3.330 \(\int \cos ^7(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx\)
Optimal. Leaf size=196 \[ -\frac{i a^2 \cos ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{12 d}-\frac{i a^3 \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{8 d}+\frac{i a^{7/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right )}{8 \sqrt{2} d}-\frac{i \cos ^7(c+d x) (a+i a \tan (c+d x))^{7/2}}{7 d}-\frac{i a \cos ^5(c+d x) (a+i a \tan (c+d x))^{5/2}}{10 d} \]
[Out]
((I/8)*a^(7/2)*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(Sqrt[2]*d) - ((I/8)*a^3*
Cos[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/d - ((I/12)*a^2*Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^(3/2))/d - ((I/
10)*a*Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^(5/2))/d - ((I/7)*Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^(7/2))/d
________________________________________________________________________________________
Rubi [A] time = 0.286647, antiderivative size = 196, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used =
{3490, 3489, 206} \[ -\frac{i a^2 \cos ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{12 d}-\frac{i a^3 \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{8 d}+\frac{i a^{7/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right )}{8 \sqrt{2} d}-\frac{i \cos ^7(c+d x) (a+i a \tan (c+d x))^{7/2}}{7 d}-\frac{i a \cos ^5(c+d x) (a+i a \tan (c+d x))^{5/2}}{10 d} \]
Antiderivative was successfully verified.
[In]
Int[Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^(7/2),x]
[Out]
((I/8)*a^(7/2)*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(Sqrt[2]*d) - ((I/8)*a^3*
Cos[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/d - ((I/12)*a^2*Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^(3/2))/d - ((I/
10)*a*Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^(5/2))/d - ((I/7)*Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^(7/2))/d
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 ^7(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx &=-\frac{i \cos ^7(c+d x) (a+i a \tan (c+d x))^{7/2}}{7 d}+\frac{1}{2} a \int \cos ^5(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx\\ &=-\frac{i a \cos ^5(c+d x) (a+i a \tan (c+d x))^{5/2}}{10 d}-\frac{i \cos ^7(c+d x) (a+i a \tan (c+d x))^{7/2}}{7 d}+\frac{1}{4} a^2 \int \cos ^3(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx\\ &=-\frac{i a^2 \cos ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{12 d}-\frac{i a \cos ^5(c+d x) (a+i a \tan (c+d x))^{5/2}}{10 d}-\frac{i \cos ^7(c+d x) (a+i a \tan (c+d x))^{7/2}}{7 d}+\frac{1}{8} a^3 \int \cos (c+d x) \sqrt{a+i a \tan (c+d x)} \, dx\\ &=-\frac{i a^3 \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{8 d}-\frac{i a^2 \cos ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{12 d}-\frac{i a \cos ^5(c+d x) (a+i a \tan (c+d x))^{5/2}}{10 d}-\frac{i \cos ^7(c+d x) (a+i a \tan (c+d x))^{7/2}}{7 d}+\frac{1}{16} a^4 \int \frac{\sec (c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx\\ &=-\frac{i a^3 \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{8 d}-\frac{i a^2 \cos ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{12 d}-\frac{i a \cos ^5(c+d x) (a+i a \tan (c+d x))^{5/2}}{10 d}-\frac{i \cos ^7(c+d x) (a+i a \tan (c+d x))^{7/2}}{7 d}+\frac{\left (i a^4\right ) \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 )}{8 d}\\ &=\frac{i a^{7/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right )}{8 \sqrt{2} d}-\frac{i a^3 \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{8 d}-\frac{i a^2 \cos ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{12 d}-\frac{i a \cos ^5(c+d x) (a+i a \tan (c+d x))^{5/2}}{10 d}-\frac{i \cos ^7(c+d x) (a+i a \tan (c+d x))^{7/2}}{7 d}\\ \end{align*}
Mathematica [A] time = 1.78194, size = 131, normalized size = 0.67 \[ -\frac{i a^3 e^{-i (c+d x)} \left (298 e^{2 i (c+d x)}+188 e^{4 i (c+d x)}+81 e^{6 i (c+d x)}+15 e^{8 i (c+d x)}-105 \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left (\sqrt{1+e^{2 i (c+d x)}}\right )+176\right ) \sqrt{a+i a \tan (c+d x)}}{1680 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^(7/2),x]
