\(\int \frac {1}{\sqrt {a+a \csc (x)}} \, dx\) [16]

Optimal result

Integrand size = 10, antiderivative size = 62 \[ \int \frac {1}{\sqrt {a+a \csc (x)}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {a} \cot (x)}{\sqrt {a+a \csc (x)}}\right )}{\sqrt {a}}+\frac {\sqrt {2} \arctan \left (\frac {\sqrt {a} \cot (x)}{\sqrt {2} \sqrt {a+a \csc (x)}}\right )}{\sqrt {a}} \]

[Out]

-2*arctan(cot(x)*a^(1/2)/(a+a*csc(x))^(1/2))/a^(1/2)+arctan(1/2*cot(x)*a^(1/2)*2^(1/2)/(a+a*csc(x))^(1/2))*2^(
1/2)/a^(1/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3861, 3859, 209, 3880} \[ \int \frac {1}{\sqrt {a+a \csc (x)}} \, dx=\frac {\sqrt {2} \arctan \left (\frac {\sqrt {a} \cot (x)}{\sqrt {2} \sqrt {a \csc (x)+a}}\right )}{\sqrt {a}}-\frac {2 \arctan \left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc (x)+a}}\right )}{\sqrt {a}} \]

[In]

Int[1/Sqrt[a + a*Csc[x]],x]

[Out]

(-2*ArcTan[(Sqrt[a]*Cot[x])/Sqrt[a + a*Csc[x]]])/Sqrt[a] + (Sqrt[2]*ArcTan[(Sqrt[a]*Cot[x])/(Sqrt[2]*Sqrt[a +
a*Csc[x]])])/Sqrt[a]

Rule 209

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

Rule 3859

Int[Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[-2*(b/d), Subst[Int[1/(a + x^2), x], x, b*(C
ot[c + d*x]/Sqrt[a + b*Csc[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 3861

Int[1/Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[1/a, Int[Sqrt[a + b*Csc[c + d*x]], x], x]
- Dist[b/a, Int[Csc[c + d*x]/Sqrt[a + b*Csc[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 3880

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

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

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\sqrt {a+a \csc (x)}} \, dx=\frac {\left (-2 \arctan \left (\sqrt {-1+\csc (x)}\right )+\sqrt {2} \arctan \left (\frac {\sqrt {-1+\csc (x)}}{\sqrt {2}}\right )\right ) \cot (x)}{\sqrt {-1+\csc (x)} \sqrt {a (1+\csc (x))}} \]

[In]

Integrate[1/Sqrt[a + a*Csc[x]],x]

[Out]

((-2*ArcTan[Sqrt[-1 + Csc[x]]] + Sqrt[2]*ArcTan[Sqrt[-1 + Csc[x]]/Sqrt[2]])*Cot[x])/(Sqrt[-1 + Csc[x]]*Sqrt[a*
(1 + Csc[x])])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(184\) vs. \(2(47)=94\).

Time = 0.53 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.98

method	result	size
default	\(\frac {\left (\sqrt {2}\, \ln \left (\frac {\csc \left (x \right )-\cot \left (x \right )+\sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \sqrt {2}+1}{-\sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \sqrt {2}+\csc \left (x \right )-\cot \left (x \right )+1}\right )+4 \sqrt {2}\, \arctan \left (\sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \sqrt {2}+1\right )+4 \sqrt {2}\, \arctan \left (\sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \sqrt {2}-1\right )+\sqrt {2}\, \ln \left (\frac {-\sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \sqrt {2}+\csc \left (x \right )-\cot \left (x \right )+1}{\csc \left (x \right )-\cot \left (x \right )+\sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \sqrt {2}+1}\right )-8 \arctan \left (\sqrt {\csc \left (x \right )-\cot \left (x \right )}\right )\right ) \left (\csc \left (x \right )-\cot \left (x \right )+1\right )}{4 \sqrt {a \left (\csc \left (x \right )+1\right )}\, \sqrt {\csc \left (x \right )-\cot \left (x \right )}}\)	\(185\)

