\(\int \frac {\cot ^3(x)}{(a+b \cot ^2(x))^{5/2}} \, dx\) [54]

Optimal result

Integrand size = 17, antiderivative size = 82 \[ \int \frac {\cot ^3(x)}{\left (a+b \cot ^2(x)\right )^{5/2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{(a-b)^{5/2}}+\frac {a}{3 (a-b) b \left (a+b \cot ^2(x)\right )^{3/2}}+\frac {1}{(a-b)^2 \sqrt {a+b \cot ^2(x)}} \]

[Out]

-arctanh((a+b*cot(x)^2)^(1/2)/(a-b)^(1/2))/(a-b)^(5/2)+1/3*a/(a-b)/b/(a+b*cot(x)^2)^(3/2)+1/(a-b)^2/(a+b*cot(x
)^2)^(1/2)

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {3751, 457, 79, 53, 65, 214} \[ \int \frac {\cot ^3(x)}{\left (a+b \cot ^2(x)\right )^{5/2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{(a-b)^{5/2}}+\frac {a}{3 b (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}+\frac {1}{(a-b)^2 \sqrt {a+b \cot ^2(x)}} \]

[In]

Int[Cot[x]^3/(a + b*Cot[x]^2)^(5/2),x]

[Out]

-(ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]]/(a - b)^(5/2)) + a/(3*(a - b)*b*(a + b*Cot[x]^2)^(3/2)) + 1/((a -
b)^2*Sqrt[a + b*Cot[x]^2])

Rule 53

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 3751

Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol]
:> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[c*(ff/f), Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2
+ ff^2*x^2)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {x^3}{\left (1+x^2\right ) \left (a+b x^2\right )^{5/2}} \, dx,x,\cot (x)\right ) \\ & = -\left (\frac {1}{2} \text {Subst}\left (\int \frac {x}{(1+x) (a+b x)^{5/2}} \, dx,x,\cot ^2(x)\right )\right ) \\ & = \frac {a}{3 (a-b) b \left (a+b \cot ^2(x)\right )^{3/2}}+\frac {\text {Subst}\left (\int \frac {1}{(1+x) (a+b x)^{3/2}} \, dx,x,\cot ^2(x)\right )}{2 (a-b)} \\ & = \frac {a}{3 (a-b) b \left (a+b \cot ^2(x)\right )^{3/2}}+\frac {1}{(a-b)^2 \sqrt {a+b \cot ^2(x)}}+\frac {\text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x}} \, dx,x,\cot ^2(x)\right )}{2 (a-b)^2} \\ & = \frac {a}{3 (a-b) b \left (a+b \cot ^2(x)\right )^{3/2}}+\frac {1}{(a-b)^2 \sqrt {a+b \cot ^2(x)}}+\frac {\text {Subst}\left (\int \frac {1}{1-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cot ^2(x)}\right )}{(a-b)^2 b} \\ & = -\frac {\text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{(a-b)^{5/2}}+\frac {a}{3 (a-b) b \left (a+b \cot ^2(x)\right )^{3/2}}+\frac {1}{(a-b)^2 \sqrt {a+b \cot ^2(x)}} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 0.11 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.84 \[ \int \frac {\cot ^3(x)}{\left (a+b \cot ^2(x)\right )^{5/2}} \, dx=\frac {a (a-b)+3 b \left (a+b \cot ^2(x)\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {a+b \cot ^2(x)}{a-b}\right )}{3 (a-b)^2 b \left (a+b \cot ^2(x)\right )^{3/2}} \]

[In]

Integrate[Cot[x]^3/(a + b*Cot[x]^2)^(5/2),x]

[Out]

(a*(a - b) + 3*b*(a + b*Cot[x]^2)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Cot[x]^2)/(a - b)])/(3*(a - b)^2*b*(a
+ b*Cot[x]^2)^(3/2))

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.07

method	result	size
derivativedivides	\(\frac {1}{3 b \left (a +b \cot \left (x \right )^{2}\right )^{\frac {3}{2}}}+\frac {1}{3 \left (a -b \right ) \left (a +b \cot \left (x \right )^{2}\right )^{\frac {3}{2}}}+\frac {1}{\left (a -b \right )^{2} \sqrt {a +b \cot \left (x \right )^{2}}}+\frac {\arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\left (a -b \right )^{2} \sqrt {-a +b}}\)	\(88\)
default	\(\frac {1}{3 b \left (a +b \cot \left (x \right )^{2}\right )^{\frac {3}{2}}}+\frac {1}{3 \left (a -b \right ) \left (a +b \cot \left (x \right )^{2}\right )^{\frac {3}{2}}}+\frac {1}{\left (a -b \right )^{2} \sqrt {a +b \cot \left (x \right )^{2}}}+\frac {\arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\left (a -b \right )^{2} \sqrt {-a +b}}\)	\(88\)

[In]

int(cot(x)^3/(a+b*cot(x)^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/3/b/(a+b*cot(x)^2)^(3/2)+1/3/(a-b)/(a+b*cot(x)^2)^(3/2)+1/(a-b)^2/(a+b*cot(x)^2)^(1/2)+1/(a-b)^2/(-a+b)^(1/2
)*arctan((a+b*cot(x)^2)^(1/2)/(-a+b)^(1/2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (70) = 140\).

