3.392 \(\int \frac{\cos (c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx\)
Optimal. Leaf size=227 \[ -\frac{315 i \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{2048 a^4 d}+\frac{105 i \cos (c+d x)}{1024 a^3 d \sqrt{a+i a \tan (c+d x)}}+\frac{21 i \cos (c+d x)}{256 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac{315 i \tanh ^{-1}\left (\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right )}{2048 \sqrt{2} a^{7/2} d}+\frac{3 i \cos (c+d x)}{32 a d (a+i a \tan (c+d x))^{5/2}}+\frac{i \cos (c+d x)}{8 d (a+i a \tan (c+d x))^{7/2}} \]
[Out]
(((315*I)/2048)*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(Sqrt[2]*a^(7/2)*d) + ((
I/8)*Cos[c + d*x])/(d*(a + I*a*Tan[c + d*x])^(7/2)) + (((3*I)/32)*Cos[c + d*x])/(a*d*(a + I*a*Tan[c + d*x])^(5
/2)) + (((21*I)/256)*Cos[c + d*x])/(a^2*d*(a + I*a*Tan[c + d*x])^(3/2)) + (((105*I)/1024)*Cos[c + d*x])/(a^3*d
*Sqrt[a + I*a*Tan[c + d*x]]) - (((315*I)/2048)*Cos[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(a^4*d)
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Rubi [A] time = 0.375353, antiderivative size = 227, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used =
{3502, 3490, 3489, 206} \[ -\frac{315 i \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{2048 a^4 d}+\frac{105 i \cos (c+d x)}{1024 a^3 d \sqrt{a+i a \tan (c+d x)}}+\frac{21 i \cos (c+d x)}{256 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac{315 i \tanh ^{-1}\left (\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right )}{2048 \sqrt{2} a^{7/2} d}+\frac{3 i \cos (c+d x)}{32 a d (a+i a \tan (c+d x))^{5/2}}+\frac{i \cos (c+d x)}{8 d (a+i a \tan (c+d x))^{7/2}} \]
Antiderivative was successfully verified.
[In]
Int[Cos[c + d*x]/(a + I*a*Tan[c + d*x])^(7/2),x]
[Out]
(((315*I)/2048)*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(Sqrt[2]*a^(7/2)*d) + ((
I/8)*Cos[c + d*x])/(d*(a + I*a*Tan[c + d*x])^(7/2)) + (((3*I)/32)*Cos[c + d*x])/(a*d*(a + I*a*Tan[c + d*x])^(5
/2)) + (((21*I)/256)*Cos[c + d*x])/(a^2*d*(a + I*a*Tan[c + d*x])^(3/2)) + (((105*I)/1024)*Cos[c + d*x])/(a^3*d
*Sqrt[a + I*a*Tan[c + d*x]]) - (((315*I)/2048)*Cos[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(a^4*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 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 (c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx &=\frac{i \cos (c+d x)}{8 d (a+i a \tan (c+d x))^{7/2}}+\frac{9 \int \frac{\cos (c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx}{16 a}\\ &=\frac{i \cos (c+d x)}{8 d (a+i a \tan (c+d x))^{7/2}}+\frac{3 i \cos (c+d x)}{32 a d (a+i a \tan (c+d x))^{5/2}}+\frac{21 \int \frac{\cos (c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx}{64 a^2}\\ &=\frac{i \cos (c+d x)}{8 d (a+i a \tan (c+d x))^{7/2}}+\frac{3 i \cos (c+d x)}{32 a d (a+i a \tan (c+d x))^{5/2}}+\frac{21 i \cos (c+d x)}{256 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac{105 \int \frac{\cos (c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx}{512 a^3}\\ &=\frac{i \cos (c+d x)}{8 d (a+i a \tan (c+d x))^{7/2}}+\frac{3 i \cos (c+d x)}{32 a d (a+i a \tan (c+d x))^{5/2}}+\frac{21 i \cos (c+d x)}{256 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac{105 i \cos (c+d x)}{1024 a^3 d \sqrt{a+i a \tan (c+d x)}}+\frac{315 \int \cos (c+d x) \sqrt{a+i a \tan (c+d x)} \, dx}{2048 a^4}\\ &=\frac{i \cos (c+d x)}{8 d (a+i a \tan (c+d x))^{7/2}}+\frac{3 i \cos (c+d x)}{32 a d (a+i a \tan (c+d x))^{5/2}}+\frac{21 i \cos (c+d x)}{256 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac{105 i \cos (c+d x)}{1024 a^3 d \sqrt{a+i a \tan (c+d x)}}-\frac{315 i \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{2048 a^4 d}+\frac{315 \int \frac{\sec (c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx}{4096 a^3}\\ &=\frac{i \cos (c+d x)}{8 d (a+i a \tan (c+d x))^{7/2}}+\frac{3 