3.49 \(\int (a \csc ^2(x))^{3/2} \, dx\)

Optimal. Leaf size=46 \[ -\frac{1}{2} a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cot (x)}{\sqrt{a \csc ^2(x)}}\right )-\frac{1}{2} a \cot (x) \sqrt{a \csc ^2(x)} \]

[Out]

-(a^(3/2)*ArcTanh[(Sqrt[a]*Cot[x])/Sqrt[a*Csc[x]^2]])/2 - (a*Cot[x]*Sqrt[a*Csc[x]^2])/2

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Rubi [A]  time = 0.0219625, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4122, 195, 217, 206} \[ -\frac{1}{2} a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cot (x)}{\sqrt{a \csc ^2(x)}}\right )-\frac{1}{2} a \cot (x) \sqrt{a \csc ^2(x)} \]

Antiderivative was successfully verified.

[In]

Int[(a*Csc[x]^2)^(3/2),x]

[Out]

-(a^(3/2)*ArcTanh[(Sqrt[a]*Cot[x])/Sqrt[a*Csc[x]^2]])/2 - (a*Cot[x]*Sqrt[a*Csc[x]^2])/2

Rule 4122

Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(b*ff)
/f, Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] &&  !IntegerQ[p
]

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

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

Rubi steps

\begin{align*} \int \left (a \csc ^2(x)\right )^{3/2} \, dx &=-\left (a \operatorname{Subst}\left (\int \sqrt{a+a x^2} \, dx,x,\cot (x)\right )\right )\\ &=-\frac{1}{2} a \cot (x) \sqrt{a \csc ^2(x)}-\frac{1}{2} a^2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+a x^2}} \, dx,x,\cot (x)\right )\\ &=-\frac{1}{2} a \cot (x) \sqrt{a \csc ^2(x)}-\frac{1}{2} a^2 \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{\cot (x)}{\sqrt{a \csc ^2(x)}}\right )\\ &=-\frac{1}{2} a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cot (x)}{\sqrt{a \csc ^2(x)}}\right )-\frac{1}{2} a \cot (x) \sqrt{a \csc ^2(x)}\\ \end{align*}

Mathematica [A]  time = 0.0719564, size = 39, normalized size = 0.85 \[ -\frac{1}{2} a \sin (x) \sqrt{a \csc ^2(x)} \left (-\log \left (\sin \left (\frac{x}{2}\right )\right )+\log \left (\cos \left (\frac{x}{2}\right )\right )+\cot (x) \csc (x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Csc[x]^2)^(3/2),x]

[Out]

-(a*Sqrt[a*Csc[x]^2]*(Cot[x]*Csc[x] + Log[Cos[x/2]] - Log[Sin[x/2]])*Sin[x])/2

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Maple [A]  time = 0.078, size = 53, normalized size = 1.2 \begin{align*} -{\frac{\sqrt{4}\sin \left ( x \right ) }{4} \left ( \left ( \cos \left ( x \right ) \right ) ^{2}\ln \left ( -{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }} \right ) +\cos \left ( x \right ) -\ln \left ( -{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }} \right ) \right ) \left ( -{\frac{a}{ \left ( \cos \left ( x \right ) \right ) ^{2}-1}} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*4^(1/2)*(cos(x)^2*ln(-(-1+cos(x))/sin(x))+cos(x)-ln(-(-1+cos(x))/sin(x)))*sin(x)*(-a/(cos(x)^2-1))^(3/2)

