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]

-1/7*cot(x)/(a*csc(x)^2)^(7/2)-6/35*cot(x)/a/(a*csc(x)^2)^(5/2)-8/35*cot(x)/a^2/(a*csc(x)^2)^(3/2)-16/35*cot(x
)/a^3/(a*csc(x)^2)^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 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]

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 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
]

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*}

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Mathematica [A]  time = 0.04, 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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fricas [A]  time = 0.81, size = 43, normalized size = 0.58 \[ \frac {{\left (5 \, \cos \relax (x)^{7} - 21 \, \cos \relax (x)^{5} + 35 \, \cos \relax (x)^{3} - 35 \, \cos \relax (x)\right )} \sqrt {-\frac {a}{\cos \relax (x)^{2} - 1}} \sin \relax (x)}{35 \, a^{4}} \]

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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giac [A]  time = 0.72, size = 69, normalized size = 0.93 \[ -\frac {32 \, {\left (\frac {35 \, \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )\right ) \tan \left (\frac {1}{2} \, x\right )^{6} + 21 \, \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )\right ) \tan \left (\frac {1}{2} \, x\right )^{4} + 7 \, \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )\right ) \tan \left (\frac {1}{2} \, x\right )^{2} + \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )\right )}{{\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}^{7}} - \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )\right )\right )}}{35 \, a^{\frac {7}{2}}} \]

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)

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maple [A]  time = 0.45, size = 45, normalized size = 0.61 \[ \frac {\sin \relax (x ) \left (5 \left (\cos ^{3}\relax (x )\right )-20 \left (\cos ^{2}\relax (x )\right )+29 \cos \relax (x )-16\right ) \sqrt {4}}{70 \left (-1+\cos \relax (x )\right )^{4} \left (-\frac {a}{-1+\cos ^{2}\relax (x )}\right )^{\frac {7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*csc(x)^2)^(7/2),x)

[Out]

1/70*sin(x)*(5*cos(x)^3-20*cos(x)^2+29*cos(x)-16)/(-1+cos(x))^4/(-1/(-1+cos(x)^2)*a)^(7/2)*4^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \csc \relax (x)^{2}\right )^{\frac {7}{2}}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (\frac {a}{{\sin \relax (x)}^2}\right )}^{7/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a/sin(x)^2)^(7/2),x)

[Out]

int(1/(a/sin(x)^2)^(7/2), x)

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sympy [A]  time = 136.16, size = 82, normalized size = 1.11 \[ - \frac {16 \cot ^{7}{\relax (x )}}{35 a^{\frac {7}{2}} \left (\csc ^{2}{\relax (x )}\right )^{\frac {7}{2}}} - \frac {8 \cot ^{5}{\relax (x )}}{5 a^{\frac {7}{2}} \left (\csc ^{2}{\relax (x )}\right )^{\frac {7}{2}}} - \frac {2 \cot ^{3}{\relax (x )}}{a^{\frac {7}{2}} \left (\csc ^{2}{\relax (x )}\right )^{\frac {7}{2}}} - \frac {\cot {\relax (x )}}{a^{\frac {7}{2}} \left (\csc ^{2}{\relax (x )}\right )^{\frac {7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csc(x)**2)**(7/2),x)

[Out]

-16*cot(x)**7/(35*a**(7/2)*(csc(x)**2)**(7/2)) - 8*cot(x)**5/(5*a**(7/2)*(csc(x)**2)**(7/2)) - 2*cot(x)**3/(a*
*(7/2)*(csc(x)**2)**(7/2)) - cot(x)/(a**(7/2)*(csc(x)**2)**(7/2))

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