\(\int \cot (c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx\) [499]

Optimal result

Integrand size = 29, antiderivative size = 191 \[ \int \cot (c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx=-\frac {A \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {a+b \tan (c+d x)}{a}\right ) (a+b \tan (c+d x))^{1+n}}{a d (1+n)}+\frac {(i A+B) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {a+b \tan (c+d x)}{a-i b}\right ) (a+b \tan (c+d x))^{1+n}}{2 (i a+b) d (1+n)}+\frac {(A+i B) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {a+b \tan (c+d x)}{a+i b}\right ) (a+b \tan (c+d x))^{1+n}}{2 (a+i b) d (1+n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.88 \[ \int \cot (c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx=\frac {\left (a (a+i b) (A-i B) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {a+b \tan (c+d x)}{a-i b}\right )+(a-i b) \left (a (A+i B) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {a+b \tan (c+d x)}{a+i b}\right )-2 A (a+i b) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b \tan (c+d x)}{a}\right )\right )\right ) (a+b \tan (c+d x))^{1+n}}{2 a (a-i b) (a+i b) d (1+n)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.03, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {3042, 4096, 3042, 4022, 3042, 4020, 25, 78, 4117, 75}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \cot (c+d x) (A+B \tan (c+d x)) (a+b \tan (c+d x))^n \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(A+B \tan (c+d x)) (a+b \tan (c+d x))^n}{\tan (c+d x)}dx\)

\(\Big \downarrow \) 4096

\(\displaystyle \int (B-A \tan (c+d x)) (a+b \tan (c+d x))^ndx+A \int \cot (c+d x) (a+b \tan (c+d x))^n \left (\tan ^2(c+d x)+1\right )dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (B-A \tan (c+d x)) (a+b \tan (c+d x))^ndx+A \int \frac {(a+b \tan (c+d x))^n \left (\tan (c+d x)^2+1\right )}{\tan (c+d x)}dx\)

\(\Big \downarrow \) 4022

\(\displaystyle -\frac {1}{2} (-B+i A) \int (1-i \tan (c+d x)) (a+b \tan (c+d x))^ndx+\frac {1}{2} (B+i A) \int (i \tan (c+d x)+1) (a+b \tan (c+d x))^ndx+A \int \frac {(a+b \tan (c+d x))^n \left (\tan (c+d x)^2+1\right )}{\tan (c+d x)}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {1}{2} (-B+i A) \int (1-i \tan (c+d x)) (a+b \tan (c+d x))^ndx+\frac {1}{2} (B+i A) \int (i \tan (c+d x)+1) (a+b \tan (c+d x))^ndx+A \int \frac {(a+b \tan (c+d x))^n \left (\tan (c+d x)^2+1\right )}{\tan (c+d x)}dx\)

\(\Big \downarrow \) 4020

\(\displaystyle \frac {i (B+i A) \int -\frac {(a+b \tan (c+d x))^n}{1-i \tan (c+d x)}d(i \tan (c+d x))}{2 d}+\frac {i (-B+i A) \int -\frac {(a+b \tan (c+d x))^n}{i \tan (c+d x)+1}d(-i \tan (c+d x))}{2 d}+A \int \frac {(a+b \tan (c+d x))^n \left (\tan (c+d x)^2+1\right )}{\tan (c+d x)}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {i (B+i A) \int \frac {(a+b \tan (c+d x))^n}{1-i \tan (c+d x)}d(i \tan (c+d x))}{2 d}-\frac {i (-B+i A) \int \frac {(a+b \tan (c+d x))^n}{i \tan (c+d x)+1}d(-i \tan (c+d x))}{2 d}+A \int \frac {(a+b \tan (c+d x))^n \left (\tan (c+d x)^2+1\right )}{\tan (c+d x)}dx\)

\(\Big \downarrow \) 78

\(\displaystyle A \int \frac {(a+b \tan (c+d x))^n \left (\tan (c+d x)^2+1\right )}{\tan (c+d x)}dx-\frac {i (B+i A) (a+b \tan (c+d x))^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {a+b \tan (c+d x)}{a-i b}\right )}{2 d (n+1) (a-i b)}-\frac {i (-B+i A) (a+b \tan (c+d x))^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {a+b \tan (c+d x)}{a+i b}\right )}{2 d (n+1) (a+i b)}\)

\(\Big \downarrow \) 4117

\(\displaystyle \frac {A \int \cot (c+d x) (a+b \tan (c+d x))^nd\tan (c+d x)}{d}-\frac {i (B+i A) (a+b \tan (c+d x))^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {a+b \tan (c+d x)}{a-i b}\right )}{2 d (n+1) (a-i b)}-\frac {i (-B+i A) (a+b \tan (c+d x))^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {a+b \tan (c+d x)}{a+i b}\right )}{2 d (n+1) (a+i b)}\)

\(\Big \downarrow \) 75

\(\displaystyle -\frac {i (B+i A) (a+b \tan (c+d x))^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {a+b \tan (c+d x)}{a-i b}\right )}{2 d (n+1) (a-i b)}-\frac {i (-B+i A) (a+b \tan (c+d x))^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {a+b \tan (c+d x)}{a+i b}\right )}{2 d (n+1) (a+i b)}-\frac {A (a+b \tan (c+d x))^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {b \tan (c+d x)}{a}+1\right )}{a d (n+1)}\)

Input:

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

Output:

((-1/2*I)*(I*A + B)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x] 
)/(a - I*b)]*(a + b*Tan[c + d*x])^(1 + n))/((a - I*b)*d*(1 + n)) - ((I/2)* 
(I*A - B)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + I*b 
)]*(a + b*Tan[c + d*x])^(1 + n))/((a + I*b)*d*(1 + n)) - (A*Hypergeometric 
2F1[1, 1 + n, 2 + n, 1 + (b*Tan[c + d*x])/a]*(a + b*Tan[c + d*x])^(1 + n)) 
/(a*d*(1 + n))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 75

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

rule 78

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] 
 &&  !IntegerQ[m] && IntegerQ[n]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4020

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + 
(f_.)*(x_)]), x_Symbol] :> Simp[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]
 

rule 4022

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

rule 4096

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

rule 4117

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

integrate(cot(d*x+c)*(a+b*tan(d*x+c))^n*(A+B*tan(d*x+c)),x, algorithm="fri 
cas")
 

Output:

integral((B*cot(d*x + c)*tan(d*x + c) + A*cot(d*x + c))*(b*tan(d*x + c) + 
a)^n, x)
 

Sympy [F]

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

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

Output:

Integral((A + B*tan(c + d*x))*(a + b*tan(c + d*x))**n*cot(c + d*x), x)
 

Maxima [F]

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

integrate(cot(d*x+c)*(a+b*tan(d*x+c))^n*(A+B*tan(d*x+c)),x, algorithm="max 
ima")
 

Output:

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

Giac [F]

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

integrate(cot(d*x+c)*(a+b*tan(d*x+c))^n*(A+B*tan(d*x+c)),x, algorithm="gia 
c")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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