Integrand size = 33, antiderivative size = 395 \[ \int \frac {\cot ^2(d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=-\frac {\sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \arctan \left (\frac {b-\left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right ) \tan (d+e x)}{\sqrt {2} \sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2+b^2-2 a c+c^2} e}+\frac {\sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \arctan \left (\frac {b-\left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right ) \tan (d+e x)}{\sqrt {2} \sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2+b^2-2 a c+c^2} e}+\frac {b \text {arctanh}\left (\frac {2 a+b \tan (d+e x)}{2 \sqrt {a} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 a^{3/2} e}-\frac {\cot (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{a e} \]
1/2*b*arctanh(1/2*(2*a+b*tan(e*x+d))/a^(1/2)/(a+b*tan(e*x+d)+c*tan(e*x+d)^ 2)^(1/2))/a^(3/2)/e-1/2*arctan(1/2*(b-(a-c-(a^2-2*a*c+b^2+c^2)^(1/2))*tan( e*x+d))*2^(1/2)/(a-c-(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)/(a+b*tan(e*x+d)+c*ta n(e*x+d)^2)^(1/2))*(a-c-(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)/e*2^(1/2)/(a^2-2* a*c+b^2+c^2)^(1/2)+1/2*arctan(1/2*(b-(a-c+(a^2-2*a*c+b^2+c^2)^(1/2))*tan(e *x+d))*2^(1/2)/(a-c+(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)/(a+b*tan(e*x+d)+c*tan (e*x+d)^2)^(1/2))*(a-c+(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)/e*2^(1/2)/(a^2-2*a *c+b^2+c^2)^(1/2)-cot(e*x+d)*(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2)/a/e
Result contains complex when optimal does not.
Time = 1.57 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.67 \[ \int \frac {\cot ^2(d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=\frac {\frac {b \text {arctanh}\left (\frac {2 a+b \tan (d+e x)}{2 \sqrt {a} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{a^{3/2}}+\frac {i \text {arctanh}\left (\frac {2 a-i b+(b-2 i c) \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {a-i b-c}}-\frac {i \text {arctanh}\left (\frac {2 a+i b+(b+2 i c) \tan (d+e x)}{2 \sqrt {a+i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {a+i b-c}}-\frac {2 \cot (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{a}}{2 e} \]
((b*ArcTanh[(2*a + b*Tan[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x] + c* Tan[d + e*x]^2])])/a^(3/2) + (I*ArcTanh[(2*a - I*b + (b - (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])]) /Sqrt[a - I*b - c] - (I*ArcTanh[(2*a + I*b + (b + (2*I)*c)*Tan[d + e*x])/( 2*Sqrt[a + I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/Sqrt[a + I*b - c] - (2*Cot[d + e*x]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/ a)/(2*e)
Time = 0.89 (sec) , antiderivative size = 387, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {3042, 4183, 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.
\(\displaystyle \int \frac {\cot ^2(d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\tan (d+e x)^2 \sqrt {a+b \tan (d+e x)+c \tan (d+e x)^2}}dx\) |
\(\Big \downarrow \) 4183 |
\(\displaystyle \frac {\int \frac {\cot ^2(d+e x)}{\left (\tan ^2(d+e x)+1\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}d\tan (d+e x)}{e}\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \frac {\int \left (\frac {\cot ^2(d+e x)}{\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}+\frac {1}{\left (-\tan ^2(d+e x)-1\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )d\tan (d+e x)}{e}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {b \text {arctanh}\left (\frac {2 a+b \tan (d+e x)}{2 \sqrt {a} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 a^{3/2}}-\frac {\sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+a-c} \arctan \left (\frac {b-\left (-\sqrt {a^2-2 a c+b^2+c^2}+a-c\right ) \tan (d+e x)}{\sqrt {2} \sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+a-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2-2 a c+b^2+c^2}}+\frac {\sqrt {\sqrt {a^2-2 a c+b^2+c^2}+a-c} \arctan \left (\frac {b-\left (\sqrt {a^2-2 a c+b^2+c^2}+a-c\right ) \tan (d+e x)}{\sqrt {2} \sqrt {\sqrt {a^2-2 a c+b^2+c^2}+a-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2-2 a c+b^2+c^2}}-\frac {\cot (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{a}}{e}\) |
(-((Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTan[(b - (a - c - Sqrt[ a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[a - c - Sqrt[a^2 + b ^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[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]]*ArcTan[(b - (a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(S qrt[2]*Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]) + (b*ArcTa nh[(2*a + b*Tan[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e *x]^2])])/(2*a^(3/2)) - (Cot[d + e*x]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/a)/e
3.1.17.3.1 Defintions of rubi rules used
Int[tan[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*((f_.)*tan[(d_.) + (e_.)*( x_)])^(n_.) + (c_.)*((f_.)*tan[(d_.) + (e_.)*(x_)])^(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*Tan[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[n 2, 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
Leaf count of result is larger than twice the leaf count of optimal. 10076 vs. \(2 (352) = 704\).
Time = 2.10 (sec) , antiderivative size = 20189, normalized size of antiderivative = 51.11 \[ \int \frac {\cot ^2(d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=\text {Too large to display} \]
\[ \int \frac {\cot ^2(d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=\int \frac {\cot ^{2}{\left (d + e x \right )}}{\sqrt {a + b \tan {\left (d + e x \right )} + c \tan ^{2}{\left (d + e x \right )}}}\, dx \]
\[ \int \frac {\cot ^2(d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=\int { \frac {\cot \left (e x + d\right )^{2}}{\sqrt {c \tan \left (e x + d\right )^{2} + b \tan \left (e x + d\right ) + a}} \,d x } \]
\[ \int \frac {\cot ^2(d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=\int { \frac {\cot \left (e x + d\right )^{2}}{\sqrt {c \tan \left (e x + d\right )^{2} + b \tan \left (e x + d\right ) + a}} \,d x } \]
Timed out. \[ \int \frac {\cot ^2(d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=\int \frac {{\mathrm {cot}\left (d+e\,x\right )}^2}{\sqrt {c\,{\mathrm {tan}\left (d+e\,x\right )}^2+b\,\mathrm {tan}\left (d+e\,x\right )+a}} \,d x \]