3/5*arctan2(sin(10*d*x +
10*c), cos(10*d*x + 10*c))) - 624*sqrt(2)*sin(2/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 128*I*sqr
t(2))*cos(3/2*arctan2(sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))), cos(1/5*arctan2(sin(10*d*x + 1
0*c), cos(10*d*x + 10*c))) + 1)) + (79*(sqrt(2)*cos(10*d*x + 10*c) - I*sqrt(2)*sin(10*d*x + 10*c))*cos(1/5*arc
tan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 79*(sqrt(2)*cos(10*d*x + 10*c) - I*sqrt(2)*sin(10*d*x + 10*c
))*sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 837*(sqrt(2)*cos(1/5*arctan2(sin(10*d*x + 10*c
), cos(10*d*x + 10*c)))^2 + sqrt(2)*sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 2*sqrt(2)*cos
(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + sqrt(2))*cos(4/5*arctan2(sin(10*d*x + 10*c), cos(10*d*
x + 10*c))) + 158*(sqrt(2)*cos(10*d*x + 10*c) - I*sqrt(2)*sin(10*d*x + 10*c))*cos(1/5*arctan2(sin(10*d*x + 10*
c), cos(10*d*x + 10*c))) + 837*(-I*sqrt(2)*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 - I*sqrt
(2)*sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 - 2*I*sqrt(2)*cos(1/5*arctan2(sin(10*d*x + 10*c
), cos(10*d*x + 10*c))) - I*sqrt(2))*sin(4/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 79*sqrt(2)*cos
(10*d*x + 10*c) - 79*I*sqrt(2)*sin(10*d*x + 10*c))*sin(7/2*arctan2(sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*
d*x + 10*c))), cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1)) + (49*sqrt(2)*cos(10*d*x + 10*c)
+ 1155*sqrt(2)*cos(4/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) - 3264*sqrt(2)*cos(3/5*arctan2(sin(10
*d*x + 10*c), cos(10*d*x + 10*c))) + 624*sqrt(2)*cos(2/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) - 49
*I*sqrt(2)*sin(10*d*x + 10*c) - 1155*I*sqrt(2)*sin(4/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 3264
*I*sqrt(2)*sin(3/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) - 624*I*sqrt(2)*sin(2/5*arctan2(sin(10*d*x
+ 10*c), cos(10*d*x + 10*c))) - 128*sqrt(2))*sin(3/2*arctan2(sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x +
10*c))), cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1)))*sqrt(a) + 4*(cos(1/5*arctan2(sin(10*
d*x + 10*c), cos(10*d*x + 10*c)))^2 + sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 2*cos(1/5*a
rctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1)^(1/4)*(105*((-I*sqrt(2)*cos(10*d*x + 10*c) - sqrt(2)*sin(
10*d*x + 10*c))*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^4 + (-I*sqrt(2)*cos(10*d*x + 10*c) -
sqrt(2)*sin(10*d*x + 10*c))*sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^4 + 4*(-I*sqrt(2)*cos(10*
d*x + 10*c) - sqrt(2)*sin(10*d*x + 10*c))*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^3 + 6*(-I*s
qrt(2)*cos(10*d*x + 10*c) - sqrt(2)*sin(10*d*x + 10*c))*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)
))^2 + 2*((-I*sqrt(2)*cos(10*d*x + 10*c) - sqrt(2)*sin(10*d*x + 10*c))*cos(1/5*arctan2(sin(10*d*x + 10*c), cos
(10*d*x + 10*c)))^2 + 2*(-I*sqrt(2)*cos(10*d*x + 10*c) - sqrt(2)*sin(10*d*x + 10*c))*cos(1/5*arctan2(sin(10*d*
x + 10*c), cos(10*d*x + 10*c))) - I*sqrt(2)*cos(10*d*x + 10*c) - sqrt(2)*sin(10*d*x + 10*c))*sin(1/5*arctan2(s
in(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 4*(-I*sqrt(2)*cos(10*d*x + 10*c) - sqrt(2)*sin(10*d*x + 10*c))*cos
(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) - I*sqrt(2)*cos(10*d*x + 10*c) - sqrt(2)*sin(10*d*x + 10
*c))*cos(9/2*arctan2(sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))), cos(1/5*arctan2(sin(10*d*x + 10
*c), cos(10*d*x + 10*c))) + 1)) + 2*(448*(I*sqrt(2)*cos(10*d*x + 10*c) + sqrt(2)*sin(10*d*x + 10*c))*cos(1/5*a
rctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 448*(I*sqrt(2)*cos(10*d*x + 10*c) + sqrt(2)*sin(10*d*x + 1
0*c))*sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 7665*(I*sqrt(2)*cos(1/5*arctan2(sin(10*d*x
+ 10*c), cos(10*d*x + 10*c)))^2 + I*sqrt(2)*sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 2*I*s
qrt(2)*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + I*sqrt(2))*cos(4/5*arctan2(sin(10*d*x + 10*c
), cos(10*d*x + 10*c))) + 10440*(-I*sqrt(2)*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 - I*sqr
t(2)*sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 - 2*I*sqrt(2)*cos(1/5*arctan2(sin(10*d*x + 10*
c), cos(10*d*x + 10*c))) - I*sqrt(2))*cos(3/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 896*(I*sqrt(2
)*cos(10*d*x + 10*c) + sqrt(2)*sin(10*d*x + 10*c))*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) +
