Integrand size = 33, antiderivative size = 156 \[ \int \frac {\cot (d+e x)}{\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}} \, dx=\frac {\text {arctanh}\left (\frac {2 a-b+(b-2 c) \cot ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 (a-b+c)^{3/2} e}-\frac {b^2-2 a c-b c+(b-2 c) c \cot ^2(d+e x)}{(a-b+c) \left (b^2-4 a c\right ) e \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \]
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))/(a-b+c)^(3/2)/e+(-b^2+2*a*c+b*c-(b-2*c)*c*cot(e* x+d)^2)/(a-b+c)/(-4*a*c+b^2)/e/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2)
Time = 10.90 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.69 \[ \int \frac {\cot (d+e x)}{\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}} \, dx=\frac {\frac {2 \sqrt {2} \left (b^2-2 c (a+c)-\left (b^2-2 b c+2 c (-a+c)\right ) \cos (2 (d+e x))\right ) \csc ^2(d+e x)}{(a-b+c) \left (-b^2+4 a c\right ) \sqrt {(3 a+b+3 c-4 (a-c) \cos (2 (d+e x))+(a-b+c) \cos (4 (d+e x))) \csc ^4(d+e x)}}-\frac {\text {arctanh}\left (\frac {-b+2 c+(-2 a+b) \tan ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \tan ^2(d+e x)}{(a-b+c)^{3/2} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}}{2 e} \]
((2*Sqrt[2]*(b^2 - 2*c*(a + c) - (b^2 - 2*b*c + 2*c*(-a + c))*Cos[2*(d + e *x)])*Csc[d + e*x]^2)/((a - b + c)*(-b^2 + 4*a*c)*Sqrt[(3*a + b + 3*c - 4* (a - c)*Cos[2*(d + e*x)] + (a - b + c)*Cos[4*(d + e*x)])*Csc[d + e*x]^4]) - (ArcTanh[(-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*Cot[d + e*x]^2 + c*C ot[d + e*x]^4]*Tan[d + e*x]^2)/((a - b + c)^(3/2)*Sqrt[c + b*Tan[d + e*x]^ 2 + a*Tan[d + e*x]^4]))/(2*e)
Time = 0.38 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {3042, 4184, 1576, 1165, 27, 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.
| \(\displaystyle \int \frac {\cot (d+e x)}{\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cot (d+e x)}{\left (a+b \cot (d+e x)^2+c \cot (d+e x)^4\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4184 |
| \(\displaystyle -\frac {\int \frac {\cot (d+e x)}{\left (\cot ^2(d+e x)+1\right ) \left (c \cot ^4(d+e x)+b \cot ^2(d+e x)+a\right )^{3/2}}d\cot (d+e x)}{e}\) |
| \(\Big \downarrow \) 1576 |
| \(\displaystyle -\frac {\int \frac {1}{\left (\cot ^2(d+e x)+1\right ) \left (c \cot ^4(d+e x)+b \cot ^2(d+e x)+a\right )^{3/2}}d\cot ^2(d+e x)}{2 e}\) |
| \(\Big \downarrow \) 1165 |
| \(\displaystyle -\frac {\frac {2 \left (-2 a c+b^2+c (b-2 c) \cot ^2(d+e x)-b c\right )}{(a-b+c) \left (b^2-4 a c\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}-\frac {2 \int -\frac {b^2-4 a c}{2 \left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot ^2(d+e x)}{(a-b+c) \left (b^2-4 a c\right )}}{2 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\int \frac {1}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot ^2(d+e x)}{a-b+c}+\frac {2 \left (-2 a c+b^2+c (b-2 c) \cot ^2(d+e x)-b c\right )}{(a-b+c) \left (b^2-4 a c\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}}{2 e}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle -\frac {\frac {2 \left (-2 a c+b^2+c (b-2 c) \cot ^2(d+e x)-b c\right )}{(a-b+c) \left (b^2-4 a c\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}-\frac {2 \int \frac {1}{4 (a-b+c)-\cot ^4(d+e x)}d\frac {(b-2 c) \cot ^2(d+e x)+2 a-b}{\sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}}{a-b+c}}{2 e}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\frac {2 \left (-2 a c+b^2+c (b-2 c) \cot ^2(d+e x)-b c\right )}{(a-b+c) \left (b^2-4 a c\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}-\frac {\text {arctanh}\left (\frac {2 a+(b-2 c) \cot ^2(d+e x)-b}{2 \sqrt {a-b+c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{(a-b+c)^{3/2}}}{2 e}\) |
-1/2*(-(ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sq rt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])]/(a - b + c)^(3/2)) + (2*(b^2 - 2*a*c - b*c + (b - 2*c)*c*Cot[d + e*x]^2))/((a - b + c)*(b^2 - 4*a*c)*S qrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]))/e
3.1.30.3.1 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/(((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_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e) *x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^ 2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m + 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]
Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^( p_.), x_Symbol] :> Simp[1/2 Subst[Int[(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[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]
Leaf count of result is larger than twice the leaf count of optimal. \(405\) vs. \(2(145)=290\).
