3.484 \(\int (e \sec (c+d x))^{-3-n} (a+i a \tan (c+d x))^n \, dx\)
Optimal. Leaf size=205 \[ -\frac{6 i (a+i a \tan (c+d x))^{n+2} (e \sec (c+d x))^{-n-3}}{a^2 d (3-n) \left (1-n^2\right )}+\frac{6 i (a+i a \tan (c+d x))^{n+3} (e \sec (c+d x))^{-n-3}}{a^3 d \left (n^4-10 n^2+9\right )}+\frac{3 i (a+i a \tan (c+d x))^{n+1} (e \sec (c+d x))^{-n-3}}{a d \left (n^2-4 n+3\right )}+\frac{i (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-n-3}}{d (3-n)} \]
[Out]
(I*(e*Sec[c + d*x])^(-3 - n)*(a + I*a*Tan[c + d*x])^n)/(d*(3 - n)) + ((3*I)*(e*Sec[c + d*x])^(-3 - n)*(a + I*a
*Tan[c + d*x])^(1 + n))/(a*d*(3 - 4*n + n^2)) - ((6*I)*(e*Sec[c + d*x])^(-3 - n)*(a + I*a*Tan[c + d*x])^(2 + n
))/(a^2*d*(3 - n)*(1 - n^2)) + ((6*I)*(e*Sec[c + d*x])^(-3 - n)*(a + I*a*Tan[c + d*x])^(3 + n))/(a^3*d*(9 - 10
*n^2 + n^4))
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Rubi [A] time = 0.304566, antiderivative size = 205, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used =
{3504, 3488} \[ -\frac{6 i (a+i a \tan (c+d x))^{n+2} (e \sec (c+d x))^{-n-3}}{a^2 d (3-n) \left (1-n^2\right )}+\frac{6 i (a+i a \tan (c+d x))^{n+3} (e \sec (c+d x))^{-n-3}}{a^3 d \left (n^4-10 n^2+9\right )}+\frac{3 i (a+i a \tan (c+d x))^{n+1} (e \sec (c+d x))^{-n-3}}{a d \left (n^2-4 n+3\right )}+\frac{i (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-n-3}}{d (3-n)} \]
Antiderivative was successfully verified.
[In]
Int[(e*Sec[c + d*x])^(-3 - n)*(a + I*a*Tan[c + d*x])^n,x]
[Out]
(I*(e*Sec[c + d*x])^(-3 - n)*(a + I*a*Tan[c + d*x])^n)/(d*(3 - n)) + ((3*I)*(e*Sec[c + d*x])^(-3 - n)*(a + I*a
*Tan[c + d*x])^(1 + n))/(a*d*(3 - 4*n + n^2)) - ((6*I)*(e*Sec[c + d*x])^(-3 - n)*(a + I*a*Tan[c + d*x])^(2 + n
))/(a^2*d*(3 - n)*(1 - n^2)) + ((6*I)*(e*Sec[c + d*x])^(-3 - n)*(a + I*a*Tan[c + d*x])^(3 + n))/(a^3*d*(9 - 10
*n^2 + n^4))
Rule 3504
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, n}, x] && EqQ[a^2 + b^2, 0] && IL
tQ[Simplify[m + n], 0] && NeQ[m + 2*n, 0]
Rule 3488
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] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0] &
& EqQ[Simplify[m + n], 0]
Rubi steps
\begin{align*} \int (e \sec (c+d x))^{-3-n} (a+i a \tan (c+d x))^n \, dx &=\frac{i (e \sec (c+d x))^{-3-n} (a+i a \tan (c+d x))^n}{d (3-n)}+\frac{3 \int (e \sec (c+d x))^{-3-n} (a+i a \tan (c+d x))^{1+n} \, dx}{a (3-n)}\\ &=\frac{i (e \sec (c+d x))^{-3-n} (a+i a \tan (c+d x))^n}{d (3-n)}+\frac{3 i (e \sec (c+d x))^{-3-n} (a+i a \tan (c+d x))^{1+n}}{a d \left (3-4 n+n^2\right )}+\frac{6 \int (e \sec (c+d x))^{-3-n} (a+i a \tan (c+d x))^{2+n} \, dx}{a^2 (1-n) (3-n)}\\ &=\frac{i (e \sec (c+d x))^{-3-n} (a+i a \tan (c+d x))^n}{d (3-n)}+\frac{3 i (e \sec (c+d x))^{-3-n} (a+i a \tan (c+d x))^{1+n}}{a d \left (3-4 n+n^2\right )}-\frac{6 i (e \sec (c+d x))^{-3-n} (a+i a \tan (c+d x))^{2+n}}{a^2 d (1-n) (3-n) (1+n)}-\frac{6 \int (e \sec (c+d x))^{-3-n} (a+i a \tan (c+d x))^{3+n} \, dx}{a^3 (1-n) (3-n) (1+n)}\\ &=\frac{i (e \sec (c+d x))^{-3-n} (a+i a \tan (c+d x))^n}{d (3-n)}+\frac{3 i (e \sec (c+d x))^{-3-n} (a+i a \tan (c+d x))^{1+n}}{a d \left (3-4 n+n^2\right )}-\frac{6 i (e \sec (c+d x))^{-3-n} (a+i a \tan (c+d x))^{2+n}}{a^2 d (1-n) (3-n) (1+n)}+\frac{6 i (e \sec (c+d x))^{-3-n} (a+i a \tan (c+d x))^{3+n}}{a^3 d \left (9-10 n^2+n^4\right )}\\ \end{align*}
Mathematica [A] time = 0.615162, size = 119, normalized size = 0.58 \[ \frac{(a+i a \tan (c+d x))^n (e \sec (c+d x))^{-n} \left (-3 i n \left (n^2-9\right ) \cos (c+d x)-i n \left (n^2-1\right ) \cos (3 (c+d x))-6 \sin (c+d x) \left (\left (n^2-1\right ) \cos (2 (c+d x))+n^2-5\right )\right )}{4 d e^3 (n-3) (n-1) (n+1) (n+3)} \]
Antiderivative was successfully verified.
[In]
Integrate[(e*Sec[c + d*x])^(-3 - n)*(a + I*a*Tan[c + d*x])^n,x]
[Out]
(((-3*I)*n*(-9 + n^2)*Cos[c + d*x] - I*n*(-1 + n^2)*Cos[3*(c + d*x)] - 6*(-5 + n^2 + (-1 + n^2)*Cos[2*(c + d*x
)])*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^n)/(4*d*e^3*(-3 + n)*(-1 + n)*(1 + n)*(3 + n)*(e*Sec[c + d*x])^n)
________________________________________________________________________________________
Maple [C] time = 1.178, size = 4990, normalized size = 24.3 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*sec(d*x+c))^(-3-n)*(a+I*a*tan(d*x+c))^n,x)
[Out]
1/8/(-3*I*d+I*n*d)*a^n*e^(-n)/e^3*exp(I*(d*x+c))^n*exp(-1/2*I*(6*c+Pi*csgn(I*exp(2*I*(d*x+c))/(exp(2*I*(d*x+c)
)+1))^3*n+6*d*x+3*Pi*csgn(I/(exp(2*I*(d*x+c))+1))*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))^2+3*Pi*csgn(I*ex
p(I*(d*x+c))/(exp(2*I*(d*x+c))+1))*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))^2+Pi*csgn(I*exp(2*I*(d*x+c)))
^3*n-Pi*csgn(I*exp(2*I*(d*x+c))/(exp(2*I*(d*x+c))+1))^2*csgn(I/(exp(2*I*(d*x+c))+1))*n-n*Pi*csgn(I/(exp(2*I*(d
*x+c))+1))*csgn(I*exp(I*(d*x+c)))*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))-n*Pi*csgn(I*e)*csgn(I*exp(I*(d*x
+c))/(exp(2*I*(d*x+c))+1))*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))+3*Pi*csgn(I*exp(I*(d*x+c)))*csgn(I*ex
p(I*(d*x+c))/(exp(2*I*(d*x+c))+1))^2+Pi*csgn(I*a/(exp(2*I*(d*x+c))+1)*exp(2*I*(d*x+c)))^3*n+3*Pi*csgn(I*e)*csg
