3.485 \(\int (e \sec (c+d x))^{-2-n} (a+i a \tan (c+d x))^n \, dx\)
Optimal. Leaf size=148 \[ \frac{2 i (a+i a \tan (c+d x))^{n+2} (e \sec (c+d x))^{-n-2}}{a^2 d n \left (4-n^2\right )}+\frac{i (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-n-2}}{d (2-n)}-\frac{2 i (a+i a \tan (c+d x))^{n+1} (e \sec (c+d x))^{-n-2}}{a d (2-n) n} \]
[Out]
(I*(e*Sec[c + d*x])^(-2 - n)*(a + I*a*Tan[c + d*x])^n)/(d*(2 - n)) - ((2*I)*(e*Sec[c + d*x])^(-2 - n)*(a + I*a
*Tan[c + d*x])^(1 + n))/(a*d*(2 - n)*n) + ((2*I)*(e*Sec[c + d*x])^(-2 - n)*(a + I*a*Tan[c + d*x])^(2 + n))/(a^
2*d*n*(4 - n^2))
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Rubi [A] time = 0.197928, antiderivative size = 148, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used =
{3504, 3488} \[ \frac{2 i (a+i a \tan (c+d x))^{n+2} (e \sec (c+d x))^{-n-2}}{a^2 d n \left (4-n^2\right )}+\frac{i (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-n-2}}{d (2-n)}-\frac{2 i (a+i a \tan (c+d x))^{n+1} (e \sec (c+d x))^{-n-2}}{a d (2-n) n} \]
Antiderivative was successfully verified.
[In]
Int[(e*Sec[c + d*x])^(-2 - n)*(a + I*a*Tan[c + d*x])^n,x]
[Out]
(I*(e*Sec[c + d*x])^(-2 - n)*(a + I*a*Tan[c + d*x])^n)/(d*(2 - n)) - ((2*I)*(e*Sec[c + d*x])^(-2 - n)*(a + I*a
*Tan[c + d*x])^(1 + n))/(a*d*(2 - n)*n) + ((2*I)*(e*Sec[c + d*x])^(-2 - n)*(a + I*a*Tan[c + d*x])^(2 + n))/(a^
2*d*n*(4 - n^2))
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))^{-2-n} (a+i a \tan (c+d x))^n \, dx &=\frac{i (e \sec (c+d x))^{-2-n} (a+i a \tan (c+d x))^n}{d (2-n)}+\frac{2 \int (e \sec (c+d x))^{-2-n} (a+i a \tan (c+d x))^{1+n} \, dx}{a (2-n)}\\ &=\frac{i (e \sec (c+d x))^{-2-n} (a+i a \tan (c+d x))^n}{d (2-n)}-\frac{2 i (e \sec (c+d x))^{-2-n} (a+i a \tan (c+d x))^{1+n}}{a d (2-n) n}-\frac{2 \int (e \sec (c+d x))^{-2-n} (a+i a \tan (c+d x))^{2+n} \, dx}{a^2 (2-n) n}\\ &=\frac{i (e \sec (c+d x))^{-2-n} (a+i a \tan (c+d x))^n}{d (2-n)}-\frac{2 i (e \sec (c+d x))^{-2-n} (a+i a \tan (c+d x))^{1+n}}{a d (2-n) n}+\frac{2 i (e \sec (c+d x))^{-2-n} (a+i a \tan (c+d x))^{2+n}}{a^2 d n \left (4-n^2\right )}\\ \end{align*}
Mathematica [A] time = 0.219606, size = 82, normalized size = 0.55 \[ -\frac{i (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-n} \left (n^2 \cos (2 (c+d x))-2 i n \sin (2 (c+d x))+n^2-4\right )}{2 d e^2 (n-2) n (n+2)} \]
Antiderivative was successfully verified.