Time = 0.09 (sec) , antiderivative size = 406, normalized size of antiderivative = 2.60
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 c \ln \left (\frac {2 a -2 b +2 c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+2 \sqrt {a -b +c}\, \sqrt {\left (\cot \left (e x +d \right )^{2}+1\right )^{2} c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+a -b +c}}{\cot \left (e x +d \right )^{2}+1}\right )}{\left (\sqrt {-4 a c +b^{2}}-b +2 c \right ) \left (\sqrt {-4 a c +b^{2}}+b -2 c \right ) \sqrt {a -b +c}}-\frac {2 c \sqrt {\left (\cot \left (e x +d \right )^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2} c -\sqrt {-4 a c +b^{2}}\, \left (\cot \left (e x +d \right )^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{\left (\sqrt {-4 a c +b^{2}}+b -2 c \right ) \left (-4 a c +b^{2}\right ) \left (\cot \left (e x +d \right )^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}+\frac {2 c \sqrt {\left (\cot \left (e x +d \right )^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2} c +\sqrt {-4 a c +b^{2}}\, \left (\cot \left (e x +d \right )^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{\left (\sqrt {-4 a c +b^{2}}-b +2 c \right ) \left (-4 a c +b^{2}\right ) \left (\cot \left (e x +d \right )^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{e}\) | \(406\) |
| default | \(\frac {-\frac {2 c \ln \left (\frac {2 a -2 b +2 c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+2 \sqrt {a -b +c}\, \sqrt {\left (\cot \left (e x +d \right )^{2}+1\right )^{2} c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+a -b +c}}{\cot \left (e x +d \right )^{2}+1}\right )}{\left (\sqrt {-4 a c +b^{2}}-b +2 c \right ) \left (\sqrt {-4 a c +b^{2}}+b -2 c \right ) \sqrt {a -b +c}}-\frac {2 c \sqrt {\left (\cot \left (e x +d \right )^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2} c -\sqrt {-4 a c +b^{2}}\, \left (\cot \left (e x +d \right )^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{\left (\sqrt {-4 a c +b^{2}}+b -2 c \right ) \left (-4 a c +b^{2}\right ) \left (\cot \left (e x +d \right )^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}+\frac {2 c \sqrt {\left (\cot \left (e x +d \right )^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2} c +\sqrt {-4 a c +b^{2}}\, \left (\cot \left (e x +d \right )^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{\left (\sqrt {-4 a c +b^{2}}-b +2 c \right ) \left (-4 a c +b^{2}\right ) \left (\cot \left (e x +d \right )^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{e}\) | \(406\) |
1/e*(-2*c/((-4*a*c+b^2)^(1/2)-b+2*c)/((-4*a*c+b^2)^(1/2)+b-2*c)/(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))-2*c/((-4 *a*c+b^2)^(1/2)+b-2*c)/(-4*a*c+b^2)/(cot(e*x+d)^2+1/2*(b+(-4*a*c+b^2)^(1/2 ))/c)*((cot(e*x+d)^2+1/2*(b+(-4*a*c+b^2)^(1/2))/c)^2*c-(-4*a*c+b^2)^(1/2)* (cot(e*x+d)^2+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)+2*c/((-4*a*c+b^2)^(1/2) -b+2*c)/(-4*a*c+b^2)/(cot(e*x+d)^2-1/2*(-b+(-4*a*c+b^2)^(1/2))/c)*((cot(e* x+d)^2-1/2*(-b+(-4*a*c+b^2)^(1/2))/c)^2*c+(-4*a*c+b^2)^(1/2)*(cot(e*x+d)^2 -1/2*(-b+(-4*a*c+b^2)^(1/2))/c))^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 887 vs. \(2 (144) = 288\).
