3.46 \(\int \frac{1}{\csc ^2(x)^{7/2}} \, dx\)

Optimal. Leaf size=57 \[ -\frac{16 \cot (x)}{35 \sqrt{\csc ^2(x)}}-\frac{8 \cot (x)}{35 \csc ^2(x)^{3/2}}-\frac{6 \cot (x)}{35 \csc ^2(x)^{5/2}}-\frac{\cot (x)}{7 \csc ^2(x)^{7/2}} \]

[Out]

-Cot[x]/(7*(Csc[x]^2)^(7/2)) - (6*Cot[x])/(35*(Csc[x]^2)^(5/2)) - (8*Cot[x])/(35*(Csc[x]^2)^(3/2)) - (16*Cot[x
])/(35*Sqrt[Csc[x]^2])

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Rubi [A]  time = 0.0188562, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {4122, 192, 191} \[ -\frac{16 \cot (x)}{35 \sqrt{\csc ^2(x)}}-\frac{8 \cot (x)}{35 \csc ^2(x)^{3/2}}-\frac{6 \cot (x)}{35 \csc ^2(x)^{5/2}}-\frac{\cot (x)}{7 \csc ^2(x)^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[(Csc[x]^2)^(-7/2),x]

[Out]

-Cot[x]/(7*(Csc[x]^2)^(7/2)) - (6*Cot[x])/(35*(Csc[x]^2)^(5/2)) - (8*Cot[x])/(35*(Csc[x]^2)^(3/2)) - (16*Cot[x
])/(35*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)^{7/2}} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right )^{9/2}} \, dx,x,\cot (x)\right )\\ &=-\frac{\cot (x)}{7 \csc ^2(x)^{7/2}}-\frac{6}{7} \operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right )^{7/2}} \, dx,x,\cot (x)\right )\\ &=-\frac{\cot (x)}{7 \csc ^2(x)^{7/2}}-\frac{6 \cot (x)}{35 \csc ^2(x)^{5/2}}-\frac{24}{35} \operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right )^{5/2}} \, dx,x,\cot (x)\right )\\ &=-\frac{\cot (x)}{7 \csc ^2(x)^{7/2}}-\frac{6 \cot (x)}{35 \csc ^2(x)^{5/2}}-\frac{8 \cot (x)}{35 \csc ^2(x)^{3/2}}-\frac{16}{35} \operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right )^{3/2}} \, dx,x,\cot (x)\right )\\ &=-\frac{\cot (x)}{7 \csc ^2(x)^{7/2}}-\frac{6 \cot (x)}{35 \csc ^2(x)^{5/2}}-\frac{8 \cot (x)}{35 \csc ^2(x)^{3/2}}-\frac{16 \cot (x)}{35 \sqrt{\csc ^2(x)}}\\ \end{align*}

Mathematica [A]  time = 0.0409032, size = 37, normalized size = 0.65 \[ \frac{(-1225 \cos (x)+245 \cos (3 x)-49 \cos (5 x)+5 \cos (7 x)) \csc (x)}{2240 \sqrt{\csc ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(Csc[x]^2)^(-7/2),x]

[Out]

((-1225*Cos[x] + 245*Cos[3*x] - 49*Cos[5*x] + 5*Cos[7*x])*Csc[x])/(2240*Sqrt[Csc[x]^2])

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Maple [A]  time = 0.08, size = 44, normalized size = 0.8 \begin{align*}{\frac{\sqrt{4}\sin \left ( x \right ) \left ( 5\, \left ( \cos \left ( x \right ) \right ) ^{3}-20\, \left ( \cos \left ( x \right ) \right ) ^{2}+29\,\cos \left ( x \right ) -16 \right ) }{70\, \left ( -1+\cos \left ( x \right ) \right ) ^{4}} \left ( - \left ( \left ( \cos \left ( x \right ) \right ) ^{2}-1 \right ) ^{-1} \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(csc(x)^2)^(7/2),x)

[Out]

1/70*4^(1/2)*sin(x)*(5*cos(x)^3-20*cos(x)^2+29*cos(x)-16)/(-1+cos(x))^4/(-1/(cos(x)^2-1))^(7/2)

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Maxima [A]  time = 1.63919, size = 31, normalized size = 0.54 \begin{align*} \frac{1}{448} \, \cos \left (7 \, x\right ) - \frac{7}{320} \, \cos \left (5 \, x\right ) + \frac{7}{64} \, \cos \left (3 \, x\right ) - \frac{35}{64} \, \cos \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(csc(x)^2)^(7/2),x, algorithm="maxima")

[Out]

1/448*cos(7*x) - 7/320*cos(5*x) + 7/64*cos(3*x) - 35/64*cos(x)

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Fricas [A]  time = 0.472471, size = 66, normalized size = 1.16 \begin{align*} \frac{1}{7} \, \cos \left (x\right )^{7} - \frac{3}{5} \, \cos \left (x\right )^{5} + \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)^(7/2),x, algorithm="fricas")

[Out]

1/7*cos(x)^7 - 3/5*cos(x)^5 + cos(x)^3 - cos(x)

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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(1/(csc(x)**2)**(7/2),x)

[Out]

Timed out

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Giac [A]  time = 1.23918, size = 105, normalized size = 1.84 \begin{align*} -\frac{32 \,{\left (\frac{7 \,{\left (\cos \left (x\right ) - 1\right )} \mathrm{sgn}\left (\sin \left (x\right )\right )}{\cos \left (x\right ) + 1} - \frac{21 \,{\left (\cos \left (x\right ) - 1\right )}^{2} \mathrm{sgn}\left (\sin \left (x\right )\right )}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{35 \,{\left (\cos \left (x\right ) - 1\right )}^{3} \mathrm{sgn}\left (\sin \left (x\right )\right )}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - \mathrm{sgn}\left (\sin \left (x\right )\right )\right )}}{35 \,{\left (\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} - 1\right )}^{7}} + \frac{32}{35} \, \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)^(7/2),x, algorithm="giac")

[Out]

-32/35*(7*(cos(x) - 1)*sgn(sin(x))/(cos(x) + 1) - 21*(cos(x) - 1)^2*sgn(sin(x))/(cos(x) + 1)^2 + 35*(cos(x) -
1)^3*sgn(sin(x))/(cos(x) + 1)^3 - sgn(sin(x)))/((cos(x) - 1)/(cos(x) + 1) - 1)^7 + 32/35*sgn(sin(x))