3.40 \(\int \csc ^2(x)^{5/2} \, dx\)

Optimal. Leaf size=36 \[ -\frac {1}{4} \cot (x) \csc ^2(x)^{3/2}-\frac {3}{8} \cot (x) \sqrt {\csc ^2(x)}-\frac {3}{8} \sinh ^{-1}(\cot (x)) \]

[Out]

-3/8*arcsinh(cot(x))-1/4*cot(x)*(csc(x)^2)^(3/2)-3/8*cot(x)*(csc(x)^2)^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4122, 195, 215} \[ -\frac {1}{4} \cot (x) \csc ^2(x)^{3/2}-\frac {3}{8} \cot (x) \sqrt {\csc ^2(x)}-\frac {3}{8} \sinh ^{-1}(\cot (x)) \]

Antiderivative was successfully verified.

[In]

Int[(Csc[x]^2)^(5/2),x]

[Out]

(-3*ArcSinh[Cot[x]])/8 - (3*Cot[x]*Sqrt[Csc[x]^2])/8 - (Cot[x]*(Csc[x]^2)^(3/2))/4

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)^{5/2} \, dx &=-\operatorname {Subst}\left (\int \left (1+x^2\right )^{3/2} \, dx,x,\cot (x)\right )\\ &=-\frac {1}{4} \cot (x) \csc ^2(x)^{3/2}-\frac {3}{4} \operatorname {Subst}\left (\int \sqrt {1+x^2} \, dx,x,\cot (x)\right )\\ &=-\frac {3}{8} \cot (x) \sqrt {\csc ^2(x)}-\frac {1}{4} \cot (x) \csc ^2(x)^{3/2}-\frac {3}{8} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\cot (x)\right )\\ &=-\frac {3}{8} \sinh ^{-1}(\cot (x))-\frac {3}{8} \cot (x) \sqrt {\csc ^2(x)}-\frac {1}{4} \cot (x) \csc ^2(x)^{3/2}\\ \end {align*}

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Mathematica [A]  time = 0.26, size = 72, normalized size = 2.00 \[ \frac {1}{64} \sin (x) \sqrt {\csc ^2(x)} \left (-\csc ^4\left (\frac {x}{2}\right )-6 \csc ^2\left (\frac {x}{2}\right )+\sec ^4\left (\frac {x}{2}\right )+6 \sec ^2\left (\frac {x}{2}\right )+24 \left (\log \left (\sin \left (\frac {x}{2}\right )\right )-\log \left (\cos \left (\frac {x}{2}\right )\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(Csc[x]^2)^(5/2),x]

[Out]

(Sqrt[Csc[x]^2]*(-6*Csc[x/2]^2 - Csc[x/2]^4 + 24*(-Log[Cos[x/2]] + Log[Sin[x/2]]) + 6*Sec[x/2]^2 + Sec[x/2]^4)
*Sin[x])/64

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fricas [B]  time = 0.66, size = 69, normalized size = 1.92 \[ \frac {6 \, \cos \relax (x)^{3} - 3 \, {\left (\cos \relax (x)^{4} - 2 \, \cos \relax (x)^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + 3 \, {\left (\cos \relax (x)^{4} - 2 \, \cos \relax (x)^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) - 10 \, \cos \relax (x)}{16 \, {\left (\cos \relax (x)^{4} - 2 \, \cos \relax (x)^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((csc(x)^2)^(5/2),x, algorithm="fricas")

[Out]

1/16*(6*cos(x)^3 - 3*(cos(x)^4 - 2*cos(x)^2 + 1)*log(1/2*cos(x) + 1/2) + 3*(cos(x)^4 - 2*cos(x)^2 + 1)*log(-1/
2*cos(x) + 1/2) - 10*cos(x))/(cos(x)^4 - 2*cos(x)^2 + 1)

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giac [B]  time = 0.51, size = 100, normalized size = 2.78 \[ -\frac {{\left (\cos \relax (x) - 1\right )} \mathrm {sgn}\left (\sin \relax (x)\right )}{8 \, {\left (\cos \relax (x) + 1\right )}} + \frac {{\left (\cos \relax (x) - 1\right )}^{2} \mathrm {sgn}\left (\sin \relax (x)\right )}{64 \, {\left (\cos \relax (x) + 1\right )}^{2}} + \frac {{\left (\frac {8 \, {\left (\cos \relax (x) - 1\right )}}{\cos \relax (x) + 1} - \frac {18 \, {\left (\cos \relax (x) - 1\right )}^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} - 1\right )} {\left (\cos \relax (x) + 1\right )}^{2}}{64 \, {\left (\cos \relax (x) - 1\right )}^{2} \mathrm {sgn}\left (\sin \relax (x)\right )} + \frac {3 \, \log \left (-\frac {\cos \relax (x) - 1}{\cos \relax (x) + 1}\right )}{16 \, \mathrm {sgn}\left (\sin \relax (x)\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((csc(x)^2)^(5/2),x, algorithm="giac")

[Out]

-1/8*(cos(x) - 1)*sgn(sin(x))/(cos(x) + 1) + 1/64*(cos(x) - 1)^2*sgn(sin(x))/(cos(x) + 1)^2 + 1/64*(8*(cos(x)
- 1)/(cos(x) + 1) - 18*(cos(x) - 1)^2/(cos(x) + 1)^2 - 1)*(cos(x) + 1)^2/((cos(x) - 1)^2*sgn(sin(x))) + 3/16*l
og(-(cos(x) - 1)/(cos(x) + 1))/sgn(sin(x))

