3.499 \(\int (e \sec (c+d x))^{-2-2 n} (a+i a \tan (c+d x))^n \, dx\)

Optimal. Leaf size=74 \[ -\frac{i (a+i a \tan (c+d x))^{n+1} (e \sec (c+d x))^{-2 (n+1)} \text{Hypergeometric2F1}\left (2,-n-1,-n,\frac{1}{2} (1-i \tan (c+d x))\right )}{4 a d (n+1)} \]

[Out]

((-I/4)*Hypergeometric2F1[2, -1 - n, -n, (1 - I*Tan[c + d*x])/2]*(a + I*a*Tan[c + d*x])^(1 + n))/(a*d*(1 + n)*
(e*Sec[c + d*x])^(2*(1 + n)))

________________________________________________________________________________________

Rubi [A]  time = 0.148941, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {3505, 3523, 7, 68} \[ -\frac{i (a+i a \tan (c+d x))^{n+1} (e \sec (c+d x))^{-2 (n+1)} \text{Hypergeometric2F1}\left (2,-n-1,-n,\frac{1}{2} (1-i \tan (c+d x))\right )}{4 a d (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[(e*Sec[c + d*x])^(-2 - 2*n)*(a + I*a*Tan[c + d*x])^n,x]

[Out]

((-I/4)*Hypergeometric2F1[2, -1 - n, -n, (1 - I*Tan[c + d*x])/2]*(a + I*a*Tan[c + d*x])^(1 + n))/(a*d*(1 + n)*
(e*Sec[c + d*x])^(2*(1 + n)))

Rule 3505

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[(d*S
ec[e + f*x])^m/((a + b*Tan[e + f*x])^(m/2)*(a - b*Tan[e + f*x])^(m/2)), Int[(a + b*Tan[e + f*x])^(m/2 + n)*(a
- b*Tan[e + f*x])^(m/2), x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0]

Rule 3523

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist
[(a*c)/f, Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^(n - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f,
m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 + b^2, 0]

Rule 7

Int[(u_.)*(Px_)^(p_), x_Symbol] :> Int[u*Px^Simplify[p], x] /; PolyQ[Px, x] &&  !RationalQ[p] && FreeQ[p, x] &
& RationalQ[Simplify[p]]

Rule 68

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((b*c - a*d)^n*(a + b*x)^(m + 1)*Hype
rgeometric2F1[-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b^(n + 1)*(m + 1)), x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rubi steps

\begin{align*} \int (e \sec (c+d x))^{-2-2 n} (a+i a \tan (c+d x))^n \, dx &=\left ((e \sec (c+d x))^{-2-2 n} (a-i a \tan (c+d x))^{\frac{1}{2} (2+2 n)} (a+i a \tan (c+d x))^{\frac{1}{2} (2+2 n)}\right ) \int (a-i a \tan (c+d x))^{\frac{1}{2} (-2-2 n)} (a+i a \tan (c+d x))^{\frac{1}{2} (-2-2 n)+n} \, dx\\ &=\frac{\left (a^2 (e \sec (c+d x))^{-2-2 n} (a-i a \tan (c+d x))^{\frac{1}{2} (2+2 n)} (a+i a \tan (c+d x))^{\frac{1}{2} (2+2 n)}\right ) \operatorname{Subst}\left (\int (a-i a x)^{-1+\frac{1}{2} (-2-2 n)} (a+i a x)^{-1+\frac{1}{2} (-2-2 n)+n} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{\left (a^2 (e \sec (c+d x))^{-2-2 n} (a-i a \tan (c+d x))^{\frac{1}{2} (2+2 n)} (a+i a \tan (c+d x))^{\frac{1}{2} (2+2 n)}\right ) \operatorname{Subst}\left (\int \frac{(a-i a x)^{-1+\frac{1}{2} (-2-2 n)}}{(a+i a x)^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac{i \, _2F_1\left (2,-1-n;-n;\frac{1}{2} (1-i \tan (c+d x))\right ) (e \sec (c+d x))^{-2 (1+n)} (a+i a \tan (c+d x))^{1+n}}{4 a d (1+n)}\\ \end{align*}

Mathematica [B]  time = 13.0512, size = 151, normalized size = 2.04 \[ -\frac{i 2^{-n-3} \left (1+e^{2 i (c+d x)}\right )^3 \left (e^{i d x}\right )^n \left (\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{-n} \sec ^n(c+d x) (\cos (d x)+i \sin (d x))^{-n} \text{Hypergeometric2F1}\left (2,n+3,n+4,1+e^{2 i (c+d x)}\right ) (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-2 n}}{d e^2 (n+3)} \]

Antiderivative was successfully verified.

[In]

Integrate[(e*Sec[c + d*x])^(-2 - 2*n)*(a + I*a*Tan[c + d*x])^n,x]

[Out]

((-I)*2^(-3 - n)*(E^(I*d*x))^n*(1 + E^((2*I)*(c + d*x)))^3*Hypergeometric2F1[2, 3 + n, 4 + n, 1 + E^((2*I)*(c
+ d*x))]*Sec[c + d*x]^n*(a + I*a*Tan[c + d*x])^n)/(d*e^2*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^n*(3 + n)
*(e*Sec[c + d*x])^(2*n)*(Cos[d*x] + I*Sin[d*x])^n)

________________________________________________________________________________________

Maple [F]  time = 1.239, size = 0, normalized size = 0. \begin{align*} \int \left ( e\sec \left ( dx+c \right ) \right ) ^{-2\,n-2} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*sec(d*x+c))^(-2*n-2)*(a+I*a*tan(d*x+c))^n,x)

[Out]

int((e*sec(d*x+c))^(-2*n-2)*(a+I*a*tan(d*x+c))^n,x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e \sec \left (d x + c\right )\right )^{-2 \, 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))^(-2-2*n)*(a+I*a*tan(d*x+c))^n,x, algorithm="maxima")

[Out]

integrate((e*sec(d*x + c))^(-2*n - 2)*(I*a*tan(d*x + c) + a)^n, x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\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 )^{-2 \, n - 2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*sec(d*x+c))^(-2-2*n)*(a+I*a*tan(d*x+c))^n,x, algorithm="fricas")

[Out]

integral((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))
^(-2*n - 2), x)

________________________________________________________________________________________

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))**(-2-2*n)*(a+I*a*tan(d*x+c))**n,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e \sec \left (d x + c\right )\right )^{-2 \, 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))^(-2-2*n)*(a+I*a*tan(d*x+c))^n,x, algorithm="giac")

[Out]

integrate((e*sec(d*x + c))^(-2*n - 2)*(I*a*tan(d*x + c) + a)^n, x)