3.361 \(\int \frac{\cos ^3(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx\)
Optimal. Leaf size=233 \[ -\frac{7 i \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{32 a^2 d}-\frac{105 i \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{256 a^2 d}+\frac{105 i \tanh ^{-1}\left (\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right )}{256 \sqrt{2} a^{3/2} d}+\frac{3 i \cos ^3(c+d x)}{16 a d \sqrt{a+i a \tan (c+d x)}}+\frac{i \cos ^3(c+d x)}{6 d (a+i a \tan (c+d x))^{3/2}}+\frac{35 i \cos (c+d x)}{128 a d \sqrt{a+i a \tan (c+d x)}} \]
[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)
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Rubi [A] time = 0.337251, antiderivative size = 233, normalized size of antiderivative = 1.,
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 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]
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 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 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 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])
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*}
Mathematica [A] time = 1.5202, 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]])
________________________________________________________________________________________
Maple [A] time = 0.301, size = 373, normalized size = 1.6 \begin{align*}{\frac{1}{3072\,{a}^{2}d}\sqrt{{\frac{a \left ( i\sin \left ( dx+c \right ) +\cos \left ( dx+c \right ) \right ) }{\cos \left ( dx+c \right ) }}} \left ( 1024\,i \left ( \cos \left ( dx+c \right ) \right ) ^{7}+1024\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}\sin \left ( dx+c \right ) +64\,i \left ( \cos \left ( dx+c \right ) \right ) ^{5}+315\,i\cos \left ( dx+c \right ) \sqrt{2}\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\arctan \left ({\frac{\sqrt{2} \left ( i\cos \left ( dx+c \right ) -i+\sin \left ( dx+c \right ) \right ) }{2\,\sin \left ( dx+c \right ) }{\frac{1}{\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}}}} \right ) +576\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}+315\,i\sqrt{2}\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\arctan \left ({\frac{\sqrt{2} \left ( i\cos \left ( dx+c \right ) -i+\sin \left ( dx+c \right ) \right ) }{2\,\sin \left ( dx+c \right ) }{\frac{1}{\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}}}} \right ) +168\,i \left ( \cos \left ( dx+c \right ) \right ) ^{3}+315\,\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\arctan \left ( 1/2\,{\frac{\sqrt{2} \left ( i\cos \left ( dx+c \right ) -i+\sin \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }{\frac{1}{\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}}}} \right ) \sqrt{2}\sin \left ( dx+c \right ) +840\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) -1260\,i\cos \left ( dx+c \right ) \right ) } \end{align*}
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^2*(a*(I*sin(d*x+c)+cos(d*x+c))/cos(d*x+c))^(1/2)*(1024*I*cos(d*x+c)^7+1024*cos(d*x+c)^6*sin(d*x+c)+
64*I*cos(d*x+c)^5+315*I*cos(d*x+c)*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/2*2^(1/2)*(I*cos(d*x+
c)-I+sin(d*x+c))/sin(d*x+c)/(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+576*sin(d*x+c)*cos(d*x+c)^4+315*I*2^(1/2)*(-
2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/2*2^(1/2)*(I*cos(d*x+c)-I+sin(d*x+c))/sin(d*x+c)/(-2*cos(d*x+c)/(c
os(d*x+c)+1))^(1/2))+168*I*cos(d*x+c)^3+315*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/2*2^(1/2)*(I*cos(d*x
+c)-I+sin(d*x+c))/sin(d*x+c)/(-2*cos(d*x+c)/(cos(d*x+c)+1))^(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))
________________________________________________________________________________________
Maxima [B] time = 2.44383, size = 3553, normalized size = 15.25 \begin{align*} \text{result too large to display} \end{align*}
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)
________________________________________________________________________________________
Fricas [A] time = 2.23513, size = 934, normalized size = 4.01 \begin{align*} \frac{{\left (315 i \, \sqrt{\frac{1}{2}} a^{2} d \sqrt{\frac{1}{a^{3} d^{2}}} e^{\left (7 i \, d x + 7 i \, c\right )} \log \left ({\left (2 \, \sqrt{\frac{1}{2}} a^{2} d \sqrt{\frac{1}{a^{3} d^{2}}} e^{\left (i \, d x + i \, c\right )} + \sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) - 315 i \, \sqrt{\frac{1}{2}} a^{2} d \sqrt{\frac{1}{a^{3} d^{2}}} e^{\left (7 i \, d x + 7 i \, c\right )} \log \left (-{\left (2 \, \sqrt{\frac{1}{2}} a^{2} d \sqrt{\frac{1}{a^{3} d^{2}}} e^{\left (i \, d x + i \, c\right )} - \sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\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 )} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-7 i \, d x - 7 i \, c\right )}}{1536 \, a^{2} d} \end{align*}
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^(7*I*d*x + 7*I*c)*log((2*sqrt(1/2)*a^2*d*sqrt(1/(a^3*d^2))*e
^(I*d*x + I*c) + sqrt(2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*(e^(2*I*d*x + 2*I*c) + 1)*e^(I*d*x + I*c))*e^(-I*d*
x - I*c)) - 315*I*sqrt(1/2)*a^2*d*sqrt(1/(a^3*d^2))*e^(7*I*d*x + 7*I*c)*log(-(2*sqrt(1/2)*a^2*d*sqrt(1/(a^3*d^
2))*e^(I*d*x + I*c) - sqrt(2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*(e^(2*I*d*x + 2*I*c) + 1)*e^(I*d*x + I*c))*e^(
-I*d*x - I*c)) + 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^(I*d*x + I*c)
)*e^(-7*I*d*x - 7*I*c)/(a^2*d)
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
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
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
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)