Integrand size = 33, antiderivative size = 79 \[ \int \frac {\cot (d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx=\frac {\text {arctanh}\left (\frac {2 a-b+(b-2 c) \cot ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 \sqrt {a-b+c} e} \]
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)
Time = 0.58 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.76 \[ \int \frac {\cot (d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx=\frac {\text {arctanh}\left (\frac {b-2 c+(2 a-b) \tan ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \tan ^2(d+e x)}{2 \sqrt {a-b+c} e \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}} \]
(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*Cot[d + e*x]^4]*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*e*Sqrt[c + b*Tan[d + e*x]^2 + a*Tan[d + e*x]^4])
Time = 0.30 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {3042, 4184, 1576, 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.
\(\displaystyle \int \frac {\cot (d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cot (d+e x)}{\sqrt {a+b \cot (d+e x)^2+c \cot (d+e x)^4}}dx\) |
\(\Big \downarrow \) 4184 |
\(\displaystyle -\frac {\int \frac {\cot (d+e x)}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot (d+e x)}{e}\) |
\(\Big \downarrow \) 1576 |
\(\displaystyle -\frac {\int \frac {1}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot ^2(d+e x)}{2 e}\) |
\(\Big \downarrow \) 1154 |
\(\displaystyle \frac {\int \frac {1}{4 (a-b+c)-\cot ^4(d+e x)}d\frac {(b-2 c) \cot ^2(d+e x)+2 a-b}{\sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}}{e}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\text {arctanh}\left (\frac {2 a+(b-2 c) \cot ^2(d+e x)-b}{2 \sqrt {a-b+c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e \sqrt {a-b+c}}\) |
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])]/(2*Sqrt[a - b + c]*e)
3.1.19.3.1 Defintions of rubi rules used
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[(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]
Time = 0.20 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.29
method | result | size |
derivativedivides | \(\frac {\ln \left (\frac {2 a -2 b +2 c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+2 \sqrt {a -b +c}\, \sqrt {\left (\cot \left (e x +d \right )^{2}+1\right )^{2} c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+a -b +c}}{\cot \left (e x +d \right )^{2}+1}\right )}{2 e \sqrt {a -b +c}}\) | \(102\) |
default | \(\frac {\ln \left (\frac {2 a -2 b +2 c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+2 \sqrt {a -b +c}\, \sqrt {\left (\cot \left (e x +d \right )^{2}+1\right )^{2} c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+a -b +c}}{\cot \left (e x +d \right )^{2}+1}\right )}{2 e \sqrt {a -b +c}}\) | \(102\) |
1/2/e/(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. 222 vs. \(2 (69) = 138\).
Time = 0.46 (sec) , antiderivative size = 433, normalized size of antiderivative = 5.48 \[ \int \frac {\cot (d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx=\left [\frac {\log \left (2 \, {\left (a^{2} - 2 \, a b + b^{2} + 2 \, {\left (a - b\right )} c + c^{2}\right )} \cos \left (2 \, e x + 2 \, d\right )^{2} + 2 \, a^{2} - b^{2} + 2 \, c^{2} + 2 \, {\left ({\left (a - b + c\right )} \cos \left (2 \, e x + 2 \, d\right )^{2} - {\left (2 \, a - b\right )} \cos \left (2 \, e x + 2 \, d\right ) + a - c\right )} \sqrt {a - b + c} \sqrt {\frac {{\left (a - b + c\right )} \cos \left (2 \, e x + 2 \, d\right )^{2} - 2 \, {\left (a - c\right )} \cos \left (2 \, e x + 2 \, d\right ) + a + b + c}{\cos \left (2 \, e x + 2 \, d\right )^{2} - 2 \, \cos \left (2 \, e x + 2 \, d\right ) + 1}} - 4 \, {\left (a^{2} - a b + b c - c^{2}\right )} \cos \left (2 \, e x + 2 \, d\right )\right )}{4 \, \sqrt {a - b + c} e}, -\frac {\sqrt {-a + b - c} \arctan \left (\frac {{\left ({\left (a - b + c\right )} \cos \left (2 \, e x + 2 \, d\right )^{2} - {\left (2 \, a - b\right )} \cos \left (2 \, e x + 2 \, d\right ) + a - c\right )} \sqrt {-a + b - c} \sqrt {\frac {{\left (a - b + c\right )} \cos \left (2 \, e x + 2 \, d\right )^{2} - 2 \, {\left (a - c\right )} \cos \left (2 \, e x + 2 \, d\right ) + a + b + c}{\cos \left (2 \, e x + 2 \, d\right )^{2} - 2 \, \cos \left (2 \, e x + 2 \, d\right ) + 1}}}{{\left (a^{2} - 2 \, a b + b^{2} + 2 \, {\left (a - b\right )} c + c^{2}\right )} \cos \left (2 \, e x + 2 \, d\right )^{2} + a^{2} - b^{2} + 2 \, a c + c^{2} - 2 \, {\left (a^{2} - a b + b c - c^{2}\right )} \cos \left (2 \, e x + 2 \, d\right )}\right )}{2 \, {\left (a - b + c\right )} e}\right ] \]
[1/4*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))/(sqrt(a - b + c) *e), -1/2*sqrt(-a + b - c)*arctan(((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))/((a^2 - 2*a*b + b^2 + 2*(a - b)*c + c^2)*cos(2*e* x + 2*d)^2 + a^2 - b^2 + 2*a*c + c^2 - 2*(a^2 - a*b + b*c - c^2)*cos(2*e*x + 2*d)))/((a - b + c)*e)]
\[ \int \frac {\cot (d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx=\int \frac {\cot {\left (d + e x \right )}}{\sqrt {a + b \cot ^{2}{\left (d + e x \right )} + c \cot ^{4}{\left (d + e x \right )}}}\, dx \]
\[ \int \frac {\cot (d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx=\int { \frac {\cot \left (e x + d\right )}{\sqrt {c \cot \left (e x + d\right )^{4} + b \cot \left (e x + d\right )^{2} + a}} \,d x } \]
Timed out. \[ \int \frac {\cot (d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\cot (d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx=\int \frac {\mathrm {cot}\left (d+e\,x\right )}{\sqrt {c\,{\mathrm {cot}\left (d+e\,x\right )}^4+b\,{\mathrm {cot}\left (d+e\,x\right )}^2+a}} \,d x \]