\(\int \cot ^3(d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx\) [23]

Optimal result

Integrand size = 35, antiderivative size = 209 \[ \int \cot ^3(d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx=-\frac {\sqrt {a-b+c} \text {arctanh}\left (\frac {2 a-b+(b-2 c) \cot ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e}+\frac {\left (b^2+4 b c-4 c (a+2 c)\right ) \text {arctanh}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{16 c^{3/2} e}-\frac {\left (b-4 c+2 c \cot ^2(d+e x)\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{8 c e} \]

[Out]

1/16*(b^2+4*b*c-4*c*(a+2*c))*arctanh(1/2*(b+2*c*cot(e*x+d)^2)/c^(1/2)/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2))
/c^(3/2)/e-1/2*arctanh(1/2*(2*a-b+(b-2*c)*cot(e*x+d)^2)/(a-b+c)^(1/2)/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2))
*(a-b+c)^(1/2)/e-1/8*(b-4*c+2*c*cot(e*x+d)^2)*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2)/c/e

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3782, 1265, 828, 857, 635, 212, 738} \[ \int \cot ^3(d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx=\frac {\left (-4 c (a+2 c)+b^2+4 b c\right ) \text {arctanh}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{16 c^{3/2} e}-\frac {\sqrt {a-b+c} \text {arctanh}\left (\frac {2 a+(b-2 c) \cot ^2(d+e x)-b}{2 \sqrt {a-b+c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e}-\frac {\left (b+2 c \cot ^2(d+e x)-4 c\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{8 c e} \]

[In]

Int[Cot[d + e*x]^3*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4],x]

[Out]

-1/2*(Sqrt[a - b + c]*ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^
2 + c*Cot[d + e*x]^4])])/e + ((b^2 + 4*b*c - 4*c*(a + 2*c))*ArcTanh[(b + 2*c*Cot[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a
+ b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(16*c^(3/2)*e) - ((b - 4*c + 2*c*Cot[d + e*x]^2)*Sqrt[a + b*Cot[d +
e*x]^2 + c*Cot[d + e*x]^4])/(8*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 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 738

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

Rule 828

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^
2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a
+ b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -
c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*
d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0
] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) &&  !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 857

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rule 1265

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

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {x^3 \sqrt {a+b x^2+c x^4}}{1+x^2} \, dx,x,\cot (d+e x)\right )}{e} \\ & = -\frac {\text {Subst}\left (\int \frac {x \sqrt {a+b x+c x^2}}{1+x} \, dx,x,\cot ^2(d+e x)\right )}{2 e} \\ & = -\frac {\left (b-4 c+2 c \cot ^2(d+e x)\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{8 c e}+\frac {\text {Subst}\left (\int \frac {\frac {1}{2} \left (b^2+4 a c-4 b c\right )+\frac {1}{2} \left (b^2+4 b c-4 c (a+2 c)\right ) x}{(1+x) \sqrt {a+b x+c x^2}} \, dx,x,\cot ^2(d+e x)\right )}{8 c e} \\ & = -\frac {\left (b-4 c+2 c \cot ^2(d+e x)\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{8 c e}+\frac {(a-b+c) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x+c x^2}} \, dx,x,\cot ^2(d+e x)\right )}{2 e}+\frac {\left (b^2+4 b c-4 c (a+2 c)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,\cot ^2(d+e x)\right )}{16 c e} \\ & = -\frac {\left (b-4 c+2 c \cot ^2(d+e x)\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{8 c e}-\frac {(a-b+c) \text {Subst}\left (\int \frac {1}{4 a-4 b+4 c-x^2} \, dx,x,\frac {2 a-b-(-b+2 c) \cot ^2(d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{e}+\frac {\left (b^2+4 b c-4 c (a+2 c)\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c \cot ^2(d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{8 c e} \\ & = -\frac {\sqrt {a-b+c} \text {arctanh}\left (\frac {2 a-b+(b-2 c) \cot ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e}+\frac {\left (b^2+4 b c-4 c (a+2 c)\right ) \text {arctanh}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{16 c^{3/2} e}-\frac {\left (b-4 c+2 c \cot ^2(d+e x)\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{8 c e} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(777\) vs. \(2(209)=418\).

