3.377 \(\int \frac{\cos ^3(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx\)
Optimal. Leaf size=270 \[ -\frac{77 i \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{512 a^3 d}+\frac{33 i \cos ^3(c+d x)}{256 a^2 d \sqrt{a+i a \tan (c+d x)}}-\frac{1155 i \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{4096 a^3 d}+\frac{385 i \cos (c+d x)}{2048 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{1155 i \tanh ^{-1}\left (\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right )}{4096 \sqrt{2} a^{5/2} d}+\frac{11 i \cos ^3(c+d x)}{96 a d (a+i a \tan (c+d x))^{3/2}}+\frac{i \cos ^3(c+d x)}{8 d (a+i a \tan (c+d x))^{5/2}} \]
[Out]
(((1155*I)/4096)*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(Sqrt[2]*a^(5/2)*d) + (
(I/8)*Cos[c + d*x]^3)/(d*(a + I*a*Tan[c + d*x])^(5/2)) + (((11*I)/96)*Cos[c + d*x]^3)/(a*d*(a + I*a*Tan[c + d*
x])^(3/2)) + (((385*I)/2048)*Cos[c + d*x])/(a^2*d*Sqrt[a + I*a*Tan[c + d*x]]) + (((33*I)/256)*Cos[c + d*x]^3)/
(a^2*d*Sqrt[a + I*a*Tan[c + d*x]]) - (((1155*I)/4096)*Cos[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(a^3*d) - (((77
*I)/512)*Cos[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]])/(a^3*d)
________________________________________________________________________________________
Rubi [A] time = 0.422262, antiderivative size = 270, normalized size of antiderivative = 1.,
number of steps used = 8, 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} \[ -\frac{77 i \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{512 a^3 d}+\frac{33 i \cos ^3(c+d x)}{256 a^2 d \sqrt{a+i a \tan (c+d x)}}-\frac{1155 i \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{4096 a^3 d}+\frac{385 i \cos (c+d x)}{2048 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{1155 i \tanh ^{-1}\left (\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right )}{4096 \sqrt{2} a^{5/2} d}+\frac{11 i \cos ^3(c+d x)}{96 a d (a+i a \tan (c+d x))^{3/2}}+\frac{i \cos ^3(c+d x)}{8 d (a+i a \tan (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
[In]
Int[Cos[c + d*x]^3/(a + I*a*Tan[c + d*x])^(5/2),x]
[Out]
(((1155*I)/4096)*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(Sqrt[2]*a^(5/2)*d) + (
(I/8)*Cos[c + d*x]^3)/(d*(a + I*a*Tan[c + d*x])^(5/2)) + (((11*I)/96)*Cos[c + d*x]^3)/(a*d*(a + I*a*Tan[c + d*
x])^(3/2)) + (((385*I)/2048)*Cos[c + d*x])/(a^2*d*Sqrt[a + I*a*Tan[c + d*x]]) + (((33*I)/256)*Cos[c + d*x]^3)/
(a^2*d*Sqrt[a + I*a*Tan[c + d*x]]) - (((1155*I)/4096)*Cos[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(a^3*d) - (((77
*I)/512)*Cos[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]])/(a^3*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))^{5/2}} \, dx &=\frac{i \cos ^3(c+d x)}{8 d (a+i a \tan (c+d x))^{5/2}}+\frac{11 \int \frac{\cos ^3(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx}{16 a}\\ &=\frac{i \cos ^3(c+d x)}{8 d (a+i a \tan (c+d x))^{5/2}}+\frac{11 i \cos ^3(c+d x)}{96 a d (a+i a \tan (c+d x))^{3/2}}+\frac{33 \int \frac{\cos ^3(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx}{64 a^2}\\ &=\frac{i \cos ^3(c+d x)}{8 d (a+i a \tan (c+d x))^{5/2}}+\frac{11 i \cos ^3(c+d x)}{96 a d (a+i a \tan (c+d x))^{3/2}}+\frac{33 i \cos ^3(c+d x)}{256 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{231 \int \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx}{512 a^3}\\ &=\frac{i \cos ^3(c+d x)}{8 d (a+i a \tan (c+d x))^{5/2}}+\frac{11 i \cos ^3(c+d x)}{96 a d (a+i a \tan (c+d x))^{3/2}}+\frac{33 i \cos ^3(c+d x)}{256 a^2 d \sqrt{a+i a \tan (c+d x)}}-\frac{77 i \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{512 a^3 d}+\frac{385 \int \frac{\cos (c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx}{1024 a^2}\\ &=\frac{i \cos ^3(c+d x)}{8 d (a+i a \tan (c+d x))^{5/2}}+\frac{11 i \cos ^3(c+d x)}{96 a d (a+i a \tan (c+d x))^{3/2}}+\frac{385 i \cos (c+d x)}{2048 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{33 i \cos ^3(c+d x)}{256 a^2 d \sqrt{a+i a \tan (c+d x)}}-\frac{77 i \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{512 