Time = 0.31 (sec) , antiderivative size = 698, normalized size of antiderivative = 8.51 \[ \int \frac {\cot ^3(x)}{\left (a+b \cot ^2(x)\right )^{5/2}} \, dx=\left [\frac {3 \, {\left (a^{2} b + 2 \, a b^{2} + b^{3} + {\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a^{2} b - b^{3}\right )} \cos \left (2 \, x\right )\right )} \sqrt {a - b} \log \left (\sqrt {a - b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} {\left (\cos \left (2 \, x\right ) - 1\right )} - {\left (a - b\right )} \cos \left (2 \, x\right ) + a\right ) + 2 \, {\left (a^{3} + a^{2} b + a b^{2} - 3 \, b^{3} + {\left (a^{3} + a^{2} b - 5 \, a b^{2} + 3 \, b^{3}\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a^{3} + a^{2} b - 2 \, a b^{2}\right )} \cos \left (2 \, x\right )\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{6 \, {\left (a^{5} b - a^{4} b^{2} - 2 \, a^{3} b^{3} + 2 \, a^{2} b^{4} + a b^{5} - b^{6} + {\left (a^{5} b - 5 \, a^{4} b^{2} + 10 \, a^{3} b^{3} - 10 \, a^{2} b^{4} + 5 \, a b^{5} - b^{6}\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a^{5} b - 3 \, a^{4} b^{2} + 2 \, a^{3} b^{3} + 2 \, a^{2} b^{4} - 3 \, a b^{5} + b^{6}\right )} \cos \left (2 \, x\right )\right )}}, -\frac {3 \, {\left (a^{2} b + 2 \, a b^{2} + b^{3} + {\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a^{2} b - b^{3}\right )} \cos \left (2 \, x\right )\right )} \sqrt {-a + b} \arctan \left (-\frac {\sqrt {-a + b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{a - b}\right ) - {\left (a^{3} + a^{2} b + a b^{2} - 3 \, b^{3} + {\left (a^{3} + a^{2} b - 5 \, a b^{2} + 3 \, b^{3}\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a^{3} + a^{2} b - 2 \, a b^{2}\right )} \cos \left (2 \, x\right )\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{3 \, {\left (a^{5} b - a^{4} b^{2} - 2 \, a^{3} b^{3} + 2 \, a^{2} b^{4} + a b^{5} - b^{6} + {\left (a^{5} b - 5 \, a^{4} b^{2} + 10 \, a^{3} b^{3} - 10 \, a^{2} b^{4} + 5 \, a b^{5} - b^{6}\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a^{5} b - 3 \, a^{4} b^{2} + 2 \, a^{3} b^{3} + 2 \, a^{2} b^{4} - 3 \, a b^{5} + b^{6}\right )} \cos \left (2 \, x\right )\right )}}\right ] \]

[In]

integrate(cot(x)^3/(a+b*cot(x)^2)^(5/2),x, algorithm="fricas")

[Out]

[1/6*(3*(a^2*b + 2*a*b^2 + b^3 + (a^2*b - 2*a*b^2 + b^3)*cos(2*x)^2 - 2*(a^2*b - b^3)*cos(2*x))*sqrt(a - b)*lo
g(sqrt(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*(cos(2*x) - 1) - (a - b)*cos(2*x) + a) + 2*(a^3
+ a^2*b + a*b^2 - 3*b^3 + (a^3 + a^2*b - 5*a*b^2 + 3*b^3)*cos(2*x)^2 - 2*(a^3 + a^2*b - 2*a*b^2)*cos(2*x))*sqr
t(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/(a^5*b - a^4*b^2 - 2*a^3*b^3 + 2*a^2*b^4 + a*b^5 - b^6 + (a^5*b
- 5*a^4*b^2 + 10*a^3*b^3 - 10*a^2*b^4 + 5*a*b^5 - b^6)*cos(2*x)^2 - 2*(a^5*b - 3*a^4*b^2 + 2*a^3*b^3 + 2*a^2*b
^4 - 3*a*b^5 + b^6)*cos(2*x)), -1/3*(3*(a^2*b + 2*a*b^2 + b^3 + (a^2*b - 2*a*b^2 + b^3)*cos(2*x)^2 - 2*(a^2*b
- b^3)*cos(2*x))*sqrt(-a + b)*arctan(-sqrt(-a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))/(a - b)) -
(a^3 + a^2*b + a*b^2 - 3*b^3 + (a^3 + a^2*b - 5*a*b^2 + 3*b^3)*cos(2*x)^2 - 2*(a^3 + a^2*b - 2*a*b^2)*cos(2*x)
)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/(a^5*b - a^4*b^2 - 2*a^3*b^3 + 2*a^2*b^4 + a*b^5 - b^6 + (a
^5*b - 5*a^4*b^2 + 10*a^3*b^3 - 10*a^2*b^4 + 5*a*b^5 - b^6)*cos(2*x)^2 - 2*(a^5*b - 3*a^4*b^2 + 2*a^3*b^3 + 2*
a^2*b^4 - 3*a*b^5 + b^6)*cos(2*x))]

