3.62 \(\int (a \csc ^4(x))^{5/2} \, dx\)

Optimal. Leaf size=118 \[ -\frac{1}{9} a^2 \cos ^2(x) \cot ^7(x) \sqrt{a \csc ^4(x)}-\frac{4}{7} a^2 \cos ^2(x) \cot ^5(x) \sqrt{a \csc ^4(x)}-\frac{6}{5} a^2 \cos ^2(x) \cot ^3(x) \sqrt{a \csc ^4(x)}-\frac{4}{3} a^2 \cos ^2(x) \cot (x) \sqrt{a \csc ^4(x)}-a^2 \sin (x) \cos (x) \sqrt{a \csc ^4(x)} \]

[Out]

(-4*a^2*Cos[x]^2*Cot[x]*Sqrt[a*Csc[x]^4])/3 - (6*a^2*Cos[x]^2*Cot[x]^3*Sqrt[a*Csc[x]^4])/5 - (4*a^2*Cos[x]^2*C
ot[x]^5*Sqrt[a*Csc[x]^4])/7 - (a^2*Cos[x]^2*Cot[x]^7*Sqrt[a*Csc[x]^4])/9 - a^2*Cos[x]*Sqrt[a*Csc[x]^4]*Sin[x]

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Rubi [A]  time = 0.027743, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4123, 3767} \[ -\frac{1}{9} a^2 \cos ^2(x) \cot ^7(x) \sqrt{a \csc ^4(x)}-\frac{4}{7} a^2 \cos ^2(x) \cot ^5(x) \sqrt{a \csc ^4(x)}-\frac{6}{5} a^2 \cos ^2(x) \cot ^3(x) \sqrt{a \csc ^4(x)}-\frac{4}{3} a^2 \cos ^2(x) \cot (x) \sqrt{a \csc ^4(x)}-a^2 \sin (x) \cos (x) \sqrt{a \csc ^4(x)} \]

Antiderivative was successfully verified.

[In]

Int[(a*Csc[x]^4)^(5/2),x]

[Out]

(-4*a^2*Cos[x]^2*Cot[x]*Sqrt[a*Csc[x]^4])/3 - (6*a^2*Cos[x]^2*Cot[x]^3*Sqrt[a*Csc[x]^4])/5 - (4*a^2*Cos[x]^2*C
ot[x]^5*Sqrt[a*Csc[x]^4])/7 - (a^2*Cos[x]^2*Cot[x]^7*Sqrt[a*Csc[x]^4])/9 - a^2*Cos[x]*Sqrt[a*Csc[x]^4]*Sin[x]

Rule 4123

Int[((b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Dist[(b^IntPart[p]*(b*(c*Sec[e + f*x])^n)^
FracPart[p])/(c*Sec[e + f*x])^(n*FracPart[p]), Int[(c*Sec[e + f*x])^(n*p), x], x] /; FreeQ[{b, c, e, f, n, p},
x] &&  !IntegerQ[p]

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps

\begin{align*} \int \left (a \csc ^4(x)\right )^{5/2} \, dx &=\left (a^2 \sqrt{a \csc ^4(x)} \sin ^2(x)\right ) \int \csc ^{10}(x) \, dx\\ &=-\left (\left (a^2 \sqrt{a \csc ^4(x)} \sin ^2(x)\right ) \operatorname{Subst}\left (\int \left (1+4 x^2+6 x^4+4 x^6+x^8\right ) \, dx,x,\cot (x)\right )\right )\\ &=-\frac{4}{3} a^2 \cos ^2(x) \cot (x) \sqrt{a \csc ^4(x)}-\frac{6}{5} a^2 \cos ^2(x) \cot ^3(x) \sqrt{a \csc ^4(x)}-\frac{4}{7} a^2 \cos ^2(x) \cot ^5(x) \sqrt{a \csc ^4(x)}-\frac{1}{9} a^2 \cos ^2(x) \cot ^7(x) \sqrt{a \csc ^4(x)}-a^2 \cos (x) \sqrt{a \csc ^4(x)} \sin (x)\\ \end{align*}

