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

Optimal. Leaf size=233 \[ \frac {105 i \tanh ^{-1}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{256 \sqrt {2} a^{3/2} d}-\frac {7 i \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{32 a^2 d}-\frac {105 i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{256 a^2 d}+\frac {3 i \cos ^3(c+d x)}{16 a d \sqrt {a+i a \tan (c+d x)}}+\frac {i \cos ^3(c+d x)}{6 d (a+i a \tan (c+d x))^{3/2}}+\frac {35 i \cos (c+d x)}{128 a d \sqrt {a+i a \tan (c+d x)}} \]

[Out]

105/512*I*arctanh(1/2*sec(d*x+c)*a^(1/2)*2^(1/2)/(a+I*a*tan(d*x+c))^(1/2))/a^(3/2)/d*2^(1/2)+35/128*I*cos(d*x+
c)/a/d/(a+I*a*tan(d*x+c))^(1/2)+3/16*I*cos(d*x+c)^3/a/d/(a+I*a*tan(d*x+c))^(1/2)-105/256*I*cos(d*x+c)*(a+I*a*t
an(d*x+c))^(1/2)/a^2/d-7/32*I*cos(d*x+c)^3*(a+I*a*tan(d*x+c))^(1/2)/a^2/d+1/6*I*cos(d*x+c)^3/d/(a+I*a*tan(d*x+
c))^(3/2)

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Rubi [A]  time = 0.34, antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {3502, 3497, 3490, 3489, 206} \[ -\frac {7 i \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{32 a^2 d}-\frac {105 i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{256 a^2 d}+\frac {105 i \tanh ^{-1}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{256 \sqrt {2} a^{3/2} d}+\frac {3 i \cos ^3(c+d x)}{16 a d \sqrt {a+i a \tan (c+d x)}}+\frac {i \cos ^3(c+d x)}{6 d (a+i a \tan (c+d x))^{3/2}}+\frac {35 i \cos (c+d x)}{128 a d \sqrt {a+i a \tan (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((105*I)/256)*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(Sqrt[2]*a^(3/2)*d) + ((I
/6)*Cos[c + d*x]^3)/(d*(a + I*a*Tan[c + d*x])^(3/2)) + (((35*I)/128)*Cos[c + d*x])/(a*d*Sqrt[a + I*a*Tan[c + d
*x]]) + (((3*I)/16)*Cos[c + d*x]^3)/(a*d*Sqrt[a + I*a*Tan[c + d*x]]) - (((105*I)/256)*Cos[c + d*x]*Sqrt[a + I*
a*Tan[c + d*x]])/(a^2*d) - (((7*I)/32)*Cos[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]])/(a^2*d)

Rule 206

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

Rule 3489

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 3490

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 3497

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*(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 3502

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*} \int \frac {\cos ^3(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx &=\frac {i \cos ^3(c+d x)}{6 d (a+i a \tan (c+d x))^{3/2}}+\frac {3 \int \frac {\cos ^3(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx}{4 a}\\ &=\frac {i \cos ^3(c+d x)}{6 d (a+i a \tan (c+d x))^{3/2}}+\frac {3 i \cos ^3(c+d x)}{16 a d \sqrt {a+i a \tan (c+d x)}}+\frac {21 \int \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx}{32 a^2}\\ &=\frac {i \cos ^3(c+d x)}{6 d (a+i a \tan (c+d x))^{3/2}}+\frac {3 i \cos ^3(c+d x)}{16 a d \sqrt {a+i a \tan (c+d x)}}-\frac {7 i \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{32 a^2 d}+\frac {35 \int \frac {\cos (c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx}{64 a}\\ &=\frac {i \cos ^3(c+d x)}{6 d (a+i a \tan (c+d x))^{3/2}}+\frac {35 i \cos (c+d x)}{128 a d \sqrt {a+i a \tan (c+d x)}}+\frac {3 i \cos ^3(c+d x)}{16 a d \sqrt {a+i a \tan (c+d x)}}-\frac {7 i \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{32 a^2 d}+\frac {105 \int \cos (c+d x) \sqrt {a+i a \tan (c+d x)} \, dx}{256 a^2}\\ &=\frac {i \cos ^3(c+d x)}{6 d (a+i a \tan (c+d x))^{3/2}}+\frac {35 i \cos (c+d x)}{128 a d \sqrt {a+i a \tan (c+d x)}}+\frac {3 i \cos ^3(c+d x)}{16 a d \sqrt {a+i a \tan (c+d x)}}-\frac {105 i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{256 a^2 d}-\frac {7 i \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{32 a^2 d}+\frac {105 \int \frac {\sec (c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx}{512 a}\\ &=\frac {i \cos ^3(c+d x)}{6 d (a+i a \tan (c+d x))^{3/2}}+\frac {35 i \cos (c+d x)}{128 a d \sqrt {a+i a \tan (c+d x)}}+\frac {3 i \cos ^3(c+d x)}{16 a d \sqrt {a+i a \tan (c+d x)}}-\frac {105 i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{256 a^2 d}-\frac {7 i \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{32 a^2 d}+\frac {(105 i) \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 )}{256 a d}\\ &=\frac {105 i \tanh ^{-1}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{256 \sqrt {2} a^{3/2} d}+\frac {i \cos ^3(c+d x)}{6 d (a+i a \tan (c+d x))^{3/2}}+\frac {35 i \cos (c+d x)}{128 a d \sqrt {a+i a \tan (c+d x)}}+\frac {3 i \cos ^3(c+d x)}{16 a d \sqrt {a+i a \tan (c+d x)}}-\frac {105 i \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{256 a^2 d}-\frac {7 i \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{32 a^2 d}\\ \end {align*}

