3.45 \(\int \frac{1}{\csc ^2(x)^{5/2}} \, dx\)

Optimal. Leaf size=43 \[ -\frac{8 \cot (x)}{15 \sqrt{\csc ^2(x)}}-\frac{4 \cot (x)}{15 \csc ^2(x)^{3/2}}-\frac{\cot (x)}{5 \csc ^2(x)^{5/2}} \]

[Out]

-Cot[x]/(5*(Csc[x]^2)^(5/2)) - (4*Cot[x])/(15*(Csc[x]^2)^(3/2)) - (8*Cot[x])/(15*Sqrt[Csc[x]^2])

________________________________________________________________________________________

Rubi [A]  time = 0.0147414, antiderivative size = 43, 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, 192, 191} \[ -\frac{8 \cot (x)}{15 \sqrt{\csc ^2(x)}}-\frac{4 \cot (x)}{15 \csc ^2(x)^{3/2}}-\frac{\cot (x)}{5 \csc ^2(x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Cot[x]/(5*(Csc[x]^2)^(5/2)) - (4*Cot[x])/(15*(Csc[x]^2)^(3/2)) - (8*Cot[x])/(15*Sqrt[Csc[x]^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 192

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p + 1
], 0] && NeQ[p, -1]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \frac{1}{\csc ^2(x)^{5/2}} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right )^{7/2}} \, dx,x,\cot (x)\right )\\ &=-\frac{\cot (x)}{5 \csc ^2(x)^{5/2}}-\frac{4}{5} \operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right )^{5/2}} \, dx,x,\cot (x)\right )\\ &=-\frac{\cot (x)}{5 \csc ^2(x)^{5/2}}-\frac{4 \cot (x)}{15 \csc ^2(x)^{3/2}}-\frac{8}{15} \operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right )^{3/2}} \, dx,x,\cot (x)\right )\\ &=-\frac{\cot (x)}{5 \csc ^2(x)^{5/2}}-\frac{4 \cot (x)}{15 \csc ^2(x)^{3/2}}-\frac{8 \cot (x)}{15 \sqrt{\csc ^2(x)}}\\ \end{align*}

Mathematica [A]  time = 0.0269639, size = 31, normalized size = 0.72 \[ -\frac{(150 \cos (x)-25 \cos (3 x)+3 \cos (5 x)) \csc (x)}{240 \sqrt{\csc ^2(x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((150*Cos[x] - 25*Cos[3*x] + 3*Cos[5*x])*Csc[x])/(240*Sqrt[Csc[x]^2])

________________________________________________________________________________________

Maple [A]  time = 0.077, size = 38, normalized size = 0.9 \begin{align*}{\frac{\sqrt{4}\sin \left ( x \right ) \left ( 3\, \left ( \cos \left ( x \right ) \right ) ^{2}-9\,\cos \left ( x \right ) +8 \right ) }{30\, \left ( -1+\cos \left ( x \right ) \right ) ^{3}} \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(1/(csc(x)^2)^(5/2),x)

[Out]

1/30*4^(1/2)*sin(x)*(3*cos(x)^2-9*cos(x)+8)/(-1+cos(x))^3/(-1/(cos(x)^2-1))^(5/2)

________________________________________________________________________________________

Maxima [A]  time = 1.66542, size = 23, normalized size = 0.53 \begin{align*} -\frac{1}{80} \, \cos \left (5 \, x\right ) + \frac{5}{48} \, \cos \left (3 \, x\right ) - \frac{5}{8} \, \cos \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(csc(x)^2)^(5/2),x, algorithm="maxima")

[Out]

-1/80*cos(5*x) + 5/48*cos(3*x) - 5/8*cos(x)

________________________________________________________________________________________

Fricas [A]  time = 0.471179, size = 53, normalized size = 1.23 \begin{align*} -\frac{1}{5} \, \cos \left (x\right )^{5} + \frac{2}{3} \, \cos \left (x\right )^{3} - \cos \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(csc(x)^2)^(5/2),x, algorithm="fricas")

[Out]

-1/5*cos(x)^5 + 2/3*cos(x)^3 - cos(x)

________________________________________________________________________________________

Sympy [A]  time = 19.7235, size = 46, normalized size = 1.07 \begin{align*} - \frac{8 \cot ^{5}{\left (x \right )}}{15 \left (\csc ^{2}{\left (x \right )}\right )^{\frac{5}{2}}} - \frac{4 \cot ^{3}{\left (x \right )}}{3 \left (\csc ^{2}{\left (x \right )}\right )^{\frac{5}{2}}} - \frac{\cot{\left (x \right )}}{\left (\csc ^{2}{\left (x \right )}\right )^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(csc(x)**2)**(5/2),x)

[Out]

-8*cot(x)**5/(15*(csc(x)**2)**(5/2)) - 4*cot(x)**3/(3*(csc(x)**2)**(5/2)) - cot(x)/(csc(x)**2)**(5/2)

________________________________________________________________________________________

Giac [A]  time = 1.30193, size = 82, normalized size = 1.91 \begin{align*} -\frac{16 \,{\left (\frac{5 \,{\left (\cos \left (x\right ) - 1\right )} \mathrm{sgn}\left (\sin \left (x\right )\right )}{\cos \left (x\right ) + 1} - \frac{10 \,{\left (\cos \left (x\right ) - 1\right )}^{2} \mathrm{sgn}\left (\sin \left (x\right )\right )}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \mathrm{sgn}\left (\sin \left (x\right )\right )\right )}}{15 \,{\left (\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} - 1\right )}^{5}} + \frac{16}{15} \, \mathrm{sgn}\left (\sin \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(csc(x)^2)^(5/2),x, algorithm="giac")

[Out]

-16/15*(5*(cos(x) - 1)*sgn(sin(x))/(cos(x) + 1) - 10*(cos(x) - 1)^2*sgn(sin(x))/(cos(x) + 1)^2 - sgn(sin(x)))/
((cos(x) - 1)/(cos(x) + 1) - 1)^5 + 16/15*sgn(sin(x))