n(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))^2-n*Pi*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))^3-n*Pi*csgn(I*
exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))^3-3*Pi*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))^3-3*Pi*csgn(I*exp(I*
(d*x+c))/(exp(2*I*(d*x+c))+1))^3-3*Pi*csgn(I*e)*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))*csgn(I*e/(exp(2*I*
(d*x+c))+1)*exp(I*(d*x+c)))-3*Pi*csgn(I/(exp(2*I*(d*x+c))+1))*csgn(I*exp(I*(d*x+c)))*csgn(I*exp(I*(d*x+c))/(ex
p(2*I*(d*x+c))+1))-Pi*csgn(I*a/(exp(2*I*(d*x+c))+1)*exp(2*I*(d*x+c)))^2*csgn(I*a)*n-2*Pi*csgn(I*exp(2*I*(d*x+c
)))^2*csgn(I*exp(I*(d*x+c)))*n+Pi*csgn(I*exp(2*I*(d*x+c))/(exp(2*I*(d*x+c))+1))*csgn(I*a/(exp(2*I*(d*x+c))+1)*
exp(2*I*(d*x+c)))*csgn(I*a)*n+Pi*csgn(I*exp(2*I*(d*x+c)))*csgn(I*exp(2*I*(d*x+c))/(exp(2*I*(d*x+c))+1))*csgn(I
/(exp(2*I*(d*x+c))+1))*n+n*Pi*csgn(I*e)*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))^2+n*Pi*csgn(I/(exp(2*I*(
d*x+c))+1))*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))^2+n*Pi*csgn(I*exp(I*(d*x+c)))*csgn(I*exp(I*(d*x+c))/(e
xp(2*I*(d*x+c))+1))^2+n*Pi*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*
x+c)))^2+Pi*csgn(I*exp(2*I*(d*x+c)))*csgn(I*exp(I*(d*x+c)))^2*n-Pi*csgn(I*exp(2*I*(d*x+c)))*csgn(I*exp(2*I*(d*
x+c))/(exp(2*I*(d*x+c))+1))^2*n-Pi*csgn(I*exp(2*I*(d*x+c))/(exp(2*I*(d*x+c))+1))*csgn(I*a/(exp(2*I*(d*x+c))+1)
*exp(2*I*(d*x+c)))^2*n))+1/8/(I*n*d+3*I*d)*a^n*e^(-n)/e^3*exp(I*(d*x+c))^n*exp(-1/2*I*(-6*c+Pi*csgn(I*exp(2*I*
(d*x+c))/(exp(2*I*(d*x+c))+1))^3*n-6*d*x+3*Pi*csgn(I/(exp(2*I*(d*x+c))+1))*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x
+c))+1))^2+3*Pi*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))^2+Pi
*csgn(I*exp(2*I*(d*x+c)))^3*n-Pi*csgn(I*exp(2*I*(d*x+c))/(exp(2*I*(d*x+c))+1))^2*csgn(I/(exp(2*I*(d*x+c))+1))*
n-n*Pi*csgn(I/(exp(2*I*(d*x+c))+1))*csgn(I*exp(I*(d*x+c)))*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))-n*Pi*cs
gn(I*e)*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))+3*Pi*csgn(I*
exp(I*(d*x+c)))*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))^2+Pi*csgn(I*a/(exp(2*I*(d*x+c))+1)*exp(2*I*(d*x+c)
))^3*n+3*Pi*csgn(I*e)*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))^2-n*Pi*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I
*(d*x+c)))^3-n*Pi*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))^3-3*Pi*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+
