3.17 \(\int \frac{1}{(a+a \csc (x))^{3/2}} \, dx\)

Optimal. Leaf size=81 \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \cot (x)}{\sqrt{a \csc (x)+a}}\right )}{a^{3/2}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt{a} \cot (x)}{\sqrt{2} \sqrt{a \csc (x)+a}}\right )}{2 \sqrt{2} a^{3/2}}+\frac{\cot (x)}{2 (a \csc (x)+a)^{3/2}} \]

[Out]

(-2*ArcTan[(Sqrt[a]*Cot[x])/Sqrt[a + a*Csc[x]]])/a^(3/2) + (5*ArcTan[(Sqrt[a]*Cot[x])/(Sqrt[2]*Sqrt[a + a*Csc[
x]])])/(2*Sqrt[2]*a^(3/2)) + Cot[x]/(2*(a + a*Csc[x])^(3/2))

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Rubi [A]  time = 0.101898, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3777, 3920, 3774, 203, 3795} \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \cot (x)}{\sqrt{a \csc (x)+a}}\right )}{a^{3/2}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt{a} \cot (x)}{\sqrt{2} \sqrt{a \csc (x)+a}}\right )}{2 \sqrt{2} a^{3/2}}+\frac{\cot (x)}{2 (a \csc (x)+a)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Csc[x])^(-3/2),x]

[Out]

(-2*ArcTan[(Sqrt[a]*Cot[x])/Sqrt[a + a*Csc[x]]])/a^(3/2) + (5*ArcTan[(Sqrt[a]*Cot[x])/(Sqrt[2]*Sqrt[a + a*Csc[
x]])])/(2*Sqrt[2]*a^(3/2)) + Cot[x]/(2*(a + a*Csc[x])^(3/2))

Rule 3777

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

Rule 3920

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[c/a,
Int[Sqrt[a + b*Csc[e + f*x]], x], x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] /
; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && 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}{(a+a \csc (x))^{3/2}} \, dx &=\frac{\cot (x)}{2 (a+a \csc (x))^{3/2}}-\frac{\int \frac{-2 a+\frac{1}{2} a \csc (x)}{\sqrt{a+a \csc (x)}} \, dx}{2 a^2}\\ &=\frac{\cot (x)}{2 (a+a \csc (x))^{3/2}}+\frac{\int \sqrt{a+a \csc (x)} \, dx}{a^2}-\frac{5 \int \frac{\csc (x)}{\sqrt{a+a \csc (x)}} \, dx}{4 a}\\ &=\frac{\cot (x)}{2 (a+a \csc (x))^{3/2}}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{a+x^2} \, dx,x,\frac{a \cot (x)}{\sqrt{a+a \csc (x)}}\right )}{a}+\frac{5 \operatorname{Subst}\left (\int \frac{1}{2 a+x^2} \, dx,x,\frac{a \cot (x)}{\sqrt{a+a \csc (x)}}\right )}{2 a}\\ &=-\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \cot (x)}{\sqrt{a+a \csc (x)}}\right )}{a^{3/2}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt{a} \cot (x)}{\sqrt{2} \sqrt{a+a \csc (x)}}\right )}{2 \sqrt{2} a^{3/2}}+\frac{\cot (x)}{2 (a+a \csc (x))^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.410785, size = 129, normalized size = 1.59 \[ -\frac{\left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right ) \left (-2 \csc (x)+8 \sqrt{\csc (x)-1} (\csc (x)+1) \tan ^{-1}\left (\sqrt{\csc (x)-1}\right )-5 \sqrt{2} \sqrt{\csc (x)-1} \csc (x) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^2 \tan ^{-1}\left (\frac{\sqrt{\csc (x)-1}}{\sqrt{2}}\right )+2\right )}{4 a (\csc (x)-1) \sqrt{a (\csc (x)+1)} \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Csc[x])^(-3/2),x]

[Out]

-((Cos[x/2] - Sin[x/2])*(2 - 2*Csc[x] + 8*ArcTan[Sqrt[-1 + Csc[x]]]*Sqrt[-1 + Csc[x]]*(1 + Csc[x]) - 5*Sqrt[2]
*ArcTan[Sqrt[-1 + Csc[x]]/Sqrt[2]]*Sqrt[-1 + Csc[x]]*Csc[x]*(Cos[x/2] + Sin[x/2])^2))/(4*a*(-1 + Csc[x])*Sqrt[
a*(1 + Csc[x])]*(Cos[x/2] + Sin[x/2]))

