\(\int \frac {\cot (x)}{(a+b \cot ^4(x))^{3/2}} \, dx\) [63]

Optimal result

Integrand size = 15, antiderivative size = 74 \[ \int \frac {\cot (x)}{\left (a+b \cot ^4(x)\right )^{3/2}} \, dx=\frac {\text {arctanh}\left (\frac {a-b \cot ^2(x)}{\sqrt {a+b} \sqrt {a+b \cot ^4(x)}}\right )}{2 (a+b)^{3/2}}-\frac {a+b \cot ^2(x)}{2 a (a+b) \sqrt {a+b \cot ^4(x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3751, 1262, 755, 12, 739, 212} \[ \int \frac {\cot (x)}{\left (a+b \cot ^4(x)\right )^{3/2}} \, dx=\frac {\text {arctanh}\left (\frac {a-b \cot ^2(x)}{\sqrt {a+b} \sqrt {a+b \cot ^4(x)}}\right )}{2 (a+b)^{3/2}}-\frac {a+b \cot ^2(x)}{2 a (a+b) \sqrt {a+b \cot ^4(x)}} \]

[In]

Int[Cot[x]/(a + b*Cot[x]^4)^(3/2),x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 755

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(a*e + c*d*x)*
((a + c*x^2)^(p + 1)/(2*a*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[1/(2*a*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^
m*Simp[c*d^2*(2*p + 3) + a*e^2*(m + 2*p + 3) + c*e*d*(m + 2*p + 4)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[
{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 1262

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(d + e*x)^q
*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x]

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 (1+x^2\right ) \left (a+b x^4\right )^{3/2}} \, dx,x,\cot (x)\right ) \\ & = -\left (\frac {1}{2} \text {Subst}\left (\int \frac {1}{(1+x) \left (a+b x^2\right )^{3/2}} \, dx,x,\cot ^2(x)\right )\right ) \\ & = -\frac {a+b \cot ^2(x)}{2 a (a+b) \sqrt {a+b \cot ^4(x)}}-\frac {\text {Subst}\left (\int \frac {a}{(1+x) \sqrt {a+b x^2}} \, dx,x,\cot ^2(x)\right )}{2 a (a+b)} \\ & = -\frac {a+b \cot ^2(x)}{2 a (a+b) \sqrt {a+b \cot ^4(x)}}-\frac {\text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x^2}} \, dx,x,\cot ^2(x)\right )}{2 (a+b)} \\ & = -\frac {a+b \cot ^2(x)}{2 a (a+b) \sqrt {a+b \cot ^4(x)}}+\frac {\text {Subst}\left (\int \frac {1}{a+b-x^2} \, dx,x,\frac {a-b \cot ^2(x)}{\sqrt {a+b \cot ^4(x)}}\right )}{2 (a+b)} \\ & = \frac {\text {arctanh}\left (\frac {a-b \cot ^2(x)}{\sqrt {a+b} \sqrt {a+b \cot ^4(x)}}\right )}{2 (a+b)^{3/2}}-\frac {a+b \cot ^2(x)}{2 a (a+b) \sqrt {a+b \cot ^4(x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.99 \[ \int \frac {\cot (x)}{\left (a+b \cot ^4(x)\right )^{3/2}} \, dx=\frac {1}{2} \left (\frac {\text {arctanh}\left (\frac {a-b \cot ^2(x)}{\sqrt {a+b} \sqrt {a+b \cot ^4(x)}}\right )}{(a+b)^{3/2}}-\frac {a+b \cot ^2(x)}{a (a+b) \sqrt {a+b \cot ^4(x)}}\right ) \]

[In]

Integrate[Cot[x]/(a + b*Cot[x]^4)^(3/2),x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(247\) vs. \(2(65)=130\).

Time = 1.02 (sec) , antiderivative size = 248, normalized size of antiderivative = 3.35

method	result	size
derivativedivides	\(-\frac {b \ln \left (\frac {2 a +2 b -2 b \left (\cot \left (x \right )^{2}+1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\cot \left (x \right )^{2}+1\right )^{2}-2 b \left (\cot \left (x \right )^{2}+1\right )+a +b}}{\cot \left (x \right )^{2}+1}\right )}{2 \left (b +\sqrt {-a b}\right ) \left (-b +\sqrt {-a b}\right ) \sqrt {a +b}}+\frac {\sqrt {\left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}}{4 a \left (-b +\sqrt {-a b}\right ) \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}-\frac {\sqrt {\left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}}{4 a \left (b +\sqrt {-a b}\right ) \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}\)	\(248\)
default	\(-\frac {b \ln \left (\frac {2 a +2 b -2 b \left (\cot \left (x \right )^{2}+1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\cot \left (x \right )^{2}+1\right )^{2}-2 b \left (\cot \left (x \right )^{2}+1\right )+a +b}}{\cot \left (x \right )^{2}+1}\right )}{2 \left (b +\sqrt {-a b}\right ) \left (-b +\sqrt {-a b}\right ) \sqrt {a +b}}+\frac {\sqrt {\left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}}{4 a \left (-b +\sqrt {-a b}\right ) \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}-\frac {\sqrt {\left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}}{4 a \left (b +\sqrt {-a b}\right ) \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}\)	\(248\)

[In]

int(cot(x)/(a+b*cot(x)^4)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 338 vs. \(2 (64) = 128\).

