Integrand size = 33, antiderivative size = 829 \[ \int \frac {\cot ^2(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=-\frac {\sqrt {2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2-(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \arctan \left (\frac {b \left (2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )+\left (b^2-(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) \tan (d+e x)}{\sqrt {2} \sqrt {2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2-(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}+\frac {\sqrt {2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2+(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \arctan \left (\frac {b \left (2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )+\left (b^2-(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) \tan (d+e x)}{\sqrt {2} \sqrt {2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2+(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}+\frac {3 b \text {arctanh}\left (\frac {2 a+b \tan (d+e x)}{2 \sqrt {a} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 a^{5/2} e}+\frac {2 \cot (d+e x) \left (b^2-2 a c+b c \tan (d+e x)\right )}{a \left (b^2-4 a c\right ) e \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}+\frac {2 \left (b \left (b^2-(3 a-c) c\right )+c \left (b^2-2 (a-c) c\right ) \tan (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}-\frac {\left (3 b^2-8 a c\right ) \cot (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{a^2 \left (b^2-4 a c\right ) e} \]
3/2*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^(5/2)/e-1/2*arctan(1/2*(b*(2*a-2*c+(a^2-2*a*c+b^2+c^2)^(1/2))+ (b^2-(a-c)*(a-c-(a^2-2*a*c+b^2+c^2)^(1/2)))*tan(e*x+d))*2^(1/2)/(2*a-2*c+( a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)/(a^2-b^2-2*a*c+c^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))*(2*a-2*c+(a^2-2*a*c +b^2+c^2)^(1/2))^(1/2)*(a^2-b^2-2*a*c+c^2-(a-c)*(a^2-2*a*c+b^2+c^2)^(1/2)) ^(1/2)/(a^2-2*a*c+b^2+c^2)^(3/2)/e*2^(1/2)+1/2*arctan(1/2*(b*(2*a-2*c-(a^2 -2*a*c+b^2+c^2)^(1/2))+(b^2-(a-c)*(a-c+(a^2-2*a*c+b^2+c^2)^(1/2)))*tan(e*x +d))*2^(1/2)/(2*a-2*c-(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)/(a^2-b^2-2*a*c+c^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))*(2*a-2*c-(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)*(a^2-b^2-2*a*c+c^2+(a-c)*(a^ 2-2*a*c+b^2+c^2)^(1/2))^(1/2)/(a^2-2*a*c+b^2+c^2)^(3/2)/e*2^(1/2)-(-8*a*c+ 3*b^2)*cot(e*x+d)*(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2)/a^2/(-4*a*c+b^2)/e +2*cot(e*x+d)*(b^2-2*a*c+b*c*tan(e*x+d))/a/(-4*a*c+b^2)/e/(a+b*tan(e*x+d)+ c*tan(e*x+d)^2)^(1/2)+2*(b*(b^2-(3*a-c)*c)+c*(b^2-2*(a-c)*c)*tan(e*x+d))/( b^2+(a-c)^2)/(-4*a*c+b^2)/e/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2)
Result contains complex when optimal does not.
Time = 6.24 (sec) , antiderivative size = 583, normalized size of antiderivative = 0.70 \[ \int \frac {\cot ^2(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=\frac {\frac {2 \left (-\frac {4 \sqrt {a-i b-c} \left (-\frac {1}{4} b \left (b^2-4 a c\right )+\frac {1}{4} i (a-c) \left (b^2-4 a c\right )\right ) \text {arctanh}\left (\frac {-2 a+i b-(b-2 i c) \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{4 a-4 i b-4 c}-\frac {4 \sqrt {a+i b-c} \left (-\frac {1}{4} b \left (b^2-4 a c\right )-\frac {1}{4} i (a-c) \left (b^2-4 a c\right )\right ) \text {arctanh}\left (\frac {-2 a-i b-(b+2 i c) \tan (d+e x)}{2 \sqrt {a+i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{4 a+4 i b-4 c}\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right )}-\frac {2 \cot (d+e x) \left (-b^2+2 a c-b c \tan (d+e x)\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}-\frac {2 \left (-b^3+b (3 a-c) c+c \left (-b^2+2 a c-2 c^2\right ) \tan (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}-\frac {2 \left (\frac {\left (2 a b c+\frac {1}{2} b \left (-3 b^2+8 a c\right )\right ) \text {arctanh}\left (\frac {2 a+b \tan (d+e x)}{2 \sqrt {a} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 a^{3/2}}+\frac {\left (3 b^2-8 a c\right ) \cot (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{2 a}\right )}{a \left (b^2-4 a c\right )}}{e} \]
((2*((-4*Sqrt[a - I*b - c]*(-1/4*(b*(b^2 - 4*a*c)) + (I/4)*(a - c)*(b^2 - 4*a*c))*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])])/(4*a - (4*I)*b - 4*c) - (4*Sqrt[a + I*b - c]*(-1/4*(b*(b^2 - 4*a*c)) - (I/4)*(a - c)*(b^2 - 4*a* c))*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])])/(4*a + (4*I)*b - 4*c)))/(( b^2 + (a - c)^2)*(b^2 - 4*a*c)) - (2*Cot[d + e*x]*(-b^2 + 2*a*c - b*c*Tan[ d + e*x]))/(a*(b^2 - 4*a*c)*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) - (2*(-b^3 + b*(3*a - c)*c + c*(-b^2 + 2*a*c - 2*c^2)*Tan[d + e*x]))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) - (2*(((2*a*b*c + (b*(-3*b^2 + 8*a*c))/2)*ArcTanh[(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)) + ((3*b ^2 - 8*a*c)*Cot[d + e*x]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(2*a )))/(a*(b^2 - 4*a*c)))/e
Time = 4.56 (sec) , antiderivative size = 815, 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.
