Integrand size = 33, antiderivative size = 547 \[ \int \frac {\cot ^5(d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=-\frac {\sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {a-c-\sqrt {a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt {2} \sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2+b^2-2 a c+c^2} e}+\frac {\sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {a-c+\sqrt {a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt {2} \sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2+b^2-2 a c+c^2} e}-\frac {b \text {arctanh}\left (\frac {b+2 c \cot (d+e x)}{2 \sqrt {c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{2 c^{3/2} e}+\frac {b \left (5 b^2-12 a c\right ) \text {arctanh}\left (\frac {b+2 c \cot (d+e x)}{2 \sqrt {c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{16 c^{7/2} e}+\frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{c e}-\frac {\cot ^2(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{3 c e}-\frac {\left (15 b^2-16 a c-10 b c \cot (d+e x)\right ) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{24 c^3 e} \]
-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))/c^(3/2)/e+1/16*b*(-12*a*c+5*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^(7/2)/e+(a+b*cot(e*x+d)+ c*cot(e*x+d)^2)^(1/2)/c/e-1/3*cot(e*x+d)^2*(a+b*cot(e*x+d)+c*cot(e*x+d)^2) ^(1/2)/c/e-1/24*(15*b^2-16*a*c-10*b*c*cot(e*x+d))*(a+b*cot(e*x+d)+c*cot(e* x+d)^2)^(1/2)/c^3/e-1/2*arctanh(1/2*(a-c+b*cot(e*x+d)-(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)/(a-c-(a^2-2*a*c+b^2+c ^2)^(1/2))^(1/2))*(a-c-(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)/e*2^(1/2)/(a^2-2*a *c+b^2+c^2)^(1/2)+1/2*arctanh(1/2*(a-c+b*cot(e*x+d)+(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)/(a-c+(a^2-2*a*c+b^2+c^2 )^(1/2))^(1/2))*(a-c+(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)/e*2^(1/2)/(a^2-2*a*c +b^2+c^2)^(1/2)
Result contains complex when optimal does not.
Time = 6.77 (sec) , antiderivative size = 735, normalized size of antiderivative = 1.34 \[ \int \frac {\cot ^5(d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=\frac {\sqrt {\cot ^2(d+e x) \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )}}{c e}-\frac {i \arctan \left (\frac {i b-2 c+(2 i a-b) \tan (d+e x)}{2 \sqrt {a+i b-c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right ) \tan (d+e x) \sqrt {\cot ^2(d+e x) \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )}}{2 \sqrt {a+i b-c} e \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}-\frac {i \arctan \left (\frac {i b+2 c+(2 i a+b) \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right ) \tan (d+e x) \sqrt {\cot ^2(d+e x) \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )}}{2 \sqrt {a-i b-c} e \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}-\frac {b \text {arctanh}\left (\frac {2 c+b \tan (d+e x)}{2 \sqrt {c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right ) \tan (d+e x) \sqrt {\cot ^2(d+e x) \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )}}{2 c^{3/2} e \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}-\frac {\cot (d+e x) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)} \left (\frac {16 \cot ^3(d+e x) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}{c}-\frac {\frac {20 b \cot ^2(d+e x) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}{c}+\frac {\frac {3 b \left (5 b^2-12 a c\right ) \text {arctanh}\left (\frac {2 c+b \tan (d+e x)}{2 \sqrt {c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )}{c^{3/2}}-\frac {2 \left (15 b^2-16 a c\right ) \cot (d+e x) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}{c}}{c}}{c}\right )}{48 e \sqrt {\cot ^2(d+e x) \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )}} \]
Sqrt[Cot[d + e*x]^2*(c + b*Tan[d + e*x] + a*Tan[d + e*x]^2)]/(c*e) - ((I/2 )*ArcTan[(I*b - 2*c + ((2*I)*a - b)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqr t[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2])]*Tan[d + e*x]*Sqrt[Cot[d + e*x]^ 2*(c + b*Tan[d + e*x] + a*Tan[d + e*x]^2)])/(Sqrt[a + I*b - c]*e*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2]) - ((I/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])]*Tan[d + e*x]*Sqrt[Cot[d + e*x]^2*(c + b*Tan[d + e*x] + a*Tan[d + e*x]^2)])/(Sqrt[a - I*b - c]*e*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x] ^2]) - (b*ArcTanh[(2*c + b*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[c + b*Tan[d + e*x ] + a*Tan[d + e*x]^2])]*Tan[d + e*x]*Sqrt[Cot[d + e*x]^2*(c + b*Tan[d + e* x] + a*Tan[d + e*x]^2)])/(2*c^(3/2)*e*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]*((16* Cot[d + e*x]^3*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2])/c - ((20*b*Cot [d + e*x]^2*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2])/c + ((3*b*(5*b^2 - 12*a*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])])/c^(3/2) - (2*(15*b^2 - 16*a*c)*Cot[d + e*x]*Sqrt[ c + b*Tan[d + e*x] + a*Tan[d + e*x]^2])/c)/c)/c))/(48*e*Sqrt[Cot[d + e*x]^ 2*(c + b*Tan[d + e*x] + a*Tan[d + e*x]^2)])
Time = 1.16 (sec) , antiderivative size = 532, normalized size of antiderivative = 0.97, 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.