a^3 d}+\frac{1155 \int \cos (c+d x) \sqrt{a+i a \tan (c+d x)} \, dx}{4096 a^3}\\ &=\frac{i \cos ^3(c+d x)}{8 d (a+i a \tan (c+d x))^{5/2}}+\frac{11 i \cos ^3(c+d x)}{96 a d (a+i a \tan (c+d x))^{3/2}}+\frac{385 i \cos (c+d x)}{2048 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{33 i \cos ^3(c+d x)}{256 a^2 d \sqrt{a+i a \tan (c+d x)}}-\frac{1155 i \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{4096 a^3 d}-\frac{77 i \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{512 a^3 d}+\frac{1155 \int \frac{\sec (c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx}{8192 a^2}\\ &=\frac{i \cos ^3(c+d x)}{8 d (a+i a \tan (c+d x))^{5/2}}+\frac{11 i \cos ^3(c+d x)}{96 a d (a+i a \tan (c+d x))^{3/2}}+\frac{385 i \cos (c+d x)}{2048 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{33 i \cos ^3(c+d x)}{256 a^2 d \sqrt{a+i a \tan (c+d x)}}-\frac{1155 i \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{4096 a^3 d}-\frac{77 i \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{512 a^3 d}+\frac{(1155 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 )}{4096 a^2 d}\\ &=\frac{1155 i \tanh ^{-1}\left (\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right )}{4096 \sqrt{2} a^{5/2} d}+\frac{i \cos ^3(c+d x)}{8 d (a+i a \tan (c+d x))^{5/2}}+\frac{11 i \cos ^3(c+d x)}{96 a d (a+i a \tan (c+d x))^{3/2}}+\frac{385 i \cos (c+d x)}{2048 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{33 i \cos ^3(c+d x)}{256 a^2 d \sqrt{a+i a \tan (c+d x)}}-\frac{1155 i \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{4096 a^3 d}-\frac{77 i \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{512 a^3 d}\\ \end{align*}
Mathematica [A] time = 1.19126, size = 165, normalized size = 0.61 \[ \frac{i \sec ^3(c+d x) \left (1111 i \sin (2 (c+d x))+2552 i \sin (4 (c+d x))+176 i \sin (6 (c+d x))-1605 \cos (2 (c+d x))+1800 \cos (4 (c+d x))+80 \cos (6 (c+d x))-3465 e^{2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left (\sqrt{1+e^{2 i (c+d x)}}\right )-3325\right )}{24576 a^2 d (\tan (c+d x)-i)^2 \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
[In]
Integrate[Cos[c + d*x]^3/(a + I*a*Tan[c + d*x])^(5/2),x]
[Out]
((I/24576)*Sec[c + d*x]^3*(-3325 - 3465*E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[1 + E^(
(2*I)*(c + d*x))]] - 1605*Cos[2*(c + d*x)] + 1800*Cos[4*(c + d*x)] + 80*Cos[6*(c + d*x)] + (1111*I)*Sin[2*(c +
d*x)] + (2552*I)*Sin[4*(c + d*x)] + (176*I)*Sin[6*(c + d*x)]))/(a^2*d*(-I + Tan[c + d*x])^2*Sqrt[a + I*a*Tan[
c + d*x]])
________________________________________________________________________________________
Maple [A] time = 0.343, size = 400, normalized size = 1.5 \begin{align*}{\frac{1}{49152\,d{a}^{3}}\sqrt{{\frac{a \left ( i\sin \left ( dx+c \right ) +\cos \left ( dx+c \right ) \right ) }{\cos \left ( dx+c \right ) }}} \left ( 24576\,i \left ( \cos \left ( dx+c \right ) \right ) ^{9}+24576\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{8}-7168\,i \left ( \cos \left ( dx+c \right ) \right ) ^{7}+5120\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}\sin \left ( dx+c \right ) +704\,i \left ( \cos \left ( dx+c \right ) \right ) ^{5}+3465\,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 ) +6336\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}+3465\,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 ) +1848\,i \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3465\,\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 ) +9240\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) -13860\,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))^(5/2),x)
[Out]
1/49152/d/a^3*(a*(I*sin(d*x+c)+cos(d*x+c))/cos(d*x+c))^(1/2)*(24576*I*cos(d*x+c)^9+24576*sin(d*x+c)*cos(d*x+c)
^8-7168*I*cos(d*x+c)^7+5120*cos(d*x+c)^6*sin(d*x+c)+704*I*cos(d*x+c)^5+3465*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))+6336*sin(d*x+c)*cos(d*x+c)^4+3465*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)/(cos(d*x+c)+1))^(1/2))+1848*I*cos(d*x+c)^3+3465*(-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)+9240*cos(d*x+c)^2*sin(d*x+c)-13860*I*cos(d*x+c))
