3.358 \(\int \cot ^3(x) \csc ^4(x) \, dx\)

Optimal. Leaf size=17 \[ \frac {\csc ^4(x)}{4}-\frac {\csc ^6(x)}{6} \]

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2606, 14} \[ \frac {\csc ^4(x)}{4}-\frac {\csc ^6(x)}{6} \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]^3*Csc[x]^4,x]

[Out]

Csc[x]^4/4 - Csc[x]^6/6

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rubi steps

\begin {align*} \int \cot ^3(x) \csc ^4(x) \, dx &=-\operatorname {Subst}\left (\int x^3 \left (-1+x^2\right ) \, dx,x,\csc (x)\right )\\ &=-\operatorname {Subst}\left (\int \left (-x^3+x^5\right ) \, dx,x,\csc (x)\right )\\ &=\frac {\csc ^4(x)}{4}-\frac {\csc ^6(x)}{6}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 17, normalized size = 1.00 \[ \frac {\csc ^4(x)}{4}-\frac {\csc ^6(x)}{6} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]^3*Csc[x]^4,x]

[Out]

Csc[x]^4/4 - Csc[x]^6/6

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cot ^3(x) \csc ^4(x) \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[Cot[x]^3*Csc[x]^4,x]

[Out]

Could not integrate

________________________________________________________________________________________

fricas [B]  time = 0.69, size = 30, normalized size = 1.76 \[ \frac {3 \, \cos \relax (x)^{2} - 1}{12 \, {\left (\cos \relax (x)^{6} - 3 \, \cos \relax (x)^{4} + 3 \, \cos \relax (x)^{2} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^3/sin(x)^7,x, algorithm="fricas")

[Out]

1/12*(3*cos(x)^2 - 1)/(cos(x)^6 - 3*cos(x)^4 + 3*cos(x)^2 - 1)

________________________________________________________________________________________

giac [A]  time = 0.61, size = 14, normalized size = 0.82 \[ \frac {3 \, \sin \relax (x)^{2} - 2}{12 \, \sin \relax (x)^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^3/sin(x)^7,x, algorithm="giac")

[Out]

1/12*(3*sin(x)^2 - 2)/sin(x)^6

________________________________________________________________________________________

maple [A]  time = 0.07, size = 22, normalized size = 1.29

method	result	size
default	\(-\frac {\cos ^{4}\relax (x )}{6 \sin \relax (x )^{6}}-\frac {\cos ^{4}\relax (x )}{12 \sin \relax (x )^{4}}\)	\(22\)
norman	\(\frac {-\frac {1}{384}+\frac {3 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{128}+\frac {3 \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{128}-\frac {\left (\tan ^{12}\left (\frac {x}{2}\right )\right )}{384}}{\tan \left (\frac {x}{2}\right )^{6}}\)	\(34\)
risch	\(\frac {4 \,{\mathrm e}^{8 i x}+\frac {8 \,{\mathrm e}^{6 i x}}{3}+4 \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{6}}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^3/sin(x)^7,x,method=_RETURNVERBOSE)

[Out]

-1/6/sin(x)^6*cos(x)^4-1/12/sin(x)^4*cos(x)^4

________________________________________________________________________________________

maxima [A]  time = 0.50, size = 14, normalized size = 0.82 \[ \frac {3 \, \sin \relax (x)^{2} - 2}{12 \, \sin \relax (x)^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^3/sin(x)^7,x, algorithm="maxima")

[Out]

1/12*(3*sin(x)^2 - 2)/sin(x)^6

________________________________________________________________________________________

mupad [B]  time = 0.07, size = 13, normalized size = 0.76 \[ \frac {\frac {{\sin \relax (x)}^2}{4}-\frac {1}{6}}{{\sin \relax (x)}^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^3/sin(x)^7,x)

[Out]

(sin(x)^2/4 - 1/6)/sin(x)^6

________________________________________________________________________________________

sympy [A]  time = 0.10, size = 15, normalized size = 0.88 \[ - \frac {2 - 3 \sin ^{2}{\relax (x )}}{12 \sin ^{6}{\relax (x )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)**3/sin(x)**7,x)

[Out]

-(2 - 3*sin(x)**2)/(12*sin(x)**6)

________________________________________________________________________________________