3.61 \(\int (a \csc ^4(x))^{7/2} \, dx\)

Optimal. Leaf size=164 \[ -\frac{1}{13} a^3 \cos ^2(x) \cot ^{11}(x) \sqrt{a \csc ^4(x)}-\frac{6}{11} a^3 \cos ^2(x) \cot ^9(x) \sqrt{a \csc ^4(x)}-\frac{5}{3} a^3 \cos ^2(x) \cot ^7(x) \sqrt{a \csc ^4(x)}-\frac{20}{7} a^3 \cos ^2(x) \cot ^5(x) \sqrt{a \csc ^4(x)}-3 a^3 \cos ^2(x) \cot ^3(x) \sqrt{a \csc ^4(x)}-2 a^3 \cos ^2(x) \cot (x) \sqrt{a \csc ^4(x)}-a^3 \sin (x) \cos (x) \sqrt{a \csc ^4(x)} \]

[Out]

-2*a^3*Cos[x]^2*Cot[x]*Sqrt[a*Csc[x]^4] - 3*a^3*Cos[x]^2*Cot[x]^3*Sqrt[a*Csc[x]^4] - (20*a^3*Cos[x]^2*Cot[x]^5
*Sqrt[a*Csc[x]^4])/7 - (5*a^3*Cos[x]^2*Cot[x]^7*Sqrt[a*Csc[x]^4])/3 - (6*a^3*Cos[x]^2*Cot[x]^9*Sqrt[a*Csc[x]^4
])/11 - (a^3*Cos[x]^2*Cot[x]^11*Sqrt[a*Csc[x]^4])/13 - a^3*Cos[x]*Sqrt[a*Csc[x]^4]*Sin[x]

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Rubi [A]  time = 0.0352601, antiderivative size = 164, 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} \[ -\frac{1}{13} a^3 \cos ^2(x) \cot ^{11}(x) \sqrt{a \csc ^4(x)}-\frac{6}{11} a^3 \cos ^2(x) \cot ^9(x) \sqrt{a \csc ^4(x)}-\frac{5}{3} a^3 \cos ^2(x) \cot ^7(x) \sqrt{a \csc ^4(x)}-\frac{20}{7} a^3 \cos ^2(x) \cot ^5(x) \sqrt{a \csc ^4(x)}-3 a^3 \cos ^2(x) \cot ^3(x) \sqrt{a \csc ^4(x)}-2 a^3 \cos ^2(x) \cot (x) \sqrt{a \csc ^4(x)}-a^3 \sin (x) \cos (x) \sqrt{a \csc ^4(x)} \]

Antiderivative was successfully verified.

[In]

Int[(a*Csc[x]^4)^(7/2),x]

[Out]

-2*a^3*Cos[x]^2*Cot[x]*Sqrt[a*Csc[x]^4] - 3*a^3*Cos[x]^2*Cot[x]^3*Sqrt[a*Csc[x]^4] - (20*a^3*Cos[x]^2*Cot[x]^5
*Sqrt[a*Csc[x]^4])/7 - (5*a^3*Cos[x]^2*Cot[x]^7*Sqrt[a*Csc[x]^4])/3 - (6*a^3*Cos[x]^2*Cot[x]^9*Sqrt[a*Csc[x]^4
])/11 - (a^3*Cos[x]^2*Cot[x]^11*Sqrt[a*Csc[x]^4])/13 - a^3*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 )^{7/2} \, dx &=\left (a^3 \sqrt{a \csc ^4(x)} \sin ^2(x)\right ) \int \csc ^{14}(x) \, dx\\ &=-\left (\left (a^3 \sqrt{a \csc ^4(x)} \sin ^2(x)\right ) \operatorname{Subst}\left (\int \left (1+6 x^2+15 x^4+20 x^6+15 x^8+6 x^{10}+x^{12}\right ) \, dx,x,\cot (x)\right )\right )\\ &=-2 a^3 \cos ^2(x) \cot (x) \sqrt{a \csc ^4(x)}-3 a^3 \cos ^2(x) \cot ^3(x) \sqrt{a \csc ^4(x)}-\frac{20}{7} a^3 \cos ^2(x) \cot ^5(x) \sqrt{a \csc ^4(x)}-\frac{5}{3} a^3 \cos ^2(x) \cot ^7(x) \sqrt{a \csc ^4(x)}-\frac{6}{11} a^3 \cos ^2(x) \cot ^9(x) \sqrt{a \csc ^4(x)}-\frac{1}{13} a^3 \cos ^2(x) \cot ^{11}(x) \sqrt{a \csc ^4(x)}-a^3 \cos (x) \sqrt{a \csc ^4(x)} \sin (x)\\ \end{align*}

Mathematica [A]  time = 0.0580877, size = 59, normalized size = 0.36 \[ -\frac{a^3 \sin (x) \cos (x) \left (231 \csc ^{12}(x)+252 \csc ^{10}(x)+280 \csc ^8(x)+320 \csc ^6(x)+384 \csc ^4(x)+512 \csc ^2(x)+1024\right ) \sqrt{a \csc ^4(x)}}{3003} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Csc[x]^4)^(7/2),x]

