\(\int (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx\) [222]

Optimal result

Integrand size = 26, antiderivative size = 78 \[ \int (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx=\frac {B (a+i a \tan (c+d x))^n}{d n}-\frac {(i A+B) \operatorname {Hypergeometric2F1}\left (1,n,1+n,\frac {1}{2} (1+i \tan (c+d x))\right ) (a+i a \tan (c+d x))^n}{2 d n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {3608, 3562, 70} \[ \int (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx=\frac {B (a+i a \tan (c+d x))^n}{d n}-\frac {(B+i A) (a+i a \tan (c+d x))^n \operatorname {Hypergeometric2F1}\left (1,n,n+1,\frac {1}{2} (i \tan (c+d x)+1)\right )}{2 d n} \]

[In]

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

[Out]

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

Rule 70

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b*c - a*d)^n*((a + b*x)^(m + 1)/(b^(
n + 1)*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rule 3562

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[-b/d, Subst[Int[(a + x)^(n - 1)/(a - x), x]
, x, b*Tan[c + d*x]], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 + b^2, 0]

Rule 3608

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

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

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.77 \[ \int (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx=\frac {\left (2 B-(i A+B) \operatorname {Hypergeometric2F1}\left (1,n,1+n,\frac {1}{2} (1+i \tan (c+d x))\right )\right ) (a+i a \tan (c+d x))^n}{2 d n} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

integrate((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/(e^(2
*I*d*x + 2*I*c) + 1), x)

Sympy [F]

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

[In]

integrate((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)), x)

Maxima [F]

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

[In]

integrate((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, x)

Giac [F]

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

[In]

integrate((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, x)

Mupad [F(-1)]

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

[In]

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

[Out]

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