i \cos (c+d x)}{32 a d (a+i a \tan (c+d x))^{5/2}}+\frac{21 i \cos (c+d x)}{256 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac{105 i \cos (c+d x)}{1024 a^3 d \sqrt{a+i a \tan (c+d x)}}-\frac{315 i \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{2048 a^4 d}+\frac{(315 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 )}{2048 a^3 d}\\ &=\frac{315 i \tanh ^{-1}\left (\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right )}{2048 \sqrt{2} a^{7/2} d}+\frac{i \cos (c+d x)}{8 d (a+i a \tan (c+d x))^{7/2}}+\frac{3 i \cos (c+d x)}{32 a d (a+i a \tan (c+d x))^{5/2}}+\frac{21 i \cos (c+d x)}{256 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac{105 i \cos (c+d x)}{1024 a^3 d \sqrt{a+i a \tan (c+d x)}}-\frac{315 i \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{2048 a^4 d}\\ \end{align*}
Mathematica [A] time = 1.9443, size = 141, normalized size = 0.62 \[ -\frac{\sec ^3(c+d x) \left (474 i \sin (2 (c+d x))-288 i \sin (4 (c+d x))+826 \cos (2 (c+d x))-224 \cos (4 (c+d x))+\frac{630 e^{4 i (c+d x)} \tanh ^{-1}\left (\sqrt{1+e^{2 i (c+d x)}}\right )}{\sqrt{1+e^{2 i (c+d x)}}}+420\right )}{4096 a^3 d (\tan (c+d x)-i)^3 \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
[In]
Integrate[Cos[c + d*x]/(a + I*a*Tan[c + d*x])^(7/2),x]
[Out]
-(Sec[c + d*x]^3*(420 + (630*E^((4*I)*(c + d*x))*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])/Sqrt[1 + E^((2*I)*(c
+ d*x))] + 826*Cos[2*(c + d*x)] - 224*Cos[4*(c + d*x)] + (474*I)*Sin[2*(c + d*x)] - (288*I)*Sin[4*(c + d*x)]))
/(4096*a^3*d*(-I + Tan[c + d*x])^3*Sqrt[a + I*a*Tan[c + d*x]])
________________________________________________________________________________________
Maple [B] time = 0.303, size = 400, normalized size = 1.8 \begin{align*}{\frac{1}{8192\,{a}^{4}d}\sqrt{{\frac{a \left ( i\sin \left ( dx+c \right ) +\cos \left ( dx+c \right ) \right ) }{\cos \left ( dx+c \right ) }}} \left ( 8192\,i \left ( \cos \left ( dx+c \right ) \right ) ^{9}+8192\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{8}-5120\,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)/(a+I*a*tan(d*x+c))^(7/2),x)
[Out]
1/8192/d/a^4*(a*(I*sin(d*x+c)+cos(d*x+c))/cos(d*x+c))^(1/2)*(8192*I*cos(d*x+c)^9+8192*sin(d*x+c)*cos(d*x+c)^8-
5120*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)/(co
s(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)/(cos(d*x+c)+1))^(1/2))+168*I*cos(d*x+c)^3+315*(-2*cos(d*x+c)/(c
os(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 = 2.25113, size = 3754, normalized size = 16.54 \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)/(a+I*a*tan(d*x+c))^(7/2),x, algorithm="maxima")
[Out]
1/16384*((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)*(((1300*I*sqrt(2)*cos(8*d*x +
8*c) + 1300*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 + (1300*I*sqrt(2)
*cos(8*d*x + 8*c) + 1300*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 + (2
600*I*sqrt(2)*cos(8*d*x + 8*c) + 2600*sqrt(2)*sin(8*d*x + 8*c))*cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x +
8*c))) + 1300*I*sqrt(2)*cos(8*d*x + 8*c) + 1300*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)) + (2572*I*sqrt(2)
*cos(8*d*x + 8*c) + 2572*sqrt(2)*sin(8*d*x + 8*c))*cos(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)) - (1300*(sqrt(2)*cos(8*d*x + 8*c) - I*sq
rt(2)*sin(8*d*x + 8*c))*cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + 1300*(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 + 2600*(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))) + 1300*sqrt(2)*co
s(8*d*x + 8*c) - 1300*I*sqrt(2)*sin(8*d*x + 8*c))*sin(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)) - 2572*(sqrt(2)*cos(8*d*x + 8*c) - I*sqrt
(2)*sin(8*d*x + 8*c))*sin(3/2*arctan2(sin(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))), cos(1/4*arctan2(si
