3.484 \(\int (e \sec (c+d x))^{-3-n} (a+i a \tan (c+d x))^n \, dx\)
Optimal. Leaf size=205 \[ \frac {6 i (a+i a \tan (c+d x))^{n+3} (e \sec (c+d x))^{-n-3}}{a^3 d \left (n^4-10 n^2+9\right )}-\frac {6 i (a+i a \tan (c+d x))^{n+2} (e \sec (c+d x))^{-n-3}}{a^2 d (3-n) \left (1-n^2\right )}+\frac {3 i (a+i a \tan (c+d x))^{n+1} (e \sec (c+d x))^{-n-3}}{a d \left (n^2-4 n+3\right )}+\frac {i (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-n-3}}{d (3-n)} \]
[Out]
I*(e*sec(d*x+c))^(-3-n)*(a+I*a*tan(d*x+c))^n/d/(3-n)+3*I*(e*sec(d*x+c))^(-3-n)*(a+I*a*tan(d*x+c))^(1+n)/a/d/(n
^2-4*n+3)-6*I*(e*sec(d*x+c))^(-3-n)*(a+I*a*tan(d*x+c))^(2+n)/a^2/d/(3-n)/(-n^2+1)+6*I*(e*sec(d*x+c))^(-3-n)*(a
+I*a*tan(d*x+c))^(3+n)/a^3/d/(n^4-10*n^2+9)
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Rubi [A] time = 0.30, antiderivative size = 205, normalized size of antiderivative = 1.00,
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 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]
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]
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*}
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Mathematica [A] time = 0.66, 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)
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fricas [A] time = 0.58, size = 263, normalized size = 1.28 \[ \frac {{\left (-i \, n^{3} - 3 i \, n^{2} + {\left (-i \, n^{3} + 3 i \, n^{2} + i \, n - 3 i\right )} e^{\left (6 i \, d x + 6 i \, c\right )} + {\left (-3 i \, n^{3} + 3 i \, n^{2} + 27 i \, n - 27 i\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + {\left (-3 i \, n^{3} - 3 i \, n^{2} + 27 i \, n + 27 i\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, n + 3 i\right )} \left (\frac {2 \, e e^{\left (i \, d x + i \, c\right )}}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{-n - 3} e^{\left (i \, d n x + i \, c n + n \log \left (\frac {2 \, e e^{\left (i \, d x + i \, c\right )}}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right ) + n \log \left (\frac {a}{e}\right )\right )}}{d n^{4} - 10 \, d n^{2} + {\left (d n^{4} - 10 \, d n^{2} + 9 \, d\right )} e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, {\left (d n^{4} - 10 \, d n^{2} + 9 \, d\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, {\left (d n^{4} - 10 \, d n^{2} + 9 \, d\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + 9 \, d} \]
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*e*e^(I*d*x + I*
c)/(e^(2*I*d*x + 2*I*c) + 1))^(-n - 3)*e^(I*d*n*x + I*c*n + n*log(2*e*e^(I*d*x + I*c)/(e^(2*I*d*x + 2*I*c) + 1
)) + n*log(a/e))/(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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e \sec \left (d x + c\right )\right )^{-n - 3} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n}\,{d x} \]
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)
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maple [C] time = 2.18, size = 4994, normalized size = 24.36 \[ \text {output too large to display} \]
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)*exp(I*(d*x+c))^n*a^n/e^3*e^(-n)*exp(-1/2*I*(6*c+6*d*x-2*csgn(I*exp(2*I*(d*x+c)))^2*csgn(I*e
xp(I*(d*x+c)))*Pi*n+n*Pi*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))^2*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c
))+1))+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))^2-3*Pi*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))*csgn(I*e
)*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))-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))-csgn(I*a/(exp(2*I*(d*x+c))+1)*exp(2*I*(d*x+c)))^2*csgn(I*a)*Pi*n-csgn(I*e
xp(2*I*(d*x+c)))*csgn(I*exp(2*I*(d*x+c))/(exp(2*I*(d*x+c))+1))^2*Pi*n+csgn(I*exp(2*I*(d*x+c)))*csgn(I*exp(I*(d
