Integrand size = 33, antiderivative size = 686 \[ \int \frac {\cot ^3(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\frac {\sqrt {2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2+(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {b^2-(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )-b \left (2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}\right ) \cot (d+e x)}{\sqrt {2} \sqrt {2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2+(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}-\frac {\sqrt {2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2-(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {b^2-(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )-b \left (2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}\right ) \cot (d+e x)}{\sqrt {2} \sqrt {2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2-(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}-\frac {2 (2 a+b \cot (d+e x))}{\left (b^2-4 a c\right ) e \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}+\frac {2 \left (a \left (b^2-2 (a-c) c\right )+b c (a+c) \cot (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \]
-2*(2*a+b*cot(e*x+d))/(-4*a*c+b^2)/e/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2) +2*(a*(b^2-2*(a-c)*c)+b*c*(a+c)*cot(e*x+d))/(b^2+(a-c)^2)/(-4*a*c+b^2)/e/( a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)-1/2*arctanh(1/2*(b^2-(a-c)*(a-c-(a^2- 2*a*c+b^2+c^2)^(1/2))-b*cot(e*x+d)*(2*a-2*c+(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)/(2*a-2*c+(a^2-2*a*c+b^2+c^2)^( 1/2))^(1/2)/(a^2-b^2-2*a*c+c^2-(a-c)*(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2))*(2* a-2*c+(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)*(a^2-b^2-2*a*c+c^2-(a-c)*(a^2-2*a*c +b^2+c^2)^(1/2))^(1/2)/(a^2-2*a*c+b^2+c^2)^(3/2)/e*2^(1/2)+1/2*arctanh(1/2 *(b^2-b*cot(e*x+d)*(2*a-2*c-(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)))*2^(1/2)/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)/(2*a-2*c -(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)/(a^2-b^2-2*a*c+c^2+(a-c)*(a^2-2*a*c+b^2+ c^2)^(1/2))^(1/2))*(2*a-2*c-(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)*(a^2-b^2-2*a* c+c^2+(a-c)*(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)/(a^2-2*a*c+b^2+c^2)^(3/2)/e*2 ^(1/2)
Result contains complex when optimal does not.
Time = 6.68 (sec) , antiderivative size = 1419, normalized size of antiderivative = 2.07 \[ \int \frac {\cot ^3(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\frac {4 \cot (d+e x) (b+2 a \tan (d+e x)) \left (-\frac {a \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )}{b^2-4 a c}\right )^{3/2}}{a e \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right ) \sqrt {\cot ^2(d+e x) \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )} \sqrt {1-\left (b^2-4 a c\right ) \left (\frac {b}{b^2-4 a c}+\frac {2 a \tan (d+e x)}{b^2-4 a c}\right )^2}}-\frac {\cot (d+e x) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)} \left (-\frac {2 \tan ^3(d+e x) \left (-b^2+2 a c-a b \tan (d+e x)\right )}{c \left (b^2-4 a c\right ) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}-\frac {2 \left (b \tan ^2(d+e x) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}+\frac {\frac {\left (-6 a^2 b^2 c+24 a^3 c^2\right ) \text {arctanh}\left (\frac {b+2 a \tan (d+e x)}{2 \sqrt {a} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )}{4 a^{5/2}}+\frac {\left (6 a^2 b c-12 a^3 c \tan (d+e x)\right ) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}{2 a^2}}{3 a}\right )}{c \left (b^2-4 a c\right )}\right )}{e \sqrt {\cot ^2(d+e x) \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )}}+\frac {\cot (d+e x) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)} \left (\frac {2 \left (-\frac {4 \sqrt {a-i b-c} \left (-\frac {1}{4} b \left (b^2-4 a c\right )+\frac {1}{4} i (a-c) \left (b^2-4 a c\right )\right ) \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 )}{-4 a+4 i b+4 c}-\frac {4 \sqrt {a+i b-c} \left (-\frac {1}{4} b \left (b^2-4 a c\right )-\frac {1}{4} i (a-c) \left (b^2-4 a c\right )\right ) \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 )}{-4 a-4 i b+4 c}\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right )}-\frac {2 \tan ^3(d+e x) \left (-b^2+2 a c-a b \tan (d+e x)\right )}{c \left (b^2-4 a c\right ) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}-\frac {2 \left (b^3+a b (a-3 c)+a \left (2 a^2+b^2-2 a c\right ) \tan (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}+\frac {4 \left (b^2-4 a c\right ) \left (\frac {a^2}{\left (b^2-4 a c\right ) \left (\frac {a^2 b^2}{\left (b^2-4 a c\right )^2}-\frac {4 a^3 c}{\left (b^2-4 a c\right )^2}\right )}\right )^{3/2} \left (-\frac {a b}{b^2-4 a c}-\frac {2 a^2 \tan (d+e x)}{b^2-4 a c}\right ) \left (-\frac {a \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )}{b^2-4 a c}\right )^{3/2}}{a^2 \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )^{3/2} \sqrt {1-\frac {\left (-\frac {a b}{b^2-4 a c}-\frac {2 a^2 \tan (d+e x)}{b^2-4 a c}\right )^2}{\frac {a^2 b^2}{\left (b^2-4 a c\right )^2}-\frac {4 a^3 c}{\left (b^2-4 a c\right )^2}}}}-\frac {2 \left (b \tan ^2(d+e x) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}+\frac {\frac {\left (-6 a^2 b^2 c+24 a^3 c^2\right ) \text {arctanh}\left (\frac {b+2 a \tan (d+e x)}{2 \sqrt {a} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )}{4 a^{5/2}}+\frac {\left (6 a^2 b c-12 a^3 c \tan (d+e x)\right ) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}{2 a^2}}{3 a}\right )}{c \left (b^2-4 a c\right )}\right )}{e \sqrt {\cot ^2(d+e x) \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )}} \]
(4*Cot[d + e*x]*(b + 2*a*Tan[d + e*x])*(-((a*(c + b*Tan[d + e*x] + a*Tan[d + e*x]^2))/(b^2 - 4*a*c)))^(3/2))/(a*e*(c + b*Tan[d + e*x] + a*Tan[d + e* x]^2)*Sqrt[Cot[d + e*x]^2*(c + b*Tan[d + e*x] + a*Tan[d + e*x]^2)]*Sqrt[1 - (b^2 - 4*a*c)*(b/(b^2 - 4*a*c) + (2*a*Tan[d + e*x])/(b^2 - 4*a*c))^2]) - (Cot[d + e*x]*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2]*((-2*Tan[d + e* x]^3*(-b^2 + 2*a*c - a*b*Tan[d + e*x]))/(c*(b^2 - 4*a*c)*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2]) - (2*(b*Tan[d + e*x]^2*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2] + (((-6*a^2*b^2*c + 24*a^3*c^2)*ArcTanh[(b + 2*a*Tan[ d + e*x])/(2*Sqrt[a]*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2])])/(4*a^( 5/2)) + ((6*a^2*b*c - 12*a^3*c*Tan[d + e*x])*Sqrt[c + b*Tan[d + e*x] + a*T an[d + e*x]^2])/(2*a^2))/(3*a)))/(c*(b^2 - 4*a*c))))/(e*Sqrt[Cot[d + e*x]^ 2*(c + b*Tan[d + e*x] + a*Tan[d + e*x]^2)]) + (Cot[d + e*x]*Sqrt[c + b*Tan [d + e*x] + a*Tan[d + e*x]^2]*((2*((-4*Sqrt[a - I*b - c]*(-1/4*(b*(b^2 - 4 *a*c)) + (I/4)*(a - c)*(b^2 - 4*a*c))*ArcTan[(I*b + 2*c - ((-2*I)*a - b)*T an[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x] ^2])])/(-4*a + (4*I)*b + 4*c) - (4*Sqrt[a + I*b - c]*(-1/4*(b*(b^2 - 4*a*c )) - (I/4)*(a - c)*(b^2 - 4*a*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 ])])/(-4*a - (4*I)*b + 4*c)))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)) - (2*Tan[d + e*x]^3*(-b^2 + 2*a*c - a*b*Tan[d + e*x]))/(c*(b^2 - 4*a*c)*Sqrt[c + ...
