3.15 \(\int \sqrt {a+a \csc (x)} \, dx\)

Optimal. Leaf size=26 \[ -2 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc (x)+a}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3774, 203} \[ -2 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc (x)+a}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + a*Csc[x]],x]

[Out]

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

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]

Rubi steps

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

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Mathematica [A]  time = 0.05, size = 32, normalized size = 1.23 \[ -\frac {2 a \cot (x) \tan ^{-1}\left (\sqrt {\csc (x)-1}\right )}{\sqrt {\csc (x)-1} \sqrt {a (\csc (x)+1)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + a*Csc[x]],x]

[Out]

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

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fricas [B]  time = 0.55, size = 120, normalized size = 4.62 \[ \left [\sqrt {-a} \log \left (\frac {2 \, a \cos \relax (x)^{2} - 2 \, {\left (\cos \relax (x)^{2} + {\left (\cos \relax (x) + 1\right )} \sin \relax (x) - 1\right )} \sqrt {-a} \sqrt {\frac {a \sin \relax (x) + a}{\sin \relax (x)}} + a \cos \relax (x) - {\left (2 \, a \cos \relax (x) + a\right )} \sin \relax (x) - a}{\cos \relax (x) + \sin \relax (x) + 1}\right ), 2 \, \sqrt {a} \arctan \left (-\frac {\sqrt {a} \sqrt {\frac {a \sin \relax (x) + a}{\sin \relax (x)}} {\left (\cos \relax (x) - \sin \relax (x) + 1\right )}}{a \cos \relax (x) + a \sin \relax (x) + a}\right )\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[sqrt(-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)), 2*sqrt(a)*arctan(-sqrt(a)*sqrt((a*sin(x) + a)/si
n(x))*(cos(x) - sin(x) + 1)/(a*cos(x) + a*sin(x) + a))]

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giac [B]  time = 1.06, size = 353, normalized size = 13.58 \[ \frac {1}{4} \, \sqrt {2} {\left (\frac {2 \, \sqrt {2} {\left (a \sqrt {{\left | a \right |}} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{3} + \tan \left (\frac {1}{2} \, x\right )^{2} + \tan \left (\frac {1}{2} \, x\right ) + 1\right ) + {\left | a \right |}^{\frac {3}{2}} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{3} + \tan \left (\frac {1}{2} \, x\right )^{2} + \tan \left (\frac {1}{2} \, x\right ) + 1\right )\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 \, \sqrt {2} {\left (a \sqrt {{\left | a \right |}} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{3} + \tan \left (\frac {1}{2} \, x\right )^{2} + \tan \left (\frac {1}{2} \, x\right ) + 1\right ) + {\left | a \right |}^{\frac {3}{2}} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{3} + \tan \left (\frac {1}{2} \, x\right )^{2} + \tan \left (\frac {1}{2} \, x\right ) + 1\right )\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 {\sqrt {2} {\left (a \sqrt {{\left | a \right |}} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{3} + \tan \left (\frac {1}{2} \, x\right )^{2} + \tan \left (\frac {1}{2} \, x\right ) + 1\right ) - {\left | a \right |}^{\frac {3}{2}} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{3} + \tan \left (\frac {1}{2} \, x\right )^{2} + \tan \left (\frac {1}{2} \, x\right ) + 1\right )\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 {\sqrt {2} {\left (a \sqrt {{\left | a \right |}} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{3} + \tan \left (\frac {1}{2} \, x\right )^{2} + \tan \left (\frac {1}{2} \, x\right ) + 1\right ) - {\left | a \right |}^{\frac {3}{2}} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{3} + \tan \left (\frac {1}{2} \, x\right )^{2} + \tan \left (\frac {1}{2} \, x\right ) + 1\right )\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}\right )} \mathrm {sgn}\left (\sin \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(2)*(2*sqrt(2)*(a*sqrt(abs(a))*sgn(tan(1/2*x)^3 + tan(1/2*x)^2 + tan(1/2*x) + 1) + abs(a)^(3/2)*sgn(ta
n(1/2*x)^3 + tan(1/2*x)^2 + tan(1/2*x) + 1))*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(abs(a)) + 2*sqrt(a*tan(1/2*x)))/
sqrt(abs(a)))/a + 2*sqrt(2)*(a*sqrt(abs(a))*sgn(tan(1/2*x)^3 + tan(1/2*x)^2 + tan(1/2*x) + 1) + abs(a)^(3/2)*s
gn(tan(1/2*x)^3 + tan(1/2*x)^2 + tan(1/2*x) + 1))*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(abs(a)) - 2*sqrt(a*tan(1/2
*x)))/sqrt(abs(a)))/a + sqrt(2)*(a*sqrt(abs(a))*sgn(tan(1/2*x)^3 + tan(1/2*x)^2 + tan(1/2*x) + 1) - abs(a)^(3/
2)*sgn(tan(1/2*x)^3 + tan(1/2*x)^2 + tan(1/2*x) + 1))*log(a*tan(1/2*x) + sqrt(2)*sqrt(a*tan(1/2*x))*sqrt(abs(a
)) + abs(a))/a - sqrt(2)*(a*sqrt(abs(a))*sgn(tan(1/2*x)^3 + tan(1/2*x)^2 + tan(1/2*x) + 1) - abs(a)^(3/2)*sgn(
tan(1/2*x)^3 + tan(1/2*x)^2 + tan(1/2*x) + 1))*log(a*tan(1/2*x) - sqrt(2)*sqrt(a*tan(1/2*x))*sqrt(abs(a)) + ab
s(a))/a)*sgn(sin(x))