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Maxima [B]  time = 1.69075, size = 429, normalized size = 9.33 \begin{align*} -\frac{{\left (4 \, a \cos \left (3 \, x\right ) \sin \left (2 \, x\right ) + 4 \, a \cos \left (x\right ) \sin \left (2 \, x\right ) - 4 \, a \cos \left (2 \, x\right ) \sin \left (x\right ) -{\left (a \cos \left (4 \, x\right )^{2} + 4 \, a \cos \left (2 \, x\right )^{2} + a \sin \left (4 \, x\right )^{2} - 4 \, a \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, a \sin \left (2 \, x\right )^{2} - 2 \,{\left (2 \, a \cos \left (2 \, x\right ) - a\right )} \cos \left (4 \, x\right ) - 4 \, a \cos \left (2 \, x\right ) + a\right )} \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right ) +{\left (a \cos \left (4 \, x\right )^{2} + 4 \, a \cos \left (2 \, x\right )^{2} + a \sin \left (4 \, x\right )^{2} - 4 \, a \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, a \sin \left (2 \, x\right )^{2} - 2 \,{\left (2 \, a \cos \left (2 \, x\right ) - a\right )} \cos \left (4 \, x\right ) - 4 \, a \cos \left (2 \, x\right ) + a\right )} \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right ) + 2 \,{\left (a \sin \left (3 \, x\right ) + a \sin \left (x\right )\right )} \cos \left (4 \, x\right ) - 2 \,{\left (a \cos \left (3 \, x\right ) + a \cos \left (x\right )\right )} \sin \left (4 \, x\right ) - 2 \,{\left (2 \, a \cos \left (2 \, x\right ) - a\right )} \sin \left (3 \, x\right ) + 2 \, a \sin \left (x\right )\right )} \sqrt{-a}}{2 \,{\left (2 \,{\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(4*a*cos(3*x)*sin(2*x) + 4*a*cos(x)*sin(2*x) - 4*a*cos(2*x)*sin(x) - (a*cos(4*x)^2 + 4*a*cos(2*x)^2 + a*s
in(4*x)^2 - 4*a*sin(4*x)*sin(2*x) + 4*a*sin(2*x)^2 - 2*(2*a*cos(2*x) - a)*cos(4*x) - 4*a*cos(2*x) + a)*arctan2
(sin(x), cos(x) + 1) + (a*cos(4*x)^2 + 4*a*cos(2*x)^2 + a*sin(4*x)^2 - 4*a*sin(4*x)*sin(2*x) + 4*a*sin(2*x)^2
- 2*(2*a*cos(2*x) - a)*cos(4*x) - 4*a*cos(2*x) + a)*arctan2(sin(x), cos(x) - 1) + 2*(a*sin(3*x) + a*sin(x))*co
s(4*x) - 2*(a*cos(3*x) + a*cos(x))*sin(4*x) - 2*(2*a*cos(2*x) - a)*sin(3*x) + 2*a*sin(x))*sqrt(-a)/(2*(2*cos(2
*x) - 1)*cos(4*x) - cos(4*x)^2 - 4*cos(2*x)^2 - sin(4*x)^2 + 4*sin(4*x)*sin(2*x) - 4*sin(2*x)^2 + 4*cos(2*x) -
1)

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Fricas [A]  time = 0.487157, size = 136, normalized size = 2.96 \begin{align*} -\frac{{\left (2 \, a \cos \left (x\right ) +{\left (a \cos \left (x\right )^{2} - a\right )} \log \left (-\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right )\right )} \sqrt{-\frac{a}{\cos \left (x\right )^{2} - 1}}}{4 \, \sin \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*a*cos(x) + (a*cos(x)^2 - a)*log(-(cos(x) - 1)/(cos(x) + 1)))*sqrt(-a/(cos(x)^2 - 1))/sin(x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a*csc(x)**2)**(3/2), x)

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Giac [B]  time = 1.32157, size = 97, normalized size = 2.11 \begin{align*} \frac{1}{8} \,{\left (2 \, \log \left (-\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right ) \mathrm{sgn}\left (\sin \left (x\right )\right ) - \frac{{\left (\frac{2 \,{\left (\cos \left (x\right ) - 1\right )} \mathrm{sgn}\left (\sin \left (x\right )\right )}{\cos \left (x\right ) + 1} - \mathrm{sgn}\left (\sin \left (x\right )\right )\right )}{\left (\cos \left (x\right ) + 1\right )}}{\cos \left (x\right ) - 1} - \frac{{\left (\cos \left (x\right ) - 1\right )} \mathrm{sgn}\left (\sin \left (x\right )\right )}{\cos \left (x\right ) + 1}\right )} a^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(2*log(-(cos(x) - 1)/(cos(x) + 1))*sgn(sin(x)) - (2*(cos(x) - 1)*sgn(sin(x))/(cos(x) + 1) - sgn(sin(x)))*(
cos(x) + 1)/(cos(x) - 1) - (cos(x) - 1)*sgn(sin(x))/(cos(x) + 1))*a^(3/2)