Integrand size = 31, antiderivative size = 294 \[ \int \frac {\cot (d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=-\frac {\sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {a-c-\sqrt {a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt {2} \sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2+b^2-2 a c+c^2} e}+\frac {\sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {a-c+\sqrt {a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt {2} \sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2+b^2-2 a c+c^2} e} \] Output:
-1/2*(a-c-(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)*arctanh(1/2*(a-c-(a^2-2*a*c+b^2 +c^2)^(1/2)+b*cot(e*x+d))*2^(1/2)/(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)^(1/2)/e+1 /2*(a-c+(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)*arctanh(1/2*(a-c+(a^2-2*a*c+b^2+c ^2)^(1/2)+b*cot(e*x+d))*2^(1/2)/(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)^(1/2)/e
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.87 \[ \int \frac {\cot (d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=-\frac {i \left (\sqrt {a-i b-c} \arctan \left (\frac {i b-2 c+(2 i a-b) \tan (d+e x)}{2 \sqrt {a+i b-c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )+\sqrt {a+i b-c} \arctan \left (\frac {i b+2 c+(2 i a+b) \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )\right ) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {a+i b-c} e \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}} \] Input:
Integrate[Cot[d + e*x]/Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2],x]
Output:
((-1/2*I)*(Sqrt[a - I*b - 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])] + Sq rt[a + I*b - 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])])*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x])/(Sqrt[a - I*b - c]*Sqrt[a + I*b - c]*e*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2])
Time = 0.58 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {3042, 4184, 1369, 25, 1363, 221}
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.
\(\displaystyle \int \frac {\cot (d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cot (d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot (d+e x)^2}}dx\) |
\(\Big \downarrow \) 4184 |
\(\displaystyle -\frac {\int \frac {\cot (d+e x)}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}d\cot (d+e x)}{e}\) |
\(\Big \downarrow \) 1369 |
\(\displaystyle -\frac {\frac {\int -\frac {b-\left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right ) \cot (d+e x)}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}d\cot (d+e x)}{2 \sqrt {a^2-2 a c+b^2+c^2}}-\frac {\int -\frac {b-\left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right ) \cot (d+e x)}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}d\cot (d+e x)}{2 \sqrt {a^2-2 a c+b^2+c^2}}}{e}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\frac {\int \frac {b-\left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right ) \cot (d+e x)}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}d\cot (d+e x)}{2 \sqrt {a^2-2 a c+b^2+c^2}}-\frac {\int \frac {b-\left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right ) \cot (d+e x)}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}d\cot (d+e x)}{2 \sqrt {a^2-2 a c+b^2+c^2}}}{e}\) |
\(\Big \downarrow \) 1363 |
\(\displaystyle -\frac {\frac {b \left (-\sqrt {a^2-2 a c+b^2+c^2}+a-c\right ) \int \frac {1}{\frac {b \left (a-c+b \cot (d+e x)-\sqrt {a^2-2 c a+b^2+c^2}\right )^2}{c \cot ^2(d+e x)+b \cot (d+e x)+a}-2 b \left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right )}d\left (-\frac {a-c+b \cot (d+e x)-\sqrt {a^2-2 c a+b^2+c^2}}{\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right )}{\sqrt {a^2-2 a c+b^2+c^2}}-\frac {b \left (\sqrt {a^2-2 a c+b^2+c^2}+a-c\right ) \int \frac {1}{\frac {b \left (a-c+b \cot (d+e x)+\sqrt {a^2-2 c a+b^2+c^2}\right )^2}{c \cot ^2(d+e x)+b \cot (d+e x)+a}-2 b \left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right )}d\left (-\frac {a-c+b \cot (d+e x)+\sqrt {a^2-2 c a+b^2+c^2}}{\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right )}{\sqrt {a^2-2 a c+b^2+c^2}}}{e}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {\frac {\sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+a-c} \text {arctanh}\left (\frac {-\sqrt {a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt {2} \sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+a-c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2-2 a c+b^2+c^2}}-\frac {\sqrt {\sqrt {a^2-2 a c+b^2+c^2}+a-c} \text {arctanh}\left (\frac {\sqrt {a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt {2} \sqrt {\sqrt {a^2-2 a c+b^2+c^2}+a-c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2-2 a c+b^2+c^2}}}{e}\) |
Input:
Int[Cot[d + e*x]/Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2],x]
Output:
-(((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]))/e)
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((g_) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f _.)*(x_)^2]), x_Symbol] :> Simp[-2*a*g*h Subst[Int[1/Simp[2*a^2*g*h*c + a *e*x^2, x], x], x, Simp[a*h - g*c*x, x]/Sqrt[d + e*x + f*x^2]], x] /; FreeQ [{a, c, d, e, f, g, h}, x] && EqQ[a*h^2*e + 2*g*h*(c*d - a*f) - g^2*c*e, 0]
Int[((g_.) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + ( f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[(c*d - a*f)^2 + a*c*e^2, 2]}, Simp [1/(2*q) Int[Simp[(-a)*h*e - g*(c*d - a*f - q) + (h*(c*d - a*f + q) - g*c *e)*x, x]/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] - Simp[1/(2*q) Int[ Simp[(-a)*h*e - g*(c*d - a*f + q) + (h*(c*d - a*f - q) - g*c*e)*x, x]/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d, e, f, g, h}, x] && NeQ[e^2 - 4*d*f, 0] && NegQ[(-a)*c]
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]
result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 0.67 (sec) , antiderivative size = 9338543, normalized size of antiderivative = 31763.75
\[\text {output too large to display}\]
Input:
int(cot(e*x+d)/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2),x)
Output:
result too large to display
Leaf count of result is larger than twice the leaf count of optimal. 5853 vs. \(2 (261) = 522\).
Time = 1.79 (sec) , antiderivative size = 5853, normalized size of antiderivative = 19.91 \[ \int \frac {\cot (d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=\text {Too large to display} \] Input:
integrate(cot(e*x+d)/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2),x, algorithm="f ricas")
Output:
Too large to include
\[ \int \frac {\cot (d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=\int \frac {\cot {\left (d + e x \right )}}{\sqrt {a + b \cot {\left (d + e x \right )} + c \cot ^{2}{\left (d + e x \right )}}}\, dx \] Input:
integrate(cot(e*x+d)/(a+b*cot(e*x+d)+c*cot(e*x+d)**2)**(1/2),x)
Output:
Integral(cot(d + e*x)/sqrt(a + b*cot(d + e*x) + c*cot(d + e*x)**2), x)
Exception generated. \[ \int \frac {\cot (d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cot(e*x+d)/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2),x, algorithm="m axima")
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. \[ \int \frac {\cot (d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(cot(e*x+d)/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2),x, algorithm="g iac")
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. \[ \int \frac {\cot (d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=\int \frac {\mathrm {cot}\left (d+e\,x\right )}{\sqrt {c\,{\mathrm {cot}\left (d+e\,x\right )}^2+b\,\mathrm {cot}\left (d+e\,x\right )+a}} \,d x \] Input:
int(cot(d + e*x)/(a + b*cot(d + e*x) + c*cot(d + e*x)^2)^(1/2),x)
Output:
int(cot(d + e*x)/(a + b*cot(d + e*x) + c*cot(d + e*x)^2)^(1/2), x)
\[ \int \frac {\cot (d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=\int \frac {\sqrt {\cot \left (e x +d \right )^{2} c +\cot \left (e x +d \right ) b +a}\, \cot \left (e x +d \right )}{\cot \left (e x +d \right )^{2} c +\cot \left (e x +d \right ) b +a}d x \] Input:
int(cot(e*x+d)/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2),x)
Output:
int((sqrt(cot(d + e*x)**2*c + cot(d + e*x)*b + a)*cot(d + e*x))/(cot(d + e *x)**2*c + cot(d + e*x)*b + a),x)