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

Optimal. Leaf size=22 \[ -\frac {1}{2} \cot (x) \sqrt {\csc ^2(x)}-\frac {1}{2} \sinh ^{-1}(\cot (x)) \]

[Out]

-1/2*arcsinh(cot(x))-1/2*cot(x)*(csc(x)^2)^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4122, 195, 215} \[ -\frac {1}{2} \cot (x) \sqrt {\csc ^2(x)}-\frac {1}{2} \sinh ^{-1}(\cot (x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-ArcSinh[Cot[x]]/2 - (Cot[x]*Sqrt[Csc[x]^2])/2

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 215

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

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
]

Rubi steps

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

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Mathematica [B]  time = 0.09, size = 51, normalized size = 2.32 \[ \frac {1}{8} \sin (x) \sqrt {\csc ^2(x)} \left (-\csc ^2\left (\frac {x}{2}\right )+\sec ^2\left (\frac {x}{2}\right )+4 \log \left (\sin \left (\frac {x}{2}\right )\right )-4 \log \left (\cos \left (\frac {x}{2}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[Csc[x]^2]*(-Csc[x/2]^2 - 4*Log[Cos[x/2]] + 4*Log[Sin[x/2]] + Sec[x/2]^2)*Sin[x])/8

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fricas [B]  time = 0.50, size = 44, normalized size = 2.00 \[ -\frac {{\left (\cos \relax (x)^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) - {\left (\cos \relax (x)^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) - 2 \, \cos \relax (x)}{4 \, {\left (\cos \relax (x)^{2} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.49, size = 69, normalized size = 3.14 \[ -\frac {{\left (\frac {2 \, {\left (\cos \relax (x) - 1\right )}}{\cos \relax (x) + 1} - 1\right )} {\left (\cos \relax (x) + 1\right )}}{8 \, {\left (\cos \relax (x) - 1\right )} \mathrm {sgn}\left (\sin \relax (x)\right )} + \frac {\log \left (-\frac {\cos \relax (x) - 1}{\cos \relax (x) + 1}\right )}{4 \, \mathrm {sgn}\left (\sin \relax (x)\right )} - \frac {\cos \relax (x) - 1}{8 \, {\left (\cos \relax (x) + 1\right )} \mathrm {sgn}\left (\sin \relax (x)\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(2*(cos(x) - 1)/(cos(x) + 1) - 1)*(cos(x) + 1)/((cos(x) - 1)*sgn(sin(x))) + 1/4*log(-(cos(x) - 1)/(cos(x)
+ 1))/sgn(sin(x)) - 1/8*(cos(x) - 1)/((cos(x) + 1)*sgn(sin(x)))

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maple [B]  time = 0.46, size = 52, normalized size = 2.36 \[ -\frac {\left (\left (\cos ^{2}\relax (x )\right ) \ln \left (-\frac {-1+\cos \relax (x )}{\sin \relax (x )}\right )-\ln \left (-\frac {-1+\cos \relax (x )}{\sin \relax (x )}\right )+\cos \relax (x )\right ) \sin \relax (x ) \left (-\frac {1}{-1+\cos ^{2}\relax (x )}\right )^{\frac {3}{2}} \sqrt {4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.57, size = 300, normalized size = 13.64 \[ -\frac {4 \, {\left (\cos \left (3 \, x\right ) + \cos \relax (x)\right )} \cos \left (4 \, x\right ) - 4 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (3 \, x\right ) - 8 \, \cos \left (2 \, x\right ) \cos \relax (x) + {\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )} \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right ) - {\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )} \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right ) + 4 \, {\left (\sin \left (3 \, x\right ) + \sin \relax (x)\right )} \sin \left (4 \, x\right ) - 8 \, \sin \left (3 \, x\right ) \sin \left (2 \, x\right ) - 8 \, \sin \left (2 \, x\right ) \sin \relax (x) + 4 \, \cos \relax (x)}{4 \, {\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(4*(cos(3*x) + cos(x))*cos(4*x) - 4*(2*cos(2*x) - 1)*cos(3*x) - 8*cos(2*x)*cos(x) + (2*(2*cos(2*x) - 1)*c
os(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)*log(co
s(x)^2 + sin(x)^2 + 2*cos(x) + 1) - (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)*log(cos(x)^2 + sin(x)^2 - 2*cos(x) + 1) + 4*(sin(3*x) + sin
(x))*sin(4*x) - 8*sin(3*x)*sin(2*x) - 8*sin(2*x)*sin(x) + 4*cos(x))/(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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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \[ \int {\left (\frac {1}{{\sin \relax (x)}^2}\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/sin(x)^2)^(3/2),x)

[Out]

int((1/sin(x)^2)^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\csc ^{2}{\relax (x )}\right )^{\frac {3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((csc(x)**2)**(3/2), x)

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