\(\int (d \tan (e+f x))^n (a+i a \tan (e+f x))^2 \, dx\) [312]

Optimal result

Integrand size = 26, antiderivative size = 75 \[ \int (d \tan (e+f x))^n (a+i a \tan (e+f x))^2 \, dx=-\frac {a^2 (d \tan (e+f x))^{1+n}}{d f (1+n)}+\frac {2 a^2 \operatorname {Hypergeometric2F1}(1,1+n,2+n,i \tan (e+f x)) (d \tan (e+f x))^{1+n}}{d f (1+n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3624, 3618, 12, 66} \[ \int (d \tan (e+f x))^n (a+i a \tan (e+f x))^2 \, dx=-\frac {a^2 (d \tan (e+f x))^{n+1}}{d f (n+1)}+\frac {2 a^2 (d \tan (e+f x))^{n+1} \operatorname {Hypergeometric2F1}(1,n+1,n+2,i \tan (e+f x))}{d f (n+1)} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.78 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.67 \[ \int (d \tan (e+f x))^n (a+i a \tan (e+f x))^2 \, dx=\frac {a^2 (-1+2 \operatorname {Hypergeometric2F1}(1,1+n,2+n,i \tan (e+f x))) \tan (e+f x) (d \tan (e+f x))^n}{f (1+n)} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

integrate((d*tan(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]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

integrate((d*tan(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*tan(f*x + e))^n, x)

Giac [F]

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

[In]

integrate((d*tan(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*tan(f*x + e))^n, x)

Mupad [F(-1)]

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

[In]

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

[Out]

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