Integrand size = 31, antiderivative size = 349 \[ \int \frac {\tan (d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=\frac {\text {arctanh}\left (\frac {2 a+b \cot (d+e x)}{2 \sqrt {a} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {a} e}+\frac {\sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {a-c-\sqrt {a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt {2} \sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2+b^2-2 a c+c^2} e}-\frac {\sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {a-c+\sqrt {a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt {2} \sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2+b^2-2 a c+c^2} e} \] Output:
arctanh(1/2*(2*a+b*cot(e*x+d))/a^(1/2)/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/ 2))/a^(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-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 = 0.42 (sec) , antiderivative size = 344, normalized size of antiderivative = 0.99 \[ \int \frac {\tan (d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=\frac {\left (i \sqrt {a} \left (\sqrt {a-i b-c} \arctan \left (\frac {i b-2 c+(2 i a-b) \tan (d+e x)}{2 \sqrt {a+i b-c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )+\sqrt {a+i b-c} \arctan \left (\frac {i b+2 c+(2 i a+b) \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )\right )+2 \sqrt {a-i b-c} \sqrt {a+i b-c} \text {arctanh}\left (\frac {b+2 a \tan (d+e x)}{2 \sqrt {a} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )\right ) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan (d+e x)}{2 \sqrt {a} \sqrt {a-i b-c} \sqrt {a+i b-c} e \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}} \] Input:
Integrate[Tan[d + e*x]/Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2],x]
Output:
((I*Sqrt[a]*(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])] + 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])]) + 2*Sqrt[a - I *b - c]*Sqrt[a + I*b - c]*ArcTanh[(b + 2*a*Tan[d + e*x])/(2*Sqrt[a]*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])/(2*Sqrt[a]*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 = 0.87 (sec) , antiderivative size = 346, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, 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.
\(\displaystyle \int \frac {\tan (d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\cot (d+e x) \sqrt {a+b \cot (d+e x)+c \cot (d+e x)^2}}dx\) |
\(\Big \downarrow \) 4184 |
\(\displaystyle -\frac {\int \frac {\tan (d+e x)}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}d\cot (d+e x)}{e}\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle -\frac {\int \left (\frac {\tan (d+e x)}{\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}-\frac {\cot (d+e x)}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right )d\cot (d+e x)}{e}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {-\frac {\sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+a-c} \text {arctanh}\left (\frac {-\sqrt {a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt {2} \sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+a-c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2-2 a c+b^2+c^2}}+\frac {\sqrt {\sqrt {a^2-2 a c+b^2+c^2}+a-c} \text {arctanh}\left (\frac {\sqrt {a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt {2} \sqrt {\sqrt {a^2-2 a c+b^2+c^2}+a-c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2-2 a c+b^2+c^2}}-\frac {\text {arctanh}\left (\frac {2 a+b \cot (d+e x)}{2 \sqrt {a} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {a}}}{e}\) |
Input:
Int[Tan[d + e*x]/Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2],x]
Output:
-((-(ArcTanh[(2*a + b*Cot[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x] + c *Cot[d + e*x]^2])]/Sqrt[a]) - (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*C ot[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[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]
Timed out.
hanged
Input:
int(tan(e*x+d)/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2),x)
Output:
int(tan(e*x+d)/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2),x)
Leaf count of result is larger than twice the leaf count of optimal. 5166 vs. \(2 (308) = 616\).
Time = 1.25 (sec) , antiderivative size = 10333, normalized size of antiderivative = 29.61 \[ \int \frac {\tan (d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=\text {Too large to display} \] Input:
integrate(tan(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
\[ \int \frac {\tan (d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=\int \frac {\tan {\left (d + e x \right )}}{\sqrt {a + b \cot {\left (d + e x \right )} + c \cot ^{2}{\left (d + e x \right )}}}\, dx \] Input:
integrate(tan(e*x+d)/(a+b*cot(e*x+d)+c*cot(e*x+d)**2)**(1/2),x)
Output:
Integral(tan(d + e*x)/sqrt(a + b*cot(d + e*x) + c*cot(d + e*x)**2), x)
\[ \int \frac {\tan (d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=\int { \frac {\tan \left (e x + d\right )}{\sqrt {c \cot \left (e x + d\right )^{2} + b \cot \left (e x + d\right ) + a}} \,d x } \] Input:
integrate(tan(e*x+d)/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2),x, algorithm="m axima")
Output:
integrate(tan(e*x + d)/sqrt(c*cot(e*x + d)^2 + b*cot(e*x + d) + a), x)
Exception generated. \[ \int \frac {\tan (d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(tan(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. \[ \int \frac {\tan (d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=\int \frac {\mathrm {tan}\left (d+e\,x\right )}{\sqrt {c\,{\mathrm {cot}\left (d+e\,x\right )}^2+b\,\mathrm {cot}\left (d+e\,x\right )+a}} \,d x \] Input:
int(tan(d + e*x)/(a + b*cot(d + e*x) + c*cot(d + e*x)^2)^(1/2),x)
Output:
int(tan(d + e*x)/(a + b*cot(d + e*x) + c*cot(d + e*x)^2)^(1/2), x)
\[ \int \frac {\tan (d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=\int \frac {\sqrt {\cot \left (e x +d \right )^{2} c +\cot \left (e x +d \right ) b +a}\, \tan \left (e x +d \right )}{\cot \left (e x +d \right )^{2} c +\cot \left (e x +d \right ) b +a}d x \] Input:
int(tan(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)*tan(d + e*x))/(cot(d + e *x)**2*c + cot(d + e*x)*b + a),x)