3.4.61 \(\int \frac {\cos ^3(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx\) [361]

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

[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 = 0.25, 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 = {3583, 3578, 3571, 3570, 212} \begin {gather*} \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)}} \end {gather*}

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 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*} \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) \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 )}{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 = 1.87, size = 145, normalized size = 0.62 \begin {gather*} \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 (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))))\right )}{1536 a d (-i+\tan (c+d x)) \sqrt {a+i a \tan (c+d x)}} \end {gather*}

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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Maple [A]
time = 0.84, size = 373, normalized size = 1.60

method	result	size
default	\(\frac {\sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (1024 i \left (\cos ^{7}\left (d x +c \right )\right )+1024 \sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )+64 i \left (\cos ^{5}\left (d x +c \right )\right )+315 i \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\left (i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )\right ) \sqrt {2}}{2 \sin \left (d x +c \right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right ) \cos \left (d x +c \right )+576 \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )+168 i \left (\cos ^{3}\left (d x +c \right )\right )+315 i \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\left (i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )\right ) \sqrt {2}}{2 \sin \left (d x +c \right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right )+315 \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\left (i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )\right ) \sqrt {2}}{2 \sin \left (d x +c \right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right ) \sin \left (d x +c \right )+840 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-1260 i \cos \left (d x +c \right )\right )}{3072 d \,a^{2}}\)	\(373\)

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,method=_RETURNVERBOSE)

[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))*cos(d*x+c)*2^(1/2)+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)*(-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))*sin(d*x+c)+840*cos(d*x+c)^2*sin(d*x+c)-
1260*I*cos(d*x+c))/a^2

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2632 vs. \(2 (178) = 356\).
time = 0.71, size = 2632, normalized size = 11.30 \begin {gather*} \text {Too large to display} \end {gather*}

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*(8*(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)*((-4*I*sqrt(2)*cos(6*d*x + 6
*c) - 45*I*sqrt(2)*cos(2/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) - 4*sqrt(2)*sin(6*d*x + 6*c) - 45*sqrt
(2)*sin(2/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 8*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)) + (4*sqrt(2)*cos(6*
d*x + 6*c) + 45*sqrt(2)*cos(2/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) - 4*I*sqrt(2)*sin(6*d*x + 6*c) -
45*I*sqrt(2)*sin(2/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) - 8*sqrt(2))*sin(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)))*sqrt(a) + 1
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)*(((-I*sqrt(2)*cos(6*d*x + 6*c) - 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 + (-I*sqrt(2)*cos(6*d*x + 6*c) -
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*(-I*sqrt(2)*cos(6*d*x + 6
*c) - sqrt(2)*sin(6*d*x + 6*c))*cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) - I*sqrt(2)*cos(6*d*x + 6
*c) - 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)) + (I*sqrt(2)*cos(6*d*x + 6*c) + 18*I*sqrt(2)*cos(2/3*arctan
2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) - 24*I*sqrt(2)*cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + s
qrt(2)*sin(6*d*x + 6*c) + 18*sqrt(2)*sin(2/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) - 24*sqrt(2)*sin(1/3
*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 64*I*sqrt(2))*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)) + ((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), cos(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))), co
s(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1)) - (sqrt(2)*cos(6*d*x + 6*c) + 18*sqrt(2)*cos(2/3*arct
an2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) - 24*sqrt(2)*cos(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) - I
*sqrt(2)*sin(6*d*x + 6*c) - 18*I*sqrt(2)*sin(2/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 24*I*sqrt(2)*s
in(1/3*arctan2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 64*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), cos(6*d*x + 6*c))) + 1)))*sqrt(a) + 315*(2*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*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)), (cos(1/3*a
rctan2(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) - 2*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(s
in(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) - I*sqrt(2)*log(sqrt(cos(1/3*arctan2(si
n(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*arct
an2(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*arctan2(sin(6*d*x + 6*c), co
s(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*arct
an2(sin(6*d*x + 6*c), cos(6*d*x + 6*c))) + 1))^...

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Fricas [A]
time = 0.38, size = 292, normalized size = 1.25 \begin {gather*} \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 {105 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (i \, a d e^{\left (2 i \, d x + 2 i \, c\right )} + 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}}} - i\right )} e^{\left (-i \, d x - i \, c\right )}}{128 \, 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 {105 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (-i \, a d e^{\left (2 i \, d x + 2 i \, c\right )} - 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}}} - i\right )} e^{\left (-i \, d x - i \, c\right )}}{128 \, 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} \end {gather*}

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(-105/128*(sqrt(2)*sqrt(1/2)*(I*a*d*e^
(2*I*d*x + 2*I*c) + I*a*d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(1/(a^3*d^2)) - 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(-105/128*(sqrt(2)*sqrt(1/2)*(-I*a*d*e^(2*I*d*x
+ 2*I*c) - I*a*d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(1/(a^3*d^2)) - 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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos ^{3}{\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}

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]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

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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