[Out]
((-I/1680)*a^3*(176 + 298*E^((2*I)*(c + d*x)) + 188*E^((4*I)*(c + d*x)) + 81*E^((6*I)*(c + d*x)) + 15*E^((8*I)
*(c + d*x)) - 105*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^(I*(c + d*x)))
________________________________________________________________________________________
Maple [B] time = 0.422, size = 1260, normalized size = 6.4 \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)^7*(a+I*a*tan(d*x+c))^(7/2),x)
[Out]
1/107520/d*a^3*(-630*I*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(13/2)*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))*cos(d*x+c)*sin(d*x+c)-1575*I*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(13
/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))*cos(d*x+c)^2*sin(d*x+c)+12
2880*sin(d*x+c)*cos(d*x+c)^13-61440*sin(d*x+c)*cos(d*x+c)^12+7168*sin(d*x+c)*cos(d*x+c)^9-18432*sin(d*x+c)*cos
(d*x+c)^11-6144*sin(d*x+c)*cos(d*x+c)^10-105*I*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(13/2)*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))*cos(d*x+c)^6*sin(d*x+c)-630*I*2^(1/2)*(-2*cos(d
*x+c)/(cos(d*x+c)+1))^(13/2)*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))*c
os(d*x+c)^5*sin(d*x+c)-1575*I*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(13/2)*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))*cos(d*x+c)^4*sin(d*x+c)-2100*I*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c
)+1))^(13/2)*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))*cos(d*x+c)^3*sin(
d*x+c)-1024*I*cos(d*x+c)^10-8960*sin(d*x+c)*cos(d*x+c)^8+13440*sin(d*x+c)*cos(d*x+c)^7-122880*I*cos(d*x+c)^14+
61440*I*cos(d*x+c)^13+79872*I*cos(d*x+c)^12-24576*I*cos(d*x+c)^11-1792*I*cos(d*x+c)^9-4480*I*cos(d*x+c)^8+1344
0*I*cos(d*x+c)^7+105*2^(1/2)*cos(d*x+c)^6*sin(d*x+c)*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))^(13/2)+630*2^(1/2)*cos(d*x+c)^5*sin(d*x+c)*arctan(1/2*2^(1/2)*(-2*cos(d*x+c)/(c
os(d*x+c)+1))^(1/2))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(13/2)+1575*2^(1/2)*cos(d*x+c)^4*sin(d*x+c)*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))^(13/2)+2100*2^(1/2)*cos(d*x+c)^3*sin
(d*x+c)*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))^(13/2)+1575*2^
(1/2)*cos(d*x+c)^2*sin(d*x+c)*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))^(13/2)+630*2^(1/2)*cos(d*x+c)*sin(d*x+c)*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))^(13/2)+105*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))^(13/2)*sin(d*x+c)-105*I*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(13/2)*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))*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)^6
________________________________________________________________________________________
Maxima [B] time = 2.46521, size = 1688, normalized size = 8.61 \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)^7*(a+I*a*tan(d*x+c))^(7/2),x, algorithm="maxima")
[Out]
1/6720*((-140*I*sqrt(2)*a^3*cos(3/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) + 140*sqrt(2)*a^3*sin(3/2
*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) + (-60*I*sqrt(2)*a^3*cos(2*d*x + 2*c)^2 - 60*I*sqrt(2)*a^3*s
in(2*d*x + 2*c)^2 - 120*I*sqrt(2)*a^3*cos(2*d*x + 2*c) - 60*I*sqrt(2)*a^3)*cos(7/2*arctan2(sin(2*d*x + 2*c), c
os(2*d*x + 2*c) + 1)) + 60*(sqrt(2)*a^3*cos(2*d*x + 2*c)^2 + sqrt(2)*a^3*sin(2*d*x + 2*c)^2 + 2*sqrt(2)*a^3*co
s(2*d*x + 2*c) + sqrt(2)*a^3)*sin(7/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)))*(cos(2*d*x + 2*c)^2 +
sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(3/4)*sqrt(a) + (-420*I*sqrt(2)*a^3*cos(1/2*arctan2(sin(2*d*x + 2
*c), cos(2*d*x + 2*c) + 1)) + 420*sqrt(2)*a^3*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) + (-84*
I*sqrt(2)*a^3*cos(2*d*x + 2*c)^2 - 84*I*sqrt(2)*a^3*sin(2*d*x + 2*c)^2 - 168*I*sqrt(2)*a^3*cos(2*d*x + 2*c) -
