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

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Rubi [A]  time = 0.0123468, antiderivative size = 36, normalized size of antiderivative = 1., 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 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 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]

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*}

Mathematica [A]  time = 0.243038, size = 72, normalized size = 2. \[ \frac{1}{64} \sin (x) \sqrt{\csc ^2(x)} \left (-\csc ^4\left (\frac{x}{2}\right )-6 \csc ^2\left (\frac{x}{2}\right )+\sec ^4\left (\frac{x}{2}\right )+6 \sec ^2\left (\frac{x}{2}\right )+24 \left (\log \left (\sin \left (\frac{x}{2}\right )\right )-\log \left (\cos \left (\frac{x}{2}\right )\right )\right )\right ) \]

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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Maple [B]  time = 0.17, size = 78, normalized size = 2.2 \begin{align*}{\frac{\sqrt{4}\sin \left ( x \right ) }{16} \left ( 3\, \left ( \cos \left ( x \right ) \right ) ^{4}\ln \left ( -{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }} \right ) +3\, \left ( \cos \left ( x \right ) \right ) ^{3}-6\, \left ( \cos \left ( x \right ) \right ) ^{2}\ln \left ( -{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }} \right ) -5\,\cos \left ( x \right ) +3\,\ln \left ( -{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }} \right ) \right ) \left ( - \left ( \left ( \cos \left ( x \right ) \right ) ^{2}-1 \right ) ^{-1} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*4^(1/2)*(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/(cos(x)^2-1))^(5/2)

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Maxima [B]  time = 1.76746, size = 1173, normalized size = 32.58 \begin{align*} \text{result too large to display} \end{align*}

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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Fricas [B]  time = 0.481315, size = 227, normalized size = 6.31 \begin{align*} \frac{6 \, \cos \left (x\right )^{3} - 3 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + 3 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) - 10 \, \cos \left (x\right )}{16 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )}} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.23869, size = 135, normalized size = 3.75 \begin{align*} -\frac{{\left (\cos \left (x\right ) - 1\right )} \mathrm{sgn}\left (\sin \left (x\right )\right )}{8 \,{\left (\cos \left (x\right ) + 1\right )}} + \frac{{\left (\cos \left (x\right ) - 1\right )}^{2} \mathrm{sgn}\left (\sin \left (x\right )\right )}{64 \,{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{{\left (\frac{8 \,{\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - \frac{18 \,{\left (\cos \left (x\right ) - 1\right )}^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - 1\right )}{\left (\cos \left (x\right ) + 1\right )}^{2}}{64 \,{\left (\cos \left (x\right ) - 1\right )}^{2} \mathrm{sgn}\left (\sin \left (x\right )\right )} + \frac{3 \, \log \left (-\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right )}{16 \, \mathrm{sgn}\left (\sin \left (x\right )\right )} \end{align*}

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))