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Maple [B]  time = 0.184, size = 1141, normalized size = 14.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+a*csc(x))^(3/2),x)

[Out]

1/8*2^(1/2)*(-1+cos(x))*(-2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)+sin(x)*cos(x)*(-(-1+cos(x))/sin(x))^(3/2)
*2^(1/2)-sin(x)*cos(x)*(-(-1+cos(x))/sin(x))^(1/2)*2^(1/2)-10*sin(x)*cos(x)*arctan((-(-1+cos(x))/sin(x))^(1/2)
)*2^(1/2)-4*sin(x)*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*sin(x)*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))-16*sin(x)*arctan(2^(1/2)*(-(-1+cos(x))/sin(x))^(
1/2)+1)+8*cos(x)*arctan(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)-1)+2*cos(x)*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))+2*cos(x)*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))+8*cos(x)*arctan(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)+1)+(-(-1+cos(x))/sin(x))^(3/2)*2^(1/2)-4*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*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)*s
in(x)-sin(x)+cos(x)-1))-16*arctan(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)+1)-16*arctan(2^(1/2)*(-(-1+cos(x))/sin(x
))^(1/2)-1)+8*cos(x)^2*arctan(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)-1)+2*cos(x)^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))+2*cos(x)^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))+8*cos(x)^2*arctan(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)+1)-16*sin(x)*arctan(2^(1/2)*(-(-1+cos(x))
/sin(x))^(1/2)-1)-10*cos(x)^2*arctan((-(-1+cos(x))/sin(x))^(1/2))*2^(1/2)+20*sin(x)*arctan((-(-1+cos(x))/sin(x
))^(1/2))*2^(1/2)-10*cos(x)*arctan((-(-1+cos(x))/sin(x))^(1/2))*2^(1/2)-2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)+20
*2^(1/2)*arctan((-(-1+cos(x))/sin(x))^(1/2))-cos(x)^2*(-(-1+cos(x))/sin(x))^(3/2)*2^(1/2)+sin(x)*(-(-1+cos(x))
/sin(x))^(3/2)*2^(1/2)+cos(x)^2*(-(-1+cos(x))/sin(x))^(1/2)*2^(1/2)+8*sin(x)*cos(x)*arctan(2^(1/2)*(-(-1+cos(x
))/sin(x))^(1/2)-1)+2*sin(x)*cos(x)*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))+2*sin(x)*cos(x)*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))+8*sin(x)*cos(x)*arctan(2
^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)+1))/(a*(sin(x)+1)/sin(x))^(3/2)/sin(x)^3/(-(-1+cos(x))/sin(x))^(3/2)

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Maxima [B]  time = 1.54373, size = 203, normalized size = 2.51 \begin{align*} -\frac{\sqrt{2} \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )^{\frac{3}{2}} - \sqrt{2} \sqrt{\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}}}{2 \,{\left (a^{\frac{3}{2}} + \frac{2 \, a^{\frac{3}{2}} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{a^{\frac{3}{2}} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right )}} + \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 )}}{a^{\frac{3}{2}}} - \frac{5 \, \sqrt{2} \arctan \left (\sqrt{\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}}\right )}{2 \, a^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*csc(x))^(3/2),x, algorithm="maxima")

[Out]

-1/2*(sqrt(2)*(sin(x)/(cos(x) + 1))^(3/2) - sqrt(2)*sqrt(sin(x)/(cos(x) + 1)))/(a^(3/2) + 2*a^(3/2)*sin(x)/(co
s(x) + 1) + a^(3/2)*sin(x)^2/(cos(x) + 1)^2) + sqrt(2)*(sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(sin(x)/(c
os(x) + 1)))) + sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(sin(x)/(cos(x) + 1)))))/a^(3/2) - 5/2*sqrt(2)*ar
ctan(sqrt(sin(x)/(cos(x) + 1)))/a^(3/2)

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Fricas [B]  time = 0.532199, size = 1299, normalized size = 16.04 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*csc(x))^(3/2),x, algorithm="fricas")