\(\displaystyle \int \frac {\cot ^2(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\tan (d+e x)^2 \left (a+b \tan (d+e x)+c \tan (d+e x)^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 4183 |
\(\displaystyle \frac {\int \frac {\cot ^2(d+e x)}{\left (\tan ^2(d+e x)+1\right ) \left (c \tan ^2(d+e x)+b \tan (d+e x)+a\right )^{3/2}}d\tan (d+e x)}{e}\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \frac {\int \left (\frac {\cot ^2(d+e x)}{\left (c \tan ^2(d+e x)+b \tan (d+e x)+a\right )^{3/2}}+\frac {1}{\left (-\tan ^2(d+e x)-1\right ) \left (c \tan ^2(d+e x)+b \tan (d+e x)+a\right )^{3/2}}\right )d\tan (d+e x)}{e}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {\sqrt {2 a-2 c+\sqrt {a^2-2 c a+b^2+c^2}} \sqrt {a^2-2 c a-b^2+c^2-(a-c) \sqrt {a^2-2 c a+b^2+c^2}} \arctan \left (\frac {b \left (2 a-2 c+\sqrt {a^2-2 c a+b^2+c^2}\right )+\left (b^2-(a-c) \left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right )\right ) \tan (d+e x)}{\sqrt {2} \sqrt {2 a-2 c+\sqrt {a^2-2 c a+b^2+c^2}} \sqrt {a^2-2 c a-b^2+c^2-(a-c) \sqrt {a^2-2 c a+b^2+c^2}} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{\sqrt {2} \left (a^2-2 c a+b^2+c^2\right )^{3/2}}+\frac {\sqrt {2 a-2 c-\sqrt {a^2-2 c a+b^2+c^2}} \sqrt {a^2-2 c a-b^2+c^2+(a-c) \sqrt {a^2-2 c a+b^2+c^2}} \arctan \left (\frac {b \left (2 a-2 c-\sqrt {a^2-2 c a+b^2+c^2}\right )+\left (b^2-(a-c) \left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right )\right ) \tan (d+e x)}{\sqrt {2} \sqrt {2 a-2 c-\sqrt {a^2-2 c a+b^2+c^2}} \sqrt {a^2-2 c a-b^2+c^2+(a-c) \sqrt {a^2-2 c a+b^2+c^2}} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{\sqrt {2} \left (a^2-2 c a+b^2+c^2\right )^{3/2}}+\frac {3 b \text {arctanh}\left (\frac {2 a+b \tan (d+e x)}{2 \sqrt {a} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{2 a^{5/2}}-\frac {\left (3 b^2-8 a c\right ) \cot (d+e x) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{a^2 \left (b^2-4 a c\right )}+\frac {2 \cot (d+e x) \left (b^2+c \tan (d+e x) b-2 a c\right )}{a \left (b^2-4 a c\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}+\frac {2 \left (b \left (b^2-(3 a-c) c\right )+c \left (b^2-2 (a-c) c\right ) \tan (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}}{e}\) |
(-((Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTan[(b*(2*a - 2*c + Sqr t[a^2 + b^2 - 2*a*c + c^2]) + (b^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a *c + c^2]))*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (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]*(a^2 + b^2 - 2*a*c + c^2)^(3/2))) + (Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sq rt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTan [(b*(2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) + (b^2 - (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]))*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c - Sq rt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (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])])/(S qrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)) + (3*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])])/(2*a^(5/2)) + (2*Cot[d + e*x]*(b^2 - 2*a*c + b*c*Tan[d + e*x]))/(a*(b^2 - 4*a*c)*Sqrt[ a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) + (2*(b*(b^2 - (3*a - c)*c) + c*(b ^2 - 2*(a - c)*c)*Tan[d + e*x]))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) - ((3*b^2 - 8*a*c)*Cot[d + e*x]*Sqrt[ a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(a^2*(b^2 - 4*a*c)))/e
3.1.25.3.1 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. 39249 vs. \(2 (764) = 1528\).
Time = 12.46 (sec) , antiderivative size = 78535, normalized size of antiderivative = 94.73 \[ \int \frac {\cot ^2(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=\text {Too large to display} \]
\[ \int \frac {\cot ^2(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=\int \frac {\cot ^{2}{\left (d + e x \right )}}{\left (a + b \tan {\left (d + e x \right )} + c \tan ^{2}{\left (d + e x \right )}\right )^{\frac {3}{2}}}\, dx \]
Timed out. \[ \int \frac {\cot ^2(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\cot ^2(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\cot ^2(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=\int \frac {{\mathrm {cot}\left (d+e\,x\right )}^2}{{\left (c\,{\mathrm {tan}\left (d+e\,x\right )}^2+b\,\mathrm {tan}\left (d+e\,x\right )+a\right )}^{3/2}} \,d x \]