3.292 \(\int \cos ^5(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx\)
Optimal. Leaf size=223 \[ -\frac{i \cos ^5(c+d x) \sqrt{a+i a \tan (c+d x)}}{5 d}-\frac{21 i \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{80 d}+\frac{9 i a \cos ^3(c+d x)}{40 d \sqrt{a+i a \tan (c+d x)}}-\frac{63 i \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{128 d}+\frac{21 i a \cos (c+d x)}{64 d \sqrt{a+i a \tan (c+d x)}}+\frac{63 i \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right )}{128 \sqrt{2} d} \]
[Out]
(((63*I)/128)*Sqrt[a]*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(Sqrt[2]*d) + (((2
1*I)/64)*a*Cos[c + d*x])/(d*Sqrt[a + I*a*Tan[c + d*x]]) + (((9*I)/40)*a*Cos[c + d*x]^3)/(d*Sqrt[a + I*a*Tan[c
+ d*x]]) - (((63*I)/128)*Cos[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/d - (((21*I)/80)*Cos[c + d*x]^3*Sqrt[a + I*a
*Tan[c + d*x]])/d - ((I/5)*Cos[c + d*x]^5*Sqrt[a + I*a*Tan[c + d*x]])/d
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Rubi [A] time = 0.390931, antiderivative size = 223, 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 =
{3497, 3502, 3490, 3489, 206} \[ -\frac{i \cos ^5(c+d x) \sqrt{a+i a \tan (c+d x)}}{5 d}-\frac{21 i \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{80 d}+\frac{9 i a \cos ^3(c+d x)}{40 d \sqrt{a+i a \tan (c+d x)}}-\frac{63 i \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{128 d}+\frac{21 i a \cos (c+d x)}{64 d \sqrt{a+i a \tan (c+d x)}}+\frac{63 i \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right )}{128 \sqrt{2} d} \]
Antiderivative was successfully verified.
[In]
Int[Cos[c + d*x]^5*Sqrt[a + I*a*Tan[c + d*x]],x]
[Out]
(((63*I)/128)*Sqrt[a]*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(Sqrt[2]*d) + (((2
1*I)/64)*a*Cos[c + d*x])/(d*Sqrt[a + I*a*Tan[c + d*x]]) + (((9*I)/40)*a*Cos[c + d*x]^3)/(d*Sqrt[a + I*a*Tan[c
+ d*x]]) - (((63*I)/128)*Cos[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/d - (((21*I)/80)*Cos[c + d*x]^3*Sqrt[a + I*a
*Tan[c + d*x]])/d - ((I/5)*Cos[c + d*x]^5*Sqrt[a + I*a*Tan[c + d*x]])/d
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]
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 \cos ^5(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx &=-\frac{i \cos ^5(c+d x) \sqrt{a+i a \tan (c+d x)}}{5 d}+\frac{1}{10} (9 a) \int \frac{\cos ^3(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx\\ &=\frac{9 i a \cos ^3(c+d x)}{40 d \sqrt{a+i a \tan (c+d x)}}-\frac{i \cos ^5(c+d x) \sqrt{a+i a \tan (c+d x)}}{5 d}+\frac{63}{80} \int \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx\\ &=\frac{9 i a \cos ^3(c+d x)}{40 d \sqrt{a+i a \tan (c+d x)}}-\frac{21 i \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{80 d}-\frac{i \cos ^5(c+d x) \sqrt{a+i a \tan (c+d x)}}{5 d}+\frac{1}{32} (21 a) \int \frac{\cos (c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx\\ &=\frac{21 i a \cos (c+d x)}{64 d \sqrt{a+i a \tan (c+d x)}}+\frac{9 i a \cos ^3(c+d x)}{40 d \sqrt{a+i a \tan (c+d x)}}-\frac{21 i \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{80 d}-\frac{i \cos ^5(c+d x) \sqrt{a+i a \tan (c+d x)}}{5 d}+\frac{63}{128} \int \cos (c+d x) \sqrt{a+i a \tan (c+d x)} \, dx\\ &=\frac{21 i a \cos (c+d x)}{64 d \sqrt{a+i a \tan (c+d x)}}+\frac{9 i a \cos ^3(c+d x)}{40 d \sqrt{a+i a \tan (c+d x)}}-\frac{63 i \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{128 d}-\frac{21 i \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{80 d}-\frac{i \cos ^5(c+d x) \sqrt{a+i a \tan (c+d x)}}{5 d}+\frac{1}{256} (63 a) \int \frac{\sec (c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx\\ &=\frac{21 i a \cos (c+d x)}{64 d \sqrt{a+i a \tan (c+d x)}}+\frac{9 i a \cos ^3(c+d x)}{40 d \sqrt{a+i a \tan (c+d x)}}-\frac{63 i \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{128 d}-\frac{21 i \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{80 d}-\frac{i \cos ^5(c+d x) \sqrt{a+i a \tan (c+d x)}}{5 d}+\frac{(63 i a) \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 )}{128 d}\\ &=\frac{63 i \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right )}{128 \sqrt{2} d}+\frac{21 i a \cos (c+d x)}{64 d \sqrt{a+i a \tan (c+d x)}}+\frac{9 i a \cos ^3(c+d x)}{40 d \sqrt{a+i a \tan (c+d x)}}-\frac{63 i \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{128 d}-\frac{21 i \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{80 d}-\frac{i \cos ^5(c+d x) \sqrt{a+i a \tan (c+d x)}}{5 d}\\ \end{align*}
