\(\int \tan ^m(c+d x) (b \tan (c+d x))^n (A+B \tan (c+d x)+C \tan ^2(c+d x)) \, dx\) [46]

Optimal result

Integrand size = 39, antiderivative size = 154 \[ \int \tan ^m(c+d x) (b \tan (c+d x))^n \left (A+B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {C \tan ^{1+m}(c+d x) (b \tan (c+d x))^n}{d (1+m+n)}+\frac {(A-C) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (1+m+n),\frac {1}{2} (3+m+n),-\tan ^2(c+d x)\right ) \tan ^{1+m}(c+d x) (b \tan (c+d x))^n}{d (1+m+n)}+\frac {B \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (2+m+n),\frac {1}{2} (4+m+n),-\tan ^2(c+d x)\right ) \tan ^{2+m}(c+d x) (b \tan (c+d x))^n}{d (2+m+n)} \] Output:

C*tan(d*x+c)^(1+m)*(b*tan(d*x+c))^n/d/(1+m+n)+(A-C)*hypergeom([1, 1/2+1/2* 
m+1/2*n],[3/2+1/2*m+1/2*n],-tan(d*x+c)^2)*tan(d*x+c)^(1+m)*(b*tan(d*x+c))^ 
n/d/(1+m+n)+B*hypergeom([1, 1+1/2*m+1/2*n],[2+1/2*m+1/2*n],-tan(d*x+c)^2)* 
tan(d*x+c)^(2+m)*(b*tan(d*x+c))^n/d/(2+m+n)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.75 \[ \int \tan ^m(c+d x) (b \tan (c+d x))^n \left (A+B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {\tan ^{1+m}(c+d x) (b \tan (c+d x))^n \left (\frac {C}{1+m+n}+\frac {(A-C) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (1+m+n),\frac {1}{2} (3+m+n),-\tan ^2(c+d x)\right )}{1+m+n}+\frac {B \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (2+m+n),\frac {1}{2} (4+m+n),-\tan ^2(c+d x)\right ) \tan (c+d x)}{2+m+n}\right )}{d} \] Input:

Integrate[Tan[c + d*x]^m*(b*Tan[c + d*x])^n*(A + B*Tan[c + d*x] + C*Tan[c 
+ d*x]^2),x]
 

Output:

(Tan[c + d*x]^(1 + m)*(b*Tan[c + d*x])^n*(C/(1 + m + n) + ((A - C)*Hyperge 
ometric2F1[1, (1 + m + n)/2, (3 + m + n)/2, -Tan[c + d*x]^2])/(1 + m + n) 
+ (B*Hypergeometric2F1[1, (2 + m + n)/2, (4 + m + n)/2, -Tan[c + d*x]^2]*T 
an[c + d*x])/(2 + m + n)))/d
 

Rubi [A] (verified)

Time = 0.90 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.96, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.205, Rules used = {2034, 3042, 4113, 3042, 4021, 3042, 3957, 278}

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.

\(\displaystyle \int \tan ^m(c+d x) (b \tan (c+d x))^n \left (A+B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx\)

\(\Big \downarrow \) 2034

\(\displaystyle \tan ^{-n}(c+d x) (b \tan (c+d x))^n \int \tan ^{m+n}(c+d x) \left (C \tan ^2(c+d x)+B \tan (c+d x)+A\right )dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \tan ^{-n}(c+d x) (b \tan (c+d x))^n \int \tan (c+d x)^{m+n} \left (C \tan (c+d x)^2+B \tan (c+d x)+A\right )dx\)

\(\Big \downarrow \) 4113

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4021

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3957

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

\(\Big \downarrow \) 278

\(\displaystyle \tan ^{-n}(c+d x) (b \tan (c+d x))^n \left (\frac {(A-C) \tan ^{m+n+1}(c+d x) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (m+n+1),\frac {1}{2} (m+n+3),-\tan ^2(c+d x)\right )}{d (m+n+1)}+\frac {B \tan ^{m+n+2}(c+d x) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (m+n+2),\frac {1}{2} (m+n+4),-\tan ^2(c+d x)\right )}{d (m+n+2)}+\frac {C \tan ^{m+n+1}(c+d x)}{d (m+n+1)}\right )\)

