\(\int \frac {\cos (e+f x)}{\sqrt {a+b \sec ^2(e+f x)}} \, dx\) [260]

Optimal result

Integrand size = 23, antiderivative size = 105 \[ \int \frac {\cos (e+f x)}{\sqrt {a+b \sec ^2(e+f x)}} \, dx=\frac {\sqrt {a+b} E\left (\arcsin \left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )|\frac {a+b}{a}\right ) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}{\sqrt {a} f \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4233, 1985, 1986, 438, 435} \[ \int \frac {\cos (e+f x)}{\sqrt {a+b \sec ^2(e+f x)}} \, dx=\frac {\sqrt {a+b} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} E\left (\arcsin \left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )|\frac {a+b}{a}\right )}{\sqrt {a} f \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}} \]

[In]

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

[Out]

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

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[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 438

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

Rule 1985

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 1986

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^(r_.))^(p_), x_Symbol] :> Dist[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 4233

Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = Fr
eeFactors[Sin[e + f*x], x]}, Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b}{1-x^2}}} \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {\text {Subst}\left (\int \frac {1}{\sqrt {\frac {a+b-a x^2}{1-x^2}}} \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {\sqrt {a+b-a \sin ^2(e+f x)} \text {Subst}\left (\int \frac {\sqrt {1-x^2}}{\sqrt {a+b-a x^2}} \, dx,x,\sin (e+f x)\right )}{f \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}} \\ & = \frac {\sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \text {Subst}\left (\int \frac {\sqrt {1-x^2}}{\sqrt {1-\frac {a x^2}{a+b}}} \, dx,x,\sin (e+f x)\right )}{f \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}} \\ & = \frac {\sqrt {a+b} E\left (\arcsin \left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )|\frac {a+b}{a}\right ) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}{\sqrt {a} f \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.87 (sec) , antiderivative size = 274, normalized size of antiderivative = 2.61 \[ \int \frac {\cos (e+f x)}{\sqrt {a+b \sec ^2(e+f x)}} \, dx=\frac {\csc (2 (e+f x)) \sin (e+f x) \left (a^2 \sqrt {-\frac {1}{b}} \sqrt {\frac {a+2 b+a \cos (2 (e+f x))}{a+b}} \operatorname {EllipticF}\left (e+f x,\frac {a}{a+b}\right )-2 i \sqrt {-\frac {a \cos ^2(e+f x)}{b}} \sqrt {a+2 b+a \cos (2 (e+f x))} \csc (2 (e+f x)) \left (2 (a+b) E\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {1}{b}} \sqrt {a+2 b+a \cos (2 (e+f x))}}{\sqrt {2}}\right )|\frac {b}{a+b}\right )-a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {1}{b}} \sqrt {a+2 b+a \cos (2 (e+f x))}}{\sqrt {2}}\right ),\frac {b}{a+b}\right )\right ) \sqrt {\frac {a \sin ^2(e+f x)}{a+b}}\right )}{\sqrt {2} a^2 \sqrt {-\frac {1}{b}} f \sqrt {a+b \sec ^2(e+f x)}} \]

[In]

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

[Out]

(Csc[2*(e + f*x)]*Sin[e + f*x]*(a^2*Sqrt[-b^(-1)]*Sqrt[(a + 2*b + a*Cos[2*(e + f*x)])/(a + b)]*EllipticF[e + f
*x, a/(a + b)] - (2*I)*Sqrt[-((a*Cos[e + f*x]^2)/b)]*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*Csc[2*(e + f*x)]*(2*(a
+ b)*EllipticE[I*ArcSinh[(Sqrt[-b^(-1)]*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]])/Sqrt[2]], b/(a + b)] - a*Elliptic
F[I*ArcSinh[(Sqrt[-b^(-1)]*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]])/Sqrt[2]], b/(a + b)])*Sqrt[(a*Sin[e + f*x]^2)/(
a + b)]))/(Sqrt[2]*a^2*Sqrt[-b^(-1)]*f*Sqrt[a + b*Sec[e + f*x]^2])