[Out]

[-1/4*(5*sqrt(2)*(cos(x)^2 - (cos(x) + 2)*sin(x) - cos(x) - 2)*sqrt(-a)*log(-(sqrt(2)*sqrt(-a)*sqrt((a*sin(x)
+ a)/sin(x))*sin(x) - a*cos(x))/(sin(x) + 1)) + 4*(cos(x)^2 - (cos(x) + 2)*sin(x) - cos(x) - 2)*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*c
os(x) + a)*sin(x) - a)/(cos(x) + sin(x) + 1)) + 2*(cos(x)^2 + (cos(x) + 1)*sin(x) - 1)*sqrt((a*sin(x) + a)/sin
(x)))/(a^2*cos(x)^2 - a^2*cos(x) - 2*a^2 - (a^2*cos(x) + 2*a^2)*sin(x)), 1/2*(5*sqrt(2)*(cos(x)^2 - (cos(x) +
2)*sin(x) - cos(x) - 2)*sqrt(a)*arctan(sqrt(2)*sqrt(a)*sqrt((a*sin(x) + a)/sin(x))*(cos(x) + 1)/(a*cos(x) + a*
sin(x) + a)) + 4*(cos(x)^2 - (cos(x) + 2)*sin(x) - cos(x) - 2)*sqrt(a)*arctan(-sqrt(a)*sqrt((a*sin(x) + a)/sin
(x))*(cos(x) - sin(x) + 1)/(a*cos(x) + a*sin(x) + a)) - (cos(x)^2 + (cos(x) + 1)*sin(x) - 1)*sqrt((a*sin(x) +
a)/sin(x)))/(a^2*cos(x)^2 - a^2*cos(x) - 2*a^2 - (a^2*cos(x) + 2*a^2)*sin(x))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \csc{\left (x \right )} + a\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*csc(x))**(3/2),x)

[Out]

Integral((a*csc(x) + a)**(-3/2), x)

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Giac [B]  time = 2.9728, size = 410, normalized size = 5.06 \begin{align*} -\frac{1}{2} \,{\left (\frac{5 \, \sqrt{2} \arctan \left (\frac{\sqrt{a \tan \left (\frac{1}{2} \, x\right )}}{\sqrt{a}}\right )}{a^{\frac{3}{2}} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, x\right ) + 1\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^{3} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, x\right ) + 1\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^{3} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, x\right ) + 1\right )} - \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^{3} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, x\right ) + 1\right )} + \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^{3} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, x\right ) + 1\right )} + \frac{\sqrt{2}{\left (\sqrt{a \tan \left (\frac{1}{2} \, x\right )} a \tan \left (\frac{1}{2} \, x\right ) - \sqrt{a \tan \left (\frac{1}{2} \, x\right )} a\right )}}{{\left (a \tan \left (\frac{1}{2} \, x\right ) + a\right )}^{2} a \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, x\right ) + 1\right )}\right )} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*csc(x))^(3/2),x, algorithm="giac")

[Out]

-1/2*(5*sqrt(2)*arctan(sqrt(a*tan(1/2*x))/sqrt(a))/(a^(3/2)*sgn(tan(1/2*x) + 1)) - 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^3*sgn(tan(1/2*x) + 1)
) - 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(a
bs(a)))/(a^3*sgn(tan(1/2*x) + 1)) - (a*sqrt(abs(a)) - abs(a)^(3/2))*log(a*tan(1/2*x) + sqrt(2)*sqrt(a*tan(1/2*
x))*sqrt(abs(a)) + abs(a))/(a^3*sgn(tan(1/2*x) + 1)) + (a*sqrt(abs(a)) - abs(a)^(3/2))*log(a*tan(1/2*x) - sqrt
(2)*sqrt(a*tan(1/2*x))*sqrt(abs(a)) + abs(a))/(a^3*sgn(tan(1/2*x) + 1)) + sqrt(2)*(sqrt(a*tan(1/2*x))*a*tan(1/
2*x) - sqrt(a*tan(1/2*x))*a)/((a*tan(1/2*x) + a)^2*a*sgn(tan(1/2*x) + 1)))*sgn(tan(1/2*x))