\(\int (e \tan (c+d x))^m (a+i a \tan (c+d x)) \, dx\) [308]

Optimal result

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

[Out]

a*hypergeom([1, 1+m],[2+m],I*tan(d*x+c))*(e*tan(d*x+c))^(1+m)/d/e/(1+m)

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} \[ \int (e \tan (c+d x))^m (a+i a \tan (c+d x)) \, dx=\frac {a (e \tan (c+d x))^{m+1} \operatorname {Hypergeometric2F1}(1,m+1,m+2,i \tan (c+d x))}{d e (m+1)} \]

[In]

Int[(e*Tan[c + d*x])^m*(a + I*a*Tan[c + d*x]),x]

[Out]

(a*Hypergeometric2F1[1, 1 + m, 2 + m, I*Tan[c + d*x]]*(e*Tan[c + d*x])^(1 + m))/(d*e*(1 + m))

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 e x}{a}\right )^m}{-a^2+a x} \, dx,x,i a \tan (c+d x)\right )}{d} \\ & = \frac {a \operatorname {Hypergeometric2F1}(1,1+m,2+m,i \tan (c+d x)) (e \tan (c+d x))^{1+m}}{d e (1+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int (e \tan (c+d x))^m (a+i a \tan (c+d x)) \, dx=\frac {a \operatorname {Hypergeometric2F1}(1,1+m,2+m,i \tan (c+d x)) (e \tan (c+d x))^{1+m}}{d e (1+m)} \]

[In]

Integrate[(e*Tan[c + d*x])^m*(a + I*a*Tan[c + d*x]),x]

[Out]

(a*Hypergeometric2F1[1, 1 + m, 2 + m, I*Tan[c + d*x]]*(e*Tan[c + d*x])^(1 + m))/(d*e*(1 + m))

Maple [F]

\[\int \left (e \tan \left (d x +c \right )\right )^{m} \left (a +i a \tan \left (d x +c \right )\right )d x\]

[In]

int((e*tan(d*x+c))^m*(a+I*a*tan(d*x+c)),x)

[Out]

int((e*tan(d*x+c))^m*(a+I*a*tan(d*x+c)),x)

Fricas [F]

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

[In]

integrate((e*tan(d*x+c))^m*(a+I*a*tan(d*x+c)),x, algorithm="fricas")

[Out]

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

Sympy [F]

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

[In]

integrate((e*tan(d*x+c))**m*(a+I*a*tan(d*x+c)),x)

[Out]

I*a*(Integral(-I*(e*tan(c + d*x))**m, x) + Integral((e*tan(c + d*x))**m*tan(c + d*x), x))

Maxima [F]

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

[In]

integrate((e*tan(d*x+c))^m*(a+I*a*tan(d*x+c)),x, algorithm="maxima")

[Out]

integrate((I*a*tan(d*x + c) + a)*(e*tan(d*x + c))^m, x)

Giac [F]

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

[In]

integrate((e*tan(d*x+c))^m*(a+I*a*tan(d*x+c)),x, algorithm="giac")

[Out]

integrate((I*a*tan(d*x + c) + a)*(e*tan(d*x + c))^m, x)

Mupad [F(-1)]

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

[In]

int((e*tan(c + d*x))^m*(a + a*tan(c + d*x)*1i),x)

[Out]

int((e*tan(c + d*x))^m*(a + a*tan(c + d*x)*1i), x)