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)*(a*csc(x)^4)^(1/2)-3*a^3*cos(x)^2*cot(x)^3*(a*csc(x)^4)^(1/2)-20/7*a^3*cos(x)^2*cot(x)^
5*(a*csc(x)^4)^(1/2)-5/3*a^3*cos(x)^2*cot(x)^7*(a*csc(x)^4)^(1/2)-6/11*a^3*cos(x)^2*cot(x)^9*(a*csc(x)^4)^(1/2
)-1/13*a^3*cos(x)^2*cot(x)^11*(a*csc(x)^4)^(1/2)-a^3*cos(x)*sin(x)*(a*csc(x)^4)^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 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]

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]

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*}

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Mathematica [A]  time = 0.07, 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]

-1/3003*(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*C
sc[x]^10 + 231*Csc[x]^12)*Sin[x])

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fricas [A]  time = 0.58, size = 118, normalized size = 0.72 \[ \frac {{\left (1024 \, a^{3} \cos \relax (x)^{13} - 6656 \, a^{3} \cos \relax (x)^{11} + 18304 \, a^{3} \cos \relax (x)^{9} - 27456 \, a^{3} \cos \relax (x)^{7} + 24024 \, a^{3} \cos \relax (x)^{5} - 12012 \, a^{3} \cos \relax (x)^{3} + 3003 \, a^{3} \cos \relax (x)\right )} \sqrt {\frac {a}{\cos \relax (x)^{4} - 2 \, \cos \relax (x)^{2} + 1}}}{3003 \, {\left (\cos \relax (x)^{10} - 5 \, \cos \relax (x)^{8} + 10 \, \cos \relax (x)^{6} - 10 \, \cos \relax (x)^{4} + 5 \, \cos \relax (x)^{2} - 1\right )} \sin \relax (x)} \]

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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giac [A]  time = 0.65, size = 69, normalized size = 0.42 \[ -\frac {{\left (3003 \, a^{3} \tan \relax (x)^{12} + 6006 \, a^{3} \tan \relax (x)^{10} + 9009 \, a^{3} \tan \relax (x)^{8} + 8580 \, a^{3} \tan \relax (x)^{6} + 5005 \, a^{3} \tan \relax (x)^{4} + 1638 \, a^{3} \tan \relax (x)^{2} + 231 \, a^{3}\right )} \sqrt {a}}{3003 \, \tan \relax (x)^{13}} \]

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

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maple [A]  time = 0.76, size = 53, normalized size = 0.32 \[ -\frac {\left (1024 \left (\cos ^{12}\relax (x )\right )-6656 \left (\cos ^{10}\relax (x )\right )+18304 \left (\cos ^{8}\relax (x )\right )-27456 \left (\cos ^{6}\relax (x )\right )+24024 \left (\cos ^{4}\relax (x )\right )-12012 \left (\cos ^{2}\relax (x )\right )+3003\right ) \cos \relax (x ) \sin \relax (x ) \left (\frac {a}{\sin \relax (x )^{4}}\right )^{\frac {7}{2}}}{3003} \]

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 = 0.44, size = 66, normalized size = 0.40 \[ -\frac {3003 \, a^{\frac {7}{2}} \tan \relax (x)^{12} + 6006 \, a^{\frac {7}{2}} \tan \relax (x)^{10} + 9009 \, a^{\frac {7}{2}} \tan \relax (x)^{8} + 8580 \, a^{\frac {7}{2}} \tan \relax (x)^{6} + 5005 \, a^{\frac {7}{2}} \tan \relax (x)^{4} + 1638 \, a^{\frac {7}{2}} \tan \relax (x)^{2} + 231 \, a^{\frac {7}{2}}}{3003 \, \tan \relax (x)^{13}} \]

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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mupad [B]  time = 5.05, size = 603, normalized size = 3.68 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a/sin(x)^4)^(7/2),x)

[Out]

(a^3*(a/((exp(-x*1i)*1i)/2 - (exp(x*1i)*1i)/2)^4)^(1/2)*(6*exp(x*4i) - 4*exp(x*2i) - 4*exp(x*6i) + exp(x*8i) +
1)*2048i)/(7*(exp(x*2i) - 1)^7*(exp(x*2i) - 2*exp(x*4i) + exp(x*6i))) + (a^3*(a/((exp(-x*1i)*1i)/2 - (exp(x*1
i)*1i)/2)^4)^(1/2)*(6*exp(x*4i) - 4*exp(x*2i) - 4*exp(x*6i) + exp(x*8i) + 1)*1536i)/((exp(x*2i) - 1)^8*(exp(x*
2i) - 2*exp(x*4i) + exp(x*6i))) + (a^3*(a/((exp(-x*1i)*1i)/2 - (exp(x*1i)*1i)/2)^4)^(1/2)*(6*exp(x*4i) - 4*exp
(x*2i) - 4*exp(x*6i) + exp(x*8i) + 1)*10240i)/(3*(exp(x*2i) - 1)^9*(exp(x*2i) - 2*exp(x*4i) + exp(x*6i))) + (a
^3*(a/((exp(-x*1i)*1i)/2 - (exp(x*1i)*1i)/2)^4)^(1/2)*(6*exp(x*4i) - 4*exp(x*2i) - 4*exp(x*6i) + exp(x*8i) + 1
)*4096i)/((exp(x*2i) - 1)^10*(exp(x*2i) - 2*exp(x*4i) + exp(x*6i))) + (a^3*(a/((exp(-x*1i)*1i)/2 - (exp(x*1i)*
1i)/2)^4)^(1/2)*(6*exp(x*4i) - 4*exp(x*2i) - 4*exp(x*6i) + exp(x*8i) + 1)*30720i)/(11*(exp(x*2i) - 1)^11*(exp(
x*2i) - 2*exp(x*4i) + exp(x*6i))) + (a^3*(a/((exp(-x*1i)*1i)/2 - (exp(x*1i)*1i)/2)^4)^(1/2)*(6*exp(x*4i) - 4*e
xp(x*2i) - 4*exp(x*6i) + exp(x*8i) + 1)*1024i)/((exp(x*2i) - 1)^12*(exp(x*2i) - 2*exp(x*4i) + exp(x*6i))) + (a
^3*(a/((exp(-x*1i)*1i)/2 - (exp(x*1i)*1i)/2)^4)^(1/2)*(6*exp(x*4i) - 4*exp(x*2i) - 4*exp(x*6i) + exp(x*8i) + 1
)*2048i)/(13*(exp(x*2i) - 1)^13*(exp(x*2i) - 2*exp(x*4i) + exp(x*6i)))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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