________________________________________________________________________________________
Maxima [B] time = 2.62346, size = 5107, normalized size = 18.91 \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))^(5/2),x, algorithm="maxima")
[Out]
1/98304*((cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x
+ 8*c)))^2 + 2*cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 1)^(3/4)*(((60*I*sqrt(2)*cos(8*d*x + 8*
c) + 60*sqrt(2)*sin(8*d*x + 8*c))*cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + (60*I*sqrt(2)*cos(8
*d*x + 8*c) + 60*sqrt(2)*sin(8*d*x + 8*c))*sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + (120*I*sqr
t(2)*cos(8*d*x + 8*c) + 120*sqrt(2)*sin(8*d*x + 8*c))*cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 6
0*I*sqrt(2)*cos(8*d*x + 8*c) + 60*sqrt(2)*sin(8*d*x + 8*c))*cos(7/2*arctan2(sin(1/4*arctan2(sin(8*d*x + 8*c),
cos(8*d*x + 8*c))), cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 1)) + (-220*I*sqrt(2)*cos(8*d*x + 8
*c) - 3840*I*sqrt(2)*cos(3/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 5184*I*sqrt(2)*cos(1/2*arctan2(sin
(8*d*x + 8*c), cos(8*d*x + 8*c))) - 220*sqrt(2)*sin(8*d*x + 8*c) - 3840*sqrt(2)*sin(3/4*arctan2(sin(8*d*x + 8*
c), cos(8*d*x + 8*c))) + 5184*sqrt(2)*sin(1/2*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) - 512*I*sqrt(2))*co
s(3/2*arctan2(sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))), cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*
x + 8*c))) + 1)) - (60*(sqrt(2)*cos(8*d*x + 8*c) - I*sqrt(2)*sin(8*d*x + 8*c))*cos(1/4*arctan2(sin(8*d*x + 8*c
), cos(8*d*x + 8*c)))^2 + 60*(sqrt(2)*cos(8*d*x + 8*c) - I*sqrt(2)*sin(8*d*x + 8*c))*sin(1/4*arctan2(sin(8*d*x
+ 8*c), cos(8*d*x + 8*c)))^2 + 120*(sqrt(2)*cos(8*d*x + 8*c) - I*sqrt(2)*sin(8*d*x + 8*c))*cos(1/4*arctan2(si
n(8*d*x + 8*c), cos(8*d*x + 8*c))) + 60*sqrt(2)*cos(8*d*x + 8*c) - 60*I*sqrt(2)*sin(8*d*x + 8*c))*sin(7/2*arct
an2(sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))), cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))
+ 1)) + (220*sqrt(2)*cos(8*d*x + 8*c) + 3840*sqrt(2)*cos(3/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) - 5
184*sqrt(2)*cos(1/2*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) - 220*I*sqrt(2)*sin(8*d*x + 8*c) - 3840*I*sqr
t(2)*sin(3/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 5184*I*sqrt(2)*sin(1/2*arctan2(sin(8*d*x + 8*c), c
os(8*d*x + 8*c))) + 512*sqrt(2))*sin(3/2*arctan2(sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))), cos(1/4
*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 1)))*sqrt(a) + (cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x +
8*c)))^2 + sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + 2*cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*
d*x + 8*c))) + 1)^(1/4)*(((292*I*sqrt(2)*cos(8*d*x + 8*c) + 292*sqrt(2)*sin(8*d*x + 8*c))*cos(1/4*arctan2(sin(
8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + (292*I*sqrt(2)*cos(8*d*x + 8*c) + 292*sqrt(2)*sin(8*d*x + 8*c))*sin(1/4*a
rctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + (3168*I*sqrt(2)*cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x +
8*c)))^2 + 3168*I*sqrt(2)*sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + 6336*I*sqrt(2)*cos(1/4*arct
an2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 3168*I*sqrt(2))*cos(3/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))
) + (584*I*sqrt(2)*cos(8*d*x + 8*c) + 584*sqrt(2)*sin(8*d*x + 8*c))*cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*
x + 8*c))) + 3168*(sqrt(2)*cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + sqrt(2)*sin(1/4*arctan2(si
n(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + 2*sqrt(2)*cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + sqrt(2
))*sin(3/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 292*I*sqrt(2)*cos(8*d*x + 8*c) + 292*sqrt(2)*sin(8*d
