3.3.39 \(\int (c (d \sec (e+f x))^p)^n (a+b \sec (e+f x)) \, dx\) [239]

3.3.39.1 Optimal result

Integrand size = 25, antiderivative size = 156 \[ \int \left (c (d \sec (e+f x))^p\right )^n (a+b \sec (e+f x)) \, dx=\frac {b \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {n p}{2},\frac {1}{2} (2-n p),\cos ^2(e+f x)\right ) \left (c (d \sec (e+f x))^p\right )^n \sin (e+f x)}{f n p \sqrt {\sin ^2(e+f x)}}-\frac {a \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1-n p),\frac {1}{2} (3-n p),\cos ^2(e+f x)\right ) \left (c (d \sec (e+f x))^p\right )^n \sin (e+f x)}{f (1-n p) \sqrt {\sin ^2(e+f x)}} \]

output

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

3.3.39.2 Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.80 \[ \int \left (c (d \sec (e+f x))^p\right )^n (a+b \sec (e+f x)) \, dx=\frac {\csc (e+f x) \left (a (1+n p) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n p}{2},1+\frac {n p}{2},\sec ^2(e+f x)\right )+b n p \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+n p),\frac {1}{2} (3+n p),\sec ^2(e+f x)\right )\right ) \left (c (d \sec (e+f x))^p\right )^n \sqrt {-\tan ^2(e+f x)}}{f n p (1+n p)} \]

input

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

output

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

3.3.39.3 Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.13, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3042, 4436, 3042, 4274, 3042, 4259, 3042, 3122}

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.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int (a+b \sec (e+f x)) \left (c (d \sec (e+f x))^p\right )^ndx\)

\(\Big \downarrow \) 4436

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

\(\Big \downarrow \) 3042

\(\displaystyle (d \sec (e+f x))^{-n p} \left (c (d \sec (e+f x))^p\right )^n \int \left (d \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{n p} \left (a+b \csc \left (e+f x+\frac {\pi }{2}\right )\right )dx\)

\(\Big \downarrow \) 4274

\(\displaystyle (d \sec (e+f x))^{-n p} \left (c (d \sec (e+f x))^p\right )^n \left (a \int (d \sec (e+f x))^{n p}dx+\frac {b \int (d \sec (e+f x))^{n p+1}dx}{d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle (d \sec (e+f x))^{-n p} \left (c (d \sec (e+f x))^p\right )^n \left (a \int \left (d \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{n p}dx+\frac {b \int \left (d \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{n p+1}dx}{d}\right )\)

\(\Big \downarrow \) 4259

\(\displaystyle (d \sec (e+f x))^{-n p} \left (c (d \sec (e+f x))^p\right )^n \left (a \left (\frac {\cos (e+f x)}{d}\right )^{n p} (d \sec (e+f x))^{n p} \int \left (\frac {\cos (e+f x)}{d}\right )^{-n p}dx+\frac {b \left (\frac {\cos (e+f x)}{d}\right )^{n p} (d \sec (e+f x))^{n p} \int \left (\frac {\cos (e+f x)}{d}\right )^{-n p-1}dx}{d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle (d \sec (e+f x))^{-n p} \left (c (d \sec (e+f x))^p\right )^n \left (a \left (\frac {\cos (e+f x)}{d}\right )^{n p} (d \sec (e+f x))^{n p} \int \left (\frac {\sin \left (e+f x+\frac {\pi }{2}\right )}{d}\right )^{-n p}dx+\frac {b \left (\frac {\cos (e+f x)}{d}\right )^{n p} (d \sec (e+f x))^{n p} \int \left (\frac {\sin \left (e+f x+\frac {\pi }{2}\right )}{d}\right )^{-n p-1}dx}{d}\right )\)

\(\Big \downarrow \) 3122

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

input

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

output

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

3.3.39.3.1 Defintions of rubi rules used

rule 3042

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

rule 3122

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( 
b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]))*Hypergeometric2 
F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2], x] /; FreeQ[{b, c, d, n}, x] 
 &&  !IntegerQ[2*n]
 

rule 4259

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] 
)^(n - 1)*((Sin[c + d*x]/b)^(n - 1)   Int[1/(Sin[c + d*x]/b)^n, x]), x] /; 
FreeQ[{b, c, d, n}, x] &&  !IntegerQ[n]
 

rule 4274

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + 
(a_)), x_Symbol] :> Simp[a   Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d   In 
t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
 

rule 4436

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

3.3.39.4 Maple [F]

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

input

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

output

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

3.3.39.5 Fricas [F]

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

input

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

output

integral((b*sec(f*x + e) + a)*((d*sec(f*x + e))^p*c)^n, x)
 

3.3.39.6 Sympy [F]

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

input

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

output

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

3.3.39.7 Maxima [F]

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

input

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

output

integrate((b*sec(f*x + e) + a)*((d*sec(f*x + e))^p*c)^n, x)
 

3.3.39.8 Giac [F]

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

input

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

output

integrate((b*sec(f*x + e) + a)*((d*sec(f*x + e))^p*c)^n, x)
 

3.3.39.9 Mupad [F(-1)]

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

input

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

output

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