Integrand size = 35, antiderivative size = 209 \[ \int \cot ^3(d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx=-\frac {\sqrt {a-b+c} \text {arctanh}\left (\frac {2 a-b+(b-2 c) \cot ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e}+\frac {\left (b^2+4 b c-4 c (a+2 c)\right ) \text {arctanh}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{16 c^{3/2} e}-\frac {\left (b-4 c+2 c \cot ^2(d+e x)\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{8 c e} \]
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
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 \[ \int \cot ^3(d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx=\frac {\left (\frac {b \text {arctanh}\left (\frac {b+2 a \tan ^2(d+e x)}{2 \sqrt {a} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}\right )}{2 \sqrt {a}}-\sqrt {c} \text {arctanh}\left (\frac {2 c+b \tan ^2(d+e x)}{2 \sqrt {c} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}\right )+\frac {1}{2} \left (\frac {(2 a-b) \text {arctanh}\left (\frac {b+2 a \tan ^2(d+e x)}{2 \sqrt {a} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}\right )}{\sqrt {a}}-\frac {4 \sqrt {a-b+c} (2 a-2 b+2 c) \text {arctanh}\left (\frac {b-2 c-(-2 a+b) \tan ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}\right )}{4 a-4 b+4 c}\right )\right ) \tan ^2(d+e x) \sqrt {\cot ^4(d+e x) \left (c+b \tan ^2(d+e x)+a \tan ^4(d+e x)\right )}}{2 e \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}-\frac {\tan ^2(d+e x) \sqrt {\cot ^4(d+e x) \left (c+b \tan ^2(d+e x)+a \tan ^4(d+e x)\right )} \left (2 \sqrt {a} \text {arctanh}\left (\frac {b+2 a \tan ^2(d+e x)}{2 \sqrt {a} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}\right )-\frac {b \text {arctanh}\left (\frac {2 c+b \tan ^2(d+e x)}{2 \sqrt {c} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}\right )}{\sqrt {c}}-2 \cot ^2(d+e x) \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}\right )}{4 e \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}+\frac {\tan ^2(d+e x) \sqrt {\cot ^4(d+e x) \left (c+b \tan ^2(d+e x)+a \tan ^4(d+e x)\right )} \left (\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {2 c+b \tan ^2(d+e x)}{2 \sqrt {c} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}\right )}{c^{3/2}}-\frac {2 \cot ^4(d+e x) \left (2 c+b \tan ^2(d+e x)\right ) \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}{c}\right )}{16 e \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}} \]
(((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)*ArcT anh[(b - 2*c - (-2*a + b)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[c + b*Ta n[d + e*x]^2 + a*Tan[d + e*x]^4])])/(4*a - 4*b + 4*c))/2)*Tan[d + e*x]^2*S qrt[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]*Sqrt[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]*S qrt[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])
Time = 0.51 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.01, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 4184, 1578, 1231, 27, 1269, 1092, 219, 1154, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \cot ^3(d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cot (d+e x)^3 \sqrt {a+b \cot (d+e x)^2+c \cot (d+e x)^4}dx\) |
| \(\Big \downarrow \) 4184 |
| \(\displaystyle -\frac {\int \frac {\cot ^3(d+e x) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}{\cot ^2(d+e x)+1}d\cot (d+e x)}{e}\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle -\frac {\int \frac {\cot ^2(d+e x) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}{\cot ^2(d+e x)+1}d\cot ^2(d+e x)}{2 e}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle -\frac {\frac {\left (b+2 c \cot ^2(d+e x)-4 c\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{4 c}-\frac {\int \frac {b^2-4 c b+\left (b^2+4 c b-4 c (a+2 c)\right ) \cot ^2(d+e x)+4 a c}{2 \left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot ^2(d+e x)}{4 c}}{2 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\left (b+2 c \cot ^2(d+e x)-4 c\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{4 c}-\frac {\int \frac {b^2-4 c b+\left (b^2+4 c b-4 c (a+2 c)\right ) \cot ^2(d+e x)+4 a c}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot ^2(d+e x)}{8 c}}{2 e}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle -\frac {\frac {\left (b+2 c \cot ^2(d+e x)-4 c\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{4 c}-\frac {\left (-4 c (a+2 c)+b^2+4 b c\right ) \int \frac {1}{\sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot ^2(d+e x)+8 c (a-b+c) \int \frac {1}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot ^2(d+e x)}{8 c}}{2 e}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle -\frac {\frac {\left (b+2 c \cot ^2(d+e x)-4 c\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{4 c}-\frac {2 \left (-4 c (a+2 c)+b^2+4 b c\right ) \int \frac {1}{4 c-\cot ^4(d+e x)}d\frac {2 c \cot ^2(d+e x)+b}{\sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}+8 c (a-b+c) \int \frac {1}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot ^2(d+e x)}{8 c}}{2 e}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\frac {\left (b+2 c \cot ^2(d+e x)-4 c\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{4 c}-\frac {8 c (a-b+c) \int \frac {1}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot ^2(d+e x)+\frac {\left (-4 c (a+2 c)+b^2+4 b c\right ) \text {arctanh}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{\sqrt {c}}}{8 c}}{2 e}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle -\frac {\frac {\left (b+2 c \cot ^2(d+e x)-4 c\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{4 c}-\frac {\frac {\left (-4 c (a+2 c)+b^2+4 b c\right ) \text {arctanh}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{\sqrt {c}}-16 c (a-b+c) \int \frac {1}{4 (a-b+c)-\cot ^4(d+e x)}d\frac {(b-2 c) \cot ^2(d+e x)+2 a-b}{\sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}}{8 c}}{2 e}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\frac {\left (b+2 c \cot ^2(d+e x)-4 c\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{4 c}-\frac {\frac {\left (-4 c (a+2 c)+b^2+4 b c\right ) \text {arctanh}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{\sqrt {c}}-8 c \sqrt {a-b+c} \text {arctanh}\left (\frac {2 a+(b-2 c) \cot ^2(d+e x)-b}{2 \sqrt {a-b+c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{8 c}}{2 e}\) |
-1/2*(-1/8*(-8*c*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])] + ( (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])])/Sqrt[c])/c + ((b - 4*c + 2*c*Cot[d + e*x]^2)*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])/(4*c))/ e
3.1.23.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-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]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(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] - Simp[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] && GtQ[p, 0] && (IntegerQ[p] || !R ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Integer Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(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] && !IGtQ[m, 0]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[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] && Int egerQ[(m - 1)/2]
Int[cot[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*(cot[(d_.) + (e_.)*(x_)]*( f_.))^(n_.) + (c_.)*(cot[(d_.) + (e_.)*(x_)]*(f_.))^(n2_.))^(p_), x_Symbol] :> Simp[-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]
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 | \(\frac {-\frac {\left (b +2 c \cot \left (e x +d \right )^{2}\right ) \sqrt {a +b \cot \left (e x +d \right )^{2}+c \cot \left (e x +d \right )^{4}}}{8 c}-\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c \cot \left (e x +d \right )^{2}}{\sqrt {c}}+\sqrt {a +b \cot \left (e x +d \right )^{2}+c \cot \left (e x +d \right )^{4}}\right )}{16 c^{\frac {3}{2}}}+\frac {\sqrt {\left (\cot \left (e x +d \right )^{2}+1\right )^{2} c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+a -b +c}}{2}+\frac {\left (b -2 c \right ) \ln \left (\frac {\frac {b}{2}-c +\left (\cot \left (e x +d \right )^{2}+1\right ) c}{\sqrt {c}}+\sqrt {\left (\cot \left (e x +d \right )^{2}+1\right )^{2} c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+a -b +c}\right )}{4 \sqrt {c}}-\frac {\sqrt {a -b +c}\, \ln \left (\frac {2 a -2 b +2 c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+2 \sqrt {a -b +c}\, \sqrt {\left (\cot \left (e x +d \right )^{2}+1\right )^{2} c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+a -b +c}}{\cot \left (e x +d \right )^{2}+1}\right )}{2}}{e}\) | \(318\) |
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)))
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 \[ \int \cot ^3(d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx=\text {Too large to display} \]
[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^...
\[ \int \cot ^3(d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx=\int \sqrt {a + b \cot ^{2}{\left (d + e x \right )} + c \cot ^{4}{\left (d + e x \right )}} \cot ^{3}{\left (d + e x \right )}\, dx \]
\[ \int \cot ^3(d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx=\int { \sqrt {c \cot \left (e x + d\right )^{4} + b \cot \left (e x + d\right )^{2} + a} \cot \left (e x + d\right )^{3} \,d x } \]
Timed out. \[ \int \cot ^3(d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx=\text {Timed out} \]
Timed out. \[ \int \cot ^3(d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx=\int {\mathrm {cot}\left (d+e\,x\right )}^3\,\sqrt {c\,{\mathrm {cot}\left (d+e\,x\right )}^4+b\,{\mathrm {cot}\left (d+e\,x\right )}^2+a} \,d x \]