Integrand size = 35, antiderivative size = 182 \[ \int \frac {\cot ^5(d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx=\frac {\text {arctanh}\left (\frac {2 a-b+(b-2 c) \cot ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 \sqrt {a-b+c} e}+\frac {(b+2 c) \text {arctanh}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{4 c^{3/2} e}-\frac {\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{2 c e} \]
1/4*(b+2*c)*arctanh(1/2*(b+2*c*cot(e*x+d)^2)/c^(1/2)/(a+b*cot(e*x+d)^2+c*c ot(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))/e/(a-b+c)^(1/2)-1/ 2*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2)/c/e
Time = 3.62 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.43 \[ \int \frac {\cot ^5(d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx=\frac {\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \tan ^2(d+e x) \left (2 c^{3/2} \text {arctanh}\left (\frac {b-2 c+(2 a-b) \tan ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}\right )+\sqrt {a-b+c} \left ((b+2 c) \text {arctanh}\left (\frac {2 c+b \tan ^2(d+e x)}{2 \sqrt {c} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}\right )-2 \sqrt {c} \cot ^2(d+e x) \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}\right )\right )}{4 c^{3/2} \sqrt {a-b+c} e \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}} \]
(Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]*Tan[d + e*x]^2*(2*c^(3/2)*A rcTanh[(b - 2*c + (2*a - b)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[c + b* Tan[d + e*x]^2 + a*Tan[d + e*x]^4])] + Sqrt[a - b + c]*((b + 2*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*Sqrt[c]*Cot[d + e*x]^2*Sqrt[c + b*Tan[d + e*x]^2 + a*Tan[d + e*x]^4])))/(4*c^(3/2)*Sqrt[a - b + c]*e*Sqrt[c + b*Tan[d + e*x]^2 + a*Tan [d + e*x]^4])
Time = 0.47 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.99, 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, 1267, 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.
| \(\displaystyle \int \frac {\cot ^5(d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cot (d+e x)^5}{\sqrt {a+b \cot (d+e x)^2+c \cot (d+e x)^4}}dx\) |
| \(\Big \downarrow \) 4184 |
| \(\displaystyle -\frac {\int \frac {\cot ^5(d+e x)}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot (d+e x)}{e}\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle -\frac {\int \frac {\cot ^4(d+e x)}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot ^2(d+e x)}{2 e}\) |
| \(\Big \downarrow \) 1267 |
| \(\displaystyle -\frac {\frac {\int -\frac {(b+2 c) \cot ^2(d+e x)+b}{2 \left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot ^2(d+e x)}{c}+\frac {\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{c}}{2 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{c}-\frac {\int \frac {(b+2 c) \cot ^2(d+e x)+b}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot ^2(d+e x)}{2 c}}{2 e}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle -\frac {\frac {\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{c}-\frac {(b+2 c) \int \frac {1}{\sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot ^2(d+e x)-2 c \int \frac {1}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot ^2(d+e x)}{2 c}}{2 e}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle -\frac {\frac {\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{c}-\frac {2 (b+2 c) \int \frac {1}{4 c-\cot ^4(d+e x)}d\frac {2 c \cot ^2(d+e x)+b}{\sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}-2 c \int \frac {1}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot ^2(d+e x)}{2 c}}{2 e}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\frac {\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{c}-\frac {\frac {(b+2 c) \text {arctanh}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{\sqrt {c}}-2 c \int \frac {1}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot ^2(d+e x)}{2 c}}{2 e}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle -\frac {\frac {\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{c}-\frac {4 c \int \frac {1}{4 (a-b+c)-\cot ^4(d+e x)}d\frac {(b-2 c) \cot ^2(d+e x)+2 a-b}{\sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}+\frac {(b+2 c) \text {arctanh}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{\sqrt {c}}}{2 c}}{2 e}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\frac {\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{c}-\frac {\frac {2 c \text {arctanh}\left (\frac {2 a+(b-2 c) \cot ^2(d+e x)-b}{2 \sqrt {a-b+c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{\sqrt {a-b+c}}+\frac {(b+2 c) \text {arctanh}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{\sqrt {c}}}{2 c}}{2 e}\) |