84*I*sqrt(2)*a^3)*cos(5/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) + 84*(sqrt(2)*a^3*cos(2*d*x + 2*c)^
2 + sqrt(2)*a^3*sin(2*d*x + 2*c)^2 + 2*sqrt(2)*a^3*cos(2*d*x + 2*c) + sqrt(2)*a^3)*sin(5/2*arctan2(sin(2*d*x +
2*c), cos(2*d*x + 2*c) + 1)))*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4)*sqrt(a
) - (210*sqrt(2)*a^3*arctan2((cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4)*sin(1/2*
arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)), (cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c
) + 1)^(1/4)*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) + 1) - 210*sqrt(2)*a^3*arctan2((cos(2*d*
x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4)*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x +
2*c) + 1)), (cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4)*cos(1/2*arctan2(sin(2*d*x
+ 2*c), cos(2*d*x + 2*c) + 1)) - 1) - 105*I*sqrt(2)*a^3*log(sqrt(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*
cos(2*d*x + 2*c) + 1)*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1))^2 + sqrt(cos(2*d*x + 2*c)^2 + s
in(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1))^2 + 2*(co
s(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4)*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*
d*x + 2*c) + 1)) + 1) + 105*I*sqrt(2)*a^3*log(sqrt(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c
) + 1)*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1))^2 + sqrt(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)
^2 + 2*cos(2*d*x + 2*c) + 1)*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1))^2 - 2*(cos(2*d*x + 2*c)^
2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4)*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)
) + 1))*sqrt(a))/d
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Fricas [B] time = 2.26525, size = 891, normalized size = 4.55 \begin{align*} -\frac{{\left (105 \, \sqrt{\frac{1}{2}} \sqrt{-\frac{a^{7}}{d^{2}}} d e^{\left (i \, d x + i \, c\right )} \log \left (\frac{{\left (2 i \, \sqrt{\frac{1}{2}} \sqrt{-\frac{a^{7}}{d^{2}}} d e^{\left (i \, d x + i \, c\right )} + \sqrt{2}{\left (a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{3}\right )} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{a^{3}}\right ) - 105 \, \sqrt{\frac{1}{2}} \sqrt{-\frac{a^{7}}{d^{2}}} d e^{\left (i \, d x + i \, c\right )} \log \left (\frac{{\left (-2 i \, \sqrt{\frac{1}{2}} \sqrt{-\frac{a^{7}}{d^{2}}} d e^{\left (i \, d x + i \, c\right )} + \sqrt{2}{\left (a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{3}\right )} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{a^{3}}\right ) - \sqrt{2}{\left (-15 i \, a^{3} e^{\left (8 i \, d x + 8 i \, c\right )} - 81 i \, a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} - 188 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} - 298 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - 176 i \, a^{3}\right )} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{1680 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^7*(a+I*a*tan(d*x+c))^(7/2),x, algorithm="fricas")
[Out]
-1/1680*(105*sqrt(1/2)*sqrt(-a^7/d^2)*d*e^(I*d*x + I*c)*log((2*I*sqrt(1/2)*sqrt(-a^7/d^2)*d*e^(I*d*x + I*c) +
sqrt(2)*(a^3*e^(2*I*d*x + 2*I*c) + a^3)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*e^(I*d*x + I*c))*e^(-I*d*x - I*c)/a^
3) - 105*sqrt(1/2)*sqrt(-a^7/d^2)*d*e^(I*d*x + I*c)*log((-2*I*sqrt(1/2)*sqrt(-a^7/d^2)*d*e^(I*d*x + I*c) + sqr
t(2)*(a^3*e^(2*I*d*x + 2*I*c) + a^3)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*e^(I*d*x + I*c))*e^(-I*d*x - I*c)/a^3)
- sqrt(2)*(-15*I*a^3*e^(8*I*d*x + 8*I*c) - 81*I*a^3*e^(6*I*d*x + 6*I*c) - 188*I*a^3*e^(4*I*d*x + 4*I*c) - 298*
I*a^3*e^(2*I*d*x + 2*I*c) - 176*I*a^3)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*e^(I*d*x + I*c))*e^(-I*d*x - 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)**7*(a+I*a*tan(d*x+c))**(7/2),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [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)^7*(a+I*a*tan(d*x+c))^(7/2),x, algorithm="giac")
[Out]
Timed out