Integrand size = 33, antiderivative size = 686 \[ \int \frac {\cot ^3(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\frac {\sqrt {2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2+(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {b^2-(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )-b \left (2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}\right ) \cot (d+e x)}{\sqrt {2} \sqrt {2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2+(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}-\frac {\sqrt {2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2-(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {b^2-(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )-b \left (2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}\right ) \cot (d+e x)}{\sqrt {2} \sqrt {2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2-(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}-\frac {2 (2 a+b \cot (d+e x))}{\left (b^2-4 a c\right ) e \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}+\frac {2 \left (a \left (b^2-2 (a-c) c\right )+b c (a+c) \cot (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \] Output:
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)*arctanh(1/2*(b^2-(a-c)*(a-c+(a^2-2*a*c+b^2+c ^2)^(1/2))-b*(2*a-2*c-(a^2-2*a*c+b^2+c^2)^(1/2))*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)/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2))*2^(1/2)/(a^2-2 *a*c+b^2+c^2)^(3/2)/e-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)*arctanh(1/2*(b^2-(a-c) *(a-c-(a^2-2*a*c+b^2+c^2)^(1/2))-b*(2*a-2*c+(a^2-2*a*c+b^2+c^2)^(1/2))*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)/(a+b*cot(e*x+d)+c*cot(e*x+d)^2) ^(1/2))*2^(1/2)/(a^2-2*a*c+b^2+c^2)^(3/2)/e-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)
Result contains complex when optimal does not.
Time = 4.46 (sec) , antiderivative size = 404, normalized size of antiderivative = 0.59 \[ \int \frac {\cot ^3(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=-\frac {-i \sqrt {a+i b-c} (-a+i b+c)^2 \left (-b^2+4 a c\right ) \arctan \left (\frac {-i b+2 c+(-2 i a+b) \tan (d+e x)}{2 \sqrt {a+i b-c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right ) \cot (d+e x) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}+(a+i b-c) \left (4 (a-i b-c) \left (a \left (2 a^2+b^2-2 a c\right )+b \left (a^2+b^2-3 a c\right ) \cot (d+e x)\right )+\sqrt {a-i b-c} \left (b^2 (b+i c)+4 i a^2 c-i a \left (b^2-4 i b c+4 c^2\right )\right ) \arctan \left (\frac {i b+2 c+(2 i a+b) \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right ) \cot (d+e x) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}\right )}{2 \left (b^2-4 a c\right ) \left (a^2+b^2-2 a c+c^2\right )^2 e \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \] Input:
Integrate[Cot[d + e*x]^3/(a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2),x]
Output:
-1/2*((-I)*Sqrt[a + I*b - c]*(-a + I*b + c)^2*(-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])]*Cot[d + e*x]*Sqrt[c + b*Tan[d + e*x] + a*T an[d + e*x]^2] + (a + I*b - c)*(4*(a - I*b - c)*(a*(2*a^2 + b^2 - 2*a*c) + b*(a^2 + b^2 - 3*a*c)*Cot[d + e*x]) + Sqrt[a - I*b - c]*(b^2*(b + I*c) + (4*I)*a^2*c - I*a*(b^2 - (4*I)*b*c + 4*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])]*Cot[d + e*x]*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2]))/((b ^2 - 4*a*c)*(a^2 + b^2 - 2*a*c + c^2)^2*e*Sqrt[a + b*Cot[d + e*x] + c*Cot[ d + e*x]^2])
Time = 3.37 (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 |
\(\displaystyle -\frac {-\frac {\sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt {(a-c) \sqrt {a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \text {arctanh}\left (\frac {-b \left (-\sqrt {a^2-2 a c+b^2+c^2}+2 a-2 c\right ) \cot (d+e x)-(a-c) \left (\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )+b^2}{\sqrt {2} \sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt {(a-c) \sqrt {a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \left (a^2-2 a c+b^2+c^2\right )^{3/2}}+\frac {\sqrt {\sqrt {a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt {-(a-c) \sqrt {a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \text {arctanh}\left (\frac {-b \left (\sqrt {a^2-2 a c+b^2+c^2}+2 a-2 c\right ) \cot (d+e x)-(a-c) \left (-\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )+b^2}{\sqrt {2} \sqrt {\sqrt {a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt {-(a-c) \sqrt {a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \left (a^2-2 a c+b^2+c^2\right )^{3/2}}+\frac {2 (2 a+b \cot (d+e x))}{\left (b^2-4 a c\right ) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}-\frac {2 \left (a \left (b^2-2 c (a-c)\right )+b c (a+c) \cot (d+e x)\right )}{\left ((a-c)^2+b^2\right ) \left (b^2-4 a c\right ) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}}{e}\) |
Input:
Int[Cot[d + e*x]^3/(a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2),x]
Output:
-((-((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)
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 = 1.80 (sec) , antiderivative size = 13066491, normalized size of antiderivative = 19047.36
\[\text {output too large to display}\]
Input:
int(cot(e*x+d)^3/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(3/2),x)
Output:
result too large to display
Leaf count of result is larger than twice the leaf count of optimal. 21092 vs. \(2 (629) = 1258\).