*x + 8*c))*cos(5/2*arctan2(sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))), cos(1/4*arctan2(sin(8*d*x + 8
*c), cos(8*d*x + 8*c))) + 1)) + (60*I*sqrt(2)*cos(8*d*x + 8*c) + 1440*I*sqrt(2)*cos(3/4*arctan2(sin(8*d*x + 8*
c), cos(8*d*x + 8*c))) - 4032*I*sqrt(2)*cos(1/2*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 768*I*sqrt(2)*c
os(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 60*sqrt(2)*sin(8*d*x + 8*c) + 1440*sqrt(2)*sin(3/4*arcta
n2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) - 4032*sqrt(2)*sin(1/2*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) +
768*sqrt(2)*sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) - 7680*I*sqrt(2))*cos(1/2*arctan2(sin(1/4*arc
tan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))), cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 1)) - (292*(
sqrt(2)*cos(8*d*x + 8*c) - I*sqrt(2)*sin(8*d*x + 8*c))*cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2
+ 292*(sqrt(2)*cos(8*d*x + 8*c) - I*sqrt(2)*sin(8*d*x + 8*c))*sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*
c)))^2 + 3168*(sqrt(2)*cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + sqrt(2)*sin(1/4*arctan2(sin(8*
d*x + 8*c), cos(8*d*x + 8*c)))^2 + 2*sqrt(2)*cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + sqrt(2))*c
os(3/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 584*(sqrt(2)*cos(8*d*x + 8*c) - I*sqrt(2)*sin(8*d*x + 8*
c))*cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) - (3168*I*sqrt(2)*cos(1/4*arctan2(sin(8*d*x + 8*c), c
os(8*d*x + 8*c)))^2 + 3168*I*sqrt(2)*sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + 6336*I*sqrt(2)*c
os(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 3168*I*sqrt(2))*sin(3/4*arctan2(sin(8*d*x + 8*c), cos(8*
d*x + 8*c))) + 292*sqrt(2)*cos(8*d*x + 8*c) - 292*I*sqrt(2)*sin(8*d*x + 8*c))*sin(5/2*arctan2(sin(1/4*arctan2(
sin(8*d*x + 8*c), cos(8*d*x + 8*c))), cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 1)) - (60*sqrt(2)
*cos(8*d*x + 8*c) + 1440*sqrt(2)*cos(3/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) - 4032*sqrt(2)*cos(1/2*a
rctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 768*sqrt(2)*cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))
- 60*I*sqrt(2)*sin(8*d*x + 8*c) - 1440*I*sqrt(2)*sin(3/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 4032*
I*sqrt(2)*sin(1/2*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) - 768*I*sqrt(2)*sin(1/4*arctan2(sin(8*d*x + 8*c
), cos(8*d*x + 8*c))) - 7680*sqrt(2))*sin(1/2*arctan2(sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))), co
s(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 1)))*sqrt(a) - (6930*sqrt(2)*arctan2((cos(1/4*arctan2(sin
(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + 2*cos(1/4*arcta
n2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 1)^(1/4)*sin(1/2*arctan2(sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x
+ 8*c))), cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 1)), (cos(1/4*arctan2(sin(8*d*x + 8*c), cos(
8*d*x + 8*c)))^2 + sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + 2*cos(1/4*arctan2(sin(8*d*x + 8*c)
, cos(8*d*x + 8*c))) + 1)^(1/4)*cos(1/2*arctan2(sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))), cos(1/4*
arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 1)) + 1) - 6930*sqrt(2)*arctan2((cos(1/4*arctan2(sin(8*d*x + 8*
c), cos(8*d*x + 8*c)))^2 + sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + 2*cos(1/4*arctan2(sin(8*d*
x + 8*c), cos(8*d*x + 8*c))) + 1)^(1/4)*sin(1/2*arctan2(sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))),
cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 1)), (cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c
)))^2 + sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + 2*cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x
+ 8*c))) + 1)^(1/4)*cos(1/2*arctan2(sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))), cos(1/4*arctan2(sin
(8*d*x + 8*c), cos(8*d*x + 8*c))) + 1)) - 1) - 3465*I*sqrt(2)*log(sqrt(cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8