*x+c)))^2*Pi*n-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))*Pi*n-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*Pi*n+n*Pi*csgn(I*e/(exp(2*I
*(d*x+c))+1)*exp(I*(d*x+c)))^2*csgn(I*e)-n*Pi*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))*csgn(I*e)*csgn(I*e
xp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))-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))+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))*Pi*n+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)*Pi*n-n*Pi*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))^3+3*Pi*csgn(I*exp(I*(d*x+c)))*csgn(I*exp
(I*(d*x+c))/(exp(2*I*(d*x+c))+1))^2+csgn(I*exp(2*I*(d*x+c))/(exp(2*I*(d*x+c))+1))^3*Pi*n-n*Pi*csgn(I*e/(exp(2*
I*(d*x+c))+1)*exp(I*(d*x+c)))^3+csgn(I*exp(2*I*(d*x+c)))^3*Pi*n-3*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*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))^2
*csgn(I*e)+3*Pi*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))^2*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))+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+csgn(I*a/(exp(2*I*(d*x+c))+1)*ex
p(2*I*(d*x+c)))^3*Pi*n))+1/8/(I*n*d+3*I*d)*exp(I*(d*x+c))^n*a^n/e^3*e^(-n)*exp(1/2*I*(6*c+6*d*x+2*csgn(I*exp(2
*I*(d*x+c)))^2*csgn(I*exp(I*(d*x+c)))*Pi*n-n*Pi*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))^2*csgn(I*exp(I*(
d*x+c))/(exp(2*I*(d*x+c))+1))-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+3*Pi*csgn(I*e/(exp(2*I*(d*x+c))+1)*e
xp(I*(d*x+c)))*csgn(I*e)*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))+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))+csgn(I*a/(exp(2*I*(d*x+c))+1)*exp(2*I*(d*x+c)))^2*
csgn(I*a)*Pi*n+csgn(I*exp(2*I*(d*x+c)))*csgn(I*exp(2*I*(d*x+c))/(exp(2*I*(d*x+c))+1))^2*Pi*n-csgn(I*exp(2*I*(d
*x+c)))*csgn(I*exp(I*(d*x+c)))^2*Pi*n+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))*Pi*n+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*Pi*n
-n*Pi*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))^2*csgn(I*e)+n*Pi*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+
c)))*csgn(I*e)*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))+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))-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))*Pi*n-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)*Pi*n+n*Pi*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))^3-3*Pi*csgn(I*exp
(I*(d*x+c)))*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))^2-csgn(I*exp(2*I*(d*x+c))/(exp(2*I*(d*x+c))+1))^3*Pi*
n+n*Pi*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))^3-csgn(I*exp(2*I*(d*x+c)))^3*Pi*n+3*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*e/(exp(2*I*(d*x+c
))+1)*exp(I*(d*x+c)))^2*csgn(I*e)-3*Pi*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))^2*csgn(I*exp(I*(d*x+c))/(
exp(2*I*(d*x+c))+1))-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-csgn(I*a/
(exp(2*I*(d*x+c))+1)*exp(2*I*(d*x+c)))^3*Pi*n))+3/8/(-I*d+I*n*d)*exp(I*(d*x+c))^n*a^n/e^3*e^(-n)*exp(-1/2*I*(2
*c+2*d*x-2*csgn(I*exp(2*I*(d*x+c)))^2*csgn(I*exp(I*(d*x+c)))*Pi*n+n*Pi*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*
x+c)))^2*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))+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))^2-3*Pi*csgn(I*e
/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))*csgn(I*e)*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))-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))-csgn(I*a/(exp(2*I*(d*x+c))+
1)*exp(2*I*(d*x+c)))^2*csgn(I*a)*Pi*n-csgn(I*exp(2*I*(d*x+c)))*csgn(I*exp(2*I*(d*x+c))/(exp(2*I*(d*x+c))+1))^2
*Pi*n+csgn(I*exp(2*I*(d*x+c)))*csgn(I*exp(I*(d*x+c)))^2*Pi*n-csgn(I*exp(2*I*(d*x+c))/(exp(2*I*(d*x+c))+1))^2*c
