\(\int \cot ^3(x) (a+b \cot ^2(x))^{3/2} \, dx\) [26]

Optimal result

Integrand size = 17, antiderivative size = 88 \[ \int \cot ^3(x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx=-(a-b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )+(a-b) \sqrt {a+b \cot ^2(x)}+\frac {1}{3} \left (a+b \cot ^2(x)\right )^{3/2}-\frac {\left (a+b \cot ^2(x)\right )^{5/2}}{5 b} \]

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

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {3751, 457, 81, 52, 65, 214} \[ \int \cot ^3(x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx=-(a-b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )-\frac {\left (a+b \cot ^2(x)\right )^{5/2}}{5 b}+\frac {1}{3} \left (a+b \cot ^2(x)\right )^{3/2}+(a-b) \sqrt {a+b \cot ^2(x)} \]

[In]

Int[Cot[x]^3*(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 - (a + b*Cot[x]^2)^(5/2)/(5*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 \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 {x (a+b x)^{3/2}}{1+x} \, dx,x,\cot ^2(x)\right )\right ) \\ & = -\frac {\left (a+b \cot ^2(x)\right )^{5/2}}{5 b}+\frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^{3/2}}{1+x} \, dx,x,\cot ^2(x)\right ) \\ & = \frac {1}{3} \left (a+b \cot ^2(x)\right )^{3/2}-\frac {\left (a+b \cot ^2(x)\right )^{5/2}}{5 b}+\frac {1}{2} (a-b) \text {Subst}\left (\int \frac {\sqrt {a+b x}}{1+x} \, dx,x,\cot ^2(x)\right ) \\ & = (a-b) \sqrt {a+b \cot ^2(x)}+\frac {1}{3} \left (a+b \cot ^2(x)\right )^{3/2}-\frac {\left (a+b \cot ^2(x)\right )^{5/2}}{5 b}+\frac {1}{2} (a-b)^2 \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x}} \, dx,x,\cot ^2(x)\right ) \\ & = (a-b) \sqrt {a+b \cot ^2(x)}+\frac {1}{3} \left (a+b \cot ^2(x)\right )^{3/2}-\frac {\left (a+b \cot ^2(x)\right )^{5/2}}{5 b}+\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}-\frac {\left (a+b \cot ^2(x)\right )^{5/2}}{5 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.61 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.03 \[ \int \cot ^3(x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx=-(a-b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )-\frac {\sqrt {a+b \cot ^2(x)} \left (3 a^2-20 a b+15 b^2+(6 a-5 b) b \cot ^2(x)+3 b^2 \cot ^4(x)\right )}{15 b} \]

[In]

Integrate[Cot[x]^3*(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]*(3*a^2 - 20*a*b + 15*b^2 +
(6*a - 5*b)*b*Cot[x]^2 + 3*b^2*Cot[x]^4))/(15*b)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(149\) vs. \(2(72)=144\).

Time = 0.04 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.70

method	result	size
derivativedivides	\(-\frac {\left (a +b \cot \left (x \right )^{2}\right )^{\frac {5}{2}}}{5 b}+\frac {b \cot \left (x \right )^{2} \sqrt {a +b \cot \left (x \right )^{2}}}{3}+\frac {4 a \sqrt {a +b \cot \left (x \right )^{2}}}{3}-b \sqrt {a +b \cot \left (x \right )^{2}}+\frac {b^{2} \arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}-\frac {2 a b \arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}+\frac {a^{2} \arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}\)	\(150\)
default	\(-\frac {\left (a +b \cot \left (x \right )^{2}\right )^{\frac {5}{2}}}{5 b}+\frac {b \cot \left (x \right )^{2} \sqrt {a +b \cot \left (x \right )^{2}}}{3}+\frac {4 a \sqrt {a +b \cot \left (x \right )^{2}}}{3}-b \sqrt {a +b \cot \left (x \right )^{2}}+\frac {b^{2} \arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}-\frac {2 a b \arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}+\frac {a^{2} \arctan \left (\frac {\sqrt {a +b \cot \left (x \right )^{2}}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}\)	\(150\)

[In]

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

[Out]

-1/5*(a+b*cot(x)^2)^(5/2)/b+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. 219 vs. \(2 (72) = 144\).

Time = 0.35 (sec) , antiderivative size = 486, normalized size of antiderivative = 5.52 \[ \int \cot ^3(x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx=\left [-\frac {15 \, {\left ({\left (a b - b^{2}\right )} \cos \left (2 \, x\right )^{2} + a b - b^{2} - 2 \, {\left (a b - b^{2}\right )} \cos \left (2 \, x\right )\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 (3 \, a^{2} - 26 \, a b + 23 \, b^{2}\right )} \cos \left (2 \, x\right )^{2} + 3 \, a^{2} - 14 \, a b + 13 \, b^{2} - 2 \, {\left (3 \, a^{2} - 20 \, a b + 12 \, 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}}}{60 \, {\left (b \cos \left (2 \, x\right )^{2} - 2 \, b \cos \left (2 \, x\right ) + b\right )}}, -\frac {15 \, {\left ({\left (a b - b^{2}\right )} \cos \left (2 \, x\right )^{2} + a b - b^{2} - 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}} {\left (\cos \left (2 \, x\right ) - 1\right )}}{{\left (a - b\right )} \cos \left (2 \, x\right ) - a}\right ) + 2 \, {\left ({\left (3 \, a^{2} - 26 \, a b + 23 \, b^{2}\right )} \cos \left (2 \, x\right )^{2} + 3 \, a^{2} - 14 \, a b + 13 \, b^{2} - 2 \, {\left (3 \, a^{2} - 20 \, a b + 12 \, 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}}}{30 \, {\left (b \cos \left (2 \, x\right )^{2} - 2 \, b \cos \left (2 \, x\right ) + b\right )}}\right ] \]

[In]

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

[Out]

[-1/60*(15*((a*b - b^2)*cos(2*x)^2 + a*b - b^2 - 2*(a*b - b^2)*cos(2*x))*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*((3*a^2 - 26*a*b + 23*b^2)*cos(2*x)^2 + 3*a^2 - 14*a*
b + 13*b^2 - 2*(3*a^2 - 20*a*b + 12*b^2)*cos(2*x))*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/(b*cos(2*x
)^2 - 2*b*cos(2*x) + b), -1/30*(15*((a*b - b^2)*cos(2*x)^2 + a*b - b^2 - 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))*(cos(2*x) - 1)/((a - b)*cos(2*x) - a)) +
2*((3*a^2 - 26*a*b + 23*b^2)*cos(2*x)^2 + 3*a^2 - 14*a*b + 13*b^2 - 2*(3*a^2 - 20*a*b + 12*b^2)*cos(2*x))*sqrt
(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/(b*cos(2*x)^2 - 2*b*cos(2*x) + b)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[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 [F(-2)]

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

[In]

integrate(cot(x)^3*(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 = 24.41 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.36 \[ \int \cot ^3(x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx=\left (\frac {a}{3\,b}-\frac {a-b}{3\,b}\right )\,{\left (b\,{\mathrm {cot}\left (x\right )}^2+a\right )}^{3/2}-\frac {{\left (b\,{\mathrm {cot}\left (x\right )}^2+a\right )}^{5/2}}{5\,b}+\left (a-b\right )\,\left (\frac {a}{b}-\frac {a-b}{b}\right )\,\sqrt {b\,{\mathrm {cot}\left (x\right )}^2+a}+\mathrm {atan}\left (\frac {{\left (a-b\right )}^{3/2}\,\sqrt {b\,{\mathrm {cot}\left (x\right )}^2+a}\,1{}\mathrm {i}}{a^2-2\,a\,b+b^2}\right )\,{\left (a-b\right )}^{3/2}\,1{}\mathrm {i} \]

[In]

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

[Out]

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