3.46 \(\int \frac {1}{\csc ^2(x)^{7/2}} \, dx\)

Optimal. Leaf size=57 \[ -\frac {16 \cot (x)}{35 \sqrt {\csc ^2(x)}}-\frac {8 \cot (x)}{35 \csc ^2(x)^{3/2}}-\frac {6 \cot (x)}{35 \csc ^2(x)^{5/2}}-\frac {\cot (x)}{7 \csc ^2(x)^{7/2}} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4122, 192, 191} \[ -\frac {16 \cot (x)}{35 \sqrt {\csc ^2(x)}}-\frac {8 \cot (x)}{35 \csc ^2(x)^{3/2}}-\frac {6 \cot (x)}{35 \csc ^2(x)^{5/2}}-\frac {\cot (x)}{7 \csc ^2(x)^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[(Csc[x]^2)^(-7/2),x]

[Out]

-Cot[x]/(7*(Csc[x]^2)^(7/2)) - (6*Cot[x])/(35*(Csc[x]^2)^(5/2)) - (8*Cot[x])/(35*(Csc[x]^2)^(3/2)) - (16*Cot[x
])/(35*Sqrt[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}{\csc ^2(x)^{7/2}} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right )^{9/2}} \, dx,x,\cot (x)\right )\\ &=-\frac {\cot (x)}{7 \csc ^2(x)^{7/2}}-\frac {6}{7} \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right )^{7/2}} \, dx,x,\cot (x)\right )\\ &=-\frac {\cot (x)}{7 \csc ^2(x)^{7/2}}-\frac {6 \cot (x)}{35 \csc ^2(x)^{5/2}}-\frac {24}{35} \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right )^{5/2}} \, dx,x,\cot (x)\right )\\ &=-\frac {\cot (x)}{7 \csc ^2(x)^{7/2}}-\frac {6 \cot (x)}{35 \csc ^2(x)^{5/2}}-\frac {8 \cot (x)}{35 \csc ^2(x)^{3/2}}-\frac {16}{35} \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right )^{3/2}} \, dx,x,\cot (x)\right )\\ &=-\frac {\cot (x)}{7 \csc ^2(x)^{7/2}}-\frac {6 \cot (x)}{35 \csc ^2(x)^{5/2}}-\frac {8 \cot (x)}{35 \csc ^2(x)^{3/2}}-\frac {16 \cot (x)}{35 \sqrt {\csc ^2(x)}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 37, normalized size = 0.65 \[ \frac {(-1225 \cos (x)+245 \cos (3 x)-49 \cos (5 x)+5 \cos (7 x)) \csc (x)}{2240 \sqrt {\csc ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(Csc[x]^2)^(-7/2),x]

[Out]

((-1225*Cos[x] + 245*Cos[3*x] - 49*Cos[5*x] + 5*Cos[7*x])*Csc[x])/(2240*Sqrt[Csc[x]^2])

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fricas [A]  time = 0.67, size = 21, normalized size = 0.37 \[ \frac {1}{7} \, \cos \relax (x)^{7} - \frac {3}{5} \, \cos \relax (x)^{5} + \cos \relax (x)^{3} - \cos \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(csc(x)^2)^(7/2),x, algorithm="fricas")

[Out]

1/7*cos(x)^7 - 3/5*cos(x)^5 + cos(x)^3 - cos(x)

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giac [A]  time = 0.55, size = 78, normalized size = 1.37 \[ -\frac {32 \, {\left (\frac {7 \, {\left (\cos \relax (x) - 1\right )} \mathrm {sgn}\left (\sin \relax (x)\right )}{\cos \relax (x) + 1} - \frac {21 \, {\left (\cos \relax (x) - 1\right )}^{2} \mathrm {sgn}\left (\sin \relax (x)\right )}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {35 \, {\left (\cos \relax (x) - 1\right )}^{3} \mathrm {sgn}\left (\sin \relax (x)\right )}{{\left (\cos \relax (x) + 1\right )}^{3}} - \mathrm {sgn}\left (\sin \relax (x)\right )\right )}}{35 \, {\left (\frac {\cos \relax (x) - 1}{\cos \relax (x) + 1} - 1\right )}^{7}} + \frac {32}{35} \, \mathrm {sgn}\left (\sin \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(csc(x)^2)^(7/2),x, algorithm="giac")

[Out]

-32/35*(7*(cos(x) - 1)*sgn(sin(x))/(cos(x) + 1) - 21*(cos(x) - 1)^2*sgn(sin(x))/(cos(x) + 1)^2 + 35*(cos(x) -
1)^3*sgn(sin(x))/(cos(x) + 1)^3 - sgn(sin(x)))/((cos(x) - 1)/(cos(x) + 1) - 1)^7 + 32/35*sgn(sin(x))

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maple [A]  time = 0.43, size = 44, normalized size = 0.77 \[ \frac {\sin \relax (x ) \left (5 \left (\cos ^{3}\relax (x )\right )-20 \left (\cos ^{2}\relax (x )\right )+29 \cos \relax (x )-16\right ) \sqrt {4}}{70 \left (-1+\cos \relax (x )\right )^{4} \left (-\frac {1}{-1+\cos ^{2}\relax (x )}\right )^{\frac {7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(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))^(7/2)*4^(1/2)

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maxima [A]  time = 0.52, size = 23, normalized size = 0.40 \[ \frac {1}{448} \, \cos \left (7 \, x\right ) - \frac {7}{320} \, \cos \left (5 \, x\right ) + \frac {7}{64} \, \cos \left (3 \, x\right ) - \frac {35}{64} \, \cos \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(csc(x)^2)^(7/2),x, algorithm="maxima")

[Out]

1/448*cos(7*x) - 7/320*cos(5*x) + 7/64*cos(3*x) - 35/64*cos(x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\left (\frac {1}{{\sin \relax (x)}^2}\right )}^{7/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(1/sin(x)^2)^(7/2), x)

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sympy [A]  time = 138.80, size = 61, normalized size = 1.07 \[ - \frac {16 \cot ^{7}{\relax (x )}}{35 \left (\csc ^{2}{\relax (x )}\right )^{\frac {7}{2}}} - \frac {8 \cot ^{5}{\relax (x )}}{5 \left (\csc ^{2}{\relax (x )}\right )^{\frac {7}{2}}} - \frac {2 \cot ^{3}{\relax (x )}}{\left (\csc ^{2}{\relax (x )}\right )^{\frac {7}{2}}} - \frac {\cot {\relax (x )}}{\left (\csc ^{2}{\relax (x )}\right )^{\frac {7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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