[In]
Integrate[(e*Sec[c + d*x])^(-2 - n)*(a + I*a*Tan[c + d*x])^n,x]
[Out]
((-I/2)*(-4 + n^2 + n^2*Cos[2*(c + d*x)] - (2*I)*n*Sin[2*(c + d*x)])*(a + I*a*Tan[c + d*x])^n)/(d*e^2*(-2 + n)
*n*(2 + n)*(e*Sec[c + d*x])^n)
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Maple [C] time = 0.823, size = 3376, normalized size = 22.8 \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))^(-2-n)*(a+I*a*tan(d*x+c))^n,x)
[Out]
1/4/(-2*I*d+I*n*d)*a^n*e^(-n)/e^2*exp(I*(d*x+c))^n*exp(1/2*I*(-4*c-Pi*csgn(I*exp(2*I*(d*x+c))/(exp(2*I*(d*x+c)
)+1))^3*n-4*d*x-2*Pi*csgn(I/(exp(2*I*(d*x+c))+1))*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))^2-2*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)))-2*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-2*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+2*Pi*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))^3+2*Pi*csgn(I*exp(I*
(d*x+c))/(exp(2*I*(d*x+c))+1))^3+2*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)))+2*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))+1/4/(2*I*d+I*n*d)*a^n*e^(-n)/e^2*exp(I*(d*x+c))^n*exp(1/2*I*(4*c-Pi*csgn(I*exp(2*I*(d
*x+c))/(exp(2*I*(d*x+c))+1))^3*n+4*d*x-2*Pi*csgn(I/(exp(2*I*(d*x+c))+1))*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c
))+1))^2-2*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*c
sgn(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)))-2*Pi*csgn(I*ex
p(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-2*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+2*Pi*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)
))^3+2*Pi*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))^3+2*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)))+2*Pi*csgn(I/(exp(2*I*(d*x+c))+1))*csgn(I*exp(I*(d*x+c)))*cs
gn(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*P
i*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)))*c
sgn(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))-1/2*I*a^n/e^2/(e^n)/n/d*exp(-1/2*n*(-I*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-I*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))-2*I*Pi*csgn(I*exp(2*I*(d*x+c)))^2*csgn(I*exp(I*(d*x+c)))+I*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)-I*Pi*csgn(I*e/(exp(2*I*(d*x+c
))+1)*exp(I*(d*x+c)))^3-I*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)))+I*Pi*csgn(I*exp(2*I*(d*x+c)))^3+I*Pi*csgn(I*a/(exp(2*I*(d*x+c))+1)*exp(2*I*(d*x+c)))^3+I*Pi
*csgn(I*exp(I*(d*x+c)))*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))^2+I*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-I*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))-I*Pi*csgn(I*a/(exp(2*I*(d*x+c))+1)*exp(2*I*(d*x+c)))^2*csgn(I*a)-I*Pi*csgn(I*ex
p(I*(d*x+c))/(exp(2*I*(d*x+c))+1))^3+I*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))+I*Pi*csgn(I*e)*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))^2+I*Pi*csgn(I*exp
(2*I*(d*x+c))/(exp(2*I*(d*x+c))+1))^3+I*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-I*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+I*Pi*csgn(I*exp(2*I*(d*x+c)))*csgn(I*exp(I*(d*x+c)))^2-2*ln(exp(I*(d*x+c)))))
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Maxima [A] time = 2.08959, size = 235, normalized size = 1.59 \begin{align*} \frac{{\left (-i \, a^{n} n^{2} + 2 i \, a^{n} n\right )} \cos \left ({\left (d x + c\right )}{\left (n + 2\right )}\right ) +{\left (-i \, a^{n} n^{2} - 2 i \, a^{n} n\right )} \cos \left ({\left (d x + c\right )}{\left (n - 2\right )}\right ) +{\left (-2 i \, a^{n} n^{2} + 8 i \, a^{n}\right )} \cos \left ({\left (d x + c\right )} n\right ) +{\left (a^{n} n^{2} - 2 \, a^{n} n\right )} \sin \left ({\left (d x + c\right )}{\left (n + 2\right )}\right ) +{\left (a^{n} n^{2} + 2 \, a^{n} n\right )} \sin \left ({\left (d x + c\right )}{\left (n - 2\right )}\right ) + 2 \,{\left (a^{n} n^{2} - 4 \, a^{n}\right )} \sin \left ({\left (d x + c\right )} n\right )}{4 \,{\left (e^{n + 2} n^{3} - 4 \, e^{n + 2} n\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*sec(d*x+c))^(-2-n)*(a+I*a*tan(d*x+c))^n,x, algorithm="maxima")
[Out]
1/4*((-I*a^n*n^2 + 2*I*a^n*n)*cos((d*x + c)*(n + 2)) + (-I*a^n*n^2 - 2*I*a^n*n)*cos((d*x + c)*(n - 2)) + (-2*I
*a^n*n^2 + 8*I*a^n)*cos((d*x + c)*n) + (a^n*n^2 - 2*a^n*n)*sin((d*x + c)*(n + 2)) + (a^n*n^2 + 2*a^n*n)*sin((d
*x + c)*(n - 2)) + 2*(a^n*n^2 - 4*a^n)*sin((d*x + c)*n))/((e^(n + 2)*n^3 - 4*e^(n + 2)*n)*d)
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Fricas [A] time = 2.39426, size = 406, normalized size = 2.74 \begin{align*} \frac{{\left (-i \, n^{2} +{\left (-i \, n^{2} + 2 i \, n\right )} e^{\left (4 i \, d x + 4 i \, c\right )} +{\left (-2 i \, n^{2} + 8 i\right )} e^{\left (2 i \, d x + 2 i \, c\right )} - 2 i \, n\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 - 2}}{d n^{3} - 4 \, d n +{\left (d n^{3} - 4 \, d n\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \,{\left (d n^{3} - 4 \, d n\right )} e^{\left (2 i \, d x + 2 i \, c\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*sec(d*x+c))^(-2-n)*(a+I*a*tan(d*x+c))^n,x, algorithm="fricas")
[Out]
(-I*n^2 + (-I*n^2 + 2*I*n)*e^(4*I*d*x + 4*I*c) + (-2*I*n^2 + 8*I)*e^(2*I*d*x + 2*I*c) - 2*I*n)*(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 - 2)/(d*n^3 - 4*d*n
+ (d*n^3 - 4*d*n)*e^(4*I*d*x + 4*I*c) + 2*(d*n^3 - 4*d*n)*e^(2*I*d*x + 2*I*c))
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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))**(-2-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 - 2}{\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))^(-2-n)*(a+I*a*tan(d*x+c))^n,x, algorithm="giac")
[Out]
integrate((e*sec(d*x + c))^(-n - 2)*(I*a*tan(d*x + c) + a)^n, x)