c)))^3-3*Pi*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))^3-3*Pi*csgn(I*e)*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c
))+1))*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))-3*Pi*csgn(I/(exp(2*I*(d*x+c))+1))*csgn(I*exp(I*(d*x+c)))*
csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))-Pi*csgn(I*a/(exp(2*I*(d*x+c))+1)*exp(2*I*(d*x+c)))^2*csgn(I*a)*n-2
*Pi*csgn(I*exp(2*I*(d*x+c)))^2*csgn(I*exp(I*(d*x+c)))*n+Pi*csgn(I*exp(2*I*(d*x+c))/(exp(2*I*(d*x+c))+1))*csgn(
I*a/(exp(2*I*(d*x+c))+1)*exp(2*I*(d*x+c)))*csgn(I*a)*n+Pi*csgn(I*exp(2*I*(d*x+c)))*csgn(I*exp(2*I*(d*x+c))/(ex
p(2*I*(d*x+c))+1))*csgn(I/(exp(2*I*(d*x+c))+1))*n+n*Pi*csgn(I*e)*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))
^2+n*Pi*csgn(I/(exp(2*I*(d*x+c))+1))*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))^2+n*Pi*csgn(I*exp(I*(d*x+c)))
*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))^2+n*Pi*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))*csgn(I*e/(exp(
2*I*(d*x+c))+1)*exp(I*(d*x+c)))^2+Pi*csgn(I*exp(2*I*(d*x+c)))*csgn(I*exp(I*(d*x+c)))^2*n-Pi*csgn(I*exp(2*I*(d*
x+c)))*csgn(I*exp(2*I*(d*x+c))/(exp(2*I*(d*x+c))+1))^2*n-Pi*csgn(I*exp(2*I*(d*x+c))/(exp(2*I*(d*x+c))+1))*csgn
(I*a/(exp(2*I*(d*x+c))+1)*exp(2*I*(d*x+c)))^2*n))+3/8/(-I*d+I*n*d)*a^n*e^(-n)/e^3*exp(I*(d*x+c))^n*exp(-1/2*I*
(2*c+Pi*csgn(I*exp(2*I*(d*x+c))/(exp(2*I*(d*x+c))+1))^3*n+2*d*x+3*Pi*csgn(I/(exp(2*I*(d*x+c))+1))*csgn(I*exp(I
*(d*x+c))/(exp(2*I*(d*x+c))+1))^2+3*Pi*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))*csgn(I*e/(exp(2*I*(d*x+c))+
1)*exp(I*(d*x+c)))^2+Pi*csgn(I*exp(2*I*(d*x+c)))^3*n-Pi*csgn(I*exp(2*I*(d*x+c))/(exp(2*I*(d*x+c))+1))^2*csgn(I
/(exp(2*I*(d*x+c))+1))*n-n*Pi*csgn(I/(exp(2*I*(d*x+c))+1))*csgn(I*exp(I*(d*x+c)))*csgn(I*exp(I*(d*x+c))/(exp(2
*I*(d*x+c))+1))-n*Pi*csgn(I*e)*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I
*(d*x+c)))+3*Pi*csgn(I*exp(I*(d*x+c)))*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))^2+Pi*csgn(I*a/(exp(2*I*(d*x
+c))+1)*exp(2*I*(d*x+c)))^3*n+3*Pi*csgn(I*e)*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))^2-n*Pi*csgn(I*e/(ex
p(2*I*(d*x+c))+1)*exp(I*(d*x+c)))^3-n*Pi*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))^3-3*Pi*csgn(I*e/(exp(2*I*
(d*x+c))+1)*exp(I*(d*x+c)))^3-3*Pi*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))^3-3*Pi*csgn(I*e)*csgn(I*exp(I*(
d*x+c))/(exp(2*I*(d*x+c))+1))*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))-3*Pi*csgn(I/(exp(2*I*(d*x+c))+1))*
