Integrand size = 33, antiderivative size = 747 \[ \int \cot ^3(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx=\frac {\sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \arctan \left (\frac {b^2+(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )-b \sqrt {a^2+b^2-2 a c+c^2} \cot (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}+\frac {b \text {arctanh}\left (\frac {b+2 c \cot (d+e x)}{2 \sqrt {c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{2 \sqrt {c} e}-\frac {b \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c \cot (d+e x)}{2 \sqrt {c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{16 c^{5/2} e}-\frac {\sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \text {arctanh}\left (\frac {b^2+(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )+b \sqrt {a^2+b^2-2 a c+c^2} \cot (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}+\frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{e}+\frac {b (b+2 c \cot (d+e x)) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{8 c^2 e}-\frac {\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}}{3 c e} \]
-1/16*b*(-4*a*c+b^2)*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^(5/2)/e-1/3*(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(3/ 2)/c/e+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))/e/c^(1/2)+(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)/e+1/8*b*( b+2*c*cot(e*x+d))*(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)/c^2/e-1/2*arctanh( 1/2*(b^2+b*cot(e*x+d)*(a^2-2*a*c+b^2+c^2)^(1/2)+(a-c)*(a-c+(a^2-2*a*c+b^2+ c^2)^(1/2)))/(a^2-2*a*c+b^2+c^2)^(1/4)*2^(1/2)/(a+b*cot(e*x+d)+c*cot(e*x+d )^2)^(1/2)/(a^2+b^2+c*(c-(a^2-2*a*c+b^2+c^2)^(1/2))-a*(2*c-(a^2-2*a*c+b^2+ c^2)^(1/2)))^(1/2))*(a^2+b^2+c*(c-(a^2-2*a*c+b^2+c^2)^(1/2))-a*(2*c-(a^2-2 *a*c+b^2+c^2)^(1/2)))^(1/2)/(a^2-2*a*c+b^2+c^2)^(1/4)/e*2^(1/2)+1/2*arctan (1/2*(b^2+(a-c)*(a-c-(a^2-2*a*c+b^2+c^2)^(1/2))-b*cot(e*x+d)*(a^2-2*a*c+b^ 2+c^2)^(1/2))/(a^2-2*a*c+b^2+c^2)^(1/4)*2^(1/2)/(a+b*cot(e*x+d)+c*cot(e*x+ d)^2)^(1/2)/(a^2+b^2+c*(c+(a^2-2*a*c+b^2+c^2)^(1/2))-a*(2*c+(a^2-2*a*c+b^2 +c^2)^(1/2)))^(1/2))*(a^2+b^2+c*(c+(a^2-2*a*c+b^2+c^2)^(1/2))-a*(2*c+(a^2- 2*a*c+b^2+c^2)^(1/2)))^(1/2)/(a^2-2*a*c+b^2+c^2)^(1/4)/e*2^(1/2)
Result contains complex when optimal does not.
Time = 6.30 (sec) , antiderivative size = 381, normalized size of antiderivative = 0.51 \[ \int \cot ^3(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx=\frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan (d+e x) \left (-3 b \left (b^2-4 c (a+2 c)\right ) \text {arctanh}\left (\frac {2 c+b \tan (d+e x)}{2 \sqrt {c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )+2 \sqrt {c} \left (12 i \sqrt {a+i b-c} c^2 \arctan \left (\frac {i b-2 c+(2 i a-b) \tan (d+e x)}{2 \sqrt {a+i b-c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )+12 i \sqrt {a-i b-c} c^2 \arctan \left (\frac {i b+2 c+(2 i a+b) \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )+\cot (d+e x) \left (3 b^2-8 a c+24 c^2-2 b c \cot (d+e x)-8 c^2 \cot ^2(d+e x)\right ) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}\right )\right )}{48 c^{5/2} e \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}} \]
(Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x]*(-3*b*(b^2 - 4*c *(a + 2*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])] + 2*Sqrt[c]*((12*I)*Sqrt[a + I*b - c]*c^2*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])] + (12*I)*Sqrt[a - I*b - c]*c^2*ArcTan[( I*b + 2*c + ((2*I)*a + b)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[c + b*Ta n[d + e*x] + a*Tan[d + e*x]^2])] + Cot[d + e*x]*(3*b^2 - 8*a*c + 24*c^2 - 2*b*c*Cot[d + e*x] - 8*c^2*Cot[d + e*x]^2)*Sqrt[c + b*Tan[d + e*x] + a*Tan [d + e*x]^2])))/(48*c^(5/2)*e*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2])
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.
\(\displaystyle \int \cot ^3(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \cot (d+e x)^3 \sqrt {a+b \cot (d+e x)+c \cot (d+e x)^2}dx\) |
\(\Big \downarrow \) 4184 |
\(\displaystyle -\frac {\int \frac {\cot ^3(d+e x) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}{\cot ^2(d+e x)+1}d\cot (d+e x)}{e}\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle -\frac {\int \left (\cot (d+e x) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}-\frac {\cot (d+e x) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}{\cot ^2(d+e x)+1}\right )d\cot (d+e x)}{e}\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle -\frac {\int \left (\cot (d+e x) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}-\frac {\cot (d+e x) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}{\cot ^2(d+e x)+1}\right )d\cot (d+e x)}{e}\) |
3.1.7.3.1 Defintions of rubi rules used
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]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 3.96 (sec) , antiderivative size = 17768080, normalized size of antiderivative = 23785.92
\[\text {output too large to display}\]
Leaf count of result is larger than twice the leaf count of optimal. 5923 vs. \(2 (670) = 1340\).
Time = 2.41 (sec) , antiderivative size = 11897, normalized size of antiderivative = 15.93 \[ \int \cot ^3(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx=\text {Too large to display} \]
\[ \int \cot ^3(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx=\int \sqrt {a + b \cot {\left (d + e x \right )} + c \cot ^{2}{\left (d + e x \right )}} \cot ^{3}{\left (d + e x \right )}\, dx \]
\[ \int \cot ^3(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx=\int { \sqrt {c \cot \left (e x + d\right )^{2} + b \cot \left (e x + d\right ) + a} \cot \left (e x + d\right )^{3} \,d x } \]
\[ \int \cot ^3(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx=\int { \sqrt {c \cot \left (e x + d\right )^{2} + b \cot \left (e x + d\right ) + a} \cot \left (e x + d\right )^{3} \,d x } \]
Timed out. \[ \int \cot ^3(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx=\int {\mathrm {cot}\left (d+e\,x\right )}^3\,\sqrt {c\,{\mathrm {cot}\left (d+e\,x\right )}^2+b\,\mathrm {cot}\left (d+e\,x\right )+a} \,d x \]