\(\int \frac {\cot ^3(x)}{(a+b \cot ^2(x))^{3/2}} \, dx\) [49]

Optimal result

Integrand size = 17, antiderivative size = 59 \[ \int \frac {\cot ^3(x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{(a-b)^{3/2}}+\frac {a}{(a-b) b \sqrt {a+b \cot ^2(x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {3751, 457, 79, 65, 214} \[ \int \frac {\cot ^3(x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=\frac {a}{b (a-b) \sqrt {a+b \cot ^2(x)}}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{(a-b)^{3/2}} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00 \[ \int \frac {\cot ^3(x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{(a-b)^{3/2}}+\frac {a}{(a-b) b \sqrt {a+b \cot ^2(x)}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.15

method	result	size
derivativedivides	\(\frac {1}{b \sqrt {a +b \cot \left (x \right )^{2}}}+\frac {1}{\left (a -b \right ) \sqrt {a +b \cot \left (x \right )^{2}}}+\frac {\arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\left (a -b \right ) \sqrt {-a +b}}\)	\(68\)
default	\(\frac {1}{b \sqrt {a +b \cot \left (x \right )^{2}}}+\frac {1}{\left (a -b \right ) \sqrt {a +b \cot \left (x \right )^{2}}}+\frac {\arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\left (a -b \right ) \sqrt {-a +b}}\)	\(68\)

[In]

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

[Out]

1/b/(a+b*cot(x)^2)^(1/2)+1/(a-b)/(a+b*cot(x)^2)^(1/2)+1/(a-b)/(-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. 186 vs. \(2 (51) = 102\).

Time = 0.31 (sec) , antiderivative size = 385, normalized size of antiderivative = 6.53 \[ \int \frac {\cot ^3(x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=\left [-\frac {{\left (a b + b^{2} - {\left (a b - b^{2}\right )} \cos \left (2 \, x\right )\right )} \sqrt {a - b} \log \left (-\sqrt {a - b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} {\left (\cos \left (2 \, x\right ) - 1\right )} - {\left (a - b\right )} \cos \left (2 \, x\right ) + a\right ) - 2 \, {\left (a^{2} - a b - {\left (a^{2} - a b\right )} \cos \left (2 \, x\right )\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{2 \, {\left (a^{3} b - a^{2} b^{2} - a b^{3} + b^{4} - {\left (a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}\right )} \cos \left (2 \, x\right )\right )}}, -\frac {{\left (a b + b^{2} - {\left (a b - b^{2}\right )} \cos \left (2 \, x\right )\right )} \sqrt {-a + b} \arctan \left (-\frac {\sqrt {-a + b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{a - b}\right ) - {\left (a^{2} - a b - {\left (a^{2} - a b\right )} \cos \left (2 \, x\right )\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{a^{3} b - a^{2} b^{2} - a b^{3} + b^{4} - {\left (a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}\right )} \cos \left (2 \, x\right )}\right ] \]

[In]

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

[Out]

[-1/2*((a*b + b^2 - (a*b - b^2)*cos(2*x))*sqrt(a - b)*log(-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^2 - a*b - (a^2 - a*b)*cos(2*x))*sqrt(((a - b)*cos(2*x)
- a - b)/(cos(2*x) - 1)))/(a^3*b - a^2*b^2 - a*b^3 + b^4 - (a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*cos(2*x)), -((a
*b + b^2 - (a*b - b^2)*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^2 - a*b - (a^2 - a*b)*cos(2*x))*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/(a^3*b - a
^2*b^2 - a*b^3 + b^4 - (a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*cos(2*x))]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate(cot(x)^3/(a+b*cot(x)^2)^(3/2),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. 109 vs. \(2 (51) = 102\).

Time = 0.32 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.85 \[ \int \frac {\cot ^3(x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=-\frac {\log \left ({\left | b \right |}\right ) \mathrm {sgn}\left (\sin \left (x\right )\right )}{2 \, {\left (\sqrt {a - b} a - \sqrt {a - b} b\right )}} + \frac {\frac {a \sin \left (x\right )}{\sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b} {\left (a b - b^{2}\right )}} + \frac {\log \left ({\left | -\sqrt {a - b} \sin \left (x\right ) + \sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b} \right |}\right )}{{\left (a - b\right )}^{\frac {3}{2}}}}{\mathrm {sgn}\left (\sin \left (x\right )\right )} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 14.40 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.88 \[ \int \frac {\cot ^3(x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=\frac {a}{\left (a\,b-b^2\right )\,\sqrt {b\,{\mathrm {cot}\left (x\right )}^2+a}}-\frac {\mathrm {atanh}\left (\frac {\sqrt {b\,{\mathrm {cot}\left (x\right )}^2+a}}{\sqrt {a-b}}\right )}{{\left (a-b\right )}^{3/2}} \]

[In]

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

[Out]

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