3.67 \(\int \frac {1}{(a \csc ^4(x))^{5/2}} \, dx\)

Optimal. Leaf size=132 \[ \frac {63 x \csc ^2(x)}{256 a^2 \sqrt {a \csc ^4(x)}}-\frac {63 \cot (x)}{256 a^2 \sqrt {a \csc ^4(x)}}-\frac {\sin ^7(x) \cos (x)}{10 a^2 \sqrt {a \csc ^4(x)}}-\frac {9 \sin ^5(x) \cos (x)}{80 a^2 \sqrt {a \csc ^4(x)}}-\frac {21 \sin ^3(x) \cos (x)}{160 a^2 \sqrt {a \csc ^4(x)}}-\frac {21 \sin (x) \cos (x)}{128 a^2 \sqrt {a \csc ^4(x)}} \]

[Out]

-63/256*cot(x)/a^2/(a*csc(x)^4)^(1/2)+63/256*x*csc(x)^2/a^2/(a*csc(x)^4)^(1/2)-21/128*cos(x)*sin(x)/a^2/(a*csc
(x)^4)^(1/2)-21/160*cos(x)*sin(x)^3/a^2/(a*csc(x)^4)^(1/2)-9/80*cos(x)*sin(x)^5/a^2/(a*csc(x)^4)^(1/2)-1/10*co
s(x)*sin(x)^7/a^2/(a*csc(x)^4)^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4123, 2635, 8} \[ \frac {63 x \csc ^2(x)}{256 a^2 \sqrt {a \csc ^4(x)}}-\frac {63 \cot (x)}{256 a^2 \sqrt {a \csc ^4(x)}}-\frac {\sin ^7(x) \cos (x)}{10 a^2 \sqrt {a \csc ^4(x)}}-\frac {9 \sin ^5(x) \cos (x)}{80 a^2 \sqrt {a \csc ^4(x)}}-\frac {21 \sin ^3(x) \cos (x)}{160 a^2 \sqrt {a \csc ^4(x)}}-\frac {21 \sin (x) \cos (x)}{128 a^2 \sqrt {a \csc ^4(x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-63*Cot[x])/(256*a^2*Sqrt[a*Csc[x]^4]) + (63*x*Csc[x]^2)/(256*a^2*Sqrt[a*Csc[x]^4]) - (21*Cos[x]*Sin[x])/(128
*a^2*Sqrt[a*Csc[x]^4]) - (21*Cos[x]*Sin[x]^3)/(160*a^2*Sqrt[a*Csc[x]^4]) - (9*Cos[x]*Sin[x]^5)/(80*a^2*Sqrt[a*
Csc[x]^4]) - (Cos[x]*Sin[x]^7)/(10*a^2*Sqrt[a*Csc[x]^4])

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

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 \frac {1}{\left (a \csc ^4(x)\right )^{5/2}} \, dx &=\frac {\csc ^2(x) \int \sin ^{10}(x) \, dx}{a^2 \sqrt {a \csc ^4(x)}}\\ &=-\frac {\cos (x) \sin ^7(x)}{10 a^2 \sqrt {a \csc ^4(x)}}+\frac {\left (9 \csc ^2(x)\right ) \int \sin ^8(x) \, dx}{10 a^2 \sqrt {a \csc ^4(x)}}\\ &=-\frac {9 \cos (x) \sin ^5(x)}{80 a^2 \sqrt {a \csc ^4(x)}}-\frac {\cos (x) \sin ^7(x)}{10 a^2 \sqrt {a \csc ^4(x)}}+\frac {\left (63 \csc ^2(x)\right ) \int \sin ^6(x) \, dx}{80 a^2 \sqrt {a \csc ^4(x)}}\\ &=-\frac {21 \cos (x) \sin ^3(x)}{160 a^2 \sqrt {a \csc ^4(x)}}-\frac {9 \cos (x) \sin ^5(x)}{80 a^2 \sqrt {a \csc ^4(x)}}-\frac {\cos (x) \sin ^7(x)}{10 a^2 \sqrt {a \csc ^4(x)}}+\frac {\left (21 \csc ^2(x)\right ) \int \sin ^4(x) \, dx}{32 a^2 \sqrt {a \csc ^4(x)}}\\ &=-\frac {21 \cos (x) \sin (x)}{128 a^2 \sqrt {a \csc ^4(x)}}-\frac {21 \cos (x) \sin ^3(x)}{160 a^2 \sqrt {a \csc ^4(x)}}-\frac {9 \cos (x) \sin ^5(x)}{80 a^2 \sqrt {a \csc ^4(x)}}-\frac {\cos (x) \sin ^7(x)}{10 a^2 \sqrt {a \csc ^4(x)}}+\frac {\left (63 \csc ^2(x)\right ) \int \sin ^2(x) \, dx}{128 a^2 \sqrt {a \csc ^4(x)}}\\ &=-\frac {63 \cot (x)}{256 a^2 \sqrt {a \csc ^4(x)}}-\frac {21 \cos (x) \sin (x)}{128 a^2 \sqrt {a \csc ^4(x)}}-\frac {21 \cos (x) \sin ^3(x)}{160 a^2 \sqrt {a \csc ^4(x)}}-\frac {9 \cos (x) \sin ^5(x)}{80 a^2 \sqrt {a \csc ^4(x)}}-\frac {\cos (x) \sin ^7(x)}{10 a^2 \sqrt {a \csc ^4(x)}}+\frac {\left (63 \csc ^2(x)\right ) \int 1 \, dx}{256 a^2 \sqrt {a \csc ^4(x)}}\\ &=-\frac {63 \cot (x)}{256 a^2 \sqrt {a \csc ^4(x)}}+\frac {63 x \csc ^2(x)}{256 a^2 \sqrt {a \csc ^4(x)}}-\frac {21 \cos (x) \sin (x)}{128 a^2 \sqrt {a \csc ^4(x)}}-\frac {21 \cos (x) \sin ^3(x)}{160 a^2 \sqrt {a \csc ^4(x)}}-\frac {9 \cos (x) \sin ^5(x)}{80 a^2 \sqrt {a \csc ^4(x)}}-\frac {\cos (x) \sin ^7(x)}{10 a^2 \sqrt {a \csc ^4(x)}}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 55, normalized size = 0.42 \[ \frac {\sin ^2(x) (2520 x-2100 \sin (2 x)+600 \sin (4 x)-150 \sin (6 x)+25 \sin (8 x)-2 \sin (10 x)) \sqrt {a \csc ^4(x)}}{10240 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a*Csc[x]^4]*Sin[x]^2*(2520*x - 2100*Sin[2*x] + 600*Sin[4*x] - 150*Sin[6*x] + 25*Sin[8*x] - 2*Sin[10*x]))
/(10240*a^3)