\(\displaystyle \int \frac {\cot ^5(d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cot (d+e x)^5}{\sqrt {a+b \cot (d+e x)+c \cot (d+e x)^2}}dx\) |
\(\Big \downarrow \) 4184 |
\(\displaystyle -\frac {\int \frac {\cot ^5(d+e x)}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}d\cot (d+e x)}{e}\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle -\frac {\int \left (\frac {\cot ^3(d+e x)}{\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}+\frac {\cot (d+e x)}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}-\frac {\cot (d+e x)}{\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right )d\cot (d+e x)}{e}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\frac {\sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+a-c} \text {arctanh}\left (\frac {-\sqrt {a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt {2} \sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+a-c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2-2 a c+b^2+c^2}}-\frac {\sqrt {\sqrt {a^2-2 a c+b^2+c^2}+a-c} \text {arctanh}\left (\frac {\sqrt {a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt {2} \sqrt {\sqrt {a^2-2 a c+b^2+c^2}+a-c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2-2 a c+b^2+c^2}}-\frac {b \left (5 b^2-12 a c\right ) \text {arctanh}\left (\frac {b+2 c \cot (d+e x)}{2 \sqrt {c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{16 c^{7/2}}+\frac {b \text {arctanh}\left (\frac {b+2 c \cot (d+e x)}{2 \sqrt {c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{2 c^{3/2}}+\frac {\left (-16 a c+15 b^2-10 b c \cot (d+e x)\right ) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{24 c^3}+\frac {\cot ^2(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{3 c}-\frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{c}}{e}\) |
-(((Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Cot[d + e*x])/(Sqrt[2]*Sqrt[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]* Sqrt[a^2 + b^2 - 2*a*c + c^2]) - (Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^ 2]]*ArcTanh[(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Cot[d + e*x])/(Sqrt [2]*Sqrt[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]*Sqrt[a^2 + b^2 - 2*a*c + c^2]) + (b*ArcTanh[ (b + 2*c*Cot[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x] ^2])])/(2*c^(3/2)) - (b*(5*b^2 - 12*a*c)*ArcTanh[(b + 2*c*Cot[d + e*x])/(2 *Sqrt[c]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(16*c^(7/2)) - Sqr t[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]/c + (Cot[d + e*x]^2*Sqrt[a + b*Co t[d + e*x] + c*Cot[d + e*x]^2])/(3*c) + ((15*b^2 - 16*a*c - 10*b*c*Cot[d + e*x])*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])/(24*c^3))/e)
3.1.1.3.1 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 = 3.55 (sec) , antiderivative size = 9581948, normalized size of antiderivative = 17517.27
\[\text {output too large to display}\]
Leaf count of result is larger than twice the leaf count of optimal. 12095 vs. \(2 (484) = 968\).
Time = 6.09 (sec) , antiderivative size = 24241, normalized size of antiderivative = 44.32 \[ \int \frac {\cot ^5(d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=\text {Too large to display} \]
\[ \int \frac {\cot ^5(d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=\int \frac {\cot ^{5}{\left (d + e x \right )}}{\sqrt {a + b \cot {\left (d + e x \right )} + c \cot ^{2}{\left (d + e x \right )}}}\, dx \]
\[ \int \frac {\cot ^5(d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=\int { \frac {\cot \left (e x + d\right )^{5}}{\sqrt {c \cot \left (e x + d\right )^{2} + b \cot \left (e x + d\right ) + a}} \,d x } \]
Exception generated. \[ \int \frac {\cot ^5(d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\cot ^5(d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=\int \frac {{\mathrm {cot}\left (d+e\,x\right )}^5}{\sqrt {c\,{\mathrm {cot}\left (d+e\,x\right )}^2+b\,\mathrm {cot}\left (d+e\,x\right )+a}} \,d x \]