-1/2*(-1/2*((2*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])])/Sqrt[a - b + c] + ((b + 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 + Sqrt[a + b*Cot[d + e*x]^2 + c*C ot[d + e*x]^4]/c)/e
3.1.17.3.1 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_))^(n_)*((a_.) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[g^n*(d + e*x)^(m + n - 1)*((a + b *x + c*x^2)^(p + 1)/(c*e^(n - 1)*(m + n + 2*p + 1))), x] + Simp[1/(c*e^n*(m + n + 2*p + 1)) Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^n*(m + n + 2*p + 1)*(f + g*x)^n - c*g^n*(m + n + 2*p + 1)*(d + e*x)^n - g^n*(d + e*x)^(n - 2)*(b*d*e*(p + 1) + a*e^2*(m + n - 1) - c*d^2*(m + n + 2*p + 1) - e*(2*c*d - b*e)*(m + n + p)*x), x], x], x] /; FreeQ[{a, b, c, d, e, f, g , m, p}, x] && IGtQ[n, 1] && IntegerQ[m] && NeQ[m + n + 2*p + 1, 0]
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 = 0.72 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.27
| method | result | size |
| derivativedivides | \(\frac {-\frac {\sqrt {a +b \cot \left (e x +d \right )^{2}+c \cot \left (e x +d \right )^{4}}}{2 c}+\frac {b \ln \left (\frac {\frac {b}{2}+c \cot \left (e x +d \right )^{2}}{\sqrt {c}}+\sqrt {a +b \cot \left (e x +d \right )^{2}+c \cot \left (e x +d \right )^{4}}\right )}{4 c^{\frac {3}{2}}}+\frac {\ln \left (\frac {\frac {b}{2}+c \cot \left (e x +d \right )^{2}}{\sqrt {c}}+\sqrt {a +b \cot \left (e x +d \right )^{2}+c \cot \left (e x +d \right )^{4}}\right )}{2 \sqrt {c}}+\frac {\ln \left (\frac {2 a -2 b +2 c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+2 \sqrt {a -b +c}\, \sqrt {\left (\cot \left (e x +d \right )^{2}+1\right )^{2} c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+a -b +c}}{\cot \left (e x +d \right )^{2}+1}\right )}{2 \sqrt {a -b +c}}}{e}\) | \(232\) |
| default | \(\frac {-\frac {\sqrt {a +b \cot \left (e x +d \right )^{2}+c \cot \left (e x +d \right )^{4}}}{2 c}+\frac {b \ln \left (\frac {\frac {b}{2}+c \cot \left (e x +d \right )^{2}}{\sqrt {c}}+\sqrt {a +b \cot \left (e x +d \right )^{2}+c \cot \left (e x +d \right )^{4}}\right )}{4 c^{\frac {3}{2}}}+\frac {\ln \left (\frac {\frac {b}{2}+c \cot \left (e x +d \right )^{2}}{\sqrt {c}}+\sqrt {a +b \cot \left (e x +d \right )^{2}+c \cot \left (e x +d \right )^{4}}\right )}{2 \sqrt {c}}+\frac {\ln \left (\frac {2 a -2 b +2 c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+2 \sqrt {a -b +c}\, \sqrt {\left (\cot \left (e x +d \right )^{2}+1\right )^{2} c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+a -b +c}}{\cot \left (e x +d \right )^{2}+1}\right )}{2 \sqrt {a -b +c}}}{e}\) | \(232\) |
1/e*(-1/2/c*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2)+1/4*b/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*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))/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. 508 vs. \(2 (158) = 316\).
Time = 1.31 (sec) , antiderivative size = 2100, normalized size of antiderivative = 11.54 \[ \int \frac {\cot ^5(d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx=\text {Too large to display} \]
[1/8*(2*sqrt(a - b + c)*c^2*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)) + (a*b - b^2 + (2*a - b)*c + 2*c^2)*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*((a - b)*c + c^2)*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^2 + c^3)*e), 1/4*(sqrt(a - b + c)*c^2*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)) + (a*b - b^2 + (2*a - b)*c + 2*c^2)*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)*s...
\[ \int \frac {\cot ^5(d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx=\int \frac {\cot ^{5}{\left (d + e x \right )}}{\sqrt {a + b \cot ^{2}{\left (d + e x \right )} + c \cot ^{4}{\left (d + e x \right )}}}\, dx \]
Timed out. \[ \int \frac {\cot ^5(d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\cot ^5(d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\cot ^5(d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx=\int \frac {{\mathrm {cot}\left (d+e\,x\right )}^5}{\sqrt {c\,{\mathrm {cot}\left (d+e\,x\right )}^4+b\,{\mathrm {cot}\left (d+e\,x\right )}^2+a}} \,d x \]