\(\int (a \cot ^4(x))^{3/2} \, dx\) [33]

Optimal result

Integrand size = 10, antiderivative size = 70 \[ \int \left (a \cot ^4(x)\right )^{3/2} \, dx=\frac {1}{3} a \cot (x) \sqrt {a \cot ^4(x)}-\frac {1}{5} a \cot ^3(x) \sqrt {a \cot ^4(x)}-a \sqrt {a \cot ^4(x)} \tan (x)-a x \sqrt {a \cot ^4(x)} \tan ^2(x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3739, 3554, 8} \[ \int \left (a \cot ^4(x)\right )^{3/2} \, dx=\frac {1}{3} a \cot (x) \sqrt {a \cot ^4(x)}-\frac {1}{5} a \cot ^3(x) \sqrt {a \cot ^4(x)}-a x \tan ^2(x) \sqrt {a \cot ^4(x)}-a \tan (x) \sqrt {a \cot ^4(x)} \]

[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 8

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

Rule 3554

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 3739

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
]])

Rubi steps \begin{align*} \text {integral}& = \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 [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.43 \[ \int \left (a \cot ^4(x)\right )^{3/2} \, dx=-\frac {1}{5} \left (a \cot ^4(x)\right )^{3/2} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},-\tan ^2(x)\right ) \tan (x) \]

[In]

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

[Out]

-1/5*((a*Cot[x]^4)^(3/2)*Hypergeometric2F1[-5/2, 1, -3/2, -Tan[x]^2]*Tan[x])

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.57

method	result	size
derivativedivides	\(\frac {\left (a \cot \left (x \right )^{4}\right )^{\frac {3}{2}} \left (-3 \cot \left (x \right )^{5}+5 \cot \left (x \right )^{3}+\frac {15 \pi }{2}-15 \,\operatorname {arccot}\left (\cot \left (x \right )\right )-15 \cot \left (x \right )\right )}{15 \cot \left (x \right )^{6}}\)	\(40\)
default	\(\frac {\left (a \cot \left (x \right )^{4}\right )^{\frac {3}{2}} \left (-3 \cot \left (x \right )^{5}+5 \cot \left (x \right )^{3}+\frac {15 \pi }{2}-15 \,\operatorname {arccot}\left (\cot \left (x \right )\right )-15 \cot \left (x \right )\right )}{15 \cot \left (x \right )^{6}}\)	\(40\)
risch	\(\frac {a \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \left ({\mathrm e}^{2 i x}+1\right )^{4}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}\, x}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}+\frac {2 i a \sqrt {\frac {a \left ({\mathrm e}^{2 i x}+1\right )^{4}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}\, \left (45 \,{\mathrm e}^{8 i x}-90 \,{\mathrm e}^{6 i x}+140 \,{\mathrm e}^{4 i x}-70 \,{\mathrm e}^{2 i x}+23\right )}{15 \left ({\mathrm e}^{2 i x}+1\right )^{2} \left ({\mathrm e}^{2 i x}-1\right )^{3}}\)	\(119\)

[In]

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

[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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.57 \[ \int \left (a \cot ^4(x)\right )^{3/2} \, dx=\frac {{\left (23 \, a \cos \left (2 \, x\right )^{3} - a \cos \left (2 \, x\right )^{2} - 11 \, a \cos \left (2 \, x\right ) + 15 \, {\left (a x \cos \left (2 \, x\right )^{2} - 2 \, a x \cos \left (2 \, x\right ) + a x\right )} \sin \left (2 \, x\right ) + 13 \, a\right )} \sqrt {\frac {a \cos \left (2 \, x\right )^{2} + 2 \, a \cos \left (2 \, x\right ) + a}{\cos \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}}}{15 \, {\left (\cos \left (2 \, x\right )^{2} - 1\right )} \sin \left (2 \, x\right )} \]

[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]

\[ \int \left (a \cot ^4(x)\right )^{3/2} \, dx=\int \left (a \cot ^{4}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \]

[In]

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

[Out]

Integral((a*cot(x)**4)**(3/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.53 \[ \int \left (a \cot ^4(x)\right )^{3/2} \, dx=-a^{\frac {3}{2}} x - \frac {15 \, a^{\frac {3}{2}} \tan \left (x\right )^{4} - 5 \, a^{\frac {3}{2}} \tan \left (x\right )^{2} + 3 \, a^{\frac {3}{2}}}{15 \, \tan \left (x\right )^{5}} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.81 \[ \int \left (a \cot ^4(x)\right )^{3/2} \, dx=\frac {1}{480} \, {\left (3 \, \tan \left (\frac {1}{2} \, x\right )^{5} - 35 \, \tan \left (\frac {1}{2} \, x\right )^{3} - 480 \, x - \frac {330 \, \tan \left (\frac {1}{2} \, x\right )^{4} - 35 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 3}{\tan \left (\frac {1}{2} \, x\right )^{5}} + 330 \, \tan \left (\frac {1}{2} \, x\right )\right )} a^{\frac {3}{2}} \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \left (a \cot ^4(x)\right )^{3/2} \, dx=\int {\left (a\,{\mathrm {cot}\left (x\right )}^4\right )}^{3/2} \,d x \]

[In]

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

[Out]

int((a*cot(x)^4)^(3/2), x)