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

Optimal. Leaf size=74 \[ -\frac{16 \cot (x)}{35 a^3 \sqrt{a \csc ^2(x)}}-\frac{8 \cot (x)}{35 a^2 \left (a \csc ^2(x)\right )^{3/2}}-\frac{6 \cot (x)}{35 a \left (a \csc ^2(x)\right )^{5/2}}-\frac{\cot (x)}{7 \left (a \csc ^2(x)\right )^{7/2}} \]

[Out]

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

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Rubi [A]  time = 0.0357382, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4122, 192, 191} \[ -\frac{16 \cot (x)}{35 a^3 \sqrt{a \csc ^2(x)}}-\frac{8 \cot (x)}{35 a^2 \left (a \csc ^2(x)\right )^{3/2}}-\frac{6 \cot (x)}{35 a \left (a \csc ^2(x)\right )^{5/2}}-\frac{\cot (x)}{7 \left (a \csc ^2(x)\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0446459, size = 42, normalized size = 0.57 \[ \frac{\sin (x) (-1225 \cos (x)+245 \cos (3 x)-49 \cos (5 x)+5 \cos (7 x)) \sqrt{a \csc ^2(x)}}{2240 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-1225*Cos[x] + 245*Cos[3*x] - 49*Cos[5*x] + 5*Cos[7*x])*Sqrt[a*Csc[x]^2]*Sin[x])/(2240*a^4)

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Maple [A]  time = 0.076, size = 45, normalized size = 0.6 \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 ( -{\frac{a}{ \left ( \cos \left ( x \right ) \right ) ^{2}-1}} \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*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/(-a/(cos(x)^2-1))^(7/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \csc \left (x\right )^{2}\right )^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*csc(x)^2)^(-7/2), x)

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Fricas [A]  time = 0.48903, size = 127, normalized size = 1.72 \begin{align*} \frac{{\left (5 \, \cos \left (x\right )^{7} - 21 \, \cos \left (x\right )^{5} + 35 \, \cos \left (x\right )^{3} - 35 \, \cos \left (x\right )\right )} \sqrt{-\frac{a}{\cos \left (x\right )^{2} - 1}} \sin \left (x\right )}{35 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csc(x)^2)^(7/2),x, algorithm="fricas")

[Out]

1/35*(5*cos(x)^7 - 21*cos(x)^5 + 35*cos(x)^3 - 35*cos(x))*sqrt(-a/(cos(x)^2 - 1))*sin(x)/a^4

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

[Out]

Timed out

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Giac [A]  time = 1.48023, size = 93, normalized size = 1.26 \begin{align*} -\frac{32 \,{\left (\frac{35 \, \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, x\right )\right ) \tan \left (\frac{1}{2} \, x\right )^{6} + 21 \, \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, x\right )\right ) \tan \left (\frac{1}{2} \, x\right )^{4} + 7 \, \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, x\right )\right ) \tan \left (\frac{1}{2} \, x\right )^{2} + \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, x\right )\right )}{{\left (\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right )}^{7}} - \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, x\right )\right )\right )}}{35 \, a^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csc(x)^2)^(7/2),x, algorithm="giac")

[Out]

-32/35*((35*sgn(tan(1/2*x))*tan(1/2*x)^6 + 21*sgn(tan(1/2*x))*tan(1/2*x)^4 + 7*sgn(tan(1/2*x))*tan(1/2*x)^2 +
sgn(tan(1/2*x)))/(tan(1/2*x)^2 + 1)^7 - sgn(tan(1/2*x)))/a^(7/2)