Mathematica [A] time = 0.555925, size = 152, normalized size = 0.68 \[ -\frac{i e^{-5 i (c+d x)} \left (-95 e^{2 i (c+d x)}+203 e^{4 i (c+d x)}+344 e^{6 i (c+d x)}+64 e^{8 i (c+d x)}+8 e^{10 i (c+d x)}-315 e^{4 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left (\sqrt{1+e^{2 i (c+d x)}}\right )-10\right ) \sqrt{a+i a \tan (c+d x)}}{1280 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Cos[c + d*x]^5*Sqrt[a + I*a*Tan[c + d*x]],x]
[Out]
((-I/1280)*(-10 - 95*E^((2*I)*(c + d*x)) + 203*E^((4*I)*(c + d*x)) + 344*E^((6*I)*(c + d*x)) + 64*E^((8*I)*(c
+ d*x)) + 8*E^((10*I)*(c + d*x)) - 315*E^((4*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[1 + E^((
2*I)*(c + d*x))]])*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^((5*I)*(c + d*x)))
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Maple [B] time = 0.352, size = 913, normalized size = 4.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(d*x+c)^5*(a+I*a*tan(d*x+c))^(1/2),x)
[Out]
-1/20480/d*(768*I*cos(d*x+c)^8+3360*I*cos(d*x+c)^6-315*arctan(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)
)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(9/2)*2^(1/2)*cos(d*x+c)^4*sin(d*x+c)+4096*I*cos(d*x+c)^10-1260*arctan(1/2*2^
(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(9/2)*2^(1/2)*cos(d*x+c)^3*sin(d*x+
c)+315*I*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(9/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x
+c)/cos(d*x+c))*2^(1/2)*cos(d*x+c)^4*sin(d*x+c)-1890*arctan(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*
(-2*cos(d*x+c)/(cos(d*x+c)+1))^(9/2)*2^(1/2)*cos(d*x+c)^2*sin(d*x+c)+1344*I*cos(d*x+c)^7-1260*arctan(1/2*2^(1/
2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(9/2)*2^(1/2)*cos(d*x+c)*sin(d*x+c)-31
5*2^(1/2)*arctan(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(9/2)*sin(d*
x+c)+512*I*cos(d*x+c)^9+1260*I*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(9/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*
x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*2^(1/2)*cos(d*x+c)^3*sin(d*x+c)-4096*sin(d*x+c)*cos(d*x+c)^9+315*I*2^(1/
2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*(-2*cos(d*x+c)/(cos(d*x+c)+
1))^(9/2)*sin(d*x+c)+4608*sin(d*x+c)*cos(d*x+c)^8+1260*I*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(9/2)*arctanh(1/2*2^(1
/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*2^(1/2)*cos(d*x+c)*sin(d*x+c)-5376*sin(d*x+c)*
cos(d*x+c)^7-10080*I*cos(d*x+c)^5+6720*cos(d*x+c)^6*sin(d*x+c)+1890*I*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(9/2)*arc
tanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*2^(1/2)*cos(d*x+c)^2*sin(d*x+c)-1
0080*cos(d*x+c)^5*sin(d*x+c))*(a*(I*sin(d*x+c)+cos(d*x+c))/cos(d*x+c))^(1/2)/(I*sin(d*x+c)+cos(d*x+c)-1)/cos(d
*x+c)^4
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Maxima [B] time = 2.80716, size = 2990, normalized size = 13.41 \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)^5*(a+I*a*tan(d*x+c))^(1/2),x, algorithm="maxima")
[Out]
-1/5120*((cos(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c)))^2 + sin(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x
+ 4*c)))^2 + 2*cos(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c))) + 1)^(3/4)*((60*I*sqrt(2)*cos(4*d*x + 4*c
) + 60*sqrt(2)*sin(4*d*x + 4*c) + 160*I*sqrt(2))*cos(3/2*arctan2(sin(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x +
