\(\int \frac {\cot ^3(d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx\) [2]

Optimal result

Integrand size = 33, antiderivative size = 384 \[ \int \frac {\cot ^3(d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=\frac {\sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {a-c-\sqrt {a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt {2} \sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2+b^2-2 a c+c^2} e}-\frac {\sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {a-c+\sqrt {a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt {2} \sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2+b^2-2 a c+c^2} e}+\frac {b \text {arctanh}\left (\frac {b+2 c \cot (d+e x)}{2 \sqrt {c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{2 c^{3/2} e}-\frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{c e} \]

[Out]

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

Rubi [A] (verified)

Time = 1.04 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {3782, 6857, 654, 635, 212, 1050, 1044, 214} \[ \int \frac {\cot ^3(d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=\frac {\sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+a-c} \text {arctanh}\left (\frac {-\sqrt {a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt {2} \sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+a-c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} e \sqrt {a^2-2 a c+b^2+c^2}}-\frac {\sqrt {\sqrt {a^2-2 a c+b^2+c^2}+a-c} \text {arctanh}\left (\frac {\sqrt {a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt {2} \sqrt {\sqrt {a^2-2 a c+b^2+c^2}+a-c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} e \sqrt {a^2-2 a c+b^2+c^2}}+\frac {b \text {arctanh}\left (\frac {b+2 c \cot (d+e x)}{2 \sqrt {c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{2 c^{3/2} e}-\frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{c e} \]

[In]

Int[Cot[d + e*x]^3/Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2],x]

[Out]

(Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Cot[d + e*x])/
(Sqrt[2]*Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*S
qrt[a^2 + b^2 - 2*a*c + c^2]*e) - (Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c + Sqrt[a^2 + b^2
- 2*a*c + c^2] + b*Cot[d + e*x])/(Sqrt[2]*Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x]
+ c*Cot[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) + (b*ArcTanh[(b + 2*c*Cot[d + e*x])/(2*Sqrt[
c]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(2*c^(3/2)*e) - Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]/
(c*e)

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 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 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 654

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

Rule 1044

Int[((g_) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[-2*
a*g*h, Subst[Int[1/Simp[2*a^2*g*h*c + a*e*x^2, x], x], x, Simp[a*h - g*c*x, x]/Sqrt[d + e*x + f*x^2]], x] /; F
reeQ[{a, c, d, e, f, g, h}, x] && EqQ[a*h^2*e + 2*g*h*(c*d - a*f) - g^2*c*e, 0]

Rule 1050

Int[((g_.) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q
= Rt[(c*d - a*f)^2 + a*c*e^2, 2]}, Dist[1/(2*q), Int[Simp[(-a)*h*e - g*(c*d - a*f - q) + (h*(c*d - a*f + q) -
g*c*e)*x, x]/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] - Dist[1/(2*q), Int[Simp[(-a)*h*e - g*(c*d - a*f + q
) + (h*(c*d - a*f - q) - g*c*e)*x, x]/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d, e, f, g,
h}, x] && NeQ[e^2 - 4*d*f, 0] && NegQ[(-a)*c]

Rule 3782

Int[cot[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*(cot[(d_.) + (e_.)*(x_)]*(f_.))^(n_.) + (c_.)*(cot[(d_.) + (e
_.)*(x_)]*(f_.))^(n2_.))^(p_), x_Symbol] :> Dist[-f/e, Subst[Int[(x/f)^m*((a + b*x^n + c*x^(2*n))^p/(f^2 + x^2
)), x], x, f*Cot[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.84 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.03 \[ \int \frac {\cot ^3(d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=\frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan (d+e x) \left (b \sqrt {a-i b-c} \sqrt {a+i b-c} \text {arctanh}\left (\frac {2 c+b \tan (d+e x)}{2 \sqrt {c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )+i \sqrt {c} \left (\sqrt {a-i b-c} c \arctan \left (\frac {i b-2 c+(2 i a-b) \tan (d+e x)}{2 \sqrt {a+i b-c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )+\sqrt {a+i b-c} \left (c \arctan \left (\frac {i b+2 c+(2 i a+b) \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )+2 i \sqrt {a-i b-c} \cot (d+e x) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}\right )\right )\right )}{2 \sqrt {a-i b-c} \sqrt {a+i b-c} c^{3/2} e \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}} \]

[In]

Integrate[Cot[d + e*x]^3/Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2],x]

[Out]

(Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x]*(b*Sqrt[a - I*b - c]*Sqrt[a + I*b - c]*ArcTanh[(2*c
+ b*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2])] + I*Sqrt[c]*(Sqrt[a - I*b - c]*c*Ar
cTan[(I*b - 2*c + ((2*I)*a - b)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2]
)] + Sqrt[a + I*b - c]*(c*ArcTan[(I*b + 2*c + ((2*I)*a + b)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[c + b*Tan[
d + e*x] + a*Tan[d + e*x]^2])] + (2*I)*Sqrt[a - I*b - c]*Cot[d + e*x]*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]
^2]))))/(2*Sqrt[a - I*b - c]*Sqrt[a + I*b - c]*c^(3/2)*e*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2])

Maple [B] (warning: unable to verify)

result has leaf size over 500,000. Avoiding possible recursion issues.

Time = 1.74 (sec) , antiderivative size = 9581108, normalized size of antiderivative = 24950.80

\[\text {output too large to display}\]

[In]

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

[Out]

result too large to display

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 11834 vs. \(2 (339) = 678\).

Time = 5.73 (sec) , antiderivative size = 23719, normalized size of antiderivative = 61.77 \[ \int \frac {\cot ^3(d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

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

[In]

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

[Out]

Integral(cot(d + e*x)**3/sqrt(a + b*cot(d + e*x) + c*cot(d + e*x)**2), x)

Maxima [F]

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

[In]

integrate(cot(e*x+d)^3/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2),x, algorithm="maxima")

[Out]

integrate(cot(e*x + d)^3/sqrt(c*cot(e*x + d)^2 + b*cot(e*x + d) + a), x)

Giac [F(-2)]

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

[In]

integrate(cot(e*x+d)^3/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Not invertible Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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