Time = 0.66 (sec) , antiderivative size = 1771, normalized size of antiderivative = 11.35 \[ \int \frac {\cot (d+e x)}{\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}} \, dx=\text {Too large to display} \]
[1/4*((a*b^2 + b^3 - 4*a*c^2 + (a*b^2 - b^3 - 4*a*c^2 - (4*a^2 - 4*a*b - b ^2)*c)*cos(2*e*x + 2*d)^2 - (4*a^2 + 4*a*b - b^2)*c - 2*(a*b^2 + 4*a*c^2 - (4*a^2 + b^2)*c)*cos(2*e*x + 2*d))*sqrt(a - b + c)*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)) - 4*(a*b^2 - b^3 - 2*(2*a - b)*c^2 - 2*c^3 + (a*b^2 - b^3 - 4*b*c^2 + 2*c^3 - (2*a^2 - 3*b^2)*c)*cos(2*e*x + 2*d)^2 - (2*a^2 - 2*a*b - b^2)*c - 2*(a*b^2 - b^3 - (2*a + b)*c^2 - (2*a^2 - a*b - 2*b^2)*c)*cos(2*e*x + 2*d))*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^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5 - 4*a*c^4 - (12*a^2 - 12*a*b - b^2)*c^3 - 3*(4*a^3 - 8*a^2*b + 3*a*b^2 + b^3)*c^2 - (4*a^4 - 12*a^3*b + 9*a^2*b^2 + 2*a*b^3 - 3*b^4)*c)*e*cos(2*e*x + 2*d)^2 - 2*(a^3*b^2 - 2*a^ 2*b^3 + a*b^4 + 4*a*c^4 + (4*a^2 - 8*a*b - b^2)*c^3 - (4*a^3 - 3*a*b^2 - 2 *b^3)*c^2 - (4*a^4 - 8*a^3*b + 3*a^2*b^2 + b^4)*c)*e*cos(2*e*x + 2*d) + (a ^3*b^2 - a^2*b^3 - a*b^4 + b^5 - 4*a*c^4 - (12*a^2 - 4*a*b - b^2)*c^3 - (1 2*a^3 - 8*a^2*b - 7*a*b^2 + b^3)*c^2 - (4*a^4 - 4*a^3*b - 7*a^2*b^2 + 6*a* b^3 + b^4)*c)*e), -1/2*((a*b^2 + b^3 - 4*a*c^2 + (a*b^2 - b^3 - 4*a*c^2...
\[ \int \frac {\cot (d+e x)}{\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}} \, dx=\int \frac {\cot {\left (d + e x \right )}}{\left (a + b \cot ^{2}{\left (d + e x \right )} + c \cot ^{4}{\left (d + e x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\cot (d+e x)}{\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}} \, dx=\int { \frac {\cot \left (e x + d\right )}{{\left (c \cot \left (e x + d\right )^{4} + b \cot \left (e x + d\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\cot (d+e x)}{\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\cot (d+e x)}{\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}} \, dx=\int \frac {\mathrm {cot}\left (d+e\,x\right )}{{\left (c\,{\mathrm {cot}\left (d+e\,x\right )}^4+b\,{\mathrm {cot}\left (d+e\,x\right )}^2+a\right )}^{3/2}} \,d x \]