Maple [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 6.41 (sec) , antiderivative size = 4095, normalized size of antiderivative = 39.00

method	result	size
default	\(\text {Expression too large to display}\)	\(4095\)

[In]

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

[Out]

-1/f/((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)/a/(2*I*a^(1/2)*b^(1/2)-a+b)*(2*I*a^(3/2)*b^(1/2)*((2*I*a^(1/2)*b^
(1/2)+a-b)/(a+b))^(1/2)*(csc(f*x+e)-cot(f*x+e))+2*I*a^(1/2)*b^(3/2)*((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)*(c
sc(f*x+e)-cot(f*x+e))+4*(-(2*I*a^(1/2)*b^(1/2)*(1-cos(f*x+e))^2*csc(f*x+e)^2+a*(1-cos(f*x+e))^2*csc(f*x+e)^2-b
*(1-cos(f*x+e))^2*csc(f*x+e)^2-a-b)/(a+b))^(1/2)*((2*I*a^(1/2)*b^(1/2)*(1-cos(f*x+e))^2*csc(f*x+e)^2-a*(1-cos(
f*x+e))^2*csc(f*x+e)^2+b*(1-cos(f*x+e))^2*csc(f*x+e)^2+a+b)/(a+b))^(1/2)*EllipticF(((2*I*a^(1/2)*b^(1/2)+a-b)/
(a+b))^(1/2)*(csc(f*x+e)-cot(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))
*a*b-2*(-(2*I*a^(1/2)*b^(1/2)*(1-cos(f*x+e))^2*csc(f*x+e)^2+a*(1-cos(f*x+e))^2*csc(f*x+e)^2-b*(1-cos(f*x+e))^2
*csc(f*x+e)^2-a-b)/(a+b))^(1/2)*((2*I*a^(1/2)*b^(1/2)*(1-cos(f*x+e))^2*csc(f*x+e)^2-a*(1-cos(f*x+e))^2*csc(f*x
+e)^2+b*(1-cos(f*x+e))^2*csc(f*x+e)^2+a+b)/(a+b))^(1/2)*EllipticE(((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)*(csc
(f*x+e)-cot(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))*a*b-4*((2*I*a^(1
/2)*b^(1/2)+a-b)/(a+b))^(1/2)*a*b*(1-cos(f*x+e))^3*csc(f*x+e)^3-(-(2*I*a^(1/2)*b^(1/2)*(1-cos(f*x+e))^2*csc(f*
x+e)^2+a*(1-cos(f*x+e))^2*csc(f*x+e)^2-b*(1-cos(f*x+e))^2*csc(f*x+e)^2-a-b)/(a+b))^(1/2)*((2*I*a^(1/2)*b^(1/2)
*(1-cos(f*x+e))^2*csc(f*x+e)^2-a*(1-cos(f*x+e))^2*csc(f*x+e)^2+b*(1-cos(f*x+e))^2*csc(f*x+e)^2+a+b)/(a+b))^(1/
2)*EllipticE(((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)*(csc(f*x+e)-cot(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))*a^2*(1-cos(f*x+e))^2*csc(f*x+e)^2-(-(2*I*a^(1/2)*b^(1/2)*(1-cos(f*x+e
))^2*csc(f*x+e)^2+a*(1-cos(f*x+e))^2*csc(f*x+e)^2-b*(1-cos(f*x+e))^2*csc(f*x+e)^2-a-b)/(a+b))^(1/2)*((2*I*a^(1
/2)*b^(1/2)*(1-cos(f*x+e))^2*csc(f*x+e)^2-a*(1-cos(f*x+e))^2*csc(f*x+e)^2+b*(1-cos(f*x+e))^2*csc(f*x+e)^2+a+b)
/(a+b))^(1/2)*EllipticE(((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)*(csc(f*x+e)-cot(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))*b^2*(1-cos(f*x+e))^2*csc(f*x+e)^2+2*I*a^(3/2)*b^(1/2)*((2*
