\(\int (d \tan (e+f x))^m (b (c \tan (e+f x))^n)^p \, dx\) [412]

Optimal result

Integrand size = 25, antiderivative size = 74 \[ \int (d \tan (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, dx=\frac {\operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (1+m+n p),\frac {1}{2} (3+m+n p),-\tan ^2(e+f x)\right ) \tan (e+f x) (d \tan (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p}{f (1+m+n p)} \] Output:

hypergeom([1, 1/2*n*p+1/2*m+1/2],[1/2*n*p+1/2*m+3/2],-tan(f*x+e)^2)*tan(f* 
x+e)*(d*tan(f*x+e))^m*(b*(c*tan(f*x+e))^n)^p/f/(n*p+m+1)
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00 \[ \int (d \tan (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, dx=\frac {\operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (1+m+n p),\frac {1}{2} (3+m+n p),-\tan ^2(e+f x)\right ) \tan (e+f x) (d \tan (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p}{f (1+m+n p)} \] Input:

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

Output:

(Hypergeometric2F1[1, (1 + m + n*p)/2, (3 + m + n*p)/2, -Tan[e + f*x]^2]*T 
an[e + f*x]*(d*Tan[e + f*x])^m*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + m + n*p))
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3042, 4061, 2034, 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.

\(\displaystyle \int (d \tan (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (d \tan (e+f x))^m \left (b (c \tan (e+f x))^n\right )^pdx\)

\(\Big \downarrow \) 4061

\(\displaystyle (c \tan (e+f x))^{-n p} \left (b (c \tan (e+f x))^n\right )^p \int (c \tan (e+f x))^{n p} (d \tan (e+f x))^mdx\)

\(\Big \downarrow \) 2034

\(\displaystyle (d \tan (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p (c \tan (e+f x))^{-m-n p} \int (c \tan (e+f x))^{m+n p}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle (d \tan (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p (c \tan (e+f x))^{-m-n p} \int (c \tan (e+f x))^{m+n p}dx\)

\(\Big \downarrow \) 3957

\(\displaystyle \frac {c (d \tan (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p (c \tan (e+f x))^{-m-n p} \int \frac {(c \tan (e+f x))^{m+n p}}{\tan ^2(e+f x) c^2+c^2}d(c \tan (e+f x))}{f}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {\tan (e+f x) (d \tan (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (m+n p+1),\frac {1}{2} (m+n p+3),-\tan ^2(e+f x)\right )}{f (m+n p+1)}\)

Input:

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

Output:

(Hypergeometric2F1[1, (1 + m + n*p)/2, (3 + m + n*p)/2, -Tan[e + f*x]^2]*T 
an[e + f*x]*(d*Tan[e + f*x])^m*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + m + n*p))
 

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 4061

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

Maple [F]

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

Input:

int((d*tan(f*x+e))^m*(b*(c*tan(f*x+e))^n)^p,x)
 

Output:

int((d*tan(f*x+e))^m*(b*(c*tan(f*x+e))^n)^p,x)
 

Fricas [F]

\[ \int (d \tan (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, dx=\int { \left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \left (d \tan \left (f x + e\right )\right )^{m} \,d x } \] Input:

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

Output:

integral(((c*tan(f*x + e))^n*b)^p*(d*tan(f*x + e))^m, x)
 

Sympy [F]

\[ \int (d \tan (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, dx=\int \left (b \left (c \tan {\left (e + f x \right )}\right )^{n}\right )^{p} \left (d \tan {\left (e + f x \right )}\right )^{m}\, dx \] Input:

integrate((d*tan(f*x+e))**m*(b*(c*tan(f*x+e))**n)**p,x)
 

Output:

Integral((b*(c*tan(e + f*x))**n)**p*(d*tan(e + f*x))**m, x)
 

Maxima [F]

\[ \int (d \tan (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, dx=\int { \left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \left (d \tan \left (f x + e\right )\right )^{m} \,d x } \] Input:

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

Output:

integrate(((c*tan(f*x + e))^n*b)^p*(d*tan(f*x + e))^m, x)
 

Giac [F]

\[ \int (d \tan (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, dx=\int { \left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \left (d \tan \left (f x + e\right )\right )^{m} \,d x } \] Input:

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

Output:

integrate(((c*tan(f*x + e))^n*b)^p*(d*tan(f*x + e))^m, x)
 

Mupad [F(-1)]

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

int((d*tan(e + f*x))^m*(b*(c*tan(e + f*x))^n)^p,x)
 

Output:

int((d*tan(e + f*x))^m*(b*(c*tan(e + f*x))^n)^p, x)
 

Reduce [F]

\[ \int (d \tan (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, dx=d^{m} c^{n p} b^{p} \left (\int \tan \left (f x +e \right )^{n p +m}d x \right ) \] Input:

int((d*tan(f*x+e))^m*(b*(c*tan(f*x+e))^n)^p,x)
 

Output:

d**m*c**(n*p)*b**p*int(tan(e + f*x)**(m + n*p),x)