\(\int \cos ^5(e+f x) (a+b \sec ^2(e+f x))^p \, dx\) [303]

Optimal result

Integrand size = 23, antiderivative size = 105 \[ \int \cos ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^p \, dx=\frac {\operatorname {AppellF1}\left (\frac {1}{2},-2+p,-p,\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right ) \cos ^2(e+f x)^p \sin (e+f x) \left (\frac {a+b-a \sin ^2(e+f x)}{a+b}\right )^{-p} \left (\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )\right )^p}{f} \] Output:

AppellF1(1/2,-2+p,-p,3/2,sin(f*x+e)^2,a*sin(f*x+e)^2/(a+b))*(cos(f*x+e)^2) 
^p*sin(f*x+e)*(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2))^p/f/(((a+b-a*sin(f*x+e)^ 
2)/(a+b))^p)
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1997\) vs. \(2(105)=210\).

Time = 15.52 (sec) , antiderivative size = 1997, normalized size of antiderivative = 19.02 \[ \int \cos ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^p \, dx =\text {Too large to display} \] Input:

Integrate[Cos[e + f*x]^5*(a + b*Sec[e + f*x]^2)^p,x]
 

Output:

(-3*(a + b)*AppellF1[1/2, 7/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x] 
^2)/(a + b))]*Cos[e + f*x]^4*(a + 2*b + a*Cos[2*(e + f*x)])^p*(Sec[e + f*x 
]^2)^(-7/2 + p)*(a + b*Sec[e + f*x]^2)^p*Sin[e + f*x])/(f*(-3*(a + b)*Appe 
llF1[1/2, 7/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + 
(-2*b*p*AppellF1[3/2, 7/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^ 
2)/(a + b))] + 7*(a + b)*AppellF1[3/2, 9/2, -p, 5/2, -Tan[e + f*x]^2, -((b 
*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2)*((-3*(a + b)*AppellF1[1/2, 7/2 
, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Co 
s[2*(e + f*x)])^p*(Sec[e + f*x]^2)^(-5/2 + p))/(-3*(a + b)*AppellF1[1/2, 7 
/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))] + (-2*b*p*App 
ellF1[3/2, 7/2, 1 - p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b)) 
] + 7*(a + b)*AppellF1[3/2, 9/2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f* 
x]^2)/(a + b))])*Tan[e + f*x]^2) + (6*a*(a + b)*p*AppellF1[1/2, 7/2, -p, 3 
/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + 2*b + a*Cos[2*(e 
+ f*x)])^(-1 + p)*(Sec[e + f*x]^2)^(-7/2 + p)*Sin[2*(e + f*x)]*Tan[e + f*x 
])/(-3*(a + b)*AppellF1[1/2, 7/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f 
*x]^2)/(a + b))] + (-2*b*p*AppellF1[3/2, 7/2, 1 - p, 5/2, -Tan[e + f*x]^2, 
 -((b*Tan[e + f*x]^2)/(a + b))] + 7*(a + b)*AppellF1[3/2, 9/2, -p, 5/2, -T 
an[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))])*Tan[e + f*x]^2) - (6*(a + b 
)*(-7/2 + p)*AppellF1[1/2, 7/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + ...
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 4636, 2057, 2058, 334, 333}

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.

\(\displaystyle \int \cos ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^p \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (a+b \sec (e+f x)^2\right )^p}{\sec (e+f x)^5}dx\)

\(\Big \downarrow \) 4636

\(\displaystyle \frac {\int \left (1-\sin ^2(e+f x)\right )^2 \left (a+\frac {b}{1-\sin ^2(e+f x)}\right )^pd\sin (e+f x)}{f}\)

\(\Big \downarrow \) 2057

\(\displaystyle \frac {\int \left (1-\sin ^2(e+f x)\right )^2 \left (\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}\right )^pd\sin (e+f x)}{f}\)

\(\Big \downarrow \) 2058

\(\displaystyle \frac {\left (1-\sin ^2(e+f x)\right )^p \left (-a \sin ^2(e+f x)+a+b\right )^{-p} \left (\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}\right )^p \int \left (1-\sin ^2(e+f x)\right )^{2-p} \left (-a \sin ^2(e+f x)+a+b\right )^pd\sin (e+f x)}{f}\)

\(\Big \downarrow \) 334

\(\displaystyle \frac {\left (1-\sin ^2(e+f x)\right )^p \left (\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}\right )^p \left (1-\frac {a \sin ^2(e+f x)}{a+b}\right )^{-p} \int \left (1-\sin ^2(e+f x)\right )^{2-p} \left (1-\frac {a \sin ^2(e+f x)}{a+b}\right )^pd\sin (e+f x)}{f}\)