I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)*(1-cos(f*x+e))^5*csc(f*x+e)^5+2*I*a^(1/2)*b^(3/2)*((2*I*a^(1/2)*b^(1/2)+a-
b)/(a+b))^(1/2)*(1-cos(f*x+e))^5*csc(f*x+e)^5-4*I*a^(3/2)*b^(1/2)*((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)*(1-c
os(f*x+e))^3*csc(f*x+e)^3+4*I*a^(1/2)*b^(3/2)*((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)*(1-cos(f*x+e))^3*csc(f*x
+e)^3+2*I*a^(3/2)*b^(1/2)*(-(2*I*a^(1/2)*b^(1/2)*(1-cos(f*x+e))^2*csc(f*x+e)^2+a*(1-cos(f*x+e))^2*csc(f*x+e)^2
-b*(1-cos(f*x+e))^2*csc(f*x+e)^2-a-b)/(a+b))^(1/2)*((2*I*a^(1/2)*b^(1/2)*(1-cos(f*x+e))^2*csc(f*x+e)^2-a*(1-co
s(f*x+e))^2*csc(f*x+e)^2+b*(1-cos(f*x+e))^2*csc(f*x+e)^2+a+b)/(a+b))^(1/2)*EllipticF(((2*I*a^(1/2)*b^(1/2)+a-b
)/(a+b))^(1/2)*(csc(f*x+e)-cot(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*I*a^(3/2)*b^(1/2)*(-(2*I*a^(1/2)*b^(1/2)*(1-cos(f*x+e))^2*csc(f*x+e)^2+a*(1-cos(f*x+e))^2*csc(f*x+e)^2-b*
(1-cos(f*x+e))^2*csc(f*x+e)^2-a-b)/(a+b))^(1/2)*((2*I*a^(1/2)*b^(1/2)*(1-cos(f*x+e))^2*csc(f*x+e)^2-a*(1-cos(f
*x+e))^2*csc(f*x+e)^2+b*(1-cos(f*x+e))^2*csc(f*x+e)^2+a+b)/(a+b))^(1/2)*EllipticF(((2*I*a^(1/2)*b^(1/2)+a-b)/(
a+b))^(1/2)*(csc(f*x+e)-cot(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))*
(1-cos(f*x+e))^2*csc(f*x+e)^2-((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)*a^2*(1-cos(f*x+e))^5*csc(f*x+e)^5-(-(2*I
*a^(1/2)*b^(1/2)*(1-cos(f*x+e))^2*csc(f*x+e)^2+a*(1-cos(f*x+e))^2*csc(f*x+e)^2-b*(1-cos(f*x+e))^2*csc(f*x+e)^2
-a-b)/(a+b))^(1/2)*((2*I*a^(1/2)*b^(1/2)*(1-cos(f*x+e))^2*csc(f*x+e)^2-a*(1-cos(f*x+e))^2*csc(f*x+e)^2+b*(1-co
s(f*x+e))^2*csc(f*x+e)^2+a+b)/(a+b))^(1/2)*EllipticE(((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)*(csc(f*x+e)-cot(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))*a^2-(-(2*I*a^(1/2)*b^(1/2)*(1
-cos(f*x+e))^2*csc(f*x+e)^2+a*(1-cos(f*x+e))^2*csc(f*x+e)^2-b*(1-cos(f*x+e))^2*csc(f*x+e)^2-a-b)/(a+b))^(1/2)*
((2*I*a^(1/2)*b^(1/2)*(1-cos(f*x+e))^2*csc(f*x+e)^2-a*(1-cos(f*x+e))^2*csc(f*x+e)^2+b*(1-cos(f*x+e))^2*csc(f*x
+e)^2+a+b)/(a+b))^(1/2)*EllipticE(((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)*(csc(f*x+e)-cot(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))*b^2+((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)*b^2*
(1-cos(f*x+e))^5*csc(f*x+e)^5+2*((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)*a^2*(1-cos(f*x+e))^3*csc(f*x+e)^3+2*((