Sympy [F]

\[ \int \frac {\cot ^3(x)}{\left (a+b \cot ^2(x)\right )^{5/2}} \, dx=\int \frac {\cot ^{3}{\left (x \right )}}{\left (a + b \cot ^{2}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(cot(x)**3/(a+b*cot(x)**2)**(5/2),x)

[Out]

Integral(cot(x)**3/(a + b*cot(x)**2)**(5/2), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cot ^3(x)}{\left (a+b \cot ^2(x)\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(cot(x)^3/(a+b*cot(x)^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a-4*b>0)', see `assume?` for
more detail

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (70) = 140\).

Time = 0.29 (sec) , antiderivative size = 219, normalized size of antiderivative = 2.67 \[ \int \frac {\cot ^3(x)}{\left (a+b \cot ^2(x)\right )^{5/2}} \, dx=-\frac {\log \left ({\left | b \right |}\right ) \mathrm {sgn}\left (\sin \left (x\right )\right )}{2 \, {\left (\sqrt {a - b} a^{2} - 2 \, \sqrt {a - b} a b + \sqrt {a - b} b^{2}\right )}} + \frac {\frac {{\left (\frac {{\left (a^{3} + a^{2} b - 5 \, a b^{2} + 3 \, b^{3}\right )} \sin \left (x\right )^{2}}{a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}} + \frac {3 \, {\left (a b^{2} - b^{3}\right )}}{a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}}\right )} \sin \left (x\right )}{{\left (a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b\right )}^{\frac {3}{2}}} + \frac {3 \, \log \left ({\left | -\sqrt {a - b} \sin \left (x\right ) + \sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b} \right |}\right )}{{\left (a^{2} - 2 \, a b + b^{2}\right )} \sqrt {a - b}}}{3 \, \mathrm {sgn}\left (\sin \left (x\right )\right )} \]

[In]

integrate(cot(x)^3/(a+b*cot(x)^2)^(5/2),x, algorithm="giac")

[Out]

-1/2*log(abs(b))*sgn(sin(x))/(sqrt(a - b)*a^2 - 2*sqrt(a - b)*a*b + sqrt(a - b)*b^2) + 1/3*(((a^3 + a^2*b - 5*
a*b^2 + 3*b^3)*sin(x)^2/(a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4) + 3*(a*b^2 - b^3)/(a^3*b - 3*a^2*b^2 + 3*a*b^3 - b
^4))*sin(x)/(a*sin(x)^2 - b*sin(x)^2 + b)^(3/2) + 3*log(abs(-sqrt(a - b)*sin(x) + sqrt(a*sin(x)^2 - b*sin(x)^2
+ b)))/((a^2 - 2*a*b + b^2)*sqrt(a - b)))/sgn(sin(x))

Mupad [B] (verification not implemented)

Time = 16.21 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.07 \[ \int \frac {\cot ^3(x)}{\left (a+b \cot ^2(x)\right )^{5/2}} \, dx=\frac {\frac {a}{3\,\left (a-b\right )}+\frac {b\,\left (b\,{\mathrm {cot}\left (x\right )}^2+a\right )}{{\left (a-b\right )}^2}}{b\,{\left (b\,{\mathrm {cot}\left (x\right )}^2+a\right )}^{3/2}}-\frac {\mathrm {atanh}\left (\frac {\sqrt {b\,{\mathrm {cot}\left (x\right )}^2+a}\,\left (2\,a^2-4\,a\,b+2\,b^2\right )}{2\,{\left (a-b\right )}^{5/2}}\right )}{{\left (a-b\right )}^{5/2}} \]

[In]

int(cot(x)^3/(a + b*cot(x)^2)^(5/2),x)

[Out]

(a/(3*(a - b)) + (b*(a + b*cot(x)^2))/(a - b)^2)/(b*(a + b*cot(x)^2)^(3/2)) - atanh(((a + b*cot(x)^2)^(1/2)*(2
*a^2 - 4*a*b + 2*b^2))/(2*(a - b)^(5/2)))/(a - b)^(5/2)