\(\int \cot ^3(x) \sqrt {a+b \cot ^2(x)} \, dx\) [19]

Optimal result

Integrand size = 17, antiderivative size = 66 \[ \int \cot ^3(x) \sqrt {a+b \cot ^2(x)} \, dx=-\sqrt {a-b} \text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )+\sqrt {a+b \cot ^2(x)}-\frac {\left (a+b \cot ^2(x)\right )^{3/2}}{3 b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 66, 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, 81, 52, 65, 214} \[ \int \cot ^3(x) \sqrt {a+b \cot ^2(x)} \, dx=-\sqrt {a-b} \text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )-\frac {\left (a+b \cot ^2(x)\right )^{3/2}}{3 b}+\sqrt {a+b \cot ^2(x)} \]

[In]

Int[Cot[x]^3*Sqrt[a + b*Cot[x]^2],x]

[Out]

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

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 81

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

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

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.98 \[ \int \cot ^3(x) \sqrt {a+b \cot ^2(x)} \, dx=-\sqrt {a-b} \text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )-\frac {\sqrt {a+b \cot ^2(x)} \left (a-3 b+b \cot ^2(x)\right )}{3 b} \]

[In]

Integrate[Cot[x]^3*Sqrt[a + b*Cot[x]^2],x]

[Out]

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

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.27

method	result	size
derivativedivides	\(-\frac {\left (a +b \cot \left (x \right )^{2}\right )^{\frac {3}{2}}}{3 b}+\sqrt {a +b \cot \left (x \right )^{2}}-\frac {b \arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}+\frac {a \arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}\)	\(84\)
default	\(-\frac {\left (a +b \cot \left (x \right )^{2}\right )^{\frac {3}{2}}}{3 b}+\sqrt {a +b \cot \left (x \right )^{2}}-\frac {b \arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}+\frac {a \arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}\)	\(84\)

[In]

int(cot(x)^3*(a+b*cot(x)^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/3*(a+b*cot(x)^2)^(3/2)/b+(a+b*cot(x)^2)^(1/2)-b/(-a+b)^(1/2)*arctan((a+b*cot(x)^2)^(1/2)/(-a+b)^(1/2))+a/(-
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. \(2 (54) = 108\).

Time = 0.34 (sec) , antiderivative size = 330, normalized size of antiderivative = 5.00 \[ \int \cot ^3(x) \sqrt {a+b \cot ^2(x)} \, dx=\left [\frac {3 \, {\left (b \cos \left (2 \, x\right ) - b\right )} \sqrt {a - b} \log \left (-2 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (2 \, x\right )^{2} - 2 \, a^{2} + b^{2} + 2 \, {\left ({\left (a - b\right )} \cos \left (2 \, x\right )^{2} - {\left (2 \, a - b\right )} \cos \left (2 \, x\right ) + a\right )} \sqrt {a - b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} + 4 \, {\left (a^{2} - a b\right )} \cos \left (2 \, x\right )\right ) - 4 \, {\left ({\left (a - 4 \, b\right )} \cos \left (2 \, x\right ) - a + 2 \, b\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{12 \, {\left (b \cos \left (2 \, x\right ) - b\right )}}, -\frac {3 \, {\left (b \cos \left (2 \, x\right ) - b\right )} \sqrt {-a + b} \arctan \left (-\frac {\sqrt {-a + b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} {\left (\cos \left (2 \, x\right ) - 1\right )}}{{\left (a - b\right )} \cos \left (2 \, x\right ) - a}\right ) + 2 \, {\left ({\left (a - 4 \, b\right )} \cos \left (2 \, x\right ) - a + 2 \, b\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{6 \, {\left (b \cos \left (2 \, x\right ) - b\right )}}\right ] \]

[In]

integrate(cot(x)^3*(a+b*cot(x)^2)^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

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

[In]

integrate(cot(x)**3*(a+b*cot(x)**2)**(1/2),x)

[Out]

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

Maxima [F(-2)]

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

[In]

integrate(cot(x)^3*(a+b*cot(x)^2)^(1/2),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. \[ \int \cot ^3(x) \sqrt {a+b \cot ^2(x)} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(cot(x)^3*(a+b*cot(x)^2)^(1/2),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 = 15.11 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00 \[ \int \cot ^3(x) \sqrt {a+b \cot ^2(x)} \, dx=\sqrt {b\,{\mathrm {cot}\left (x\right )}^2+a}-\frac {{\left (b\,{\mathrm {cot}\left (x\right )}^2+a\right )}^{3/2}}{3\,b}+2\,\mathrm {atan}\left (\frac {2\,\sqrt {b\,{\mathrm {cot}\left (x\right )}^2+a}\,\sqrt {\frac {b}{4}-\frac {a}{4}}}{a-b}\right )\,\sqrt {\frac {b}{4}-\frac {a}{4}} \]

[In]

int(cot(x)^3*(a + b*cot(x)^2)^(1/2),x)

[Out]

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