3.864 \(\int \cot ^3(x) \csc (x) \sqrt {1+\csc (x)} \, dx\)

Optimal. Leaf size=25 \[ \frac {4}{5} (\csc (x)+1)^{5/2}-\frac {2}{7} (\csc (x)+1)^{7/2} \]

[Out]

4/5*(1+csc(x))^(5/2)-2/7*(1+csc(x))^(7/2)

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Rubi [A]  time = 0.08, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4372, 1570, 1469, 627, 43} \[ \frac {4}{5} (\csc (x)+1)^{5/2}-\frac {2}{7} (\csc (x)+1)^{7/2} \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]^3*Csc[x]*Sqrt[1 + Csc[x]],x]

[Out]

(4*(1 + Csc[x])^(5/2))/5 - (2*(1 + Csc[x])^(7/2))/7

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 627

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c*x)/e)^
p, x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && I
ntegerQ[m + p]))

Rule 1469

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[I
nt[(d + e*x)^q*(a + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[Sim
plify[m - n + 1], 0]

Rule 1570

Int[(x_)^(m_.)*((a_.) + (c_.)*(x_)^(mn2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_.))^(q_.), x_Symbol] :> Int[x^(m - 2*n
*p)*(d + e*x^n)^q*(c + a*x^(2*n))^p, x] /; FreeQ[{a, c, d, e, m, n, q}, x] && EqQ[mn2, -2*n] && IntegerQ[p]

Rule 4372

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))]^(n_), x_Symbol] :> With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, Dist
[1/(b*c*d^(n - 1)), Subst[Int[SubstFor[(1 - d^2*x^2)^((n - 1)/2)/x^n, Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*
(a + b*x)]/d], x] /; FunctionOfQ[Sin[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && N
onsumQ[u] && (EqQ[F, Cot] || EqQ[F, cot])

Rubi steps

\begin {align*} \int \cot ^3(x) \csc (x) \sqrt {1+\csc (x)} \, dx &=\operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {1}{x}} \left (1-x^2\right )}{x^4} \, dx,x,\sin (x)\right )\\ &=\operatorname {Subst}\left (\int \frac {\left (-1+\frac {1}{x^2}\right ) \sqrt {1+\frac {1}{x}}}{x^2} \, dx,x,\sin (x)\right )\\ &=-\operatorname {Subst}\left (\int \sqrt {1+x} \left (-1+x^2\right ) \, dx,x,\csc (x)\right )\\ &=-\operatorname {Subst}\left (\int (-1+x) (1+x)^{3/2} \, dx,x,\csc (x)\right )\\ &=-\operatorname {Subst}\left (\int \left (-2 (1+x)^{3/2}+(1+x)^{5/2}\right ) \, dx,x,\csc (x)\right )\\ &=\frac {4}{5} (1+\csc (x))^{5/2}-\frac {2}{7} (1+\csc (x))^{7/2}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 18, normalized size = 0.72 \[ -\frac {2}{35} (\csc (x)+1)^{5/2} (5 \csc (x)-9) \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]^3*Csc[x]*Sqrt[1 + Csc[x]],x]

[Out]

(-2*(1 + Csc[x])^(5/2)*(-9 + 5*Csc[x]))/35

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fricas [B]  time = 0.41, size = 44, normalized size = 1.76 \[ \frac {2 \, {\left (13 \, \cos \relax (x)^{2} + {\left (9 \, \cos \relax (x)^{2} - 8\right )} \sin \relax (x) - 8\right )} \sqrt {\frac {\sin \relax (x) + 1}{\sin \relax (x)}}}{35 \, {\left (\cos \relax (x)^{2} - 1\right )} \sin \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^3*csc(x)*(1+csc(x))^(1/2),x, algorithm="fricas")

[Out]

2/35*(13*cos(x)^2 + (9*cos(x)^2 - 8)*sin(x) - 8)*sqrt((sin(x) + 1)/sin(x))/((cos(x)^2 - 1)*sin(x))