csgn(I*exp(I*(d*x+c)))*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))-Pi*csgn(I*a/(exp(2*I*(d*x+c))+1)*exp(2*I*(d
*x+c)))^2*csgn(I*a)*n-2*Pi*csgn(I*exp(2*I*(d*x+c)))^2*csgn(I*exp(I*(d*x+c)))*n+Pi*csgn(I*exp(2*I*(d*x+c))/(exp
(2*I*(d*x+c))+1))*csgn(I*a/(exp(2*I*(d*x+c))+1)*exp(2*I*(d*x+c)))*csgn(I*a)*n+Pi*csgn(I*exp(2*I*(d*x+c)))*csgn
(I*exp(2*I*(d*x+c))/(exp(2*I*(d*x+c))+1))*csgn(I/(exp(2*I*(d*x+c))+1))*n+n*Pi*csgn(I*e)*csgn(I*e/(exp(2*I*(d*x
+c))+1)*exp(I*(d*x+c)))^2+n*Pi*csgn(I/(exp(2*I*(d*x+c))+1))*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))^2+n*Pi
*csgn(I*exp(I*(d*x+c)))*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))^2+n*Pi*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x
+c))+1))*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))^2+Pi*csgn(I*exp(2*I*(d*x+c)))*csgn(I*exp(I*(d*x+c)))^2*
n-Pi*csgn(I*exp(2*I*(d*x+c)))*csgn(I*exp(2*I*(d*x+c))/(exp(2*I*(d*x+c))+1))^2*n-Pi*csgn(I*exp(2*I*(d*x+c))/(ex
p(2*I*(d*x+c))+1))*csgn(I*a/(exp(2*I*(d*x+c))+1)*exp(2*I*(d*x+c)))^2*n))+3/8/(I*d+I*n*d)*a^n*e^(-n)/e^3*exp(I*
(d*x+c))^n*exp(-1/2*I*(-2*c+Pi*csgn(I*exp(2*I*(d*x+c))/(exp(2*I*(d*x+c))+1))^3*n-2*d*x+3*Pi*csgn(I/(exp(2*I*(d
*x+c))+1))*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))^2+3*Pi*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))*csgn
(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))^2+Pi*csgn(I*exp(2*I*(d*x+c)))^3*n-Pi*csgn(I*exp(2*I*(d*x+c))/(exp(2*
I*(d*x+c))+1))^2*csgn(I/(exp(2*I*(d*x+c))+1))*n-n*Pi*csgn(I/(exp(2*I*(d*x+c))+1))*csgn(I*exp(I*(d*x+c)))*csgn(
I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))-n*Pi*csgn(I*e)*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))*csgn(I*e/(ex
p(2*I*(d*x+c))+1)*exp(I*(d*x+c)))+3*Pi*csgn(I*exp(I*(d*x+c)))*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))^2+Pi
*csgn(I*a/(exp(2*I*(d*x+c))+1)*exp(2*I*(d*x+c)))^3*n+3*Pi*csgn(I*e)*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c
)))^2-n*Pi*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))^3-n*Pi*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))^3-
3*Pi*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))^3-3*Pi*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))^3-3*Pi*c
sgn(I*e)*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))-3*Pi*csgn(I
/(exp(2*I*(d*x+c))+1))*csgn(I*exp(I*(d*x+c)))*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))-Pi*csgn(I*a/(exp(2*I
*(d*x+c))+1)*exp(2*I*(d*x+c)))^2*csgn(I*a)*n-2*Pi*csgn(I*exp(2*I*(d*x+c)))^2*csgn(I*exp(I*(d*x+c)))*n+Pi*csgn(
