3.16 \(\int \frac {1}{\sqrt {a+a \csc (x)}} \, dx\)

Optimal. Leaf size=62 \[ \frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {2} \sqrt {a \csc (x)+a}}\right )}{\sqrt {a}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc (x)+a}}\right )}{\sqrt {a}} \]

[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)

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Rubi [A]  time = 0.06, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3776, 3774, 203, 3795} \[ \frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {2} \sqrt {a \csc (x)+a}}\right )}{\sqrt {a}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc (x)+a}}\right )}{\sqrt {a}} \]

Antiderivative was successfully verified.

[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 203

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

Rule 3774

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 3776

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 3795

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*} \int \frac {1}{\sqrt {a+a \csc (x)}} \, dx &=\frac {\int \sqrt {a+a \csc (x)} \, dx}{a}-\int \frac {\csc (x)}{\sqrt {a+a \csc (x)}} \, dx\\ &=-\left (2 \operatorname {Subst}\left (\int \frac {1}{a+x^2} \, dx,x,\frac {a \cot (x)}{\sqrt {a+a \csc (x)}}\right )\right )+2 \operatorname {Subst}\left (\int \frac {1}{2 a+x^2} \, dx,x,\frac {a \cot (x)}{\sqrt {a+a \csc (x)}}\right )\\ &=-\frac {2 \tan ^{-1}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {a+a \csc (x)}}\right )}{\sqrt {a}}+\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {2} \sqrt {a+a \csc (x)}}\right )}{\sqrt {a}}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 54, normalized size = 0.87 \[ \frac {\cot (x) \left (\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {\csc (x)-1}}{\sqrt {2}}\right )-2 \tan ^{-1}\left (\sqrt {\csc (x)-1}\right )\right )}{\sqrt {\csc (x)-1} \sqrt {a (\csc (x)+1)}} \]

Antiderivative was successfully verified.

[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])])

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fricas [A]  time = 0.54, size = 219, normalized size = 3.53 \[ \left [\frac {\sqrt {2} a \sqrt {-\frac {1}{a}} \log \left (\frac {\sqrt {2} \sqrt {\frac {a \sin \relax (x) + a}{\sin \relax (x)}} \sqrt {-\frac {1}{a}} \sin \relax (x) + \cos \relax (x)}{\sin \relax (x) + 1}\right ) - \sqrt {-a} \log \left (\frac {2 \, a \cos \relax (x)^{2} + 2 \, {\left (\cos \relax (x)^{2} + {\left (\cos \relax (x) + 1\right )} \sin \relax (x) - 1\right )} \sqrt {-a} \sqrt {\frac {a \sin \relax (x) + a}{\sin \relax (x)}} + a \cos \relax (x) - {\left (2 \, a \cos \relax (x) + a\right )} \sin \relax (x) - a}{\cos \relax (x) + \sin \relax (x) + 1}\right )}{a}, -\frac {2 \, {\left (\sqrt {2} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a \sin \relax (x) + a}{\sin \relax (x)}} \sin \relax (x)}{\sqrt {a} {\left (\cos \relax (x) + \sin \relax (x) + 1\right )}}\right ) - \sqrt {a} \arctan \left (-\frac {\sqrt {a} \sqrt {\frac {a \sin \relax (x) + a}{\sin \relax (x)}} {\left (\cos \relax (x) - \sin \relax (x) + 1\right )}}{a \cos \relax (x) + a \sin \relax (x) + a}\right )\right )}}{a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[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]

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \csc \relax (x) + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(a*csc(x) + a), x)

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maple [B]  time = 0.74, size = 221, normalized size = 3.56 \[ -\frac {\left (4 \sqrt {2}\, \arctan \left (\sqrt {-\frac {-1+\cos \relax (x )}{\sin \relax (x )}}\right )-\ln \left (-\frac {\sqrt {2}\, \sqrt {-\frac {-1+\cos \relax (x )}{\sin \relax (x )}}\, \sin \relax (x )+\sin \relax (x )-\cos \relax (x )+1}{\sqrt {2}\, \sqrt {-\frac {-1+\cos \relax (x )}{\sin \relax (x )}}\, \sin \relax (x )-\sin \relax (x )+\cos \relax (x )-1}\right )-4 \arctan \left (\sqrt {2}\, \sqrt {-\frac {-1+\cos \relax (x )}{\sin \relax (x )}}+1\right )-4 \arctan \left (\sqrt {2}\, \sqrt {-\frac {-1+\cos \relax (x )}{\sin \relax (x )}}-1\right )-\ln \left (-\frac {\sqrt {2}\, \sqrt {-\frac {-1+\cos \relax (x )}{\sin \relax (x )}}\, \sin \relax (x )-\sin \relax (x )+\cos \relax (x )-1}{\sqrt {2}\, \sqrt {-\frac {-1+\cos \relax (x )}{\sin \relax (x )}}\, \sin \relax (x )+\sin \relax (x )-\cos \relax (x )+1}\right )\right ) \left (1-\cos \relax (x )+\sin \relax (x )\right ) \sqrt {2}}{4 \sqrt {\frac {a \left (1+\sin \relax (x )\right )}{\sin \relax (x )}}\, \sin \relax (x ) \sqrt {-\frac {-1+\cos \relax (x )}{\sin \relax (x )}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(4*2^(1/2)*arctan((-(-1+cos(x))/sin(x))^(1/2))-ln(-(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)+sin(x)-cos
(x)+1)/(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)-sin(x)+cos(x)-1))-4*arctan(2^(1/2)*(-(-1+cos(x))/sin(x))^(1
/2)+1)-4*arctan(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)-1)-ln(-(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)-sin(x)+
cos(x)-1)/(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)+sin(x)-cos(x)+1)))*(1-cos(x)+sin(x))/(a*(1+sin(x))/sin(x
))^(1/2)/sin(x)/(-(-1+cos(x))/sin(x))^(1/2)*2^(1/2)

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maxima [A]  time = 0.45, size = 83, normalized size = 1.34 \[ \frac {\sqrt {2} {\left (\sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\frac {\sin \relax (x)}{\cos \relax (x) + 1}}\right )}\right ) + \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\frac {\sin \relax (x)}{\cos \relax (x) + 1}}\right )}\right )\right )}}{\sqrt {a}} - \frac {2 \, \sqrt {2} \arctan \left (\sqrt {\frac {\sin \relax (x)}{\cos \relax (x) + 1}}\right )}{\sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {a+\frac {a}{\sin \relax (x)}}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + a/sin(x))^(1/2),x)

[Out]

int(1/(a + a/sin(x))^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \csc {\relax (x )} + a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(a*csc(x) + a), x)

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