\(\int (d \tan (e+f x))^n (a-i a \tan (e+f x)) \, dx\) [318]

Optimal result

Integrand size = 24, antiderivative size = 43 \[ \int (d \tan (e+f x))^n (a-i a \tan (e+f x)) \, dx=\frac {a \operatorname {Hypergeometric2F1}(1,1+n,2+n,-i \tan (e+f x)) (d \tan (e+f x))^{1+n}}{d f (1+n)} \]

[Out]

a*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 = 0.06 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3618, 66} \[ \int (d \tan (e+f x))^n (a-i a \tan (e+f x)) \, dx=\frac {a (d \tan (e+f x))^{n+1} \operatorname {Hypergeometric2F1}(1,n+1,n+2,-i \tan (e+f x))}{d f (n+1)} \]

[In]

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

[Out]

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

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (i a^2\right ) \text {Subst}\left (\int \frac {\left (\frac {i d x}{a}\right )^n}{-a^2+a x} \, dx,x,-i a \tan (e+f x)\right )}{f} \\ & = \frac {a \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 = 0.07 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.02 \[ \int (d \tan (e+f x))^n (a-i a \tan (e+f x)) \, dx=\frac {a \operatorname {Hypergeometric2F1}(1,1+n,2+n,-i \tan (e+f x)) \tan (e+f x) (d \tan (e+f x))^n}{f (1+n)} \]

[In]

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

[Out]

(a*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]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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