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giac [B]  time = 0.14, size = 128, normalized size = 5.12 \[ \frac {2 \, {\left (35 \, {\left (\sqrt {\sin \relax (x)^{2} + \sin \relax (x)} - \sin \relax (x)\right )}^{6} - 35 \, {\left (\sqrt {\sin \relax (x)^{2} + \sin \relax (x)} - \sin \relax (x)\right )}^{5} - 35 \, {\left (\sqrt {\sin \relax (x)^{2} + \sin \relax (x)} - \sin \relax (x)\right )}^{4} + 105 \, {\left (\sqrt {\sin \relax (x)^{2} + \sin \relax (x)} - \sin \relax (x)\right )}^{3} - 91 \, {\left (\sqrt {\sin \relax (x)^{2} + \sin \relax (x)} - \sin \relax (x)\right )}^{2} + 35 \, \sqrt {\sin \relax (x)^{2} + \sin \relax (x)} - 35 \, \sin \relax (x) - 5\right )} \mathrm {sgn}\left (\sin \relax (x)\right )}{35 \, {\left (\sqrt {\sin \relax (x)^{2} + \sin \relax (x)} - \sin \relax (x)\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^3*csc(x)*(1+csc(x))^(1/2),x, algorithm="giac")

[Out]

2/35*(35*(sqrt(sin(x)^2 + sin(x)) - sin(x))^6 - 35*(sqrt(sin(x)^2 + sin(x)) - sin(x))^5 - 35*(sqrt(sin(x)^2 +
sin(x)) - sin(x))^4 + 105*(sqrt(sin(x)^2 + sin(x)) - sin(x))^3 - 91*(sqrt(sin(x)^2 + sin(x)) - sin(x))^2 + 35*
sqrt(sin(x)^2 + sin(x)) - 35*sin(x) - 5)*sgn(sin(x))/(sqrt(sin(x)^2 + sin(x)) - sin(x))^7

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maple [B]  time = 0.18, size = 38, normalized size = 1.52 \[ -\frac {2 \left (9 \left (\cos ^{2}\relax (x )\right ) \sin \relax (x )+13 \left (\cos ^{2}\relax (x )\right )-8 \sin \relax (x )-8\right ) \sqrt {\frac {1+\sin \relax (x )}{\sin \relax (x )}}}{35 \sin \relax (x )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)^3*csc(x)*(1+csc(x))^(1/2),x)

[Out]

-2/35*(9*cos(x)^2*sin(x)+13*cos(x)^2-8*sin(x)-8)*((1+sin(x))/sin(x))^(1/2)/sin(x)^3

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maxima [A]  time = 0.32, size = 21, normalized size = 0.84 \[ -\frac {2}{7} \, {\left (\frac {1}{\sin \relax (x)} + 1\right )}^{\frac {7}{2}} + \frac {4}{5} \, {\left (\frac {1}{\sin \relax (x)} + 1\right )}^{\frac {5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^3*csc(x)*(1+csc(x))^(1/2),x, algorithm="maxima")

[Out]

-2/7*(1/sin(x) + 1)^(7/2) + 4/5*(1/sin(x) + 1)^(5/2)

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mupad [B]  time = 3.42, size = 24, normalized size = 0.96 \[ \frac {2\,{\left (\sin \relax (x)+1\right )}^{5/2}\,\sqrt {\frac {1}{\sin \relax (x)}}\,\left (9\,\sin \relax (x)-5\right )}{35\,{\sin \relax (x)}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cot(x)^3*(1/sin(x) + 1)^(1/2))/sin(x),x)

[Out]

(2*(sin(x) + 1)^(5/2)*(1/sin(x))^(1/2)*(9*sin(x) - 5))/(35*sin(x)^3)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\csc {\relax (x )} + 1} \cot ^{3}{\relax (x )} \csc {\relax (x )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)**3*csc(x)*(1+csc(x))**(1/2),x)

[Out]

Integral(sqrt(csc(x) + 1)*cot(x)**3*csc(x), x)

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