\(\int (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^n \, dx\) [1173]

Optimal result

Integrand size = 28, antiderivative size = 114 \[ \int (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^n \, dx=-\frac {i \operatorname {AppellF1}\left (m,-n,1,1+m,-\frac {d (1+i \tan (e+f x))}{i c-d},\frac {1}{2} (1+i \tan (e+f x))\right ) (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^n \left (\frac {c+d \tan (e+f x)}{c+i d}\right )^{-n}}{2 f m} \]

[Out]

-1/2*I*AppellF1(m,-n,1,1+m,-d*(1+I*tan(f*x+e))/(I*c-d),1/2+1/2*I*tan(f*x+e))*(a+I*a*tan(f*x+e))^m*(c+d*tan(f*x
+e))^n/f/m/(((c+d*tan(f*x+e))/(c+I*d))^n)

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {3645, 142, 141} \[ \int (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^n \, dx=-\frac {i (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^n \left (\frac {c+d \tan (e+f x)}{c+i d}\right )^{-n} \operatorname {AppellF1}\left (m,-n,1,m+1,-\frac {d (i \tan (e+f x)+1)}{i c-d},\frac {1}{2} (i \tan (e+f x)+1)\right )}{2 f m} \]

[In]

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

[Out]

((-1/2*I)*AppellF1[m, -n, 1, 1 + m, -((d*(1 + I*Tan[e + f*x]))/(I*c - d)), (1 + I*Tan[e + f*x])/2]*(a + I*a*Ta
n[e + f*x])^m*(c + d*Tan[e + f*x])^n)/(f*m*((c + d*Tan[e + f*x])/(c + I*d))^n)

Rule 141

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[(b*e - a*f
)^p*((a + b*x)^(m + 1)/(b^(p + 1)*(m + 1)*(b/(b*c - a*d))^n))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/(
b*c - a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] &&  !IntegerQ[m] &&  !Int
egerQ[n] && IntegerQ[p] && GtQ[b/(b*c - a*d), 0] &&  !(GtQ[d/(d*a - c*b), 0] && SimplerQ[c + d*x, a + b*x])

Rule 142

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[(c + d*x)^
FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d)))^FracPart[n]), Int[(a + b*x)^m*(b*(c/(b*c -
a*d)) + b*d*(x/(b*c - a*d)))^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] &&  !IntegerQ[m] &&
!IntegerQ[n] && IntegerQ[p] &&  !GtQ[b/(b*c - a*d), 0] &&  !SimplerQ[c + d*x, a + b*x]

Rule 3645

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

Rubi steps \begin{align*} \text {integral}& = \frac {\left (i a^2\right ) \text {Subst}\left (\int \frac {(a+x)^{-1+m} \left (c-\frac {i d x}{a}\right )^n}{-a^2+a x} \, dx,x,i a \tan (e+f x)\right )}{f} \\ & = \frac {\left (i a^2 (c+d \tan (e+f x))^n \left (\frac {c+d \tan (e+f x)}{c+i d}\right )^{-n}\right ) \text {Subst}\left (\int \frac {(a+x)^{-1+m} \left (\frac {c}{c+i d}-\frac {i d x}{a (c+i d)}\right )^n}{-a^2+a x} \, dx,x,i a \tan (e+f x)\right )}{f} \\ & = -\frac {i \operatorname {AppellF1}\left (m,-n,1,1+m,-\frac {d (1+i \tan (e+f x))}{i c-d},\frac {1}{2} (1+i \tan (e+f x))\right ) (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^n \left (\frac {c+d \tan (e+f x)}{c+i d}\right )^{-n}}{2 f m} \\ \end{align*}

Mathematica [F]

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

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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