[In]

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

[Out]

1/4*(2^(1/2)*ln((csc(x)-cot(x)+(csc(x)-cot(x))^(1/2)*2^(1/2)+1)/(-(csc(x)-cot(x))^(1/2)*2^(1/2)+csc(x)-cot(x)+
1))+4*2^(1/2)*arctan((csc(x)-cot(x))^(1/2)*2^(1/2)+1)+4*2^(1/2)*arctan((csc(x)-cot(x))^(1/2)*2^(1/2)-1)+2^(1/2
)*ln((-(csc(x)-cot(x))^(1/2)*2^(1/2)+csc(x)-cot(x)+1)/(csc(x)-cot(x)+(csc(x)-cot(x))^(1/2)*2^(1/2)+1))-8*arcta
n((csc(x)-cot(x))^(1/2)))/(a*(csc(x)+1))^(1/2)/(csc(x)-cot(x))^(1/2)*(csc(x)-cot(x)+1)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 219, normalized size of antiderivative = 3.53 \[ \int \frac {1}{\sqrt {a+a \csc (x)}} \, dx=\left [\frac {\sqrt {2} a \sqrt {-\frac {1}{a}} \log \left (\frac {\sqrt {2} \sqrt {\frac {a \sin \left (x\right ) + a}{\sin \left (x\right )}} \sqrt {-\frac {1}{a}} \sin \left (x\right ) + \cos \left (x\right )}{\sin \left (x\right ) + 1}\right ) - \sqrt {-a} \log \left (\frac {2 \, a \cos \left (x\right )^{2} + 2 \, {\left (\cos \left (x\right )^{2} + {\left (\cos \left (x\right ) + 1\right )} \sin \left (x\right ) - 1\right )} \sqrt {-a} \sqrt {\frac {a \sin \left (x\right ) + a}{\sin \left (x\right )}} + a \cos \left (x\right ) - {\left (2 \, a \cos \left (x\right ) + a\right )} \sin \left (x\right ) - a}{\cos \left (x\right ) + \sin \left (x\right ) + 1}\right )}{a}, -\frac {2 \, {\left (\sqrt {2} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a \sin \left (x\right ) + a}{\sin \left (x\right )}} \sin \left (x\right )}{\sqrt {a} {\left (\cos \left (x\right ) + \sin \left (x\right ) + 1\right )}}\right ) - \sqrt {a} \arctan \left (-\frac {\sqrt {a} \sqrt {\frac {a \sin \left (x\right ) + a}{\sin \left (x\right )}} {\left (\cos \left (x\right ) - \sin \left (x\right ) + 1\right )}}{a \cos \left (x\right ) + a \sin \left (x\right ) + a}\right )\right )}}{a}\right ] \]

[In]

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

[Out]

[(sqrt(2)*a*sqrt(-1/a)*log((sqrt(2)*sqrt((a*sin(x) + a)/sin(x))*sqrt(-1/a)*sin(x) + cos(x))/(sin(x) + 1)) - sq
rt(-a)*log((2*a*cos(x)^2 + 2*(cos(x)^2 + (cos(x) + 1)*sin(x) - 1)*sqrt(-a)*sqrt((a*sin(x) + a)/sin(x)) + a*cos
(x) - (2*a*cos(x) + a)*sin(x) - a)/(cos(x) + sin(x) + 1)))/a, -2*(sqrt(2)*sqrt(a)*arctan(sqrt(2)*sqrt((a*sin(x
) + a)/sin(x))*sin(x)/(sqrt(a)*(cos(x) + sin(x) + 1))) - sqrt(a)*arctan(-sqrt(a)*sqrt((a*sin(x) + a)/sin(x))*(
cos(x) - sin(x) + 1)/(a*cos(x) + a*sin(x) + a)))/a]

Sympy [F]