Time = 6.36 (sec) , antiderivative size = 777, normalized size of antiderivative = 3.72 \[ \int \cot ^3(d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx=\frac {\left (\frac {b \text {arctanh}\left (\frac {b+2 a \tan ^2(d+e x)}{2 \sqrt {a} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}\right )}{2 \sqrt {a}}-\sqrt {c} \text {arctanh}\left (\frac {2 c+b \tan ^2(d+e x)}{2 \sqrt {c} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}\right )+\frac {1}{2} \left (\frac {(2 a-b) \text {arctanh}\left (\frac {b+2 a \tan ^2(d+e x)}{2 \sqrt {a} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}\right )}{\sqrt {a}}-\frac {4 \sqrt {a-b+c} (2 a-2 b+2 c) \text {arctanh}\left (\frac {b-2 c-(-2 a+b) \tan ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}\right )}{4 a-4 b+4 c}\right )\right ) \tan ^2(d+e x) \sqrt {\cot ^4(d+e x) \left (c+b \tan ^2(d+e x)+a \tan ^4(d+e x)\right )}}{2 e \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}-\frac {\tan ^2(d+e x) \sqrt {\cot ^4(d+e x) \left (c+b \tan ^2(d+e x)+a \tan ^4(d+e x)\right )} \left (2 \sqrt {a} \text {arctanh}\left (\frac {b+2 a \tan ^2(d+e x)}{2 \sqrt {a} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}\right )-\frac {b \text {arctanh}\left (\frac {2 c+b \tan ^2(d+e x)}{2 \sqrt {c} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}\right )}{\sqrt {c}}-2 \cot ^2(d+e x) \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}\right )}{4 e \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}+\frac {\tan ^2(d+e x) \sqrt {\cot ^4(d+e x) \left (c+b \tan ^2(d+e x)+a \tan ^4(d+e x)\right )} \left (\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {2 c+b \tan ^2(d+e x)}{2 \sqrt {c} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}\right )}{c^{3/2}}-\frac {2 \cot ^4(d+e x) \left (2 c+b \tan ^2(d+e x)\right ) \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}{c}\right )}{16 e \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}} \]

[In]

Integrate[Cot[d + e*x]^3*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4],x]

[Out]

(((b*ArcTanh[(b + 2*a*Tan[d + e*x]^2)/(2*Sqrt[a]*Sqrt[c + b*Tan[d + e*x]^2 + a*Tan[d + e*x]^4])])/(2*Sqrt[a])
- Sqrt[c]*ArcTanh[(2*c + b*Tan[d + e*x]^2)/(2*Sqrt[c]*Sqrt[c + b*Tan[d + e*x]^2 + a*Tan[d + e*x]^4])] + (((2*a
- b)*ArcTanh[(b + 2*a*Tan[d + e*x]^2)/(2*Sqrt[a]*Sqrt[c + b*Tan[d + e*x]^2 + a*Tan[d + e*x]^4])])/Sqrt[a] - (
4*Sqrt[a - b + c]*(2*a - 2*b + 2*c)*ArcTanh[(b - 2*c - (-2*a + b)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[c +
b*Tan[d + e*x]^2 + a*Tan[d + e*x]^4])])/(4*a - 4*b + 4*c))/2)*Tan[d + e*x]^2*Sqrt[Cot[d + e*x]^4*(c + b*Tan[d
+ e*x]^2 + a*Tan[d + e*x]^4)])/(2*e*Sqrt[c + b*Tan[d + e*x]^2 + a*Tan[d + e*x]^4]) - (Tan[d + e*x]^2*Sqrt[Cot[
d + e*x]^4*(c + b*Tan[d + e*x]^2 + a*Tan[d + e*x]^4)]*(2*Sqrt[a]*ArcTanh[(b + 2*a*Tan[d + e*x]^2)/(2*Sqrt[a]*S
qrt[c + b*Tan[d + e*x]^2 + a*Tan[d + e*x]^4])] - (b*ArcTanh[(2*c + b*Tan[d + e*x]^2)/(2*Sqrt[c]*Sqrt[c + b*Tan
[d + e*x]^2 + a*Tan[d + e*x]^4])])/Sqrt[c] - 2*Cot[d + e*x]^2*Sqrt[c + b*Tan[d + e*x]^2 + a*Tan[d + e*x]^4]))/
(4*e*Sqrt[c + b*Tan[d + e*x]^2 + a*Tan[d + e*x]^4]) + (Tan[d + e*x]^2*Sqrt[Cot[d + e*x]^4*(c + b*Tan[d + e*x]^
2 + a*Tan[d + e*x]^4)]*(((b^2 - 4*a*c)*ArcTanh[(2*c + b*Tan[d + e*x]^2)/(2*Sqrt[c]*Sqrt[c + b*Tan[d + e*x]^2 +
a*Tan[d + e*x]^4])])/c^(3/2) - (2*Cot[d + e*x]^4*(2*c + b*Tan[d + e*x]^2)*Sqrt[c + b*Tan[d + e*x]^2 + a*Tan[d
+ e*x]^4])/c))/(16*e*Sqrt[c + b*Tan[d + e*x]^2 + a*Tan[d + e*x]^4])