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maple [B]  time = 0.44, size = 78, normalized size = 2.17 \[ \frac {\left (3 \left (\cos ^{4}\relax (x )\right ) \ln \left (-\frac {-1+\cos \relax (x )}{\sin \relax (x )}\right )+3 \left (\cos ^{3}\relax (x )\right )-6 \left (\cos ^{2}\relax (x )\right ) \ln \left (-\frac {-1+\cos \relax (x )}{\sin \relax (x )}\right )-5 \cos \relax (x )+3 \ln \left (-\frac {-1+\cos \relax (x )}{\sin \relax (x )}\right )\right ) \sin \relax (x ) \left (-\frac {1}{-1+\cos ^{2}\relax (x )}\right )^{\frac {5}{2}} \sqrt {4}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((csc(x)^2)^(5/2),x)

[Out]

1/16*(3*cos(x)^4*ln(-(-1+cos(x))/sin(x))+3*cos(x)^3-6*cos(x)^2*ln(-(-1+cos(x))/sin(x))-5*cos(x)+3*ln(-(-1+cos(
x))/sin(x)))*sin(x)*(-1/(-1+cos(x)^2))^(5/2)*4^(1/2)

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maxima [B]  time = 0.60, size = 869, normalized size = 24.14 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((csc(x)^2)^(5/2),x, algorithm="maxima")

[Out]

-1/16*(4*(3*cos(7*x) - 11*cos(5*x) - 11*cos(3*x) + 3*cos(x))*cos(8*x) - 12*(4*cos(6*x) - 6*cos(4*x) + 4*cos(2*
x) - 1)*cos(7*x) + 16*(11*cos(5*x) + 11*cos(3*x) - 3*cos(x))*cos(6*x) - 44*(6*cos(4*x) - 4*cos(2*x) + 1)*cos(5
*x) - 24*(11*cos(3*x) - 3*cos(x))*cos(4*x) + 44*(4*cos(2*x) - 1)*cos(3*x) - 48*cos(2*x)*cos(x) + 3*(2*(4*cos(6
*x) - 6*cos(4*x) + 4*cos(2*x) - 1)*cos(8*x) - cos(8*x)^2 + 8*(6*cos(4*x) - 4*cos(2*x) + 1)*cos(6*x) - 16*cos(6
*x)^2 + 12*(4*cos(2*x) - 1)*cos(4*x) - 36*cos(4*x)^2 - 16*cos(2*x)^2 + 4*(2*sin(6*x) - 3*sin(4*x) + 2*sin(2*x)
)*sin(8*x) - sin(8*x)^2 + 16*(3*sin(4*x) - 2*sin(2*x))*sin(6*x) - 16*sin(6*x)^2 - 36*sin(4*x)^2 + 48*sin(4*x)*
sin(2*x) - 16*sin(2*x)^2 + 8*cos(2*x) - 1)*log(cos(x)^2 + sin(x)^2 + 2*cos(x) + 1) - 3*(2*(4*cos(6*x) - 6*cos(
4*x) + 4*cos(2*x) - 1)*cos(8*x) - cos(8*x)^2 + 8*(6*cos(4*x) - 4*cos(2*x) + 1)*cos(6*x) - 16*cos(6*x)^2 + 12*(
4*cos(2*x) - 1)*cos(4*x) - 36*cos(4*x)^2 - 16*cos(2*x)^2 + 4*(2*sin(6*x) - 3*sin(4*x) + 2*sin(2*x))*sin(8*x) -
sin(8*x)^2 + 16*(3*sin(4*x) - 2*sin(2*x))*sin(6*x) - 16*sin(6*x)^2 - 36*sin(4*x)^2 + 48*sin(4*x)*sin(2*x) - 1
6*sin(2*x)^2 + 8*cos(2*x) - 1)*log(cos(x)^2 + sin(x)^2 - 2*cos(x) + 1) + 4*(3*sin(7*x) - 11*sin(5*x) - 11*sin(
3*x) + 3*sin(x))*sin(8*x) - 24*(2*sin(6*x) - 3*sin(4*x) + 2*sin(2*x))*sin(7*x) + 16*(11*sin(5*x) + 11*sin(3*x)
- 3*sin(x))*sin(6*x) - 88*(3*sin(4*x) - 2*sin(2*x))*sin(5*x) - 24*(11*sin(3*x) - 3*sin(x))*sin(4*x) + 176*sin
(3*x)*sin(2*x) - 48*sin(2*x)*sin(x) + 12*cos(x))/(2*(4*cos(6*x) - 6*cos(4*x) + 4*cos(2*x) - 1)*cos(8*x) - cos(
8*x)^2 + 8*(6*cos(4*x) - 4*cos(2*x) + 1)*cos(6*x) - 16*cos(6*x)^2 + 12*(4*cos(2*x) - 1)*cos(4*x) - 36*cos(4*x)
^2 - 16*cos(2*x)^2 + 4*(2*sin(6*x) - 3*sin(4*x) + 2*sin(2*x))*sin(8*x) - sin(8*x)^2 + 16*(3*sin(4*x) - 2*sin(2
*x))*sin(6*x) - 16*sin(6*x)^2 - 36*sin(4*x)^2 + 48*sin(4*x)*sin(2*x) - 16*sin(2*x)^2 + 8*cos(2*x) - 1)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int {\left (\frac {1}{{\sin \relax (x)}^2}\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/sin(x)^2)^(5/2),x)

[Out]

int((1/sin(x)^2)^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\csc ^{2}{\relax (x )}\right )^{\frac {5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((csc(x)**2)**(5/2),x)

[Out]

Integral((csc(x)**2)**(5/2), x)

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