\(\int (d \sec (e+f x))^m (b (c \tan (e+f x))^n)^p \, dx\) [478]

Optimal result

Integrand size = 25, antiderivative size = 97 \[ \int (d \sec (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, dx=\frac {\cos ^2(e+f x)^{\frac {1}{2} (1+m+n p)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (1+n p),\frac {1}{2} (1+m+n p),\frac {1}{2} (3+n p),\sin ^2(e+f x)\right ) (d \sec (e+f x))^m \tan (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (1+n p)} \] Output:

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.92 \[ \int (d \sec (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, dx=\frac {\cot (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {m}{2},\frac {1}{2} (1-n p),\frac {2+m}{2},\sec ^2(e+f x)\right ) (d \sec (e+f x))^m \left (-\tan ^2(e+f x)\right )^{\frac {1}{2} (1-n p)} \left (b (c \tan (e+f x))^n\right )^p}{f m} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3042, 4142, 3042, 3097}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4142

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3097

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

Input:

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

Output:

((Cos[e + f*x]^2)^((1 + m + n*p)/2)*Hypergeometric2F1[(1 + n*p)/2, (1 + m 
+ n*p)/2, (3 + n*p)/2, Sin[e + f*x]^2]*(d*Sec[e + f*x])^m*Tan[e + f*x]*(b* 
(c*Tan[e + f*x])^n)^p)/(f*(1 + n*p))
 

Defintions of rubi rules used

rule 3042

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

rule 3097

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( 
n_), x_Symbol] :> Simp[(a*Sec[e + f*x])^m*(b*Tan[e + f*x])^(n + 1)*((Cos[e 
+ f*x]^2)^((m + n + 1)/2)/(b*f*(n + 1)))*Hypergeometric2F1[(n + 1)/2, (m + 
n + 1)/2, (n + 3)/2, Sin[e + f*x]^2], x] /; FreeQ[{a, b, e, f, m, n}, x] && 
  !IntegerQ[(n - 1)/2] &&  !IntegerQ[m/2]
 

rule 4142

Int[(u_.)*((b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> S 
imp[b^IntPart[p]*((b*(c*Tan[e + f*x])^n)^FracPart[p]/(c*Tan[e + f*x])^(n*Fr 
acPart[p]))   Int[ActivateTrig[u]*(c*Tan[e + f*x])^(n*p), x], x] /; FreeQ[{ 
b, c, e, f, n, p}, x] &&  !IntegerQ[p] &&  !IntegerQ[n] && (EqQ[u, 1] || Ma 
tchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, 
 cos, tan, cot, sec, csc}, trig]])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

integrate((d*sec(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*sec(f*x + e))^m, x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((d*sec(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*sec(f*x + e))^m, x)
 

Giac [F]

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

integrate((d*sec(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*sec(f*x + e))^m, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

int((d*sec(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)**(n*p)*sec(e + f*x)**m,x)