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

Optimal. Leaf size=46 \[ \frac{\sec (e+f x) \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)} \log (\sin (e+f x))}{f} \]

[Out]

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

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Rubi [A]  time = 0.170772, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {2948, 3475} \[ \frac{\sec (e+f x) \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)} \log (\sin (e+f x))}{f} \]

Antiderivative was successfully verified.

[In]

Int[Csc[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]],x]

[Out]

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

Rule 2948

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

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \csc (e+f x) \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)} \, dx &=\left (\sec (e+f x) \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}\right ) \int \cot (e+f x) \, dx\\ &=\frac{\log (\sin (e+f x)) \sec (e+f x) \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}{f}\\ \end{align*}

Mathematica [A]  time = 0.108767, size = 62, normalized size = 1.35 \[ \frac{\sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} \left (\log \left (\sin \left (\frac{1}{2} (e+f x)\right )\right )+\log \left (\cos \left (\frac{1}{2} (e+f x)\right )\right )\right )}{f} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]],x]

[Out]

((Log[Cos[(e + f*x)/2]] + Log[Sin[(e + f*x)/2]])*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*
x]])/f

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Maple [A]  time = 0.339, size = 73, normalized size = 1.6 \begin{align*}{\frac{1}{f\cos \left ( fx+e \right ) } \left ( -\ln \left ( 2\, \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1} \right ) +\ln \left ( -{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) \right ) \sqrt{a \left ( 1+\sin \left ( fx+e \right ) \right ) }\sqrt{-c \left ( -1+\sin \left ( fx+e \right ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a \sin \left (f x + e\right ) + a} \sqrt{-c \sin \left (f x + e\right ) + c}}{\sin \left (f x + e\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 3.69213, size = 539, normalized size = 11.72 \begin{align*} \left [\frac{\sqrt{a c} \log \left (\frac{4 \,{\left (256 \, a c \cos \left (f x + e\right )^{5} - 512 \, a c \cos \left (f x + e\right )^{3} + 337 \, a c \cos \left (f x + e\right ) +{\left (256 \, \cos \left (f x + e\right )^{4} - 512 \, \cos \left (f x + e\right )^{2} + 175\right )} \sqrt{a c} \sqrt{a \sin \left (f x + e\right ) + a} \sqrt{-c \sin \left (f x + e\right ) + c}\right )}}{\cos \left (f x + e\right )^{3} - \cos \left (f x + e\right )}\right )}{2 \, f}, -\frac{\sqrt{-a c} \arctan \left (\frac{\sqrt{-a c}{\left (16 \, \cos \left (f x + e\right )^{2} - 7\right )} \sqrt{a \sin \left (f x + e\right ) + a} \sqrt{-c \sin \left (f x + e\right ) + c}}{16 \, a c \cos \left (f x + e\right )^{3} - 25 \, a c \cos \left (f x + e\right )}\right )}{f}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*sqrt(a*c)*log(4*(256*a*c*cos(f*x + e)^5 - 512*a*c*cos(f*x + e)^3 + 337*a*c*cos(f*x + e) + (256*cos(f*x +
e)^4 - 512*cos(f*x + e)^2 + 175)*sqrt(a*c)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c))/(cos(f*x + e)^3
- cos(f*x + e)))/f, -sqrt(-a*c)*arctan(sqrt(-a*c)*(16*cos(f*x + e)^2 - 7)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*si
n(f*x + e) + c)/(16*a*c*cos(f*x + e)^3 - 25*a*c*cos(f*x + e)))/f]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a \left (\sin{\left (e + f x \right )} + 1\right )} \sqrt{- c \left (\sin{\left (e + f x \right )} - 1\right )}}{\sin{\left (e + f x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError