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[(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]

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Rubi [A]  time = 0.0575193, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, 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 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 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 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 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*}

Mathematica [A]  time = 0.12595, 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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Maple [B]  time = 0.139, size = 221, normalized size = 3.6 \begin{align*} -{\frac{\sqrt{2} \left ( 1-\cos \left ( x \right ) +\sin \left ( x \right ) \right ) }{4\,\sin \left ( x \right ) } \left ( 4\,\sqrt{2}\arctan \left ( \sqrt{-{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}} \right ) -\ln \left ( -{ \left ( \sqrt{2}\sqrt{-{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}}\sin \left ( x \right ) +\sin \left ( x \right ) -\cos \left ( x \right ) +1 \right ) \left ( \sqrt{2}\sqrt{-{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}}\sin \left ( x \right ) -\sin \left ( x \right ) +\cos \left ( x \right ) -1 \right ) ^{-1}} \right ) -4\,\arctan \left ( \sqrt{2}\sqrt{-{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}}+1 \right ) -4\,\arctan \left ( \sqrt{2}\sqrt{-{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}}-1 \right ) -\ln \left ( -{ \left ( \sqrt{2}\sqrt{-{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}}\sin \left ( x \right ) -\sin \left ( x \right ) +\cos \left ( x \right ) -1 \right ) \left ( \sqrt{2}\sqrt{-{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}}\sin \left ( x \right ) +\sin \left ( x \right ) -\cos \left ( x \right ) +1 \right ) ^{-1}} \right ) \right ){\frac{1}{\sqrt{{\frac{a \left ( \sin \left ( x \right ) +1 \right ) }{\sin \left ( x \right ) }}}}}{\frac{1}{\sqrt{-{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*2^(1/2)*(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)+si
n(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))/si
n(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*(sin(x)+
1)/sin(x))^(1/2)/sin(x)/(-(-1+cos(x))/sin(x))^(1/2)

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Maxima [A]  time = 1.57662, size = 112, normalized size = 1.81 \begin{align*} \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}} \end{align*}

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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Fricas [A]  time = 0.506052, size = 668, normalized size = 10.77 \begin{align*} \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 ] \end{align*}

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

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a \csc \left (x\right ) + a}}\,{d x} \end{align*}

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)