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

Optimal. Leaf size=29 \[ -\frac {2 \sec (e+f x) \sqrt {c-c \sin (e+f x)}}{a f} \]

[Out]

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

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Rubi [A]  time = 0.13, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2736, 2673} \[ -\frac {2 \sec (e+f x) \sqrt {c-c \sin (e+f x)}}{a f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2673

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(b*(g*
Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1))/(f*g*(m - 1)), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && Eq
Q[a^2 - b^2, 0] && EqQ[2*m + p - 1, 0] && NeQ[m, 1]

Rule 2736

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Di
st[a^m*c^m, Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&
EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] &&  !(IntegerQ[n] && ((LtQ[m, 0] && GtQ[n, 0]) || LtQ[0,
n, m] || LtQ[m, n, 0]))

Rubi steps

\begin {align*} \int \frac {\sqrt {c-c \sin (e+f x)}}{a+a \sin (e+f x)} \, dx &=\frac {\int \sec ^2(e+f x) (c-c \sin (e+f x))^{3/2} \, dx}{a c}\\ &=-\frac {2 \sec (e+f x) \sqrt {c-c \sin (e+f x)}}{a f}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 29, normalized size = 1.00 \[ -\frac {2 \sec (e+f x) \sqrt {c-c \sin (e+f x)}}{a f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.44, size = 29, normalized size = 1.00 \[ -\frac {2 \, \sqrt {-c \sin \left (f x + e\right ) + c}}{a f \cos \left (f x + e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(-c*sin(f*x + e) + c)/(a*f*cos(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((c-c*sin(f*x+e))^(1/2)/(a+a*sin(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)-2*sqrt(2*c)*sign(sin(1/2*(f*x+exp(1))-1/4*pi))/a/f/((-cos(1/4*(2*f*x+2*exp(1)-pi))+1)/(cos(1/4*(2*f*x+2*e
xp(1)-pi))+1)-1)

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maple [A]  time = 0.61, size = 39, normalized size = 1.34 \[ \frac {2 c \left (\sin \left (f x +e \right )-1\right )}{a \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.78, size = 76, normalized size = 2.62 \[ \frac {2 \, {\left (\sqrt {c} + \frac {\sqrt {c} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )}}{{\left (a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )} f \sqrt {\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.20, size = 40, normalized size = 1.38 \[ -\frac {4\,\cos \left (e+f\,x\right )\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}}{a\,f\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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