\(\int \cos (e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx\) [231]

Optimal result

Integrand size = 23, antiderivative size = 82 \[ \int \cos (e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx=\frac {\sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |\frac {a}{a+b}\right .\right ) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}{f \sqrt {\frac {a+b-a \sin ^2(e+f x)}{a+b}}} \] Output:

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

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.84 \[ \int \cos (e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx=\frac {\sqrt {2} \cos (e+f x) E\left (e+f x\left |\frac {a}{a+b}\right .\right ) \sqrt {a+b \sec ^2(e+f x)}}{f \sqrt {\frac {a+2 b+a \cos (2 (e+f x))}{a+b}}} \] Input:

Integrate[Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2],x]
 

Output:

(Sqrt[2]*Cos[e + f*x]*EllipticE[e + f*x, a/(a + b)]*Sqrt[a + b*Sec[e + f*x 
]^2])/(f*Sqrt[(a + 2*b + a*Cos[2*(e + f*x)])/(a + b)])
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.10, 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, 330, 327}

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.

\(\displaystyle \int \cos (e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sqrt {a+b \sec (e+f x)^2}}{\sec (e+f x)}dx\)

\(\Big \downarrow \) 4636

\(\displaystyle \frac {\int \sqrt {a+\frac {b}{1-\sin ^2(e+f x)}}d\sin (e+f x)}{f}\)

\(\Big \downarrow \) 2057

\(\displaystyle \frac {\int \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}d\sin (e+f x)}{f}\)

\(\Big \downarrow \) 2058

\(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \int \frac {\sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\)

\(\Big \downarrow \) 330

\(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \int \frac {\sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{f \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} E\left (\arcsin (\sin (e+f x))\left |\frac {a}{a+b}\right .\right )}{f \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}\)

Input:

Int[Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2],x]
 

Output:

(EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[1 - Sin[e + f*x]^2]*Sqrt[ 
(a + b - a*Sin[e + f*x]^2)/(1 - Sin[e + f*x]^2)])/(f*Sqrt[1 - (a*Sin[e + f 
*x]^2)/(a + b)])
 

Defintions of rubi rules used

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 330

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2]   Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^ 
2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &&  !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 [C] (verified)

Result contains complex when optimal does not.

Time = 3.95 (sec) , antiderivative size = 1770, normalized size of antiderivative = 21.59

method	result	size
default	\(\text {Expression too large to display}\)	\(1770\)

Input:

int(cos(f*x+e)*(a+b*sec(f*x+e)^2)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/f/((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)/(2*I*a^(1/2)*b^(1/2)-a+b)*cos( 
f*x+e)*((cos(f*x+e)^2+2*cos(f*x+e)+1)*(1/(a+b)*(I*a^(1/2)*b^(1/2)*cos(f*x+ 
e)-I*a^(1/2)*b^(1/2)+cos(f*x+e)*a+b)/(1+cos(f*x+e)))^(1/2)*(-1/(a+b)*(I*a^ 
(1/2)*b^(1/2)*cos(f*x+e)-I*a^(1/2)*b^(1/2)-cos(f*x+e)*a-b)/(1+cos(f*x+e))) 
^(1/2)*a^2*EllipticE(((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)*(cot(f*x+e)-c 
sc(f*x+e)),(-(4*I*a^(3/2)*b^(1/2)-4*I*a^(1/2)*b^(3/2)-a^2+6*a*b-b^2)/(a+b) 
^2)^(1/2))+(2*cos(f*x+e)^2+4*cos(f*x+e)+2)*(1/(a+b)*(I*a^(1/2)*b^(1/2)*cos 
(f*x+e)-I*a^(1/2)*b^(1/2)+cos(f*x+e)*a+b)/(1+cos(f*x+e)))^(1/2)*(-1/(a+b)* 
(I*a^(1/2)*b^(1/2)*cos(f*x+e)-I*a^(1/2)*b^(1/2)-cos(f*x+e)*a-b)/(1+cos(f*x 
+e)))^(1/2)*a*b*EllipticE(((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)*(cot(f*x 
+e)-csc(f*x+e)),(-(4*I*a^(3/2)*b^(1/2)-4*I*a^(1/2)*b^(3/2)-a^2+6*a*b-b^2)/ 
(a+b)^2)^(1/2))+(cos(f*x+e)^2+2*cos(f*x+e)+1)*(1/(a+b)*(I*a^(1/2)*b^(1/2)* 
cos(f*x+e)-I*a^(1/2)*b^(1/2)+cos(f*x+e)*a+b)/(1+cos(f*x+e)))^(1/2)*(-1/(a+ 
b)*(I*a^(1/2)*b^(1/2)*cos(f*x+e)-I*a^(1/2)*b^(1/2)-cos(f*x+e)*a-b)/(1+cos( 
f*x+e)))^(1/2)*b^2*EllipticE(((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)*(cot( 
f*x+e)-csc(f*x+e)),(-(4*I*a^(3/2)*b^(1/2)-4*I*a^(1/2)*b^(3/2)-a^2+6*a*b-b^ 
2)/(a+b)^2)^(1/2))+I*(-2*cos(f*x+e)^2-4*cos(f*x+e)-2)*a^(3/2)*b^(1/2)*(1/( 
a+b)*(I*a^(1/2)*b^(1/2)*cos(f*x+e)-I*a^(1/2)*b^(1/2)+cos(f*x+e)*a+b)/(1+co 
s(f*x+e)))^(1/2)*(-1/(a+b)*(I*a^(1/2)*b^(1/2)*cos(f*x+e)-I*a^(1/2)*b^(1/2) 
-cos(f*x+e)*a-b)/(1+cos(f*x+e)))^(1/2)*EllipticF(((2*I*a^(1/2)*b^(1/2)+...
 

Fricas [F]

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

integrate(cos(f*x+e)*(a+b*sec(f*x+e)^2)^(1/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate(cos(f*x+e)*(a+b*sec(f*x+e)**2)**(1/2),x)
 

Output:

Integral(sqrt(a + b*sec(e + f*x)**2)*cos(e + f*x), x)
 

Maxima [F]

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

integrate(cos(f*x+e)*(a+b*sec(f*x+e)^2)^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate(cos(f*x+e)*(a+b*sec(f*x+e)^2)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int(cos(e + f*x)*(a + b/cos(e + f*x)^2)^(1/2),x)
 

Output:

int(cos(e + f*x)*(a + b/cos(e + f*x)^2)^(1/2), x)
 

Reduce [F]

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

int(cos(f*x+e)*(a+b*sec(f*x+e)^2)^(1/2),x)
 

Output:

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