[Out]

-(a^3*Cos[x]*Sqrt[a*Csc[x]^4]*(1024 + 512*Csc[x]^2 + 384*Csc[x]^4 + 320*Csc[x]^6 + 280*Csc[x]^8 + 252*Csc[x]^1
0 + 231*Csc[x]^12)*Sin[x])/3003

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Maple [A]  time = 0.273, size = 53, normalized size = 0.3 \begin{align*} -{\frac{ \left ( 1024\, \left ( \cos \left ( x \right ) \right ) ^{12}-6656\, \left ( \cos \left ( x \right ) \right ) ^{10}+18304\, \left ( \cos \left ( x \right ) \right ) ^{8}-27456\, \left ( \cos \left ( x \right ) \right ) ^{6}+24024\, \left ( \cos \left ( x \right ) \right ) ^{4}-12012\, \left ( \cos \left ( x \right ) \right ) ^{2}+3003 \right ) \cos \left ( x \right ) \sin \left ( x \right ) }{3003} \left ({\frac{a}{ \left ( \sin \left ( x \right ) \right ) ^{4}}} \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*csc(x)^4)^(7/2),x)

[Out]

-1/3003*(1024*cos(x)^12-6656*cos(x)^10+18304*cos(x)^8-27456*cos(x)^6+24024*cos(x)^4-12012*cos(x)^2+3003)*cos(x
)*sin(x)*(a/sin(x)^4)^(7/2)

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Maxima [A]  time = 1.50019, size = 89, normalized size = 0.54 \begin{align*} -\frac{3003 \, a^{\frac{7}{2}} \tan \left (x\right )^{12} + 6006 \, a^{\frac{7}{2}} \tan \left (x\right )^{10} + 9009 \, a^{\frac{7}{2}} \tan \left (x\right )^{8} + 8580 \, a^{\frac{7}{2}} \tan \left (x\right )^{6} + 5005 \, a^{\frac{7}{2}} \tan \left (x\right )^{4} + 1638 \, a^{\frac{7}{2}} \tan \left (x\right )^{2} + 231 \, a^{\frac{7}{2}}}{3003 \, \tan \left (x\right )^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*csc(x)^4)^(7/2),x, algorithm="maxima")

[Out]

-1/3003*(3003*a^(7/2)*tan(x)^12 + 6006*a^(7/2)*tan(x)^10 + 9009*a^(7/2)*tan(x)^8 + 8580*a^(7/2)*tan(x)^6 + 500
5*a^(7/2)*tan(x)^4 + 1638*a^(7/2)*tan(x)^2 + 231*a^(7/2))/tan(x)^13

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Fricas [A]  time = 0.547037, size = 360, normalized size = 2.2 \begin{align*} \frac{{\left (1024 \, a^{3} \cos \left (x\right )^{13} - 6656 \, a^{3} \cos \left (x\right )^{11} + 18304 \, a^{3} \cos \left (x\right )^{9} - 27456 \, a^{3} \cos \left (x\right )^{7} + 24024 \, a^{3} \cos \left (x\right )^{5} - 12012 \, a^{3} \cos \left (x\right )^{3} + 3003 \, a^{3} \cos \left (x\right )\right )} \sqrt{\frac{a}{\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1}}}{3003 \,{\left (\cos \left (x\right )^{10} - 5 \, \cos \left (x\right )^{8} + 10 \, \cos \left (x\right )^{6} - 10 \, \cos \left (x\right )^{4} + 5 \, \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)^(7/2),x, algorithm="fricas")

[Out]

1/3003*(1024*a^3*cos(x)^13 - 6656*a^3*cos(x)^11 + 18304*a^3*cos(x)^9 - 27456*a^3*cos(x)^7 + 24024*a^3*cos(x)^5
- 12012*a^3*cos(x)^3 + 3003*a^3*cos(x))*sqrt(a/(cos(x)^4 - 2*cos(x)^2 + 1))/((cos(x)^10 - 5*cos(x)^8 + 10*cos
(x)^6 - 10*cos(x)^4 + 5*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)**(7/2),x)

[Out]

Timed out

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Giac [A]  time = 1.29849, size = 93, normalized size = 0.57 \begin{align*} -\frac{{\left (3003 \, a^{3} \tan \left (x\right )^{12} + 6006 \, a^{3} \tan \left (x\right )^{10} + 9009 \, a^{3} \tan \left (x\right )^{8} + 8580 \, a^{3} \tan \left (x\right )^{6} + 5005 \, a^{3} \tan \left (x\right )^{4} + 1638 \, a^{3} \tan \left (x\right )^{2} + 231 \, a^{3}\right )} \sqrt{a}}{3003 \, \tan \left (x\right )^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*csc(x)^4)^(7/2),x, algorithm="giac")

[Out]

-1/3003*(3003*a^3*tan(x)^12 + 6006*a^3*tan(x)^10 + 9009*a^3*tan(x)^8 + 8580*a^3*tan(x)^6 + 5005*a^3*tan(x)^4 +
1638*a^3*tan(x)^2 + 231*a^3)*sqrt(a)/tan(x)^13