n(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)*(((-3060*I*sqrt(2)*cos(8*d*x + 8*c) - 3060*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 + (-3060*I*sqrt(2)*cos(8*d*x + 8*c) - 3060*sqrt(2)*sin(8*d*x + 8*c))*sin(1/4*arctan
2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + (-6120*I*sqrt(2)*cos(8*d*x + 8*c) - 6120*sqrt(2)*sin(8*d*x + 8*c))*
cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) - 3060*I*sqrt(2)*cos(8*d*x + 8*c) - 3060*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)) + (-748*I*sqrt(2)*cos(8*d*x + 8*c) - 748*sqrt(2)*sin(8*d*x + 8*c) - 512*I*sqrt(2)
)*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)) + (3060*(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 + 3060*(sqrt(2)*cos(8*d*x + 8*c) - I*sqrt(2)*sin(8*d*x + 8*c))*sin(1/4*arctan2(s
in(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + 6120*(sqrt(2)*cos(8*d*x + 8*c) - I*sqrt(2)*sin(8*d*x + 8*c))*cos(1/4*a
rctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c))) + 3060*sqrt(2)*cos(8*d*x + 8*c) - 3060*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)) + 4*(187*sqrt(2)*cos(8*d*x + 8*c) - 187*I*sqrt(2)*sin(8*d*x + 8*c) + 128*sqrt(2))*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)))*sqrt(a) - (630*sqrt(2)*arctan2((cos(1/4*arctan2(sin(8*d*x + 8*c), cos(8*d*x + 8*c)))^2 + sin(1/4*a
rctan2(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), co
s(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) - 630*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*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)) - 1) - 3
15*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 + s
qrt(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*ar
ctan2(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) + 315*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*c
os(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))), c
os(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^4*d)
________________________________________________________________________________________
Fricas [A] time = 2.12115, size = 940, normalized size = 4.14 \begin{align*} \frac{{\left (315 i \, \sqrt{\frac{1}{2}} a^{4} d \sqrt{\frac{1}{a^{7} d^{2}}} e^{\left (9 i \, d x + 9 i \, c\right )} \log \left ({\left (2 \, \sqrt{\frac{1}{2}} a^{4} d \sqrt{\frac{1}{a^{7} 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^{4} d \sqrt{\frac{1}{a^{7} d^{2}}} e^{\left (9 i \, d x + 9 i \, c\right )} \log \left (-{\left (2 \, \sqrt{\frac{1}{2}} a^{4} d \sqrt{\frac{1}{a^{7} 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 (10 i \, d x + 10 i \, c\right )} + 197 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 535 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 298 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 104 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 16 i\right )} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-9 i \, d x - 9 i \, c\right )}}{4096 \, a^{4} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)/(a+I*a*tan(d*x+c))^(7/2),x, algorithm="fricas")
[Out]
1/4096*(315*I*sqrt(1/2)*a^4*d*sqrt(1/(a^7*d^2))*e^(9*I*d*x + 9*I*c)*log((2*sqrt(1/2)*a^4*d*sqrt(1/(a^7*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^4*d*sqrt(1/(a^7*d^2))*e^(9*I*d*x + 9*I*c)*log(-(2*sqrt(1/2)*a^4*d*sqrt(1/(a^7*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^(10*I*d*x + 10*I*c) + 197*I*e^(8*I*d*x +
8*I*c) + 535*I*e^(6*I*d*x + 6*I*c) + 298*I*e^(4*I*d*x + 4*I*c) + 104*I*e^(2*I*d*x + 2*I*c) + 16*I)*e^(I*d*x +
I*c))*e^(-9*I*d*x - 9*I*c)/(a^4*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)/(a+I*a*tan(d*x+c))**(7/2),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (d x + c\right )}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)/(a+I*a*tan(d*x+c))^(7/2),x, algorithm="giac")
[Out]
integrate(cos(d*x + c)/(I*a*tan(d*x + c) + a)^(7/2), x)