4*c))), cos(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c))) + 1)) - 20*(3*sqrt(2)*cos(4*d*x + 4*c) - 3*I*sqr
t(2)*sin(4*d*x + 4*c) + 8*sqrt(2))*sin(3/2*arctan2(sin(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c))), cos(1
/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c))) + 1)))*sqrt(a) + (cos(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x
+ 4*c)))^2 + sin(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c)))^2 + 2*cos(1/2*arctan2(sin(4*d*x + 4*c), cos(
4*d*x + 4*c))) + 1)^(1/4)*((32*I*sqrt(2)*cos(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c)))^2 + 32*I*sqrt(2)
*sin(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c)))^2 + 64*I*sqrt(2)*cos(1/2*arctan2(sin(4*d*x + 4*c), cos(4
*d*x + 4*c))) + 32*I*sqrt(2))*cos(5/2*arctan2(sin(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c))), cos(1/2*ar
ctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c))) + 1)) + (-100*I*sqrt(2)*cos(4*d*x + 4*c) - 400*I*sqrt(2)*cos(1/2*ar
ctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c))) - 100*sqrt(2)*sin(4*d*x + 4*c) - 400*sqrt(2)*sin(1/2*arctan2(sin(4*
d*x + 4*c), cos(4*d*x + 4*c))) + 960*I*sqrt(2))*cos(1/2*arctan2(sin(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x +
4*c))), cos(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c))) + 1)) - 32*(sqrt(2)*cos(1/2*arctan2(sin(4*d*x + 4
*c), cos(4*d*x + 4*c)))^2 + sqrt(2)*sin(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c)))^2 + 2*sqrt(2)*cos(1/2
*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c))) + sqrt(2))*sin(5/2*arctan2(sin(1/2*arctan2(sin(4*d*x + 4*c), cos
(4*d*x + 4*c))), cos(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c))) + 1)) + (100*sqrt(2)*cos(4*d*x + 4*c) +
400*sqrt(2)*cos(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c))) - 100*I*sqrt(2)*sin(4*d*x + 4*c) - 400*I*sqrt
(2)*sin(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c))) - 960*sqrt(2))*sin(1/2*arctan2(sin(1/2*arctan2(sin(4*
d*x + 4*c), cos(4*d*x + 4*c))), cos(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c))) + 1)))*sqrt(a) + (630*sqr
t(2)*arctan2((cos(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c)))^2 + sin(1/2*arctan2(sin(4*d*x + 4*c), cos(4
*d*x + 4*c)))^2 + 2*cos(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c))) + 1)^(1/4)*sin(1/2*arctan2(sin(1/2*ar
ctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c))), cos(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c))) + 1)), (cos(1
/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c)))^2 + sin(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c)))^2 + 2
*cos(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c))) + 1)^(1/4)*cos(1/2*arctan2(sin(1/2*arctan2(sin(4*d*x + 4
*c), cos(4*d*x + 4*c))), cos(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c))) + 1)) + 1) - 630*sqrt(2)*arctan2
((cos(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c)))^2 + sin(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c))
)^2 + 2*cos(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c))) + 1)^(1/4)*sin(1/2*arctan2(sin(1/2*arctan2(sin(4*
d*x + 4*c), cos(4*d*x + 4*c))), cos(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c))) + 1)), (cos(1/2*arctan2(s
in(4*d*x + 4*c), cos(4*d*x + 4*c)))^2 + sin(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c)))^2 + 2*cos(1/2*arc
tan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c))) + 1)^(1/4)*cos(1/2*arctan2(sin(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d
*x + 4*c))), cos(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c))) + 1)) - 1) - 315*I*sqrt(2)*log(sqrt(cos(1/2*
arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c)))^2 + sin(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c)))^2 + 2*co
s(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c))) + 1)*cos(1/2*arctan2(sin(1/2*arctan2(sin(4*d*x + 4*c), cos(
4*d*x + 4*c))), cos(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c))) + 1))^2 + sqrt(cos(1/2*arctan2(sin(4*d*x