Input:

Int[Tan[c + d*x]^m*(b*Tan[c + d*x])^n*(A + B*Tan[c + d*x] + C*Tan[c + d*x] 
^2),x]
 

Output:

((b*Tan[c + d*x])^n*((C*Tan[c + d*x]^(1 + m + n))/(d*(1 + m + n)) + ((A - 
C)*Hypergeometric2F1[1, (1 + m + n)/2, (3 + m + n)/2, -Tan[c + d*x]^2]*Tan 
[c + d*x]^(1 + m + n))/(d*(1 + m + n)) + (B*Hypergeometric2F1[1, (2 + m + 
n)/2, (4 + m + n)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(2 + m + n))/(d*(2 + m 
+ n))))/Tan[c + d*x]^n
 

Defintions of rubi rules used

rule 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

rule 2034

Int[(Fx_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Simp[b^IntPart 
[n]*((b*v)^FracPart[n]/(a^IntPart[n]*(a*v)^FracPart[n]))   Int[(a*v)^(m + n 
)*Fx, x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] && 
  !IntegerQ[m + n]
 

rule 3042

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

rule 3957

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d   Subst[Int 
[x^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{b, c, d, n}, x] && 
!IntegerQ[n]
 

rule 4021

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

rule 4113

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) 
+ (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*((a + 
 b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])^m*Si 
mp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && 
NeQ[A*b^2 - a*b*B + a^2*C, 0] &&  !LeQ[m, -1]
 

Maple [F]

\[\int \tan \left (d x +c \right )^{m} \left (b \tan \left (d x +c \right )\right )^{n} \left (A +B \tan \left (d x +c \right )+C \tan \left (d x +c \right )^{2}\right )d x\]

Input:

int(tan(d*x+c)^m*(b*tan(d*x+c))^n*(A+B*tan(d*x+c)+C*tan(d*x+c)^2),x)
 

Output:

int(tan(d*x+c)^m*(b*tan(d*x+c))^n*(A+B*tan(d*x+c)+C*tan(d*x+c)^2),x)
 

Fricas [F]

\[ \int \tan ^m(c+d x) (b \tan (c+d x))^n \left (A+B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\int { {\left (C \tan \left (d x + c\right )^{2} + B \tan \left (d x + c\right ) + A\right )} \left (b \tan \left (d x + c\right )\right )^{n} \tan \left (d x + c\right )^{m} \,d x } \] Input:

integrate(tan(d*x+c)^m*(b*tan(d*x+c))^n*(A+B*tan(d*x+c)+C*tan(d*x+c)^2),x, 
 algorithm="fricas")
 

Output:

integral((C*tan(d*x + c)^2 + B*tan(d*x + c) + A)*(b*tan(d*x + c))^n*tan(d* 
x + c)^m, x)
 

Sympy [F]

\[ \int \tan ^m(c+d x) (b \tan (c+d x))^n \left (A+B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\int \left (b \tan {\left (c + d x \right )}\right )^{n} \left (A + B \tan {\left (c + d x \right )} + C \tan ^{2}{\left (c + d x \right )}\right ) \tan ^{m}{\left (c + d x \right )}\, dx \] Input:

integrate(tan(d*x+c)**m*(b*tan(d*x+c))**n*(A+B*tan(d*x+c)+C*tan(d*x+c)**2) 
,x)
 

Output:

Integral((b*tan(c + d*x))**n*(A + B*tan(c + d*x) + C*tan(c + d*x)**2)*tan( 
c + d*x)**m, x)
 