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.52

method	result	size
derivativedivides	\(\frac {-\frac {\left (b +2 c \cot \left (e x +d \right )^{2}\right ) \sqrt {a +b \cot \left (e x +d \right )^{2}+c \cot \left (e x +d \right )^{4}}}{8 c}-\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c \cot \left (e x +d \right )^{2}}{\sqrt {c}}+\sqrt {a +b \cot \left (e x +d \right )^{2}+c \cot \left (e x +d \right )^{4}}\right )}{16 c^{\frac {3}{2}}}+\frac {\sqrt {\left (\cot \left (e x +d \right )^{2}+1\right )^{2} c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+a -b +c}}{2}+\frac {\left (b -2 c \right ) \ln \left (\frac {\frac {b}{2}-c +\left (\cot \left (e x +d \right )^{2}+1\right ) c}{\sqrt {c}}+\sqrt {\left (\cot \left (e x +d \right )^{2}+1\right )^{2} c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+a -b +c}\right )}{4 \sqrt {c}}-\frac {\sqrt {a -b +c}\, \ln \left (\frac {2 a -2 b +2 c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+2 \sqrt {a -b +c}\, \sqrt {\left (\cot \left (e x +d \right )^{2}+1\right )^{2} c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+a -b +c}}{\cot \left (e x +d \right )^{2}+1}\right )}{2}}{e}\)	\(318\)
default	\(\frac {-\frac {\left (b +2 c \cot \left (e x +d \right )^{2}\right ) \sqrt {a +b \cot \left (e x +d \right )^{2}+c \cot \left (e x +d \right )^{4}}}{8 c}-\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c \cot \left (e x +d \right )^{2}}{\sqrt {c}}+\sqrt {a +b \cot \left (e x +d \right )^{2}+c \cot \left (e x +d \right )^{4}}\right )}{16 c^{\frac {3}{2}}}+\frac {\sqrt {\left (\cot \left (e x +d \right )^{2}+1\right )^{2} c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+a -b +c}}{2}+\frac {\left (b -2 c \right ) \ln \left (\frac {\frac {b}{2}-c +\left (\cot \left (e x +d \right )^{2}+1\right ) c}{\sqrt {c}}+\sqrt {\left (\cot \left (e x +d \right )^{2}+1\right )^{2} c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+a -b +c}\right )}{4 \sqrt {c}}-\frac {\sqrt {a -b +c}\, \ln \left (\frac {2 a -2 b +2 c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+2 \sqrt {a -b +c}\, \sqrt {\left (\cot \left (e x +d \right )^{2}+1\right )^{2} c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+a -b +c}}{\cot \left (e x +d \right )^{2}+1}\right )}{2}}{e}\)	\(318\)

[In]

int(cot(e*x+d)^3*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/e*(-1/8*(b+2*c*cot(e*x+d)^2)/c*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2)-1/16*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*
cot(e*x+d)^2)/c^(1/2)+(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2))+1/2*((cot(e*x+d)^2+1)^2*c+(b-2*c)*(cot(e*x+d)^2
+1)+a-b+c)^(1/2)+1/4*(b-2*c)*ln((1/2*b-c+(cot(e*x+d)^2+1)*c)/c^(1/2)+((cot(e*x+d)^2+1)^2*c+(b-2*c)*(cot(e*x+d)
^2+1)+a-b+c)^(1/2))/c^(1/2)-1/2*(a-b+c)^(1/2)*ln((2*a-2*b+2*c+(b-2*c)*(cot(e*x+d)^2+1)+2*(a-b+c)^(1/2)*((cot(e
*x+d)^2+1)^2*c+(b-2*c)*(cot(e*x+d)^2+1)+a-b+c)^(1/2))/(cot(e*x+d)^2+1)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 569 vs. \(2 (185) = 370\).

Time = 2.34 (sec) , antiderivative size = 2344, normalized size of antiderivative = 11.22 \[ \int \cot ^3(d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx=\text {Too large to display} \]

[In]

integrate(cot(e*x+d)^3*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2),x, algorithm="fricas")

[Out]