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maple [B]  time = 0.72, size = 199, normalized size = 7.65 \[ \frac {\sqrt {2}\, \sqrt {\frac {a \left (1+\sin \relax (x )\right )}{\sin \relax (x )}}\, \sin \relax (x ) \sqrt {-\frac {-1+\cos \relax (x )}{\sin \relax (x )}}\, \left (\ln \left (-\frac {\sqrt {2}\, \sqrt {-\frac {-1+\cos \relax (x )}{\sin \relax (x )}}\, \sin \relax (x )+\sin \relax (x )-\cos \relax (x )+1}{\sqrt {2}\, \sqrt {-\frac {-1+\cos \relax (x )}{\sin \relax (x )}}\, \sin \relax (x )-\sin \relax (x )+\cos \relax (x )-1}\right )+4 \arctan \left (\sqrt {2}\, \sqrt {-\frac {-1+\cos \relax (x )}{\sin \relax (x )}}+1\right )+4 \arctan \left (\sqrt {2}\, \sqrt {-\frac {-1+\cos \relax (x )}{\sin \relax (x )}}-1\right )+\ln \left (-\frac {\sqrt {2}\, \sqrt {-\frac {-1+\cos \relax (x )}{\sin \relax (x )}}\, \sin \relax (x )-\sin \relax (x )+\cos \relax (x )-1}{\sqrt {2}\, \sqrt {-\frac {-1+\cos \relax (x )}{\sin \relax (x )}}\, \sin \relax (x )+\sin \relax (x )-\cos \relax (x )+1}\right )\right )}{2-2 \cos \relax (x )+2 \sin \relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*2^(1/2)*(a*(1+sin(x))/sin(x))^(1/2)*sin(x)*(-(-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))

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maxima [B]  time = 0.44, size = 148, normalized size = 5.69 \[ -\frac {2}{3} \, \sqrt {2} \sqrt {a} \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right )^{\frac {3}{2}} + \sqrt {2} {\left (\sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\frac {\sin \relax (x)}{\cos \relax (x) + 1}}\right )}\right ) + \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\frac {\sin \relax (x)}{\cos \relax (x) + 1}}\right )}\right )\right )} \sqrt {a} - 2 \, \sqrt {2} \sqrt {a} \sqrt {\frac {\sin \relax (x)}{\cos \relax (x) + 1}} + \frac {2 \, {\left (\frac {3 \, \sqrt {2} \sqrt {a} \sin \relax (x)}{\cos \relax (x) + 1} + \frac {\sqrt {2} \sqrt {a} \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}}\right )}}{3 \, \sqrt {\frac {\sin \relax (x)}{\cos \relax (x) + 1}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*sqrt(2)*sqrt(a)*(sin(x)/(cos(x) + 1))^(3/2) + 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)*(sqrt(2) - 2*sqrt(sin(x)/(cos(x) + 1)))))*sqrt(a) - 2*sqrt(2)*
sqrt(a)*sqrt(sin(x)/(cos(x) + 1)) + 2/3*(3*sqrt(2)*sqrt(a)*sin(x)/(cos(x) + 1) + sqrt(2)*sqrt(a)*sin(x)^2/(cos
(x) + 1)^2)/sqrt(sin(x)/(cos(x) + 1))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \sqrt {a+\frac {a}{\sin \relax (x)}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a/sin(x))^(1/2),x)

[Out]

int((a + a/sin(x))^(1/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a*csc(x) + a), x)

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