Maxima [F]

\[ \int \tan ^m(c+d x) (b \tan (c+d x))^n \left (A+B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\int { {\left (C \tan \left (d x + c\right )^{2} + B \tan \left (d x + c\right ) + A\right )} \left (b \tan \left (d x + c\right )\right )^{n} \tan \left (d x + c\right )^{m} \,d x } \] Input:

integrate(tan(d*x+c)^m*(b*tan(d*x+c))^n*(A+B*tan(d*x+c)+C*tan(d*x+c)^2),x, 
 algorithm="maxima")
 

Output:

integrate((C*tan(d*x + c)^2 + B*tan(d*x + c) + A)*(b*tan(d*x + c))^n*tan(d 
*x + c)^m, x)
 

Giac [F]

\[ \int \tan ^m(c+d x) (b \tan (c+d x))^n \left (A+B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\int { {\left (C \tan \left (d x + c\right )^{2} + B \tan \left (d x + c\right ) + A\right )} \left (b \tan \left (d x + c\right )\right )^{n} \tan \left (d x + c\right )^{m} \,d x } \] Input:

integrate(tan(d*x+c)^m*(b*tan(d*x+c))^n*(A+B*tan(d*x+c)+C*tan(d*x+c)^2),x, 
 algorithm="giac")
 

Output:

integrate((C*tan(d*x + c)^2 + B*tan(d*x + c) + A)*(b*tan(d*x + c))^n*tan(d 
*x + c)^m, x)
 

Mupad [F(-1)]

Timed out. \[ \int \tan ^m(c+d x) (b \tan (c+d x))^n \left (A+B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\int {\mathrm {tan}\left (c+d\,x\right )}^m\,{\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^n\,\left (C\,{\mathrm {tan}\left (c+d\,x\right )}^2+B\,\mathrm {tan}\left (c+d\,x\right )+A\right ) \,d x \] Input:

int(tan(c + d*x)^m*(b*tan(c + d*x))^n*(A + B*tan(c + d*x) + C*tan(c + d*x) 
^2),x)
 

Output:

int(tan(c + d*x)^m*(b*tan(c + d*x))^n*(A + B*tan(c + d*x) + C*tan(c + d*x) 
^2), x)
 

Reduce [F]

\[ \int \tan ^m(c+d x) (b \tan (c+d x))^n \left (A+B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {b^{n} \left (\tan \left (d x +c \right )^{m +n} b +\left (\int \tan \left (d x +c \right )^{m +n}d x \right ) a d m +\left (\int \tan \left (d x +c \right )^{m +n}d x \right ) a d n -\left (\int \frac {\tan \left (d x +c \right )^{m +n}}{\tan \left (d x +c \right )}d x \right ) b d m -\left (\int \frac {\tan \left (d x +c \right )^{m +n}}{\tan \left (d x +c \right )}d x \right ) b d n +\left (\int \tan \left (d x +c \right )^{m +n} \tan \left (d x +c \right )^{2}d x \right ) c d m +\left (\int \tan \left (d x +c \right )^{m +n} \tan \left (d x +c \right )^{2}d x \right ) c d n \right )}{d \left (m +n \right )} \] Input:

int(tan(d*x+c)^m*(b*tan(d*x+c))^n*(A+B*tan(d*x+c)+C*tan(d*x+c)^2),x)
                                                                                    
                                                                                    
 

Output:

(b**n*(tan(c + d*x)**(m + n)*b + int(tan(c + d*x)**(m + n),x)*a*d*m + int( 
tan(c + d*x)**(m + n),x)*a*d*n - int(tan(c + d*x)**(m + n)/tan(c + d*x),x) 
*b*d*m - int(tan(c + d*x)**(m + n)/tan(c + d*x),x)*b*d*n + int(tan(c + d*x 
)**(m + n)*tan(c + d*x)**2,x)*c*d*m + int(tan(c + d*x)**(m + n)*tan(c + d* 
x)**2,x)*c*d*n))/(d*(m + n))