\(\int \sqrt {-\csc (e+f x)} \sqrt {a-a \csc (e+f x)} \, dx\) [20]

Optimal result

Integrand size = 28, antiderivative size = 38 \[ \int \sqrt {-\csc (e+f x)} \sqrt {a-a \csc (e+f x)} \, dx=-\frac {2 \sqrt {a} \text {arcsinh}\left (\frac {\sqrt {a} \cot (e+f x)}{\sqrt {a-a \csc (e+f x)}}\right )}{f} \]

[Out]

-2*arcsinh(cot(f*x+e)*a^(1/2)/(a-a*csc(f*x+e))^(1/2))*a^(1/2)/f

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {3886, 221} \[ \int \sqrt {-\csc (e+f x)} \sqrt {a-a \csc (e+f x)} \, dx=-\frac {2 \sqrt {a} \text {arcsinh}\left (\frac {\sqrt {a} \cot (e+f x)}{\sqrt {a-a \csc (e+f x)}}\right )}{f} \]

[In]

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

[Out]

(-2*Sqrt[a]*ArcSinh[(Sqrt[a]*Cot[e + f*x])/Sqrt[a - a*Csc[e + f*x]]])/f

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt[a])]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 3886

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[-2*(a/(b
*f))*Sqrt[a*(d/b)], Subst[Int[1/Sqrt[1 + x^2/a], x], x, b*(Cot[e + f*x]/Sqrt[a + b*Csc[e + f*x]])], x] /; Free
Q[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[a*(d/b), 0]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{a}}} \, dx,x,-\frac {a \cot (e+f x)}{\sqrt {a-a \csc (e+f x)}}\right )}{f} \\ & = -\frac {2 \sqrt {a} \text {arcsinh}\left (\frac {\sqrt {a} \cot (e+f x)}{\sqrt {a-a \csc (e+f x)}}\right )}{f} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(101\) vs. \(2(38)=76\).

Time = 3.14 (sec) , antiderivative size = 101, normalized size of antiderivative = 2.66 \[ \int \sqrt {-\csc (e+f x)} \sqrt {a-a \csc (e+f x)} \, dx=\frac {2 \left (\text {arcsinh}\left (\tan \left (\frac {1}{2} (e+f x)\right )\right )+\text {arctanh}\left (\sqrt {\sec ^2\left (\frac {1}{2} (e+f x)\right )}\right )\right ) \sqrt {-\csc (e+f x)} \sqrt {a-a \csc (e+f x)} \tan \left (\frac {1}{2} (e+f x)\right )}{f \sqrt {\sec ^2\left (\frac {1}{2} (e+f x)\right )} \left (-1+\tan \left (\frac {1}{2} (e+f x)\right )\right )} \]

[In]

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

[Out]

(2*(ArcSinh[Tan[(e + f*x)/2]] + ArcTanh[Sqrt[Sec[(e + f*x)/2]^2]])*Sqrt[-Csc[e + f*x]]*Sqrt[a - a*Csc[e + f*x]
]*Tan[(e + f*x)/2])/(f*Sqrt[Sec[(e + f*x)/2]^2]*(-1 + Tan[(e + f*x)/2]))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(126\) vs. \(2(32)=64\).

Time = 2.75 (sec) , antiderivative size = 127, normalized size of antiderivative = 3.34

method	result	size
default	\(-\frac {\sin \left (f x +e \right ) \left (\arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {1}{1+\cos \left (f x +e \right )}}}\right )+\arctan \left (\frac {\sqrt {2}\, \sin \left (f x +e \right )}{2 \left (1+\cos \left (f x +e \right )\right ) \sqrt {-\frac {1}{1+\cos \left (f x +e \right )}}}\right )\right ) \sqrt {-a \left (-1+\csc \left (f x +e \right )\right )}\, \sqrt {-\csc \left (f x +e \right )}\, \sqrt {2}}{f \left (-\cos \left (f x +e \right )+\sin \left (f x +e \right )-1\right ) \sqrt {-\frac {1}{1+\cos \left (f x +e \right )}}}\)	\(127\)

[In]

int((-csc(f*x+e))^(1/2)*(a-a*csc(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (32) = 64\).