Time = 10.69 (sec) , antiderivative size = 21092, normalized size of antiderivative = 30.75 \[ \int \frac {\cot ^3(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate(cot(e*x+d)^3/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(3/2),x, algorithm= "fricas")
Output:
Too large to include
\[ \int \frac {\cot ^3(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\int \frac {\cot ^{3}{\left (d + e x \right )}}{\left (a + b \cot {\left (d + e x \right )} + c \cot ^{2}{\left (d + e x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(cot(e*x+d)**3/(a+b*cot(e*x+d)+c*cot(e*x+d)**2)**(3/2),x)
Output:
Integral(cot(d + e*x)**3/(a + b*cot(d + e*x) + c*cot(d + e*x)**2)**(3/2), x)
Exception generated. \[ \int \frac {\cot ^3(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cot(e*x+d)^3/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(3/2),x, algorithm= "maxima")
Output:
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. \[ \int \frac {\cot ^3(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(cot(e*x+d)^3/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(3/2),x, algorithm= "giac")
Output:
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. \[ \int \frac {\cot ^3(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\int \frac {{\mathrm {cot}\left (d+e\,x\right )}^3}{{\left (c\,{\mathrm {cot}\left (d+e\,x\right )}^2+b\,\mathrm {cot}\left (d+e\,x\right )+a\right )}^{3/2}} \,d x \] Input:
int(cot(d + e*x)^3/(a + b*cot(d + e*x) + c*cot(d + e*x)^2)^(3/2),x)
Output:
int(cot(d + e*x)^3/(a + b*cot(d + e*x) + c*cot(d + e*x)^2)^(3/2), x)
\[ \int \frac {\cot ^3(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int(cot(e*x+d)^3/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(3/2),x)
Output:
( - 4*cot(d + e*x)**2*int((sqrt(cot(d + e*x)**2*c + cot(d + e*x)*b + a)*co t(d + e*x))/(cot(d + e*x)**4*c**2 + 2*cot(d + e*x)**3*b*c + 2*cot(d + e*x) **2*a*c + cot(d + e*x)**2*b**2 + 2*cot(d + e*x)*a*b + a**2),x)*a*c**2*e + cot(d + e*x)**2*int((sqrt(cot(d + e*x)**2*c + cot(d + e*x)*b + a)*cot(d + e*x))/(cot(d + e*x)**4*c**2 + 2*cot(d + e*x)**3*b*c + 2*cot(d + e*x)**2*a* c + cot(d + e*x)**2*b**2 + 2*cot(d + e*x)*a*b + a**2),x)*b**2*c*e + 2*sqrt (cot(d + e*x)**2*c + cot(d + e*x)*b + a)*cot(d + e*x)*b - 4*cot(d + e*x)*i nt((sqrt(cot(d + e*x)**2*c + cot(d + e*x)*b + a)*cot(d + e*x))/(cot(d + e* x)**4*c**2 + 2*cot(d + e*x)**3*b*c + 2*cot(d + e*x)**2*a*c + cot(d + e*x)* *2*b**2 + 2*cot(d + e*x)*a*b + a**2),x)*a*b*c*e + cot(d + e*x)*int((sqrt(c ot(d + e*x)**2*c + cot(d + e*x)*b + a)*cot(d + e*x))/(cot(d + e*x)**4*c**2 + 2*cot(d + e*x)**3*b*c + 2*cot(d + e*x)**2*a*c + cot(d + e*x)**2*b**2 + 2*cot(d + e*x)*a*b + a**2),x)*b**3*e + 4*sqrt(cot(d + e*x)**2*c + cot(d + e*x)*b + a)*a - 4*int((sqrt(cot(d + e*x)**2*c + cot(d + e*x)*b + a)*cot(d + e*x))/(cot(d + e*x)**4*c**2 + 2*cot(d + e*x)**3*b*c + 2*cot(d + e*x)**2* a*c + cot(d + e*x)**2*b**2 + 2*cot(d + e*x)*a*b + a**2),x)*a**2*c*e + int( (sqrt(cot(d + e*x)**2*c + cot(d + e*x)*b + a)*cot(d + e*x))/(cot(d + e*x)* *4*c**2 + 2*cot(d + e*x)**3*b*c + 2*cot(d + e*x)**2*a*c + cot(d + e*x)**2* b**2 + 2*cot(d + e*x)*a*b + a**2),x)*a*b**2*e)/(e*(4*cot(d + e*x)**2*a*c** 2 - cot(d + e*x)**2*b**2*c + 4*cot(d + e*x)*a*b*c - cot(d + e*x)*b**3 +...