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fricas [A]  time = 0.47, size = 73, normalized size = 0.55 \[ -\frac {{\left (315 \, x \cos \relax (x)^{2} - {\left (128 \, \cos \relax (x)^{11} - 784 \, \cos \relax (x)^{9} + 2024 \, \cos \relax (x)^{7} - 2858 \, \cos \relax (x)^{5} + 2455 \, \cos \relax (x)^{3} - 965 \, \cos \relax (x)\right )} \sin \relax (x) - 315 \, x\right )} \sqrt {\frac {a}{\cos \relax (x)^{4} - 2 \, \cos \relax (x)^{2} + 1}}}{1280 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csc(x)^4)^(5/2),x, algorithm="fricas")

[Out]

-1/1280*(315*x*cos(x)^2 - (128*cos(x)^11 - 784*cos(x)^9 + 2024*cos(x)^7 - 2858*cos(x)^5 + 2455*cos(x)^3 - 965*
cos(x))*sin(x) - 315*x)*sqrt(a/(cos(x)^4 - 2*cos(x)^2 + 1))/a^3

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giac [A]  time = 0.27, size = 49, normalized size = 0.37 \[ \frac {63 \, x}{256 \, a^{\frac {5}{2}}} - \frac {965 \, \tan \relax (x)^{9} + 2370 \, \tan \relax (x)^{7} + 2688 \, \tan \relax (x)^{5} + 1470 \, \tan \relax (x)^{3} + 315 \, \tan \relax (x)}{1280 \, {\left (\tan \relax (x)^{2} + 1\right )}^{5} a^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csc(x)^4)^(5/2),x, algorithm="giac")

[Out]

63/256*x/a^(5/2) - 1/1280*(965*tan(x)^9 + 2370*tan(x)^7 + 2688*tan(x)^5 + 1470*tan(x)^3 + 315*tan(x))/((tan(x)
^2 + 1)^5*a^(5/2))

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maple [A]  time = 0.78, size = 57, normalized size = 0.43 \[ -\frac {128 \sin \relax (x ) \left (\cos ^{9}\relax (x )\right )-656 \sin \relax (x ) \left (\cos ^{7}\relax (x )\right )+1368 \sin \relax (x ) \left (\cos ^{5}\relax (x )\right )-1490 \left (\cos ^{3}\relax (x )\right ) \sin \relax (x )+965 \cos \relax (x ) \sin \relax (x )-315 x}{1280 \left (\frac {a}{\sin \relax (x )^{4}}\right )^{\frac {5}{2}} \sin \relax (x )^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*csc(x)^4)^(5/2),x)

[Out]

-1/1280*(128*sin(x)*cos(x)^9-656*sin(x)*cos(x)^7+1368*sin(x)*cos(x)^5-1490*cos(x)^3*sin(x)+965*cos(x)*sin(x)-3
15*x)/(a/sin(x)^4)^(5/2)/sin(x)^10

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maxima [A]  time = 0.51, size = 88, normalized size = 0.67 \[ -\frac {965 \, \tan \relax (x)^{9} + 2370 \, \tan \relax (x)^{7} + 2688 \, \tan \relax (x)^{5} + 1470 \, \tan \relax (x)^{3} + 315 \, \tan \relax (x)}{1280 \, {\left (a^{\frac {5}{2}} \tan \relax (x)^{10} + 5 \, a^{\frac {5}{2}} \tan \relax (x)^{8} + 10 \, a^{\frac {5}{2}} \tan \relax (x)^{6} + 10 \, a^{\frac {5}{2}} \tan \relax (x)^{4} + 5 \, a^{\frac {5}{2}} \tan \relax (x)^{2} + a^{\frac {5}{2}}\right )}} + \frac {63 \, x}{256 \, a^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csc(x)^4)^(5/2),x, algorithm="maxima")

[Out]

-1/1280*(965*tan(x)^9 + 2370*tan(x)^7 + 2688*tan(x)^5 + 1470*tan(x)^3 + 315*tan(x))/(a^(5/2)*tan(x)^10 + 5*a^(
5/2)*tan(x)^8 + 10*a^(5/2)*tan(x)^6 + 10*a^(5/2)*tan(x)^4 + 5*a^(5/2)*tan(x)^2 + a^(5/2)) + 63/256*x/a^(5/2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (\frac {a}{{\sin \relax (x)}^4}\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a/sin(x)^4)^(5/2),x)

[Out]

int(1/(a/sin(x)^4)^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \csc ^{4}{\relax (x )}\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csc(x)**4)**(5/2),x)

[Out]

Integral((a*csc(x)**4)**(-5/2), x)

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