Time = 0.26 (sec) , antiderivative size = 296, normalized size of antiderivative = 7.79 \[ \int \sqrt {-\csc (e+f x)} \sqrt {a-a \csc (e+f x)} \, dx=\left [\frac {\sqrt {a} \log \left (\frac {a \cos \left (f x + e\right )^{3} - 7 \, a \cos \left (f x + e\right )^{2} - 4 \, {\left (\cos \left (f x + e\right )^{3} + 3 \, \cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right ) - 3\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 3\right )} \sqrt {a} \sqrt {\frac {a \sin \left (f x + e\right ) - a}{\sin \left (f x + e\right )}} \sqrt {-\frac {1}{\sin \left (f x + e\right )}} - 9 \, a \cos \left (f x + e\right ) - {\left (a \cos \left (f x + e\right )^{2} + 8 \, a \cos \left (f x + e\right ) - a\right )} \sin \left (f x + e\right ) - a}{\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right )^{2} - 1\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 1}\right )}{2 \, f}, \frac {\sqrt {-a} \arctan \left (-\frac {{\left (\cos \left (f x + e\right )^{2} - 2 \, \sin \left (f x + e\right ) - 1\right )} \sqrt {-a} \sqrt {\frac {a \sin \left (f x + e\right ) - a}{\sin \left (f x + e\right )}} \sqrt {-\frac {1}{\sin \left (f x + e\right )}}}{2 \, a \cos \left (f x + e\right )}\right )}{f}\right ] \]

[In]

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

[Out]

[1/2*sqrt(a)*log((a*cos(f*x + e)^3 - 7*a*cos(f*x + e)^2 - 4*(cos(f*x + e)^3 + 3*cos(f*x + e)^2 + (cos(f*x + e)
^2 - 2*cos(f*x + e) - 3)*sin(f*x + e) - cos(f*x + e) - 3)*sqrt(a)*sqrt((a*sin(f*x + e) - a)/sin(f*x + e))*sqrt
(-1/sin(f*x + e)) - 9*a*cos(f*x + e) - (a*cos(f*x + e)^2 + 8*a*cos(f*x + e) - a)*sin(f*x + e) - a)/(cos(f*x +
e)^3 + cos(f*x + e)^2 - (cos(f*x + e)^2 - 1)*sin(f*x + e) - cos(f*x + e) - 1))/f, sqrt(-a)*arctan(-1/2*(cos(f*
x + e)^2 - 2*sin(f*x + e) - 1)*sqrt(-a)*sqrt((a*sin(f*x + e) - a)/sin(f*x + e))*sqrt(-1/sin(f*x + e))/(a*cos(f
*x + e)))/f]

Sympy [F]

\[ \int \sqrt {-\csc (e+f x)} \sqrt {a-a \csc (e+f x)} \, dx=\int \sqrt {- \csc {\left (e + f x \right )}} \sqrt {- a \left (\csc {\left (e + f x \right )} - 1\right )}\, dx \]

[In]

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

[Out]

Integral(sqrt(-csc(e + f*x))*sqrt(-a*(csc(e + f*x) - 1)), x)

Maxima [F]

\[ \int \sqrt {-\csc (e+f x)} \sqrt {a-a \csc (e+f x)} \, dx=\int { \sqrt {-a \csc \left (f x + e\right ) + a} \sqrt {-\csc \left (f x + e\right )} \,d x } \]

[In]

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

[Out]

integrate(sqrt(-a*csc(f*x + e) + a)*sqrt(-csc(f*x + e)), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 101 vs. \(2 (32) = 64\).

Time = 0.66 (sec) , antiderivative size = 101, normalized size of antiderivative = 2.66 \[ \int \sqrt {-\csc (e+f x)} \sqrt {a-a \csc (e+f x)} \, dx=-\frac {\frac {2 \, a \arctan \left (\frac {a^{\frac {3}{2}} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + \sqrt {a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a^{3}}}{\sqrt {-a} a}\right )}{\sqrt {-a}} - \sqrt {a} \log \left ({\left | a^{\frac {3}{2}} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + \sqrt {a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a^{3}} \right |}\right )}{f} \]

[In]

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

[Out]

-(2*a*arctan((a^(3/2)*tan(1/2*f*x + 1/2*e) + sqrt(a^3*tan(1/2*f*x + 1/2*e)^2 + a^3))/(sqrt(-a)*a))/sqrt(-a) -
sqrt(a)*log(abs(a^(3/2)*tan(1/2*f*x + 1/2*e) + sqrt(a^3*tan(1/2*f*x + 1/2*e)^2 + a^3))))/f

Mupad [F(-1)]

Timed out. \[ \int \sqrt {-\csc (e+f x)} \sqrt {a-a \csc (e+f x)} \, dx=\int \sqrt {a-\frac {a}{\sin \left (e+f\,x\right )}}\,\sqrt {-\frac {1}{\sin \left (e+f\,x\right )}} \,d x \]

[In]

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

[Out]

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