3.89 \(\int \frac {\sec (e+f x) \sqrt {c-c \sec (e+f x)}}{a+a \sec (e+f x)} \, dx\)

Optimal. Leaf size=39 \[ \frac {2 c \tan (e+f x)}{f (a \sec (e+f x)+a) \sqrt {c-c \sec (e+f x)}} \]

[Out]

2*c*tan(f*x+e)/f/(a+a*sec(f*x+e))/(c-c*sec(f*x+e))^(1/2)

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Rubi [A]  time = 0.11, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {3953} \[ \frac {2 c \tan (e+f x)}{f (a \sec (e+f x)+a) \sqrt {c-c \sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Int[(Sec[e + f*x]*Sqrt[c - c*Sec[e + f*x]])/(a + a*Sec[e + f*x]),x]

[Out]

(2*c*Tan[e + f*x])/(f*(a + a*Sec[e + f*x])*Sqrt[c - c*Sec[e + f*x]])

Rule 3953

Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.) +
(c_)], x_Symbol] :> Simp[(2*a*c*Cot[e + f*x]*(a + b*Csc[e + f*x])^m)/(b*f*(2*m + 1)*Sqrt[c + d*Csc[e + f*x]]),
x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[m, -2^(-1)]

Rubi steps

\begin {align*} \int \frac {\sec (e+f x) \sqrt {c-c \sec (e+f x)}}{a+a \sec (e+f x)} \, dx &=\frac {2 c \tan (e+f x)}{f (a+a \sec (e+f x)) \sqrt {c-c \sec (e+f x)}}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 29, normalized size = 0.74 \[ -\frac {2 \cot (e+f x) \sqrt {c-c \sec (e+f x)}}{a f} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sec[e + f*x]*Sqrt[c - c*Sec[e + f*x]])/(a + a*Sec[e + f*x]),x]

[Out]

(-2*Cot[e + f*x]*Sqrt[c - c*Sec[e + f*x]])/(a*f)

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fricas [A]  time = 0.44, size = 45, normalized size = 1.15 \[ -\frac {2 \, \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{a f \sin \left (f x + e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt((c*cos(f*x + e) - c)/cos(f*x + e))*cos(f*x + e)/(a*f*sin(f*x + e))

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)-sqrt(2)*sqrt(c*tan(1/2*(f*x+exp(1)))^2-c)*sign(tan(1/2*(f*x+e
xp(1)))^3+tan(1/2*(f*x+exp(1))))*sign(cos(f*x+exp(1)))/a/f

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maple [A]  time = 2.12, size = 43, normalized size = 1.10 \[ -\frac {2 \sqrt {\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}}\, \cos \left (f x +e \right )}{a f \sin \left (f x +e \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(f*x+e)*(c-c*sec(f*x+e))^(1/2)/(a+a*sec(f*x+e)),x)

[Out]

-2/a/f*(c*(-1+cos(f*x+e))/cos(f*x+e))^(1/2)*cos(f*x+e)/sin(f*x+e)

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maxima [B]  time = 0.75, size = 84, normalized size = 2.15 \[ -\frac {\sqrt {2} \sqrt {c} - \frac {\sqrt {2} \sqrt {c} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}}{a f \sqrt {\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1} \sqrt {\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(sqrt(2)*sqrt(c) - sqrt(2)*sqrt(c)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2)/(a*f*sqrt(sin(f*x + e)/(cos(f*x + e)
+ 1) + 1)*sqrt(sin(f*x + e)/(cos(f*x + e) + 1) - 1))

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mupad [B]  time = 1.77, size = 40, normalized size = 1.03 \[ -\frac {\sin \left (2\,e+2\,f\,x\right )\,\sqrt {c-\frac {c}{\cos \left (e+f\,x\right )}}}{a\,f\,{\sin \left (e+f\,x\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(sin(2*e + 2*f*x)*(c - c/cos(e + f*x))^(1/2))/(a*f*sin(e + f*x)^2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\sqrt {- c \sec {\left (e + f x \right )} + c} \sec {\left (e + f x \right )}}{\sec {\left (e + f x \right )} + 1}\, dx}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(c-c*sec(f*x+e))**(1/2)/(a+a*sec(f*x+e)),x)

[Out]

Integral(sqrt(-c*sec(e + f*x) + c)*sec(e + f*x)/(sec(e + f*x) + 1), x)/a

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