\(\int \sqrt {\csc (e+f x)} \sqrt {a+a \csc (e+f x)} \, dx\) [19]

Optimal result

Integrand size = 25, antiderivative size = 37 \[ \int \sqrt {\csc (e+f x)} \sqrt {a+a \csc (e+f x)} \, dx=-\frac {2 \sqrt {a} \text {arcsinh}\left (\frac {\sqrt {a} \cot (e+f x)}{\sqrt {a+a \csc (e+f x)}}\right )}{f} \]

[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 = 0.07 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {3886, 221} \[ \int \sqrt {\csc (e+f x)} \sqrt {a+a \csc (e+f x)} \, dx=-\frac {2 \sqrt {a} \text {arcsinh}\left (\frac {\sqrt {a} \cot (e+f x)}{\sqrt {a \csc (e+f x)+a}}\right )}{f} \]

[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. \(108\) vs. \(2(37)=74\).

Time = 0.89 (sec) , antiderivative size = 108, normalized size of antiderivative = 2.92 \[ \int \sqrt {\csc (e+f x)} \sqrt {a+a \csc (e+f x)} \, dx=\frac {2 \cot (e+f x) \sqrt {a (1+\csc (e+f x))} \left (\log (1+\csc (e+f x))-\log \left (\sqrt {\csc (e+f x)}+\csc ^{\frac {3}{2}}(e+f x)+\sqrt {\cot ^2(e+f x)} \sqrt {1+\csc (e+f x)}\right )\right )}{f \sqrt {\cot ^2(e+f x)} \sqrt {1+\csc (e+f x)}} \]

[In]

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

[Out]

(2*Cot[e + f*x]*Sqrt[a*(1 + Csc[e + f*x])]*(Log[1 + Csc[e + f*x]] - Log[Sqrt[Csc[e + f*x]] + Csc[e + f*x]^(3/2
) + Sqrt[Cot[e + f*x]^2]*Sqrt[1 + Csc[e + f*x]]]))/(f*Sqrt[Cot[e + f*x]^2]*Sqrt[1 + Csc[e + f*x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(97\) vs. \(2(31)=62\).

Time = 2.51 (sec) , antiderivative size = 98, normalized size of antiderivative = 2.65

method	result	size
default	\(-\frac {\sin \left (f x +e \right ) \left (\operatorname {arcsinh}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )+\operatorname {arctanh}\left (\frac {\sqrt {2}}{2 \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}}\right )\right ) \sqrt {\csc \left (f x +e \right )}\, \sqrt {a \left (1+\csc \left (f x +e \right )\right )}\, \sqrt {2}}{f \left (\cos \left (f x +e \right )+\sin \left (f x +e \right )+1\right ) \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}}\)	\(98\)

[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)*(arcsinh(cot(f*x+e)-csc(f*x+e))+arctanh(1/2*2^(1/2)/(1/(1+cos(f*x+e)))^(1/2)))*csc(f*x+e)^(1/2
)*(a*(1+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. 75 vs. \(2 (31) = 62\).

Time = 0.28 (sec) , antiderivative size = 283, normalized size of antiderivative = 7.65 \[ \int \sqrt {\csc (e+f x)} \sqrt {a+a \csc (e+f x)} \, dx=\left [\frac {\sqrt {a} \log \left (\frac {a \cos \left (f x + e\right )^{3} - 7 \, a \cos \left (f x + e\right )^{2} - 9 \, a \cos \left (f x + e\right ) + {\left (a \cos \left (f x + e\right )^{2} + 8 \, a \cos \left (f x + e\right ) - a\right )} \sin \left (f x + e\right ) + \frac {4 \, {\left (\cos \left (f x + e\right )^{3} + 3 \, \cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right ) - 3\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 3\right )} \sqrt {a} \sqrt {\frac {a \sin \left (f x + e\right ) + a}{\sin \left (f x + e\right )}}}{\sqrt {\sin \left (f x + e\right )}} - a}{\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right )^{2} - 1\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 1}\right )}{2 \, f}, \frac {\sqrt {-a} \arctan \left (\frac {{\left (\cos \left (f x + e\right )^{2} + 2 \, \sin \left (f x + e\right ) - 1\right )} \sqrt {-a} \sqrt {\frac {a \sin \left (f x + e\right ) + a}{\sin \left (f x + e\right )}}}{2 \, a \cos \left (f x + e\right ) \sqrt {\sin \left (f x + e\right )}}\right )}{f}\right ] \]

[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 - 9*a*cos(f*x + e) + (a*cos(f*x + e)^2 + 8*a*cos(f*x +
e) - a)*sin(f*x + e) + 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(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))/(a*cos(f*x + e)*sqrt(sin(f*x + e))
))/f]

Sympy [F]

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

[In]

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

[Out]

Integral(sqrt(a*(csc(e + f*x) + 1))*sqrt(csc(e + f*x)), 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 [F(-2)]

Exception generated. \[ \int \sqrt {\csc (e+f x)} \sqrt {a+a \csc (e+f x)} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Limit: Max order reached or unable to make series expansion Error: Bad Argument Value

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)