[1/32*(8*(c^2*cos(2*e*x + 2*d) - c^2)*sqrt(a - b + c)*log(2*(a^2 - 2*a*b + b^2 + 2*(a - b)*c + c^2)*cos(2*e*x
+ 2*d)^2 + 2*a^2 - b^2 + 2*c^2 - 2*((a - b + c)*cos(2*e*x + 2*d)^2 - (2*a - b)*cos(2*e*x + 2*d) + a - c)*sqrt(
a - b + c)*sqrt(((a - b + c)*cos(2*e*x + 2*d)^2 - 2*(a - c)*cos(2*e*x + 2*d) + a + b + c)/(cos(2*e*x + 2*d)^2
- 2*cos(2*e*x + 2*d) + 1)) - 4*(a^2 - a*b + b*c - c^2)*cos(2*e*x + 2*d)) + (b^2 - 4*(a - b)*c - 8*c^2 - (b^2 -
4*(a - b)*c - 8*c^2)*cos(2*e*x + 2*d))*sqrt(c)*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*cos(2*e*x + 2*d)^2 + b^2 +
4*(a + 2*b)*c + 8*c^2 - 4*((b - 2*c)*cos(2*e*x + 2*d)^2 - 2*b*cos(2*e*x + 2*d) + b + 2*c)*sqrt(c)*sqrt(((a - b
+ c)*cos(2*e*x + 2*d)^2 - 2*(a - c)*cos(2*e*x + 2*d) + a + b + c)/(cos(2*e*x + 2*d)^2 - 2*cos(2*e*x + 2*d) +
1)) - 2*(b^2 + 4*a*c - 8*c^2)*cos(2*e*x + 2*d))/(cos(2*e*x + 2*d)^2 - 2*cos(2*e*x + 2*d) + 1)) + 4*(b*c - 2*c^
2 - (b*c - 6*c^2)*cos(2*e*x + 2*d))*sqrt(((a - b + c)*cos(2*e*x + 2*d)^2 - 2*(a - c)*cos(2*e*x + 2*d) + a + b
+ c)/(cos(2*e*x + 2*d)^2 - 2*cos(2*e*x + 2*d) + 1)))/(c^2*e*cos(2*e*x + 2*d) - c^2*e), -1/16*((b^2 - 4*(a - b)
*c - 8*c^2 - (b^2 - 4*(a - b)*c - 8*c^2)*cos(2*e*x + 2*d))*sqrt(-c)*arctan(-1/2*((b - 2*c)*cos(2*e*x + 2*d)^2
- 2*b*cos(2*e*x + 2*d) + b + 2*c)*sqrt(-c)*sqrt(((a - b + c)*cos(2*e*x + 2*d)^2 - 2*(a - c)*cos(2*e*x + 2*d) +
a + b + c)/(cos(2*e*x + 2*d)^2 - 2*cos(2*e*x + 2*d) + 1))/(((a - b)*c + c^2)*cos(2*e*x + 2*d)^2 + (a + b)*c +
c^2 - 2*(a*c - c^2)*cos(2*e*x + 2*d))) - 4*(c^2*cos(2*e*x + 2*d) - c^2)*sqrt(a - b + c)*log(2*(a^2 - 2*a*b +
b^2 + 2*(a - b)*c + c^2)*cos(2*e*x + 2*d)^2 + 2*a^2 - b^2 + 2*c^2 - 2*((a - b + c)*cos(2*e*x + 2*d)^2 - (2*a -
b)*cos(2*e*x + 2*d) + a - c)*sqrt(a - b + c)*sqrt(((a - b + c)*cos(2*e*x + 2*d)^2 - 2*(a - c)*cos(2*e*x + 2*d
) + a + b + c)/(cos(2*e*x + 2*d)^2 - 2*cos(2*e*x + 2*d) + 1)) - 4*(a^2 - a*b + b*c - c^2)*cos(2*e*x + 2*d)) -
2*(b*c - 2*c^2 - (b*c - 6*c^2)*cos(2*e*x + 2*d))*sqrt(((a - b + c)*cos(2*e*x + 2*d)^2 - 2*(a - c)*cos(2*e*x +
2*d) + a + b + c)/(cos(2*e*x + 2*d)^2 - 2*cos(2*e*x + 2*d) + 1)))/(c^2*e*cos(2*e*x + 2*d) - c^2*e), 1/32*(16*(
c^2*cos(2*e*x + 2*d) - c^2)*sqrt(-a + b - c)*arctan(((a - b + c)*cos(2*e*x + 2*d)^2 - (2*a - b)*cos(2*e*x + 2*
d) + a - c)*sqrt(-a + b - c)*sqrt(((a - b + c)*cos(2*e*x + 2*d)^2 - 2*(a - c)*cos(2*e*x + 2*d) + a + b + c)/(c
os(2*e*x + 2*d)^2 - 2*cos(2*e*x + 2*d) + 1))/((a^2 - 2*a*b + b^2 + 2*(a - b)*c + c^2)*cos(2*e*x + 2*d)^2 + a^2
- b^2 + 2*a*c + c^2 - 2*(a^2 - a*b + b*c - c^2)*cos(2*e*x + 2*d))) + (b^2 - 4*(a - b)*c - 8*c^2 - (b^2 - 4*(a
