\(\int \cot (x) (a+b \cot ^2(x))^{3/2} \, dx\) [28]

Optimal result

Integrand size = 15, antiderivative size = 69 \[ \int \cot (x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx=(a-b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )-(a-b) \sqrt {a+b \cot ^2(x)}-\frac {1}{3} \left (a+b \cot ^2(x)\right )^{3/2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3751, 455, 52, 65, 214} \[ \int \cot (x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx=(a-b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )-(a-b) \sqrt {a+b \cot ^2(x)}-\frac {1}{3} \left (a+b \cot ^2(x)\right )^{3/2} \]

[In]

Int[Cot[x]*(a + b*Cot[x]^2)^(3/2),x]

[Out]

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

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && 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 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 455

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(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] && EqQ[m
- n + 1, 0]

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

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.91 \[ \int \cot (x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx=(a-b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )-\frac {1}{3} \sqrt {a+b \cot ^2(x)} \left (4 a-3 b+b \cot ^2(x)\right ) \]

[In]

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

[Out]

(a - b)^(3/2)*ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]] - (Sqrt[a + b*Cot[x]^2]*(4*a - 3*b + b*Cot[x]^2))/3

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(135\) vs. \(2(57)=114\).

Time = 0.02 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.97

method	result	size
derivativedivides	\(-\frac {b \cot \left (x \right )^{2} \sqrt {a +b \cot \left (x \right )^{2}}}{3}-\frac {4 a \sqrt {a +b \cot \left (x \right )^{2}}}{3}+b \sqrt {a +b \cot \left (x \right )^{2}}-\frac {b^{2} \arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}+\frac {2 a b \arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}-\frac {a^{2} \arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}\)	\(136\)
default	\(-\frac {b \cot \left (x \right )^{2} \sqrt {a +b \cot \left (x \right )^{2}}}{3}-\frac {4 a \sqrt {a +b \cot \left (x \right )^{2}}}{3}+b \sqrt {a +b \cot \left (x \right )^{2}}-\frac {b^{2} \arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}+\frac {2 a b \arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}-\frac {a^{2} \arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}\)	\(136\)

[In]

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

[Out]

-1/3*b*cot(x)^2*(a+b*cot(x)^2)^(1/2)-4/3*a*(a+b*cot(x)^2)^(1/2)+b*(a+b*cot(x)^2)^(1/2)-b^2/(-a+b)^(1/2)*arctan
((a+b*cot(x)^2)^(1/2)/(-a+b)^(1/2))+2*a*b/(-a+b)^(1/2)*arctan((a+b*cot(x)^2)^(1/2)/(-a+b)^(1/2))-a^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. 141 vs. \(2 (57) = 114\).

Time = 0.33 (sec) , antiderivative size = 330, normalized size of antiderivative = 4.78 \[ \int \cot (x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx=\left [-\frac {3 \, {\left ({\left (a - b\right )} \cos \left (2 \, x\right ) - a + b\right )} \sqrt {a - b} \log \left (-2 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (2 \, x\right )^{2} - 2 \, a^{2} + b^{2} + 2 \, {\left ({\left (a - b\right )} \cos \left (2 \, x\right )^{2} - {\left (2 \, a - b\right )} \cos \left (2 \, x\right ) + a\right )} \sqrt {a - b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} + 4 \, {\left (a^{2} - a b\right )} \cos \left (2 \, x\right )\right ) + 8 \, {\left (2 \, {\left (a - b\right )} \cos \left (2 \, x\right ) - 2 \, a + b\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{12 \, {\left (\cos \left (2 \, x\right ) - 1\right )}}, \frac {3 \, {\left ({\left (a - b\right )} \cos \left (2 \, x\right ) - a + b\right )} \sqrt {-a + b} \arctan \left (-\frac {\sqrt {-a + b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} {\left (\cos \left (2 \, x\right ) - 1\right )}}{{\left (a - b\right )} \cos \left (2 \, x\right ) - a}\right ) - 4 \, {\left (2 \, {\left (a - b\right )} \cos \left (2 \, x\right ) - 2 \, a + b\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{6 \, {\left (\cos \left (2 \, x\right ) - 1\right )}}\right ] \]

[In]

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

[Out]

[-1/12*(3*((a - b)*cos(2*x) - a + b)*sqrt(a - b)*log(-2*(a^2 - 2*a*b + b^2)*cos(2*x)^2 - 2*a^2 + b^2 + 2*((a -
b)*cos(2*x)^2 - (2*a - b)*cos(2*x) + a)*sqrt(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)) + 4*(a^2
- a*b)*cos(2*x)) + 8*(2*(a - b)*cos(2*x) - 2*a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/(cos(2*x)
- 1), 1/6*(3*((a - b)*cos(2*x) - a + b)*sqrt(-a + b)*arctan(-sqrt(-a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(co
s(2*x) - 1))*(cos(2*x) - 1)/((a - b)*cos(2*x) - a)) - 4*(2*(a - b)*cos(2*x) - 2*a + b)*sqrt(((a - b)*cos(2*x)
- a - b)/(cos(2*x) - 1)))/(cos(2*x) - 1)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate(cot(x)*(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 [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to convert to real sageVARb Error: Bad Argument ValueUnable to convert to real sageVARb Error: Bad A
rgument Val

Mupad [B] (verification not implemented)

Time = 16.99 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.01 \[ \int \cot (x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx=\mathrm {atanh}\left (\frac {{\left (a-b\right )}^{3/2}\,\sqrt {b\,{\mathrm {cot}\left (x\right )}^2+a}}{a^2-2\,a\,b+b^2}\right )\,{\left (a-b\right )}^{3/2}-\frac {{\left (b\,{\mathrm {cot}\left (x\right )}^2+a\right )}^{3/2}}{3}-\left (a-b\right )\,\sqrt {b\,{\mathrm {cot}\left (x\right )}^2+a} \]

[In]

int(cot(x)*(a + b*cot(x)^2)^(3/2),x)

[Out]

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