\(\int \cot ^m(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx\) [567]

Optimal result

Integrand size = 34, antiderivative size = 179 \[ \int \cot ^m(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx=\frac {(A-i B) \operatorname {AppellF1}(1-m,1-n,1,2-m,-i \tan (c+d x),i \tan (c+d x)) \cot ^{-1+m}(c+d x) (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n}{d (1-m)}+\frac {i B \cot ^{-1+m}(c+d x) \operatorname {Hypergeometric2F1}(1-m,1-n,2-m,-i \tan (c+d x)) (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n}{d (1-m)} \]

[Out]

(A-I*B)*AppellF1(1-m,1-n,1,2-m,-I*tan(d*x+c),I*tan(d*x+c))*cot(d*x+c)^(-1+m)*(a+I*a*tan(d*x+c))^n/d/(1-m)/((1+
I*tan(d*x+c))^n)+I*B*cot(d*x+c)^(-1+m)*hypergeom([1-m, 1-n],[2-m],-I*tan(d*x+c))*(a+I*a*tan(d*x+c))^n/d/(1-m)/
((1+I*tan(d*x+c))^n)

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {4326, 3682, 3645, 140, 138, 3680, 68, 66} \[ \int \cot ^m(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx=\frac {(A-i B) \cot ^{m-1}(c+d x) (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n \operatorname {AppellF1}(1-m,1-n,1,2-m,-i \tan (c+d x),i \tan (c+d x))}{d (1-m)}+\frac {i B \cot ^{m-1}(c+d x) (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n \operatorname {Hypergeometric2F1}(1-m,1-n,2-m,-i \tan (c+d x))}{d (1-m)} \]

[In]

Int[Cot[c + d*x]^m*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]),x]

[Out]

((A - I*B)*AppellF1[1 - m, 1 - n, 1, 2 - m, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Cot[c + d*x]^(-1 + m)*(a + I*a*
Tan[c + d*x])^n)/(d*(1 - m)*(1 + I*Tan[c + d*x])^n) + (I*B*Cot[c + d*x]^(-1 + m)*Hypergeometric2F1[1 - m, 1 -
n, 2 - m, (-I)*Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/(d*(1 - m)*(1 + I*Tan[c + d*x])^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 68

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

Rule 138

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

Rule 140

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[c^IntPart[n]*((c +
d*x)^FracPart[n]/(1 + d*(x/c))^FracPart[n]), Int[(b*x)^m*(1 + d*(x/c))^n*(e + f*x)^p, x], x] /; FreeQ[{b, c, d
, e, f, m, n, p}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] &&  !GtQ[c, 0]

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]

Rule 3680

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

Rule 3682

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

Rule 4326

Int[(cot[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Dist[(c*Cot[a + b*x])^m*(c*Tan[a + b*x])^m, Int[A
ctivateTrig[u]/(c*Tan[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownTangentIntegrandQ
[u, x]

Rubi steps \begin{align*} \text {integral}& = \left (\cot ^m(c+d x) \tan ^m(c+d x)\right ) \int \tan ^{-m}(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx \\ & = -\left (\left ((-A+i B) \cot ^m(c+d x) \tan ^m(c+d x)\right ) \int \tan ^{-m}(c+d x) (a+i a \tan (c+d x))^n \, dx\right )+\frac {\left (i B \cot ^m(c+d x) \tan ^m(c+d x)\right ) \int \tan ^{-m}(c+d x) (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^n \, dx}{a} \\ & = -\frac {\left (i a^2 (-A+i B) \cot ^m(c+d x) \tan ^m(c+d x)\right ) \text {Subst}\left (\int \frac {\left (-\frac {i x}{a}\right )^{-m} (a+x)^{-1+n}}{-a^2+a x} \, dx,x,i a \tan (c+d x)\right )}{d}+\frac {\left (i a B \cot ^m(c+d x) \tan ^m(c+d x)\right ) \text {Subst}\left (\int x^{-m} (a+i a x)^{-1+n} \, dx,x,\tan (c+d x)\right )}{d} \\ & = -\frac {\left (i a (-A+i B) \cot ^m(c+d x) (1+i \tan (c+d x))^{-n} \tan ^m(c+d x) (a+i a \tan (c+d x))^n\right ) \text {Subst}\left (\int \frac {\left (-\frac {i x}{a}\right )^{-m} \left (1+\frac {x}{a}\right )^{-1+n}}{-a^2+a x} \, dx,x,i a \tan (c+d x)\right )}{d}+\frac {\left (i B \cot ^m(c+d x) (1+i \tan (c+d x))^{-n} \tan ^m(c+d x) (a+i a \tan (c+d x))^n\right ) \text {Subst}\left (\int (1+i x)^{-1+n} x^{-m} \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {(A-i B) \operatorname {AppellF1}(1-m,1-n,1,2-m,-i \tan (c+d x),i \tan (c+d x)) \cot ^{-1+m}(c+d x) (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n}{d (1-m)}+\frac {i B \cot ^{-1+m}(c+d x) \operatorname {Hypergeometric2F1}(1-m,1-n,2-m,-i \tan (c+d x)) (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n}{d (1-m)} \\ \end{align*}

Mathematica [F]

\[ \int \cot ^m(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx=\int \cot ^m(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx \]

[In]

Integrate[Cot[c + d*x]^m*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]),x]

[Out]

Integrate[Cot[c + d*x]^m*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]), x]

Maple [F]

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

[In]

int(cot(d*x+c)^m*(a+I*a*tan(d*x+c))^n*(A+B*tan(d*x+c)),x)

[Out]

int(cot(d*x+c)^m*(a+I*a*tan(d*x+c))^n*(A+B*tan(d*x+c)),x)

Fricas [F]

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

[In]

integrate(cot(d*x+c)^m*(a+I*a*tan(d*x+c))^n*(A+B*tan(d*x+c)),x, algorithm="fricas")

[Out]

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

Sympy [F]

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

[In]

integrate(cot(d*x+c)**m*(a+I*a*tan(d*x+c))**n*(A+B*tan(d*x+c)),x)

[Out]

Integral((I*a*(tan(c + d*x) - I))**n*(A + B*tan(c + d*x))*cot(c + d*x)**m, x)

Maxima [F]

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

[In]

integrate(cot(d*x+c)^m*(a+I*a*tan(d*x+c))^n*(A+B*tan(d*x+c)),x, algorithm="maxima")

[Out]

integrate((B*tan(d*x + c) + A)*(I*a*tan(d*x + c) + a)^n*cot(d*x + c)^m, x)

Giac [F]

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

[In]

integrate(cot(d*x+c)^m*(a+I*a*tan(d*x+c))^n*(A+B*tan(d*x+c)),x, algorithm="giac")

[Out]

integrate((B*tan(d*x + c) + A)*(I*a*tan(d*x + c) + a)^n*cot(d*x + c)^m, x)

Mupad [F(-1)]

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

[In]

int(cot(c + d*x)^m*(A + B*tan(c + d*x))*(a + a*tan(c + d*x)*1i)^n,x)

[Out]

int(cot(c + d*x)^m*(A + B*tan(c + d*x))*(a + a*tan(c + d*x)*1i)^n, x)