\(\Big \downarrow \) 333

\(\displaystyle \frac {\sin (e+f x) \left (1-\sin ^2(e+f x)\right )^p \left (\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}\right )^p \left (1-\frac {a \sin ^2(e+f x)}{a+b}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},p-2,-p,\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )}{f}\)

Input:

Int[Cos[e + f*x]^5*(a + b*Sec[e + f*x]^2)^p,x]
 

Output:

(AppellF1[1/2, -2 + p, -p, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b) 
]*Sin[e + f*x]*(1 - Sin[e + f*x]^2)^p*((a + b - a*Sin[e + f*x]^2)/(1 - Sin 
[e + f*x]^2))^p)/(f*(1 - (a*Sin[e + f*x]^2)/(a + b))^p)
 

Defintions of rubi rules used

rule 333

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[a^p*c^q*x*AppellF1[1/2, -p, -q, 3/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; F 
reeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[p] || GtQ[a, 
0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 334

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[a^IntPart[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p])   Int[ 
(1 + b*(x^2/a))^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, p, q}, x] && 
NeQ[b*c - a*d, 0] &&  !(IntegerQ[p] || GtQ[a, 0])
 

rule 2057

Int[(u_.)*((a_) + (b_.)/((c_) + (d_.)*(x_)^(n_)))^(p_), x_Symbol] :> Int[u* 
((b + a*c + a*d*x^n)/(c + d*x^n))^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
 

rule 2058

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ 
(r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + 
 b*x^n)^(p*q)*(c + d*x^n)^(p*r))]   Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* 
r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
 

rule 3042

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

rule 4636

Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_ 
))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f 
 Subst[Int[(a + b/(1 - ff^2*x^2)^(n/2))^p/(1 - ff^2*x^2)^((m + 1)/2), x], x 
, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2] 
 && IntegerQ[n/2] &&  !IntegerQ[p]
 

Maple [F]

\[\int \cos \left (f x +e \right )^{5} \left (a +b \sec \left (f x +e \right )^{2}\right )^{p}d x\]

Input:

int(cos(f*x+e)^5*(a+b*sec(f*x+e)^2)^p,x)
 

Output:

int(cos(f*x+e)^5*(a+b*sec(f*x+e)^2)^p,x)
 

Fricas [F]

\[ \int \cos ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^p \, dx=\int { {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \cos \left (f x + e\right )^{5} \,d x } \] Input:

integrate(cos(f*x+e)^5*(a+b*sec(f*x+e)^2)^p,x, algorithm="fricas")
 

Output:

integral((b*sec(f*x + e)^2 + a)^p*cos(f*x + e)^5, x)
 

Sympy [F(-1)]

Timed out. \[ \int \cos ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^p \, dx=\text {Timed out} \] Input:

integrate(cos(f*x+e)**5*(a+b*sec(f*x+e)**2)**p,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \cos ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^p \, dx=\int { {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \cos \left (f x + e\right )^{5} \,d x } \] Input:

integrate(cos(f*x+e)^5*(a+b*sec(f*x+e)^2)^p,x, algorithm="maxima")
 

Output:

integrate((b*sec(f*x + e)^2 + a)^p*cos(f*x + e)^5, x)
 

Giac [F]

\[ \int \cos ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^p \, dx=\int { {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \cos \left (f x + e\right )^{5} \,d x } \] Input:

integrate(cos(f*x+e)^5*(a+b*sec(f*x+e)^2)^p,x, algorithm="giac")
 

Output:

integrate((b*sec(f*x + e)^2 + a)^p*cos(f*x + e)^5, x)
 

Mupad [F(-1)]

Timed out. \[ \int \cos ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^p \, dx=\int {\cos \left (e+f\,x\right )}^5\,{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^p \,d x \] Input:

int(cos(e + f*x)^5*(a + b/cos(e + f*x)^2)^p,x)
 

Output:

int(cos(e + f*x)^5*(a + b/cos(e + f*x)^2)^p, x)
 

Reduce [F]

\[ \int \cos ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^p \, dx=\int \left (\sec \left (f x +e \right )^{2} b +a \right )^{p} \cos \left (f x +e \right )^{5}d x \] Input:

int(cos(f*x+e)^5*(a+b*sec(f*x+e)^2)^p,x)
                                                                                    
                                                                                    
 

Output:

int((sec(e + f*x)**2*b + a)**p*cos(e + f*x)**5,x)