7665*(sqrt(2)*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + sqrt(2)*sin(1/5*arctan2(sin(10*d*x
+ 10*c), cos(10*d*x + 10*c)))^2 + 2*sqrt(2)*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + sqrt(2)
)*sin(4/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) - 10440*(sqrt(2)*cos(1/5*arctan2(sin(10*d*x + 10*c)
, cos(10*d*x + 10*c)))^2 + sqrt(2)*sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 2*sqrt(2)*cos(
1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + sqrt(2))*sin(3/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x
+ 10*c))) + 448*I*sqrt(2)*cos(10*d*x + 10*c) + 448*sqrt(2)*sin(10*d*x + 10*c))*cos(5/2*arctan2(sin(1/5*arctan
2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))), cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1)) + 1
5*(7*I*sqrt(2)*cos(10*d*x + 10*c) + 210*I*sqrt(2)*cos(4/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) - 9
12*I*sqrt(2)*cos(3/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 352*I*sqrt(2)*cos(2/5*arctan2(sin(10*d
*x + 10*c), cos(10*d*x + 10*c))) + 7*sqrt(2)*sin(10*d*x + 10*c) + 210*sqrt(2)*sin(4/5*arctan2(sin(10*d*x + 10*
c), cos(10*d*x + 10*c))) - 912*sqrt(2)*sin(3/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 352*sqrt(2)*
sin(2/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1536*I*sqrt(2))*cos(1/2*arctan2(sin(1/5*arctan2(sin
(10*d*x + 10*c), cos(10*d*x + 10*c))), cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1)) + 105*((
sqrt(2)*cos(10*d*x + 10*c) - I*sqrt(2)*sin(10*d*x + 10*c))*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10
*c)))^4 + (sqrt(2)*cos(10*d*x + 10*c) - I*sqrt(2)*sin(10*d*x + 10*c))*sin(1/5*arctan2(sin(10*d*x + 10*c), cos(
10*d*x + 10*c)))^4 + 4*(sqrt(2)*cos(10*d*x + 10*c) - I*sqrt(2)*sin(10*d*x + 10*c))*cos(1/5*arctan2(sin(10*d*x
+ 10*c), cos(10*d*x + 10*c)))^3 + 6*(sqrt(2)*cos(10*d*x + 10*c) - I*sqrt(2)*sin(10*d*x + 10*c))*cos(1/5*arctan
2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 2*((sqrt(2)*cos(10*d*x + 10*c) - I*sqrt(2)*sin(10*d*x + 10*c))*
cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 2*(sqrt(2)*cos(10*d*x + 10*c) - I*sqrt(2)*sin(10*
d*x + 10*c))*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + sqrt(2)*cos(10*d*x + 10*c) - I*sqrt(2)
*sin(10*d*x + 10*c))*sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 4*(sqrt(2)*cos(10*d*x + 10*c
) - I*sqrt(2)*sin(10*d*x + 10*c))*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + sqrt(2)*cos(10*d*
x + 10*c) - I*sqrt(2)*sin(10*d*x + 10*c))*sin(9/2*arctan2(sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*
c))), cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1)) - 2*(448*(sqrt(2)*cos(10*d*x + 10*c) - I*
sqrt(2)*sin(10*d*x + 10*c))*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 448*(sqrt(2)*cos(10*d
*x + 10*c) - I*sqrt(2)*sin(10*d*x + 10*c))*sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 7665*(
sqrt(2)*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + sqrt(2)*sin(1/5*arctan2(sin(10*d*x + 10*c
), cos(10*d*x + 10*c)))^2 + 2*sqrt(2)*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + sqrt(2))*cos(
4/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) - 10440*(sqrt(2)*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(
10*d*x + 10*c)))^2 + sqrt(2)*sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 2*sqrt(2)*cos(1/5*ar
ctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + sqrt(2))*cos(3/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*
c))) + 896*(sqrt(2)*cos(10*d*x + 10*c) - I*sqrt(2)*sin(10*d*x + 10*c))*cos(1/5*arctan2(sin(10*d*x + 10*c), cos
(10*d*x + 10*c))) - 7665*(I*sqrt(2)*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + I*sqrt(2)*sin
(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 2*I*sqrt(2)*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(
10*d*x + 10*c))) + I*sqrt(2))*sin(4/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) - 10440*(-I*sqrt(2)*cos
(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 - I*sqrt(2)*sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10
*d*x + 10*c)))^2 - 2*I*sqrt(2)*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) - I*sqrt(2))*sin(3/5*a
rctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 448*sqrt(2)*cos(10*d*x + 10*c) - 448*I*sqrt(2)*sin(10*d*x +
10*c))*sin(5/2*arctan2(sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))), cos(1/5*arctan2(sin(10*d*x +
10*c), cos(10*d*x + 10*c))) + 1)) - 15*(7*sqrt(2)*cos(10*d*x + 10*c) + 210*sqrt(2)*cos(4/5*arctan2(sin(10*d*x