I*exp(2*I*(d*x+c))/(exp(2*I*(d*x+c))+1))*csgn(I*a/(exp(2*I*(d*x+c))+1)*exp(2*I*(d*x+c)))*csgn(I*a)*n+Pi*csgn(I
*exp(2*I*(d*x+c)))*csgn(I*exp(2*I*(d*x+c))/(exp(2*I*(d*x+c))+1))*csgn(I/(exp(2*I*(d*x+c))+1))*n+n*Pi*csgn(I*e)
*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))^2+n*Pi*csgn(I/(exp(2*I*(d*x+c))+1))*csgn(I*exp(I*(d*x+c))/(exp(
2*I*(d*x+c))+1))^2+n*Pi*csgn(I*exp(I*(d*x+c)))*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))^2+n*Pi*csgn(I*exp(I
*(d*x+c))/(exp(2*I*(d*x+c))+1))*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))^2+Pi*csgn(I*exp(2*I*(d*x+c)))*cs
gn(I*exp(I*(d*x+c)))^2*n-Pi*csgn(I*exp(2*I*(d*x+c)))*csgn(I*exp(2*I*(d*x+c))/(exp(2*I*(d*x+c))+1))^2*n-Pi*csgn
(I*exp(2*I*(d*x+c))/(exp(2*I*(d*x+c))+1))*csgn(I*a/(exp(2*I*(d*x+c))+1)*exp(2*I*(d*x+c)))^2*n))
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Maxima [A] time = 1.9043, size = 464, normalized size = 2.26 \begin{align*} \frac{{\left (-i \, a^{n} n^{3} + 3 i \, a^{n} n^{2} + i \, a^{n} n - 3 i \, a^{n}\right )} \cos \left ({\left (d x + c\right )}{\left (n + 3\right )}\right ) +{\left (-3 i \, a^{n} n^{3} + 3 i \, a^{n} n^{2} + 27 i \, a^{n} n - 27 i \, a^{n}\right )} \cos \left ({\left (d x + c\right )}{\left (n + 1\right )}\right ) +{\left (-3 i \, a^{n} n^{3} - 3 i \, a^{n} n^{2} + 27 i \, a^{n} n + 27 i \, a^{n}\right )} \cos \left ({\left (d x + c\right )}{\left (n - 1\right )}\right ) +{\left (-i \, a^{n} n^{3} - 3 i \, a^{n} n^{2} + i \, a^{n} n + 3 i \, a^{n}\right )} \cos \left ({\left (d x + c\right )}{\left (n - 3\right )}\right ) +{\left (a^{n} n^{3} - 3 \, a^{n} n^{2} - a^{n} n + 3 \, a^{n}\right )} \sin \left ({\left (d x + c\right )}{\left (n + 3\right )}\right ) + 3 \,{\left (a^{n} n^{3} - a^{n} n^{2} - 9 \, a^{n} n + 9 \, a^{n}\right )} \sin \left ({\left (d x + c\right )}{\left (n + 1\right )}\right ) + 3 \,{\left (a^{n} n^{3} + a^{n} n^{2} - 9 \, a^{n} n - 9 \, a^{n}\right )} \sin \left ({\left (d x + c\right )}{\left (n - 1\right )}\right ) +{\left (a^{n} n^{3} + 3 \, a^{n} n^{2} - a^{n} n - 3 \, a^{n}\right )} \sin \left ({\left (d x + c\right )}{\left (n - 3\right )}\right )}{8 \,{\left (e^{n + 3} n^{4} - 10 \, e^{n + 3} n^{2} + 9 \, e^{n + 3}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*sec(d*x+c))^(-3-n)*(a+I*a*tan(d*x+c))^n,x, algorithm="maxima")
[Out]
1/8*((-I*a^n*n^3 + 3*I*a^n*n^2 + I*a^n*n - 3*I*a^n)*cos((d*x + c)*(n + 3)) + (-3*I*a^n*n^3 + 3*I*a^n*n^2 + 27*
I*a^n*n - 27*I*a^n)*cos((d*x + c)*(n + 1)) + (-3*I*a^n*n^3 - 3*I*a^n*n^2 + 27*I*a^n*n + 27*I*a^n)*cos((d*x + c