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Mathematica [A]  time = 1.70, size = 145, normalized size = 0.62 \[ \frac {\sec (c+d x) \left (\frac {630 e^{2 i (c+d x)} \tanh ^{-1}\left (\sqrt {1+e^{2 i (c+d x)}}\right )}{\sqrt {1+e^{2 i (c+d x)}}}-2 (3 i (86 \sin (2 (c+d x))+8 \sin (4 (c+d x))+55 i)+158 \cos (2 (c+d x))+8 \cos (4 (c+d x)))\right )}{1536 a d (\tan (c+d x)-i) \sqrt {a+i a \tan (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sec[c + d*x]*((630*E^((2*I)*(c + d*x))*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])/Sqrt[1 + E^((2*I)*(c + d*x))]
- 2*(158*Cos[2*(c + d*x)] + 8*Cos[4*(c + d*x)] + (3*I)*(55*I + 86*Sin[2*(c + d*x)] + 8*Sin[4*(c + d*x)]))))/(1
536*a*d*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])

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fricas [A]  time = 0.58, size = 292, normalized size = 1.25 \[ \frac {{\left (315 i \, \sqrt {\frac {1}{2}} a^{2} d \sqrt {\frac {1}{a^{3} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (\frac {{\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (13440 i \, a d e^{\left (2 i \, d x + 2 i \, c\right )} + 13440 i \, a d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{3} d^{2}}} + 13440 i\right )} e^{\left (-i \, d x - i \, c\right )}}{16384 \, a d}\right ) - 315 i \, \sqrt {\frac {1}{2}} a^{2} d \sqrt {\frac {1}{a^{3} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (\frac {{\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (-13440 i \, a d e^{\left (2 i \, d x + 2 i \, c\right )} - 13440 i \, a d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{3} d^{2}}} + 13440 i\right )} e^{\left (-i \, d x - i \, c\right )}}{16384 \, a d}\right ) + \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-16 i \, e^{\left (10 i \, d x + 10 i \, c\right )} - 224 i \, e^{\left (8 i \, d x + 8 i \, c\right )} - 43 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 215 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 58 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 8 i\right )}\right )} e^{\left (-6 i \, d x - 6 i \, c\right )}}{1536 \, a^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1536*(315*I*sqrt(1/2)*a^2*d*sqrt(1/(a^3*d^2))*e^(6*I*d*x + 6*I*c)*log(1/16384*(sqrt(2)*sqrt(1/2)*(13440*I*a*
d*e^(2*I*d*x + 2*I*c) + 13440*I*a*d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(1/(a^3*d^2)) + 13440*I)*e^(-I*d*x
- I*c)/(a*d)) - 315*I*sqrt(1/2)*a^2*d*sqrt(1/(a^3*d^2))*e^(6*I*d*x + 6*I*c)*log(1/16384*(sqrt(2)*sqrt(1/2)*(-1
3440*I*a*d*e^(2*I*d*x + 2*I*c) - 13440*I*a*d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(1/(a^3*d^2)) + 13440*I)*e
^(-I*d*x - I*c)/(a*d)) + sqrt(2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*(-16*I*e^(10*I*d*x + 10*I*c) - 224*I*e^(8*I
*d*x + 8*I*c) - 43*I*e^(6*I*d*x + 6*I*c) + 215*I*e^(4*I*d*x + 4*I*c) + 58*I*e^(2*I*d*x + 2*I*c) + 8*I))*e^(-6*
I*d*x - 6*I*c)/(a^2*d)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (d x + c\right )^{3}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 1.16, size = 373, normalized size = 1.60 \[ \frac {\sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (1024 i \left (\cos ^{7}\left (d x +c \right )\right )+1024 \sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )+64 i \left (\cos ^{5}\left (d x +c \right )\right )+315 i \cos \left (d x +c \right ) \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\left (i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )\right ) \sqrt {2}}{2 \sin \left (d x +c \right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right )+576 \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )+168 i \left (\cos ^{3}\left (d x +c \right )\right )+315 i \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\left (i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )\right ) \sqrt {2}}{2 \sin \left (d x +c \right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right )+315 \sqrt {2}\, \sin \left (d x +c \right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\left (i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )\right ) \sqrt {2}}{2 \sin \left (d x +c \right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right )+840 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-1260 i \cos \left (d x +c \right )\right )}{3072 d \,a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3072/d*(a*(I*sin(d*x+c)+cos(d*x+c))/cos(d*x+c))^(1/2)*(1024*I*cos(d*x+c)^7+1024*sin(d*x+c)*cos(d*x+c)^6+64*I
*cos(d*x+c)^5+315*I*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/2*(I*cos(d*x+c)-I+sin(d*x+c))/sin(d*x+c)/(-2
*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))*2^(1/2)*cos(d*x+c)+576*sin(d*x+c)*cos(d*x+c)^4+168*I*cos(d*x+c)^3+3
15*I*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/2*(I*cos(d*x+c)-I+sin(d*x+c))/sin(d*x+c)/(-2*cos(d*x+c)/(1+
cos(d*x+c)))^(1/2)*2^(1/2))*2^(1/2)+315*2^(1/2)*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/2*(I*
cos(d*x+c)-I+sin(d*x+c))/sin(d*x+c)/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))+840*cos(d*x+c)^2*sin(d*x+c)-
1260*I*cos(d*x+c))/a^2