- b)*c - 8*c^2)*cos(2*e*x + 2*d))*sqrt(c)*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*cos(2*e*x + 2*d)^2 + b^2 + 4*(a
+ 2*b)*c + 8*c^2 - 4*((b - 2*c)*cos(2*e*x + 2*d)^2 - 2*b*cos(2*e*x + 2*d) + b + 2*c)*sqrt(c)*sqrt(((a - b + c)
*cos(2*e*x + 2*d)^2 - 2*(a - c)*cos(2*e*x + 2*d) + a + b + c)/(cos(2*e*x + 2*d)^2 - 2*cos(2*e*x + 2*d) + 1)) -
2*(b^2 + 4*a*c - 8*c^2)*cos(2*e*x + 2*d))/(cos(2*e*x + 2*d)^2 - 2*cos(2*e*x + 2*d) + 1)) + 4*(b*c - 2*c^2 - (
b*c - 6*c^2)*cos(2*e*x + 2*d))*sqrt(((a - b + c)*cos(2*e*x + 2*d)^2 - 2*(a - c)*cos(2*e*x + 2*d) + a + b + c)/
(cos(2*e*x + 2*d)^2 - 2*cos(2*e*x + 2*d) + 1)))/(c^2*e*cos(2*e*x + 2*d) - c^2*e), 1/16*(8*(c^2*cos(2*e*x + 2*d
) - c^2)*sqrt(-a + b - c)*arctan(((a - b + c)*cos(2*e*x + 2*d)^2 - (2*a - b)*cos(2*e*x + 2*d) + a - c)*sqrt(-a
+ b - c)*sqrt(((a - b + c)*cos(2*e*x + 2*d)^2 - 2*(a - c)*cos(2*e*x + 2*d) + a + b + c)/(cos(2*e*x + 2*d)^2 -
2*cos(2*e*x + 2*d) + 1))/((a^2 - 2*a*b + b^2 + 2*(a - b)*c + c^2)*cos(2*e*x + 2*d)^2 + a^2 - b^2 + 2*a*c + c^
2 - 2*(a^2 - a*b + b*c - c^2)*cos(2*e*x + 2*d))) - (b^2 - 4*(a - b)*c - 8*c^2 - (b^2 - 4*(a - b)*c - 8*c^2)*co
s(2*e*x + 2*d))*sqrt(-c)*arctan(-1/2*((b - 2*c)*cos(2*e*x + 2*d)^2 - 2*b*cos(2*e*x + 2*d) + b + 2*c)*sqrt(-c)*
sqrt(((a - b + c)*cos(2*e*x + 2*d)^2 - 2*(a - c)*cos(2*e*x + 2*d) + a + b + c)/(cos(2*e*x + 2*d)^2 - 2*cos(2*e
*x + 2*d) + 1))/(((a - b)*c + c^2)*cos(2*e*x + 2*d)^2 + (a + b)*c + c^2 - 2*(a*c - c^2)*cos(2*e*x + 2*d))) + 2
*(b*c - 2*c^2 - (b*c - 6*c^2)*cos(2*e*x + 2*d))*sqrt(((a - b + c)*cos(2*e*x + 2*d)^2 - 2*(a - c)*cos(2*e*x + 2
*d) + a + b + c)/(cos(2*e*x + 2*d)^2 - 2*cos(2*e*x + 2*d) + 1)))/(c^2*e*cos(2*e*x + 2*d) - c^2*e)]

Sympy [F]

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

[In]

integrate(cot(e*x+d)**3*(a+b*cot(e*x+d)**2+c*cot(e*x+d)**4)**(1/2),x)

[Out]

Integral(sqrt(a + b*cot(d + e*x)**2 + c*cot(d + e*x)**4)*cot(d + e*x)**3, x)

Maxima [F]

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

[In]

integrate(cot(e*x+d)^3*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(c*cot(e*x + d)^4 + b*cot(e*x + d)^2 + a)*cot(e*x + d)^3, x)

Giac [F(-1)]

Timed out. \[ \int \cot ^3(d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx=\text {Timed out} \]

[In]

integrate(cot(e*x+d)^3*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

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

[In]

int(cot(d + e*x)^3*(a + b*cot(d + e*x)^2 + c*cot(d + e*x)^4)^(1/2),x)

[Out]

int(cot(d + e*x)^3*(a + b*cot(d + e*x)^2 + c*cot(d + e*x)^4)^(1/2), x)