Time = 0.45 (sec) , antiderivative size = 670, normalized size of antiderivative = 9.05 \[ \int \frac {\cot (x)}{\left (a+b \cot ^4(x)\right )^{3/2}} \, dx=\left [\frac {{\left ({\left (a^{2} + a b\right )} \cos \left (2 \, x\right )^{2} + a^{2} + a b - 2 \, {\left (a^{2} - a b\right )} \cos \left (2 \, x\right )\right )} \sqrt {a + b} \log \left (\frac {1}{2} \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (2 \, x\right )^{2} + \frac {1}{2} \, a^{2} + \frac {1}{2} \, b^{2} + \frac {1}{2} \, {\left ({\left (a + b\right )} \cos \left (2 \, x\right )^{2} - 2 \, a \cos \left (2 \, x\right ) + a - b\right )} \sqrt {a + b} \sqrt {\frac {{\left (a + b\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a - b\right )} \cos \left (2 \, x\right ) + a + b}{\cos \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}} - {\left (a^{2} - b^{2}\right )} \cos \left (2 \, x\right )\right ) - 2 \, {\left ({\left (a^{2} - b^{2}\right )} \cos \left (2 \, x\right )^{2} + a^{2} + 2 \, a b + b^{2} - 2 \, {\left (a^{2} + a b\right )} \cos \left (2 \, x\right )\right )} \sqrt {\frac {{\left (a + b\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a - b\right )} \cos \left (2 \, x\right ) + a + b}{\cos \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}}}{4 \, {\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3} + {\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a^{4} + a^{3} b - a^{2} b^{2} - a b^{3}\right )} \cos \left (2 \, x\right )\right )}}, -\frac {{\left ({\left (a^{2} + a b\right )} \cos \left (2 \, x\right )^{2} + a^{2} + a b - 2 \, {\left (a^{2} - a b\right )} \cos \left (2 \, x\right )\right )} \sqrt {-a - b} \arctan \left (\frac {{\left ({\left (a + b\right )} \cos \left (2 \, x\right )^{2} - 2 \, a \cos \left (2 \, x\right ) + a - b\right )} \sqrt {-a - b} \sqrt {\frac {{\left (a + b\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a - b\right )} \cos \left (2 \, x\right ) + a + b}{\cos \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}}}{{\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (2 \, x\right )^{2} + a^{2} + 2 \, a b + b^{2} - 2 \, {\left (a^{2} - b^{2}\right )} \cos \left (2 \, x\right )}\right ) + {\left ({\left (a^{2} - b^{2}\right )} \cos \left (2 \, x\right )^{2} + a^{2} + 2 \, a b + b^{2} - 2 \, {\left (a^{2} + a b\right )} \cos \left (2 \, x\right )\right )} \sqrt {\frac {{\left (a + b\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a - b\right )} \cos \left (2 \, x\right ) + a + b}{\cos \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}}}{2 \, {\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3} + {\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a^{4} + a^{3} b - a^{2} b^{2} - a b^{3}\right )} \cos \left (2 \, x\right )\right )}}\right ] \]

[In]

integrate(cot(x)/(a+b*cot(x)^4)^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {\cot (x)}{\left (a+b \cot ^4(x)\right )^{3/2}} \, dx=\int \frac {\cot {\left (x \right )}}{\left (a + b \cot ^{4}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(cot(x)/(a+b*cot(x)**4)**(3/2),x)

[Out]

Integral(cot(x)/(a + b*cot(x)**4)**(3/2), x)

Maxima [F]

\[ \int \frac {\cot (x)}{\left (a+b \cot ^4(x)\right )^{3/2}} \, dx=\int { \frac {\cot \left (x\right )}{{\left (b \cot \left (x\right )^{4} + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(cot(x)/(a+b*cot(x)^4)^(3/2),x, algorithm="maxima")

[Out]

integrate(cot(x)/(b*cot(x)^4 + a)^(3/2), x)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.50 \[ \int \frac {\cot (x)}{\left (a+b \cot ^4(x)\right )^{3/2}} \, dx=-\frac {\frac {{\left (a - b\right )} \sin \left (x\right )^{2}}{a^{2} + a b} + \frac {b}{a^{2} + a b}}{2 \, \sqrt {a \sin \left (x\right )^{4} + b \sin \left (x\right )^{4} - 2 \, b \sin \left (x\right )^{2} + b}} - \frac {\log \left ({\left | -{\left (\sqrt {a + b} \sin \left (x\right )^{2} - \sqrt {a \sin \left (x\right )^{4} + b \sin \left (x\right )^{4} - 2 \, b \sin \left (x\right )^{2} + b}\right )} \sqrt {a + b} + b \right |}\right )}{2 \, {\left (a + b\right )}^{\frac {3}{2}}} \]

[In]

integrate(cot(x)/(a+b*cot(x)^4)^(3/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot (x)}{\left (a+b \cot ^4(x)\right )^{3/2}} \, dx=\int \frac {\mathrm {cot}\left (x\right )}{{\left (b\,{\mathrm {cot}\left (x\right )}^4+a\right )}^{3/2}} \,d x \]

[In]

int(cot(x)/(a + b*cot(x)^4)^(3/2),x)

[Out]

int(cot(x)/(a + b*cot(x)^4)^(3/2), x)