\(\int (d \cot (e+f x))^n (a+i a \tan (e+f x))^2 \, dx\) [789]

Optimal result

Integrand size = 26, antiderivative size = 72 \[ \int (d \cot (e+f x))^n (a+i a \tan (e+f x))^2 \, dx=\frac {a^2 d (d \cot (e+f x))^{-1+n}}{f (1-n)}-\frac {2 a^2 d (d \cot (e+f x))^{-1+n} \operatorname {Hypergeometric2F1}(1,-1+n,n,-i \cot (e+f x))}{f (1-n)} \]

[Out]

a^2*d*(d*cot(f*x+e))^(-1+n)/f/(1-n)-2*a^2*d*(d*cot(f*x+e))^(-1+n)*hypergeom([1, -1+n],[n],-I*cot(f*x+e))/f/(1-
n)

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {3754, 3624, 3618, 12, 66} \[ \int (d \cot (e+f x))^n (a+i a \tan (e+f x))^2 \, dx=\frac {a^2 d (d \cot (e+f x))^{n-1}}{f (1-n)}-\frac {2 a^2 d (d \cot (e+f x))^{n-1} \operatorname {Hypergeometric2F1}(1,n-1,n,-i \cot (e+f x))}{f (1-n)} \]

[In]

Int[(d*Cot[e + f*x])^n*(a + I*a*Tan[e + f*x])^2,x]

[Out]

(a^2*d*(d*Cot[e + f*x])^(-1 + n))/(f*(1 - n)) - (2*a^2*d*(d*Cot[e + f*x])^(-1 + n)*Hypergeometric2F1[1, -1 + n
, n, (-I)*Cot[e + f*x]])/(f*(1 - n))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 66

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x)^(m + 1)/(b*(m + 1)))*Hypergeometr
ic2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[
c, 0] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))

Rule 3618

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

Rule 3624

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

Rule 3754

Int[(cot[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^(n_.))^(p_.), x_Symbol] :> Dist
[d^(n*p), Int[(d*Cot[e + f*x])^(m - n*p)*(b + a*Cot[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x
] &&  !IntegerQ[m] && IntegersQ[n, p]

Rubi steps \begin{align*} \text {integral}& = d^2 \int (d \cot (e+f x))^{-2+n} (i a+a \cot (e+f x))^2 \, dx \\ & = \frac {a^2 d (d \cot (e+f x))^{-1+n}}{f (1-n)}+d^2 \int (d \cot (e+f x))^{-2+n} \left (-2 a^2+2 i a^2 \cot (e+f x)\right ) \, dx \\ & = \frac {a^2 d (d \cot (e+f x))^{-1+n}}{f (1-n)}+\frac {\left (4 i a^4 d^2\right ) \text {Subst}\left (\int \frac {2^{2-n} \left (-\frac {i d x}{a^2}\right )^{-2+n}}{-4 a^4-2 a^2 x} \, dx,x,2 i a^2 \cot (e+f x)\right )}{f} \\ & = \frac {a^2 d (d \cot (e+f x))^{-1+n}}{f (1-n)}+\frac {\left (i 2^{4-n} a^4 d^2\right ) \text {Subst}\left (\int \frac {\left (-\frac {i d x}{a^2}\right )^{-2+n}}{-4 a^4-2 a^2 x} \, dx,x,2 i a^2 \cot (e+f x)\right )}{f} \\ & = \frac {a^2 d (d \cot (e+f x))^{-1+n}}{f (1-n)}-\frac {2 a^2 d (d \cot (e+f x))^{-1+n} \operatorname {Hypergeometric2F1}(1,-1+n,n,-i \cot (e+f x))}{f (1-n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.02 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.72 \[ \int (d \cot (e+f x))^n (a+i a \tan (e+f x))^2 \, dx=-\frac {a^2 d (d \cot (e+f x))^{-1+n} (-1+2 \operatorname {Hypergeometric2F1}(1,1-n,2-n,i \tan (e+f x)))}{f (-1+n)} \]

[In]

Integrate[(d*Cot[e + f*x])^n*(a + I*a*Tan[e + f*x])^2,x]

[Out]

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

Maple [F]

\[\int \left (d \cot \left (f x +e \right )\right )^{n} \left (a +i a \tan \left (f x +e \right )\right )^{2}d x\]

[In]

int((d*cot(f*x+e))^n*(a+I*a*tan(f*x+e))^2,x)

[Out]

int((d*cot(f*x+e))^n*(a+I*a*tan(f*x+e))^2,x)

Fricas [F]

\[ \int (d \cot (e+f x))^n (a+i a \tan (e+f x))^2 \, dx=\int { {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{2} \left (d \cot \left (f x + e\right )\right )^{n} \,d x } \]

[In]

integrate((d*cot(f*x+e))^n*(a+I*a*tan(f*x+e))^2,x, algorithm="fricas")

[Out]

integral(4*a^2*((I*d*e^(2*I*f*x + 2*I*e) + I*d)/(e^(2*I*f*x + 2*I*e) - 1))^n*e^(4*I*f*x + 4*I*e)/(e^(4*I*f*x +
4*I*e) + 2*e^(2*I*f*x + 2*I*e) + 1), x)

Sympy [F]

\[ \int (d \cot (e+f x))^n (a+i a \tan (e+f x))^2 \, dx=- a^{2} \left (\int \left (- \left (d \cot {\left (e + f x \right )}\right )^{n}\right )\, dx + \int \left (d \cot {\left (e + f x \right )}\right )^{n} \tan ^{2}{\left (e + f x \right )}\, dx + \int \left (- 2 i \left (d \cot {\left (e + f x \right )}\right )^{n} \tan {\left (e + f x \right )}\right )\, dx\right ) \]

[In]

integrate((d*cot(f*x+e))**n*(a+I*a*tan(f*x+e))**2,x)

[Out]

-a**2*(Integral(-(d*cot(e + f*x))**n, x) + Integral((d*cot(e + f*x))**n*tan(e + f*x)**2, x) + Integral(-2*I*(d
*cot(e + f*x))**n*tan(e + f*x), x))

Maxima [F]

\[ \int (d \cot (e+f x))^n (a+i a \tan (e+f x))^2 \, dx=\int { {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{2} \left (d \cot \left (f x + e\right )\right )^{n} \,d x } \]

[In]

integrate((d*cot(f*x+e))^n*(a+I*a*tan(f*x+e))^2,x, algorithm="maxima")

[Out]

integrate((I*a*tan(f*x + e) + a)^2*(d*cot(f*x + e))^n, x)

Giac [F]

\[ \int (d \cot (e+f x))^n (a+i a \tan (e+f x))^2 \, dx=\int { {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{2} \left (d \cot \left (f x + e\right )\right )^{n} \,d x } \]

[In]

integrate((d*cot(f*x+e))^n*(a+I*a*tan(f*x+e))^2,x, algorithm="giac")

[Out]

integrate((I*a*tan(f*x + e) + a)^2*(d*cot(f*x + e))^n, x)

Mupad [F(-1)]

Timed out. \[ \int (d \cot (e+f x))^n (a+i a \tan (e+f x))^2 \, dx=\int {\left (d\,\mathrm {cot}\left (e+f\,x\right )\right )}^n\,{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2 \,d x \]

[In]

int((d*cot(e + f*x))^n*(a + a*tan(e + f*x)*1i)^2,x)

[Out]

int((d*cot(e + f*x))^n*(a + a*tan(e + f*x)*1i)^2, x)