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maxima [B]  time = 1.10, size = 2632, normalized size = 11.30 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6144*((cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x
+ 6*c)))^2 + 2*cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1)^(3/4)*((32*I*sqrt(2)*cos(6*d*x + 6*c)
+ 360*I*sqrt(2)*cos(2/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 32*sqrt(2)*sin(6*d*x + 6*c) + 360*sqrt
(2)*sin(2/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) - 64*I*sqrt(2))*cos(3/2*arctan2(sin(1/3*arctan2(sin(6
*d*x + 6*c), cos(6*d*x + 6*c))), cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1)) - (32*sqrt(2)*cos(
6*d*x + 6*c) + 360*sqrt(2)*cos(2/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) - 32*I*sqrt(2)*sin(6*d*x + 6*c
) - 360*I*sqrt(2)*sin(2/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) - 64*sqrt(2))*sin(3/2*arctan2(sin(1/3*a
rctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))), cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1)))*sqrt(
a) + (cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6
*c)))^2 + 2*cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1)^(1/4)*(((12*I*sqrt(2)*cos(6*d*x + 6*c) +
12*sqrt(2)*sin(6*d*x + 6*c))*cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + (12*I*sqrt(2)*cos(6*d*x
+ 6*c) + 12*sqrt(2)*sin(6*d*x + 6*c))*sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + (24*I*sqrt(2)*
cos(6*d*x + 6*c) + 24*sqrt(2)*sin(6*d*x + 6*c))*cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 12*I*sq
rt(2)*cos(6*d*x + 6*c) + 12*sqrt(2)*sin(6*d*x + 6*c))*cos(5/2*arctan2(sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*
d*x + 6*c))), cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1)) + (-12*I*sqrt(2)*cos(6*d*x + 6*c) - 2
16*I*sqrt(2)*cos(2/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 288*I*sqrt(2)*cos(1/3*arctan2(sin(6*d*x +
6*c), cos(6*d*x + 6*c))) - 12*sqrt(2)*sin(6*d*x + 6*c) - 216*sqrt(2)*sin(2/3*arctan2(sin(6*d*x + 6*c), cos(6*d
*x + 6*c))) + 288*sqrt(2)*sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) - 768*I*sqrt(2))*cos(1/2*arctan
2(sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))), cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) +
1)) - 12*((sqrt(2)*cos(6*d*x + 6*c) - I*sqrt(2)*sin(6*d*x + 6*c))*cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x
+ 6*c)))^2 + (sqrt(2)*cos(6*d*x + 6*c) - I*sqrt(2)*sin(6*d*x + 6*c))*sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*
d*x + 6*c)))^2 + 2*(sqrt(2)*cos(6*d*x + 6*c) - I*sqrt(2)*sin(6*d*x + 6*c))*cos(1/3*arctan2(sin(6*d*x + 6*c), c
os(6*d*x + 6*c))) + sqrt(2)*cos(6*d*x + 6*c) - I*sqrt(2)*sin(6*d*x + 6*c))*sin(5/2*arctan2(sin(1/3*arctan2(sin
(6*d*x + 6*c), cos(6*d*x + 6*c))), cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1)) + (12*sqrt(2)*co
s(6*d*x + 6*c) + 216*sqrt(2)*cos(2/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) - 288*sqrt(2)*cos(1/3*arctan
2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) - 12*I*sqrt(2)*sin(6*d*x + 6*c) - 216*I*sqrt(2)*sin(2/3*arctan2(sin(6*d
*x + 6*c), cos(6*d*x + 6*c))) + 288*I*sqrt(2)*sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 768*sqrt(
2))*sin(1/2*arctan2(sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))), cos(1/3*arctan2(sin(6*d*x + 6*c), co
s(6*d*x + 6*c))) + 1)))*sqrt(a) - (630*sqrt(2)*arctan2((cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2
+ sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + 2*cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*
c))) + 1)^(1/4)*sin(1/2*arctan2(sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))), cos(1/3*arctan2(sin(6*d*
x + 6*c), cos(6*d*x + 6*c))) + 1)), (cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + sin(1/3*arctan2(
sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + 2*cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1)^(1/4)*cos
(1/2*arctan2(sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))), cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x
+ 6*c))) + 1)) + 1) - 630*sqrt(2)*arctan2((cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + sin(1/3*a
rctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + 2*cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1)^(1
/4)*sin(1/2*arctan2(sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))), cos(1/3*arctan2(sin(6*d*x + 6*c), co
s(6*d*x + 6*c))) + 1)), (cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + sin(1/3*arctan2(sin(6*d*x +
6*c), cos(6*d*x + 6*c)))^2 + 2*cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1)^(1/4)*cos(1/2*arctan2
(sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))), cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) +
1)) - 1) - 315*I*sqrt(2)*log(sqrt(cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + sin(1/3*arctan2(sin
(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + 2*cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1)*cos(1/2*arct
an2(sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))), cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))
+ 1))^2 + sqrt(cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + sin(1/3*arctan2(sin(6*d*x + 6*c), cos
(6*d*x + 6*c)))^2 + 2*cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1)*sin(1/2*arctan2(sin(1/3*arctan
2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))), cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1))^2 + 2*(cos(
1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 +
2*cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1)^(1/4)*cos(1/2*arctan2(sin(1/3*arctan2(sin(6*d*x +
6*c), cos(6*d*x + 6*c))), cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1)) + 1) + 315*I*sqrt(2)*log(
sqrt(cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*
c)))^2 + 2*cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1)*cos(1/2*arctan2(sin(1/3*arctan2(sin(6*d*x
+ 6*c), cos(6*d*x + 6*c))), cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1))^2 + sqrt(cos(1/3*arcta
n2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + 2*cos(1/3
*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1)*sin(1/2*arctan2(sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x
+ 6*c))), cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1))^2 - 2*(cos(1/3*arctan2(sin(6*d*x + 6*c),
cos(6*d*x + 6*c)))^2 + sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c)))^2 + 2*cos(1/3*arctan2(sin(6*d*x +
6*c), cos(6*d*x + 6*c))) + 1)^(1/4)*cos(1/2*arctan2(sin(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))), cos
(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1)) + 1))*sqrt(a))/(a^2*d)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\cos \left (c+d\,x\right )}^3}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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