+ 10*c), cos(10*d*x + 10*c))) - 912*sqrt(2)*cos(3/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 352*sqr
t(2)*cos(2/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) - 7*I*sqrt(2)*sin(10*d*x + 10*c) - 210*I*sqrt(2)
*sin(4/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 912*I*sqrt(2)*sin(3/5*arctan2(sin(10*d*x + 10*c),
cos(10*d*x + 10*c))) - 352*I*sqrt(2)*sin(2/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1536*sqrt(2))*
sin(1/2*arctan2(sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))), cos(1/5*arctan2(sin(10*d*x + 10*c),
cos(10*d*x + 10*c))) + 1)))*sqrt(a) + 45045*(2*sqrt(2)*arctan2((cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x
+ 10*c)))^2 + sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 2*cos(1/5*arctan2(sin(10*d*x + 10*
c), cos(10*d*x + 10*c))) + 1)^(1/4)*sin(1/2*arctan2(sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))),
cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1)), (cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*
x + 10*c)))^2 + sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 2*cos(1/5*arctan2(sin(10*d*x + 10
*c), cos(10*d*x + 10*c))) + 1)^(1/4)*cos(1/2*arctan2(sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))),
cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1)) + 1) - 2*sqrt(2)*arctan2((cos(1/5*arctan2(sin(
10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 2*cos(1/
5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1)^(1/4)*sin(1/2*arctan2(sin(1/5*arctan2(sin(10*d*x + 10*
c), cos(10*d*x + 10*c))), cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1)), (cos(1/5*arctan2(sin
(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 2*cos(1
/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1)^(1/4)*cos(1/2*arctan2(sin(1/5*arctan2(sin(10*d*x + 10
*c), cos(10*d*x + 10*c))), cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1)) - 1) - I*sqrt(2)*log
(sqrt(cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*
d*x + 10*c)))^2 + 2*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1)*cos(1/2*arctan2(sin(1/5*arct
an2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))), cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1))^2
+ sqrt(cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + sin(1/5*arctan2(sin(10*d*x + 10*c), cos(1
0*d*x + 10*c)))^2 + 2*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1)*sin(1/2*arctan2(sin(1/5*ar
ctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))), cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1))
^2 + 2*(cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + sin(1/5*arctan2(sin(10*d*x + 10*c), cos(1
0*d*x + 10*c)))^2 + 2*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1)^(1/4)*cos(1/2*arctan2(sin(
1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))), cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))
+ 1)) + 1) + I*sqrt(2)*log(sqrt(cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + sin(1/5*arctan2(
sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 2*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1)*c
os(1/2*arctan2(sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))), cos(1/5*arctan2(sin(10*d*x + 10*c), c
os(10*d*x + 10*c))) + 1))^2 + sqrt(cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + sin(1/5*arctan
2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 2*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1)
*sin(1/2*arctan2(sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))), cos(1/5*arctan2(sin(10*d*x + 10*c),
cos(10*d*x + 10*c))) + 1))^2 - 2*(cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + sin(1/5*arctan
2(sin(10*d*x + 10*c), cos(10*d*x + 10*c)))^2 + 2*cos(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))) + 1)
^(1/4)*cos(1/2*arctan2(sin(1/5*arctan2(sin(10*d*x + 10*c), cos(10*d*x + 10*c))), cos(1/5*arctan2(sin(10*d*x +
10*c), cos(10*d*x + 10*c))) + 1)) + 1))*sqrt(a))/(a^4*d)

Giac [F]

\[ \int \frac {\cos ^3(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\int { \frac {\cos \left (d x + c\right )^{3}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate(cos(d*x+c)^3/(a+I*a*tan(d*x+c))^(7/2),x, algorithm="giac")

[Out]

integrate(cos(d*x + c)^3/(I*a*tan(d*x + c) + a)^(7/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^3(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^3}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{7/2}} \,d x \]

[In]

int(cos(c + d*x)^3/(a + a*tan(c + d*x)*1i)^(7/2),x)

[Out]

int(cos(c + d*x)^3/(a + a*tan(c + d*x)*1i)^(7/2), x)