Mathematica [A]  time = 0.0359674, size = 47, normalized size = 0.4 \[ -\frac{1}{315} a^2 \sin (x) \cos (x) \left (35 \csc ^8(x)+40 \csc ^6(x)+48 \csc ^4(x)+64 \csc ^2(x)+128\right ) \sqrt{a \csc ^4(x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Csc[x]^4)^(5/2),x]

[Out]

-(a^2*Cos[x]*Sqrt[a*Csc[x]^4]*(128 + 64*Csc[x]^2 + 48*Csc[x]^4 + 40*Csc[x]^6 + 35*Csc[x]^8)*Sin[x])/315

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Maple [A]  time = 0.175, size = 41, normalized size = 0.4 \begin{align*} -{\frac{ \left ( 128\, \left ( \cos \left ( x \right ) \right ) ^{8}-576\, \left ( \cos \left ( x \right ) \right ) ^{6}+1008\, \left ( \cos \left ( x \right ) \right ) ^{4}-840\, \left ( \cos \left ( x \right ) \right ) ^{2}+315 \right ) \cos \left ( x \right ) \sin \left ( x \right ) }{315} \left ({\frac{a}{ \left ( \sin \left ( x \right ) \right ) ^{4}}} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*csc(x)^4)^(5/2),x)

[Out]

-1/315*(128*cos(x)^8-576*cos(x)^6+1008*cos(x)^4-840*cos(x)^2+315)*cos(x)*sin(x)*(a/sin(x)^4)^(5/2)

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Maxima [A]  time = 1.49922, size = 65, normalized size = 0.55 \begin{align*} -\frac{315 \, a^{\frac{5}{2}} \tan \left (x\right )^{8} + 420 \, a^{\frac{5}{2}} \tan \left (x\right )^{6} + 378 \, a^{\frac{5}{2}} \tan \left (x\right )^{4} + 180 \, a^{\frac{5}{2}} \tan \left (x\right )^{2} + 35 \, a^{\frac{5}{2}}}{315 \, \tan \left (x\right )^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*csc(x)^4)^(5/2),x, algorithm="maxima")

[Out]

-1/315*(315*a^(5/2)*tan(x)^8 + 420*a^(5/2)*tan(x)^6 + 378*a^(5/2)*tan(x)^4 + 180*a^(5/2)*tan(x)^2 + 35*a^(5/2)
)/tan(x)^9

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Fricas [A]  time = 0.492788, size = 252, normalized size = 2.14 \begin{align*} \frac{{\left (128 \, a^{2} \cos \left (x\right )^{9} - 576 \, a^{2} \cos \left (x\right )^{7} + 1008 \, a^{2} \cos \left (x\right )^{5} - 840 \, a^{2} \cos \left (x\right )^{3} + 315 \, a^{2} \cos \left (x\right )\right )} \sqrt{\frac{a}{\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1}}}{315 \,{\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*csc(x)^4)^(5/2),x, algorithm="fricas")

[Out]

1/315*(128*a^2*cos(x)^9 - 576*a^2*cos(x)^7 + 1008*a^2*cos(x)^5 - 840*a^2*cos(x)^3 + 315*a^2*cos(x))*sqrt(a/(co
s(x)^4 - 2*cos(x)^2 + 1))/((cos(x)^6 - 3*cos(x)^4 + 3*cos(x)^2 - 1)*sin(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((a*csc(x)**4)**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 1.27646, size = 69, normalized size = 0.58 \begin{align*} -\frac{{\left (315 \, a^{2} \tan \left (x\right )^{8} + 420 \, a^{2} \tan \left (x\right )^{6} + 378 \, a^{2} \tan \left (x\right )^{4} + 180 \, a^{2} \tan \left (x\right )^{2} + 35 \, a^{2}\right )} \sqrt{a}}{315 \, \tan \left (x\right )^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*csc(x)^4)^(5/2),x, algorithm="giac")

[Out]

-1/315*(315*a^2*tan(x)^8 + 420*a^2*tan(x)^6 + 378*a^2*tan(x)^4 + 180*a^2*tan(x)^2 + 35*a^2)*sqrt(a)/tan(x)^9