sgn(I/(exp(2*I*(d*x+c))+1))*Pi*n-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)*e
xp(2*I*(d*x+c)))^2*Pi*n+n*Pi*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))^2*csgn(I*e)-n*Pi*csgn(I*e/(exp(2*I*
(d*x+c))+1)*exp(I*(d*x+c)))*csgn(I*e)*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))-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))+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))*Pi*n+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)*Pi*n-n*Pi*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c)
)+1))^3+3*Pi*csgn(I*exp(I*(d*x+c)))*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))^2+csgn(I*exp(2*I*(d*x+c))/(exp
(2*I*(d*x+c))+1))^3*Pi*n-n*Pi*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))^3+csgn(I*exp(2*I*(d*x+c)))^3*Pi*n-
3*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*c
sgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))^2*csgn(I*e)+3*Pi*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))^2*
csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))+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+csgn(I*a/(exp(2*I*(d*x+c))+1)*exp(2*I*(d*x+c)))^3*Pi*n))+3/8/(I*d+I*n*d)*exp(I*(d*x+c))^n*a^n/e^
3*e^(-n)*exp(1/2*I*(2*c+2*d*x+2*csgn(I*exp(2*I*(d*x+c)))^2*csgn(I*exp(I*(d*x+c)))*Pi*n-n*Pi*csgn(I*e/(exp(2*I*
(d*x+c))+1)*exp(I*(d*x+c)))^2*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))-n*Pi*csgn(I/(exp(2*I*(d*x+c))+1))*cs
gn(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+3*Pi*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))*csgn(I*e)*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1
))+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))+csgn(I
*a/(exp(2*I*(d*x+c))+1)*exp(2*I*(d*x+c)))^2*csgn(I*a)*Pi*n+csgn(I*exp(2*I*(d*x+c)))*csgn(I*exp(2*I*(d*x+c))/(e
xp(2*I*(d*x+c))+1))^2*Pi*n-csgn(I*exp(2*I*(d*x+c)))*csgn(I*exp(I*(d*x+c)))^2*Pi*n+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))*Pi*n+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*Pi*n-n*Pi*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))^2*csgn(I*e)+n*
Pi*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))*csgn(I*e)*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))+n*Pi*cs
gn(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))-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))*Pi*n-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)*Pi*n+n*Pi*csgn(I*exp(I*(d*x
+c))/(exp(2*I*(d*x+c))+1))^3-3*Pi*csgn(I*exp(I*(d*x+c)))*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))^2-csgn(I*
exp(2*I*(d*x+c))/(exp(2*I*(d*x+c))+1))^3*Pi*n+n*Pi*csgn(I*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))^3-csgn(I*exp(
2*I*(d*x+c)))^3*Pi*n+3*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*e/(exp(2*I*(d*x+c))+1)*exp(I*(d*x+c)))^2*csgn(I*e)-3*Pi*csgn(I*e/(exp(2*I*(d*x+c))+
1)*exp(I*(d*x+c)))^2*csgn(I*exp(I*(d*x+c))/(exp(2*I*(d*x+c))+1))-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-csgn(I*a/(exp(2*I*(d*x+c))+1)*exp(2*I*(d*x+c)))^3*Pi*n))
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maxima [A] time = 1.09, size = 344, normalized size = 1.68 \[ \frac {{\left (-i \, a^{n} n^{3} + 3 i \, a^{n} n^{2} + i \, a^{n} n - 3 i \, a^{n}\right )} \cos \left ({\left (d x + c\right )} {\left (n + 3\right )}\right ) + {\left (-3 i \, a^{n} n^{3} + 3 i \, a^{n} n^{2} + 27 i \, a^{n} n - 27 i \, a^{n}\right )} \cos \left ({\left (d x + c\right )} {\left (n + 1\right )}\right ) + {\left (-3 i \, a^{n} n^{3} - 3 i \, a^{n} n^{2} + 27 i \, a^{n} n + 27 i \, a^{n}\right )} \cos \left ({\left (d x + c\right )} {\left (n - 1\right )}\right ) + {\left (-i \, a^{n} n^{3} - 3 i \, a^{n} n^{2} + i \, a^{n} n + 3 i \, a^{n}\right )} \cos \left ({\left (d x + c\right )} {\left (n - 3\right )}\right ) + {\left (a^{n} n^{3} - 3 \, a^{n} n^{2} - a^{n} n + 3 \, a^{n}\right )} \sin \left ({\left (d x + c\right )} {\left (n + 3\right )}\right ) + 3 \, {\left (a^{n} n^{3} - a^{n} n^{2} - 9 \, a^{n} n + 9 \, a^{n}\right )} \sin \left ({\left (d x + c\right )} {\left (n + 1\right )}\right ) + 3 \, {\left (a^{n} n^{3} + a^{n} n^{2} - 9 \, a^{n} n - 9 \, a^{n}\right )} \sin \left ({\left (d x + c\right )} {\left (n - 1\right )}\right ) + {\left (a^{n} n^{3} + 3 \, a^{n} n^{2} - a^{n} n - 3 \, a^{n}\right )} \sin \left ({\left (d x + c\right )} {\left (n - 3\right )}\right )}{8 \, {\left (e^{n + 3} n^{4} - 10 \, e^{n + 3} n^{2} + 9 \, e^{n + 3}\right )} d} \]
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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mupad [B] time = 9.24, size = 425, normalized size = 2.07 \[ -\frac {\left (2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )\,\left (2\,{\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}^2+\sin \left (3\,c+3\,d\,x\right )\,1{}\mathrm {i}-1\right )\,\left (\frac {{\left (a-\frac {a\,\sin \left (c+d\,x\right )\,1{}\mathrm {i}}{2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1}\right )}^n\,\left (-n^3-3\,n^2+n+3\right )}{d\,\left (n^4\,1{}\mathrm {i}-n^2\,10{}\mathrm {i}+9{}\mathrm {i}\right )}+\frac {{\left (a-\frac {a\,\sin \left (c+d\,x\right )\,1{}\mathrm {i}}{2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1}\right )}^n\,\left (-2\,{\sin \left (3\,c+3\,d\,x\right )}^2+\sin \left (6\,c+6\,d\,x\right )\,1{}\mathrm {i}+1\right )\,\left (-n^3+3\,n^2+n-3\right )}{d\,\left (n^4\,1{}\mathrm {i}-n^2\,10{}\mathrm {i}+9{}\mathrm {i}\right )}+\frac {{\left (a-\frac {a\,\sin \left (c+d\,x\right )\,1{}\mathrm {i}}{2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1}\right )}^n\,\left (-2\,{\sin \left (c+d\,x\right )}^2+\sin \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}+1\right )\,\left (-3\,n^3-3\,n^2+27\,n+27\right )}{d\,\left (n^4\,1{}\mathrm {i}-n^2\,10{}\mathrm {i}+9{}\mathrm {i}\right )}+\frac {{\left (a-\frac {a\,\sin \left (c+d\,x\right )\,1{}\mathrm {i}}{2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1}\right )}^n\,\left (-2\,{\sin \left (2\,c+2\,d\,x\right )}^2+\sin \left (4\,c+4\,d\,x\right )\,1{}\mathrm {i}+1\right )\,\left (-3\,n^3+3\,n^2+27\,n-27\right )}{d\,\left (n^4\,1{}\mathrm {i}-n^2\,10{}\mathrm {i}+9{}\mathrm {i}\right )}\right )}{8\,{\left (-\frac {e}{2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1}\right )}^{n+3}\,{\left ({\sin \left (c+d\,x\right )}^2-1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + a*tan(c + d*x)*1i)^n/(e/cos(c + d*x))^(n + 3),x)
[Out]
-((2*sin(c/2 + (d*x)/2)^2 - 1)*(sin(3*c + 3*d*x)*1i + 2*sin((3*c)/2 + (3*d*x)/2)^2 - 1)*(((a - (a*sin(c + d*x)
*1i)/(2*sin(c/2 + (d*x)/2)^2 - 1))^n*(n - 3*n^2 - n^3 + 3))/(d*(n^4*1i - n^2*10i + 9i)) + ((a - (a*sin(c + d*x
)*1i)/(2*sin(c/2 + (d*x)/2)^2 - 1))^n*(sin(6*c + 6*d*x)*1i - 2*sin(3*c + 3*d*x)^2 + 1)*(n + 3*n^2 - n^3 - 3))/
(d*(n^4*1i - n^2*10i + 9i)) + ((a - (a*sin(c + d*x)*1i)/(2*sin(c/2 + (d*x)/2)^2 - 1))^n*(sin(2*c + 2*d*x)*1i -
2*sin(c + d*x)^2 + 1)*(27*n - 3*n^2 - 3*n^3 + 27))/(d*(n^4*1i - n^2*10i + 9i)) + ((a - (a*sin(c + d*x)*1i)/(2
*sin(c/2 + (d*x)/2)^2 - 1))^n*(sin(4*c + 4*d*x)*1i - 2*sin(2*c + 2*d*x)^2 + 1)*(27*n + 3*n^2 - 3*n^3 - 27))/(d
*(n^4*1i - n^2*10i + 9i))))/(8*(-e/(2*sin(c/2 + (d*x)/2)^2 - 1))^(n + 3)*(sin(c + d*x)^2 - 1)^2)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e \sec {\left (c + d x \right )}\right )^{- n - 3} \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{n}\, dx \]
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]
Integral((e*sec(c + d*x))**(-n - 3)*(I*a*(tan(c + d*x) - I))**n, x)
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