)*(n - 1)) + (-I*a^n*n^3 - 3*I*a^n*n^2 + I*a^n*n + 3*I*a^n)*cos((d*x + c)*(n - 3)) + (a^n*n^3 - 3*a^n*n^2 - a^
n*n + 3*a^n)*sin((d*x + c)*(n + 3)) + 3*(a^n*n^3 - a^n*n^2 - 9*a^n*n + 9*a^n)*sin((d*x + c)*(n + 1)) + 3*(a^n*
n^3 + a^n*n^2 - 9*a^n*n - 9*a^n)*sin((d*x + c)*(n - 1)) + (a^n*n^3 + 3*a^n*n^2 - a^n*n - 3*a^n)*sin((d*x + c)*
(n - 3)))/((e^(n + 3)*n^4 - 10*e^(n + 3)*n^2 + 9*e^(n + 3))*d)
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Fricas [A] time = 2.13417, size = 653, normalized size = 3.19 \begin{align*} \frac{{\left (-i \, n^{3} - 3 i \, n^{2} +{\left (-i \, n^{3} + 3 i \, n^{2} + i \, n - 3 i\right )} e^{\left (6 i \, d x + 6 i \, c\right )} +{\left (-3 i \, n^{3} + 3 i \, n^{2} + 27 i \, n - 27 i\right )} e^{\left (4 i \, d x + 4 i \, c\right )} +{\left (-3 i \, n^{3} - 3 i \, n^{2} + 27 i \, n + 27 i\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, n + 3 i\right )} \left (\frac{2 \, a e^{\left (2 i \, d x + 2 i \, c\right )}}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{n} \left (\frac{2 \, e e^{\left (i \, d x + i \, c\right )}}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{-n - 3}}{d n^{4} - 10 \, d n^{2} +{\left (d n^{4} - 10 \, d n^{2} + 9 \, d\right )} e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \,{\left (d n^{4} - 10 \, d n^{2} + 9 \, d\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \,{\left (d n^{4} - 10 \, d n^{2} + 9 \, d\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + 9 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*sec(d*x+c))^(-3-n)*(a+I*a*tan(d*x+c))^n,x, algorithm="fricas")
[Out]
(-I*n^3 - 3*I*n^2 + (-I*n^3 + 3*I*n^2 + I*n - 3*I)*e^(6*I*d*x + 6*I*c) + (-3*I*n^3 + 3*I*n^2 + 27*I*n - 27*I)*
e^(4*I*d*x + 4*I*c) + (-3*I*n^3 - 3*I*n^2 + 27*I*n + 27*I)*e^(2*I*d*x + 2*I*c) + I*n + 3*I)*(2*a*e^(2*I*d*x +
2*I*c)/(e^(2*I*d*x + 2*I*c) + 1))^n*(2*e*e^(I*d*x + I*c)/(e^(2*I*d*x + 2*I*c) + 1))^(-n - 3)/(d*n^4 - 10*d*n^2
+ (d*n^4 - 10*d*n^2 + 9*d)*e^(6*I*d*x + 6*I*c) + 3*(d*n^4 - 10*d*n^2 + 9*d)*e^(4*I*d*x + 4*I*c) + 3*(d*n^4 -
10*d*n^2 + 9*d)*e^(2*I*d*x + 2*I*c) + 9*d)
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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((e*sec(d*x+c))**(-3-n)*(a+I*a*tan(d*x+c))**n,x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e \sec \left (d x + c\right )\right )^{-n - 3}{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*sec(d*x+c))^(-3-n)*(a+I*a*tan(d*x+c))^n,x, algorithm="giac")
[Out]
integrate((e*sec(d*x + c))^(-n - 3)*(I*a*tan(d*x + c) + a)^n, x)