*d*x + 8*c)))^2 + sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + 2*cos(1/4*arctan2(sin(8*d*x + 8*c),
cos(8*d*x + 8*c))) + 1)*cos(1/2*arctan2(sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))), cos(1/4*arctan2
(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 1))^2 + sqrt(cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 +
sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + 2*cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))
) + 1)*sin(1/2*arctan2(sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))), cos(1/4*arctan2(sin(8*d*x + 8*c),
cos(8*d*x + 8*c))) + 1))^2 + 2*(cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + sin(1/4*arctan2(sin(
8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + 2*cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 1)^(1/4)*cos(1/2
*arctan2(sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))), cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8
*c))) + 1)) + 1) + 3465*I*sqrt(2)*log(sqrt(cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + sin(1/4*ar
ctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + 2*cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 1)*cos
(1/2*arctan2(sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))), cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x
+ 8*c))) + 1))^2 + sqrt(cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + sin(1/4*arctan2(sin(8*d*x +
8*c), cos(8*d*x + 8*c)))^2 + 2*cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 1)*sin(1/2*arctan2(sin(1
/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))), cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 1))^2
- 2*(cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*
c)))^2 + 2*cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 1)^(1/4)*cos(1/2*arctan2(sin(1/4*arctan2(sin
(8*d*x + 8*c), cos(8*d*x + 8*c))), cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 1)) + 1))*sqrt(a))/(
a^3*d)
________________________________________________________________________________________
Fricas [A] time = 2.1253, size = 988, normalized size = 3.66 \begin{align*} \frac{{\left (3465 i \, \sqrt{\frac{1}{2}} a^{3} d \sqrt{\frac{1}{a^{5} d^{2}}} e^{\left (9 i \, d x + 9 i \, c\right )} \log \left ({\left (2 \, \sqrt{\frac{1}{2}} a^{3} d \sqrt{\frac{1}{a^{5} 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 ) - 3465 i \, \sqrt{\frac{1}{2}} a^{3} d \sqrt{\frac{1}{a^{5} d^{2}}} e^{\left (9 i \, d x + 9 i \, c\right )} \log \left (-{\left (2 \, \sqrt{\frac{1}{2}} a^{3} d \sqrt{\frac{1}{a^{5} 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 (-128 i \, e^{\left (12 i \, d x + 12 i \, c\right )} - 2176 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 247 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 3325 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 1358 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 376 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 48 i\right )} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-9 i \, d x - 9 i \, c\right )}}{24576 \, a^{3} 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))^(5/2),x, algorithm="fricas")
[Out]
1/24576*(3465*I*sqrt(1/2)*a^3*d*sqrt(1/(a^5*d^2))*e^(9*I*d*x + 9*I*c)*log((2*sqrt(1/2)*a^3*d*sqrt(1/(a^5*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)) - 3465*I*sqrt(1/2)*a^3*d*sqrt(1/(a^5*d^2))*e^(9*I*d*x + 9*I*c)*log(-(2*sqrt(1/2)*a^3*d*sqrt(1/(a^5
*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))*(-128*I*e^(12*I*d*x + 12*I*c) - 2176*I*e^(10*I*d
*x + 10*I*c) + 247*I*e^(8*I*d*x + 8*I*c) + 3325*I*e^(6*I*d*x + 6*I*c) + 1358*I*e^(4*I*d*x + 4*I*c) + 376*I*e^(
2*I*d*x + 2*I*c) + 48*I)*e^(I*d*x + I*c))*e^(-9*I*d*x - 9*I*c)/(a^3*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))**(5/2),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{5}{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))^(5/2),x, algorithm="giac")
[Out]
integrate(cos(d*x + c)^3/(I*a*tan(d*x + c) + a)^(5/2), x)