3.33 \(\int (a \cot ^4(x))^{3/2} \, dx\)

Optimal. Leaf size=70 \[ -\frac{1}{5} a \cot ^3(x) \sqrt{a \cot ^4(x)}+\frac{1}{3} a \cot (x) \sqrt{a \cot ^4(x)}-a x \tan ^2(x) \sqrt{a \cot ^4(x)}-a \tan (x) \sqrt{a \cot ^4(x)} \]

[Out]

(a*Cot[x]*Sqrt[a*Cot[x]^4])/3 - (a*Cot[x]^3*Sqrt[a*Cot[x]^4])/5 - a*Sqrt[a*Cot[x]^4]*Tan[x] - a*x*Sqrt[a*Cot[x
]^4]*Tan[x]^2

________________________________________________________________________________________

Rubi [A]  time = 0.0271455, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3658, 3473, 8} \[ -\frac{1}{5} a \cot ^3(x) \sqrt{a \cot ^4(x)}+\frac{1}{3} a \cot (x) \sqrt{a \cot ^4(x)}-a x \tan ^2(x) \sqrt{a \cot ^4(x)}-a \tan (x) \sqrt{a \cot ^4(x)} \]

Antiderivative was successfully verified.

[In]

Int[(a*Cot[x]^4)^(3/2),x]

[Out]

(a*Cot[x]*Sqrt[a*Cot[x]^4])/3 - (a*Cot[x]^3*Sqrt[a*Cot[x]^4])/5 - a*Sqrt[a*Cot[x]^4]*Tan[x] - a*x*Sqrt[a*Cot[x
]^4]*Tan[x]^2

Rule 3658

Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Di
st[((b*ff^n)^IntPart[p]*(b*Tan[e + f*x]^n)^FracPart[p])/(Tan[e + f*x]/ff)^(n*FracPart[p]), Int[ActivateTrig[u]
*(Tan[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 8

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

Rubi steps

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

Mathematica [A]  time = 0.135134, size = 39, normalized size = 0.56 \[ -\frac{1}{15} \tan ^6(x) \left (a \cot ^4(x)\right )^{3/2} \left (15 x+\cot (x) \left (3 \csc ^4(x)-11 \csc ^2(x)+23\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Cot[x]^4)^(3/2),x]

[Out]

-((a*Cot[x]^4)^(3/2)*(15*x + Cot[x]*(23 - 11*Csc[x]^2 + 3*Csc[x]^4))*Tan[x]^6)/15

________________________________________________________________________________________

Maple [A]  time = 0.06, size = 40, normalized size = 0.6 \begin{align*}{\frac{1}{15\, \left ( \cot \left ( x \right ) \right ) ^{6}} \left ( a \left ( \cot \left ( x \right ) \right ) ^{4} \right ) ^{{\frac{3}{2}}} \left ( -3\, \left ( \cot \left ( x \right ) \right ) ^{5}+5\, \left ( \cot \left ( x \right ) \right ) ^{3}+{\frac{15\,\pi }{2}}-15\,{\rm arccot} \left (\cot \left ( x \right ) \right )-15\,\cot \left ( x \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*cot(x)^4)^(3/2),x)

[Out]

1/15*(a*cot(x)^4)^(3/2)*(-3*cot(x)^5+5*cot(x)^3+15/2*Pi-15*arccot(cot(x))-15*cot(x))/cot(x)^6

________________________________________________________________________________________

Maxima [A]  time = 1.65532, size = 50, normalized size = 0.71 \begin{align*} -a^{\frac{3}{2}} x - \frac{15 \, a^{\frac{3}{2}} \tan \left (x\right )^{4} - 5 \, a^{\frac{3}{2}} \tan \left (x\right )^{2} + 3 \, a^{\frac{3}{2}}}{15 \, \tan \left (x\right )^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cot(x)^4)^(3/2),x, algorithm="maxima")

[Out]

-a^(3/2)*x - 1/15*(15*a^(3/2)*tan(x)^4 - 5*a^(3/2)*tan(x)^2 + 3*a^(3/2))/tan(x)^5

________________________________________________________________________________________

Fricas [A]  time = 2.17736, size = 289, normalized size = 4.13 \begin{align*} \frac{{\left (23 \, a \cos \left (2 \, x\right )^{3} - a \cos \left (2 \, x\right )^{2} - 11 \, a \cos \left (2 \, x\right ) + 15 \,{\left (a x \cos \left (2 \, x\right )^{2} - 2 \, a x \cos \left (2 \, x\right ) + a x\right )} \sin \left (2 \, x\right ) + 13 \, a\right )} \sqrt{\frac{a \cos \left (2 \, x\right )^{2} + 2 \, a \cos \left (2 \, x\right ) + a}{\cos \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}}}{15 \,{\left (\cos \left (2 \, x\right )^{2} - 1\right )} \sin \left (2 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cot(x)^4)^(3/2),x, algorithm="fricas")

[Out]

1/15*(23*a*cos(2*x)^3 - a*cos(2*x)^2 - 11*a*cos(2*x) + 15*(a*x*cos(2*x)^2 - 2*a*x*cos(2*x) + a*x)*sin(2*x) + 1
3*a)*sqrt((a*cos(2*x)^2 + 2*a*cos(2*x) + a)/(cos(2*x)^2 - 2*cos(2*x) + 1))/((cos(2*x)^2 - 1)*sin(2*x))

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \cot ^{4}{\left (x \right )}\right )^{\frac{3}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cot(x)**4)**(3/2),x)

[Out]

Integral((a*cot(x)**4)**(3/2), x)

________________________________________________________________________________________

Giac [A]  time = 1.23163, size = 77, normalized size = 1.1 \begin{align*} \frac{1}{480} \,{\left (3 \, \tan \left (\frac{1}{2} \, x\right )^{5} - 35 \, \tan \left (\frac{1}{2} \, x\right )^{3} - 480 \, x - \frac{330 \, \tan \left (\frac{1}{2} \, x\right )^{4} - 35 \, \tan \left (\frac{1}{2} \, x\right )^{2} + 3}{\tan \left (\frac{1}{2} \, x\right )^{5}} + 330 \, \tan \left (\frac{1}{2} \, x\right )\right )} a^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cot(x)^4)^(3/2),x, algorithm="giac")

[Out]

1/480*(3*tan(1/2*x)^5 - 35*tan(1/2*x)^3 - 480*x - (330*tan(1/2*x)^4 - 35*tan(1/2*x)^2 + 3)/tan(1/2*x)^5 + 330*
tan(1/2*x))*a^(3/2)