+ 4*c), cos(4*d*x + 4*c)))^2 + sin(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c)))^2 + 2*cos(1/2*arctan2(sin(
4*d*x + 4*c), cos(4*d*x + 4*c))) + 1)*sin(1/2*arctan2(sin(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c))), co
s(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c))) + 1))^2 + 2*(cos(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x +
4*c)))^2 + sin(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c)))^2 + 2*cos(1/2*arctan2(sin(4*d*x + 4*c), cos(4*
d*x + 4*c))) + 1)^(1/4)*cos(1/2*arctan2(sin(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c))), cos(1/2*arctan2(
sin(4*d*x + 4*c), cos(4*d*x + 4*c))) + 1)) + 1) + 315*I*sqrt(2)*log(sqrt(cos(1/2*arctan2(sin(4*d*x + 4*c), cos
(4*d*x + 4*c)))^2 + sin(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c)))^2 + 2*cos(1/2*arctan2(sin(4*d*x + 4*c
), cos(4*d*x + 4*c))) + 1)*cos(1/2*arctan2(sin(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c))), cos(1/2*arcta
n2(sin(4*d*x + 4*c), cos(4*d*x + 4*c))) + 1))^2 + sqrt(cos(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c)))^2
+ sin(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c)))^2 + 2*cos(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c
))) + 1)*sin(1/2*arctan2(sin(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c))), cos(1/2*arctan2(sin(4*d*x + 4*c
), cos(4*d*x + 4*c))) + 1))^2 - 2*(cos(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c)))^2 + sin(1/2*arctan2(si
n(4*d*x + 4*c), cos(4*d*x + 4*c)))^2 + 2*cos(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c))) + 1)^(1/4)*cos(1
/2*arctan2(sin(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c))), cos(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x +
4*c))) + 1)) + 1))*sqrt(a))/d
________________________________________________________________________________________
Fricas [A] time = 2.44112, size = 906, normalized size = 4.06 \begin{align*} -\frac{{\left (315 \, \sqrt{\frac{1}{2}} d \sqrt{-\frac{a}{d^{2}}} e^{\left (5 i \, d x + 5 i \, c\right )} \log \left (\frac{1}{63} \,{\left (126 i \, \sqrt{\frac{1}{2}} d \sqrt{-\frac{a}{d^{2}}} e^{\left (i \, d x + i \, c\right )} + 63 \, \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 \, \sqrt{\frac{1}{2}} d \sqrt{-\frac{a}{d^{2}}} e^{\left (5 i \, d x + 5 i \, c\right )} \log \left (\frac{1}{63} \,{\left (-126 i \, \sqrt{\frac{1}{2}} d \sqrt{-\frac{a}{d^{2}}} e^{\left (i \, d x + i \, c\right )} + 63 \, \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 (-8 i \, e^{\left (10 i \, d x + 10 i \, c\right )} - 64 i \, e^{\left (8 i \, d x + 8 i \, c\right )} - 344 i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 203 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 95 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 10 i\right )} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-5 i \, d x - 5 i \, c\right )}}{1280 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^5*(a+I*a*tan(d*x+c))^(1/2),x, algorithm="fricas")
[Out]
-1/1280*(315*sqrt(1/2)*d*sqrt(-a/d^2)*e^(5*I*d*x + 5*I*c)*log(1/63*(126*I*sqrt(1/2)*d*sqrt(-a/d^2)*e^(I*d*x +
I*c) + 63*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*sqrt(1/2)*d*sqrt(-a/d^2)*e^(5*I*d*x + 5*I*c)*log(1/63*(-126*I*sqrt(1/2)*d*sqrt(-a/d^2)*e^(I*d*x + I*c
) + 63*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))*(-8*I*e^(10*I*d*x + 10*I*c) - 64*I*e^(8*I*d*x + 8*I*c) - 344*I*e^(
6*I*d*x + 6*I*c) - 203*I*e^(4*I*d*x + 4*I*c) + 95*I*e^(2*I*d*x + 2*I*c) + 10*I)*e^(I*d*x + I*c))*e^(-5*I*d*x -
5*I*c)/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)**5*(a+I*a*tan(d*x+c))**(1/2),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{i \, a \tan \left (d x + c\right ) + a} \cos \left (d x + c\right )^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^5*(a+I*a*tan(d*x+c))^(1/2),x, algorithm="giac")
[Out]
integrate(sqrt(I*a*tan(d*x + c) + a)*cos(d*x + c)^5, x)