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

[In]

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

[Out]

Integral(1/sqrt(a*csc(x) + a), x)

Maxima [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.34 \[ \int \frac {1}{\sqrt {a+a \csc (x)}} \, dx=\frac {\sqrt {2} {\left (\sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}}\right )}\right ) + \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}}\right )}\right )\right )}}{\sqrt {a}} - \frac {2 \, \sqrt {2} \arctan \left (\sqrt {\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}}\right )}{\sqrt {a}} \]

[In]

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

[Out]

sqrt(2)*(sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(sin(x)/(cos(x) + 1)))) + sqrt(2)*arctan(-1/2*sqrt(2)*(sq
rt(2) - 2*sqrt(sin(x)/(cos(x) + 1)))))/sqrt(a) - 2*sqrt(2)*arctan(sqrt(sin(x)/(cos(x) + 1)))/sqrt(a)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 205 vs. \(2 (47) = 94\).

Time = 0.43 (sec) , antiderivative size = 205, normalized size of antiderivative = 3.31 \[ \int \frac {1}{\sqrt {a+a \csc (x)}} \, dx=-\frac {4 \, \sqrt {2} \sqrt {a} \arctan \left (\frac {\sqrt {a \tan \left (\frac {1}{2} \, x\right )}}{\sqrt {a}}\right ) - \frac {2 \, {\left (a \sqrt {{\left | a \right |}} + {\left | a \right |}^{\frac {3}{2}}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | a \right |}} + 2 \, \sqrt {a \tan \left (\frac {1}{2} \, x\right )}\right )}}{2 \, \sqrt {{\left | a \right |}}}\right )}{a} - \frac {2 \, {\left (a \sqrt {{\left | a \right |}} + {\left | a \right |}^{\frac {3}{2}}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | a \right |}} - 2 \, \sqrt {a \tan \left (\frac {1}{2} \, x\right )}\right )}}{2 \, \sqrt {{\left | a \right |}}}\right )}{a} - \frac {{\left (a \sqrt {{\left | a \right |}} - {\left | a \right |}^{\frac {3}{2}}\right )} \log \left (a \tan \left (\frac {1}{2} \, x\right ) + \sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, x\right )} \sqrt {{\left | a \right |}} + {\left | a \right |}\right )}{a} + \frac {{\left (a \sqrt {{\left | a \right |}} - {\left | a \right |}^{\frac {3}{2}}\right )} \log \left (a \tan \left (\frac {1}{2} \, x\right ) - \sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, x\right )} \sqrt {{\left | a \right |}} + {\left | a \right |}\right )}{a}}{2 \, a} \]

[In]

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

[Out]

-1/2*(4*sqrt(2)*sqrt(a)*arctan(sqrt(a*tan(1/2*x))/sqrt(a)) - 2*(a*sqrt(abs(a)) + abs(a)^(3/2))*arctan(1/2*sqrt
(2)*(sqrt(2)*sqrt(abs(a)) + 2*sqrt(a*tan(1/2*x)))/sqrt(abs(a)))/a - 2*(a*sqrt(abs(a)) + abs(a)^(3/2))*arctan(-
1/2*sqrt(2)*(sqrt(2)*sqrt(abs(a)) - 2*sqrt(a*tan(1/2*x)))/sqrt(abs(a)))/a - (a*sqrt(abs(a)) - abs(a)^(3/2))*lo
g(a*tan(1/2*x) + sqrt(2)*sqrt(a*tan(1/2*x))*sqrt(abs(a)) + abs(a))/a + (a*sqrt(abs(a)) - abs(a)^(3/2))*log(a*t
an(1/2*x) - sqrt(2)*sqrt(a*tan(1/2*x))*sqrt(abs(a)) + abs(a))/a)/a

Mupad [F(-1)]

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

[In]

int(1/(a + a/sin(x))^(1/2),x)

[Out]

int(1/(a + a/sin(x))^(1/2), x)