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

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Rubi [A]  time = 0.0155253, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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 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])

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*}

Mathematica [A]  time = 0.0518105, 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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Maple [B]  time = 0.151, size = 199, normalized size = 7.7 \begin{align*}{\frac{\sin \left ( x \right ) \sqrt{2}}{2-2\,\cos \left ( x \right ) +2\,\sin \left ( x \right ) }\sqrt{{\frac{a \left ( \sin \left ( x \right ) +1 \right ) }{\sin \left ( x \right ) }}}\sqrt{-{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}} \left ( \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 ) } \end{align*}

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*(sin(x)+1)/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 = 1.58212, size = 200, normalized size = 7.69 \begin{align*} -\frac{2}{3} \, \sqrt{2} \sqrt{a} \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )^{\frac{3}{2}} + \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} - 2 \, \sqrt{2} \sqrt{a} \sqrt{\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}} + \frac{2 \,{\left (\frac{3 \, \sqrt{2} \sqrt{a} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{\sqrt{2} \sqrt{a} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right )}}{3 \, \sqrt{\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}}} \end{align*}

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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Fricas [B]  time = 0.488618, size = 377, normalized size = 14.5 \begin{align*} \left [\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 ), 2 \, \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 ] \end{align*}

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

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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Giac [B]  time = 2.0381, size = 477, normalized size = 18.35 \begin{align*} \text{result too large to display} \end{align*}

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))