2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)*b^2*(1-cos(f*x+e))^3*csc(f*x+e)^3-2*I*a^(1/2)*b^(3/2)*(-(2*I*a^(1/2)*b^(
1/2)*(1-cos(f*x+e))^2*csc(f*x+e)^2+a*(1-cos(f*x+e))^2*csc(f*x+e)^2-b*(1-cos(f*x+e))^2*csc(f*x+e)^2-a-b)/(a+b))
^(1/2)*((2*I*a^(1/2)*b^(1/2)*(1-cos(f*x+e))^2*csc(f*x+e)^2-a*(1-cos(f*x+e))^2*csc(f*x+e)^2+b*(1-cos(f*x+e))^2*
csc(f*x+e)^2+a+b)/(a+b))^(1/2)*EllipticF(((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)*(csc(f*x+e)-cot(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*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)*a
^2*(csc(f*x+e)-cot(f*x+e))+((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)*b^2*(csc(f*x+e)-cot(f*x+e))-2*(-(2*I*a^(1/2
)*b^(1/2)*(1-cos(f*x+e))^2*csc(f*x+e)^2+a*(1-cos(f*x+e))^2*csc(f*x+e)^2-b*(1-cos(f*x+e))^2*csc(f*x+e)^2-a-b)/(
a+b))^(1/2)*((2*I*a^(1/2)*b^(1/2)*(1-cos(f*x+e))^2*csc(f*x+e)^2-a*(1-cos(f*x+e))^2*csc(f*x+e)^2+b*(1-cos(f*x+e
))^2*csc(f*x+e)^2+a+b)/(a+b))^(1/2)*EllipticE(((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)*(csc(f*x+e)-cot(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))*a*b*(1-cos(f*x+e))^2*csc(f*x+e)^2-2*
I*a^(1/2)*b^(3/2)*(-(2*I*a^(1/2)*b^(1/2)*(1-cos(f*x+e))^2*csc(f*x+e)^2+a*(1-cos(f*x+e))^2*csc(f*x+e)^2-b*(1-co
s(f*x+e))^2*csc(f*x+e)^2-a-b)/(a+b))^(1/2)*((2*I*a^(1/2)*b^(1/2)*(1-cos(f*x+e))^2*csc(f*x+e)^2-a*(1-cos(f*x+e)
)^2*csc(f*x+e)^2+b*(1-cos(f*x+e))^2*csc(f*x+e)^2+a+b)/(a+b))^(1/2)*EllipticF(((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))
^(1/2)*(csc(f*x+e)-cot(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))*(1-co
s(f*x+e))^2*csc(f*x+e)^2+4*(-(2*I*a^(1/2)*b^(1/2)*(1-cos(f*x+e))^2*csc(f*x+e)^2+a*(1-cos(f*x+e))^2*csc(f*x+e)^
2-b*(1-cos(f*x+e))^2*csc(f*x+e)^2-a-b)/(a+b))^(1/2)*((2*I*a^(1/2)*b^(1/2)*(1-cos(f*x+e))^2*csc(f*x+e)^2-a*(1-c
os(f*x+e))^2*csc(f*x+e)^2+b*(1-cos(f*x+e))^2*csc(f*x+e)^2+a+b)/(a+b))^(1/2)*EllipticF(((2*I*a^(1/2)*b^(1/2)+a-
b)/(a+b))^(1/2)*(csc(f*x+e)-cot(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))*a*b*(1-cos(f*x+e))^2*csc(f*x+e)^2)/((a*(1-cos(f*x+e))^4*csc(f*x+e)^4+b*(1-cos(f*x+e))^4*csc(f*x+e)^4-2*a*(
1-cos(f*x+e))^2*csc(f*x+e)^2+2*b*(1-cos(f*x+e))^2*csc(f*x+e)^2+a+b)/((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^2)^(1/2)
/((1-cos(f*x+e))^2*csc(f*x+e)^2-1)/((1-cos(f*x+e))^2*csc(f*x+e)^2+1)

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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