Time = 4.29 (sec) , antiderivative size = 679, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {3042, 4184, 7276, 2009}
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.
\(\displaystyle \int \frac {\cot ^3(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cot (d+e x)^3}{\left (a+b \cot (d+e x)+c \cot (d+e x)^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 4184 |
\(\displaystyle -\frac {\int \frac {\cot ^3(d+e x)}{\left (\cot ^2(d+e x)+1\right ) \left (c \cot ^2(d+e x)+b \cot (d+e x)+a\right )^{3/2}}d\cot (d+e x)}{e}\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle -\frac {\int \left (\frac {\cot (d+e x)}{\left (c \cot ^2(d+e x)+b \cot (d+e x)+a\right )^{3/2}}-\frac {\cot (d+e x)}{\left (\cot ^2(d+e x)+1\right ) \left (c \cot ^2(d+e x)+b \cot (d+e x)+a\right )^{3/2}}\right )d\cot (d+e x)}{e}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {-\frac {\sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt {(a-c) \sqrt {a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \text {arctanh}\left (\frac {-b \left (-\sqrt {a^2-2 a c+b^2+c^2}+2 a-2 c\right ) \cot (d+e x)-(a-c) \left (\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )+b^2}{\sqrt {2} \sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt {(a-c) \sqrt {a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \left (a^2-2 a c+b^2+c^2\right )^{3/2}}+\frac {\sqrt {\sqrt {a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt {-(a-c) \sqrt {a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \text {arctanh}\left (\frac {-b \left (\sqrt {a^2-2 a c+b^2+c^2}+2 a-2 c\right ) \cot (d+e x)-(a-c) \left (-\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )+b^2}{\sqrt {2} \sqrt {\sqrt {a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt {-(a-c) \sqrt {a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \left (a^2-2 a c+b^2+c^2\right )^{3/2}}+\frac {2 (2 a+b \cot (d+e x))}{\left (b^2-4 a c\right ) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}-\frac {2 \left (a \left (b^2-2 c (a-c)\right )+b c (a+c) \cot (d+e x)\right )}{\left ((a-c)^2+b^2\right ) \left (b^2-4 a c\right ) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}}{e}\) |
-((-((Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a *c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*( a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c - Sqrt[a^2 + b^2 - 2 *a*c + c^2])*Cot[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a* c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (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]*(a^2 + b^2 - 2*a*c + c^2)^(3/2))) + (Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*S qrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTa nh[(b^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])*Cot[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c + Sq rt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (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])])/(S qrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)) + (2*(2*a + b*Cot[d + e*x]))/((b^2 - 4*a*c)*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]) - (2*(a*(b^2 - 2*(a - c)*c) + b*c*(a + c)*Cot[d + e*x]))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)*Sqr t[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]))/e)
3.1.13.3.1 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 = 1.53 (sec) , antiderivative size = 13067316, normalized size of antiderivative = 19048.57
\[\text {output too large to display}\]
Leaf count of result is larger than twice the leaf count of optimal. 21092 vs. \(2 (629) = 1258\).
Time = 12.13 (sec) , antiderivative size = 21092, normalized size of antiderivative = 30.75 \[ \int \frac {\cot ^3(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\text {Too large to display} \]
\[ \int \frac {\cot ^3(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\int \frac {\cot ^{3}{\left (d + e x \right )}}{\left (a + b \cot {\left (d + e x \right )} + c \cot ^{2}{\left (d + e x \right )}\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {\cot ^3(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Exception generated. \[ \int \frac {\cot ^3(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Not invertible Error: Bad Argument Value
Timed out. \[ \int \frac {\cot ^3(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\int \frac {{\mathrm {cot}\left (d+e\,x\right )}^3}{{\left (c\,{\mathrm {cot}\left (d+e\,x\right )}^2+b\,\mathrm {cot}\left (d+e\,x\right )+a\right )}^{3/2}} \,d x \]