Integrand size = 33, antiderivative size = 1007 \[ \int \frac {\cot ^3(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=\frac {\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} e}-\frac {3 \left (5 b^2-4 a c\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 )}{8 a^{7/2} 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}} \text {arctanh}\left (\frac {b^2-(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )-b \left (2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}\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}} \text {arctanh}\left (\frac {b^2-(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )-b \left (2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}\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 {2 \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 \cot ^2(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 (a \left (b^2-2 (a-c) c\right )+b c (a+c) \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 {b \left (15 b^2-52 a c\right ) \cot (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 a^3 \left (b^2-4 a c\right ) e}-\frac {\left (5 b^2-12 a c\right ) \cot ^2(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{2 a^2 \left (b^2-4 a c\right ) e} \]
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-3/8*(-4*a*c+5*b^2)*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^(7/2)/e-1/2*arctanh(1/2*(b^2-(a-c)* (a-c-(a^2-2*a*c+b^2+c^2)^(1/2))-b*(2*a-2*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*a rctanh(1/2*(b^2-(a-c)*(a-c+(a^2-2*a*c+b^2+c^2)^(1/2))-b*(2*a-2*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)+1/4*b*(-52*a*c+15*b^2)*cot(e*x+d)*(a+b*tan(e*x+d)+c*tan(e *x+d)^2)^(1/2)/a^3/(-4*a*c+b^2)/e-1/2*(-12*a*c+5*b^2)*cot(e*x+d)^2*(a+b*ta n(e*x+d)+c*tan(e*x+d)^2)^(1/2)/a^2/(-4*a*c+b^2)/e-2*(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*cot(e*x+d)^2 *(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*(a*(b^2-2*(a-c)*c)+b*c*(a+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 = 6.21 (sec) , antiderivative size = 786, normalized size of antiderivative = 0.78 \[ \int \frac {\cot ^3(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=\frac {-\frac {2 \left (-\frac {b^2}{2}+2 a c\right ) \text {arctanh}\left (\frac {2 a+b \tan (d+e x)}{2 \sqrt {a} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{a^{3/2} \left (b^2-4 a c\right )}-\frac {2 \left (-\frac {4 \sqrt {a+i b-c} \left (\frac {1}{4} i b \left (b^2-4 a c\right )-\frac {1}{4} (a-c) \left (b^2-4 a c\right )\right ) \text {arctanh}\left (\frac {2 a+i b-(-b-2 i c) \tan (d+e x)}{2 \sqrt {a+i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{4 a+4 i b-4 c}-\frac {4 \sqrt {a-i b-c} \left (-\frac {1}{4} i b \left (b^2-4 a c\right )-\frac {1}{4} (a-c) \left (b^2-4 a c\right )\right ) \text {arctanh}\left (\frac {2 a-i b-(-b+2 i c) \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{4 a-4 i b-4 c}\right )}{\left (b^2-4 a c\right ) \left (b^2+(-a+c)^2\right )}+\frac {2 \left (-b^2+2 a c-b c \tan (d+e x)\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}-\frac {2 \cot ^2(d+e x) \left (-b^2+2 a c-b c \tan (d+e x)\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}-\frac {2 \left (-a \left (b^2-2 a c+2 c^2\right )+c (-a b-b c) \tan (d+e x)\right )}{\left (b^2-4 a c\right ) \left (b^2+(-a+c)^2\right ) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}-\frac {2 \left (-\frac {\left (-5 b^2+12 a c\right ) \cot ^2(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 a}-\frac {\frac {\left (-\frac {1}{4} b^2 \left (15 b^2-52 a c\right )+a c \left (5 b^2-12 a c\right )\right ) \text {arctanh}\left (\frac {2 a+b \tan (d+e x)}{2 \sqrt {a} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 a^{3/2}}+\frac {b \left (15 b^2-52 a c\right ) \cot (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 a}}{2 a}\right )}{a \left (b^2-4 a c\right )}}{e} \]
((-2*(-1/2*b^2 + 2*a*c)*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)*(b^2 - 4*a*c)) - (2*((-4*S qrt[a + I*b - c]*((I/4)*b*(b^2 - 4*a*c) - ((a - c)*(b^2 - 4*a*c))/4)*ArcTa nh[(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*I)*b*(b^2 - 4*a*c) - ((a - c)*(b^2 - 4*a*c))/4)*ArcTanh[( 2*a - I*b - (-b + (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[a + b*T an[d + e*x] + c*Tan[d + e*x]^2])])/(4*a - (4*I)*b - 4*c)))/((b^2 - 4*a*c)* (b^2 + (-a + c)^2)) + (2*(-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*Cot[d + e*x]^2*(-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*(-(a*(b^2 - 2*a*c + 2*c^2)) + c*(-(a*b) - b*c)*Tan[d + e*x]))/((b^2 - 4*a*c)*(b^2 + (-a + c)^2)*Sqrt[a + b*Tan[d + e*x] + c*Ta n[d + e*x]^2]) - (2*(-1/4*((-5*b^2 + 12*a*c)*Cot[d + e*x]^2*Sqrt[a + b*Tan [d + e*x] + c*Tan[d + e*x]^2])/a - (((-1/4*(b^2*(15*b^2 - 52*a*c)) + a*c*( 5*b^2 - 12*a*c))*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)) + (b*(15*b^2 - 52*a*c)*Cot[d + e*x]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(4*a))/(2*a)))/(a*(b^2 - 4*a*c)))/e
Time = 4.82 (sec) , antiderivative size = 984, 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.
| \(\displaystyle \int \frac {\cot ^3(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (d+e x)^3 \left (a+b \tan (d+e x)+c \tan (d+e x)^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4183 |
| \(\displaystyle \frac {\int \frac {\cot ^3(d+e x)}{\left (\tan ^2(d+e x)+1\right ) \left (c \tan ^2(d+e x)+b \tan (d+e x)+a\right )^{3/2}}d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {\int \left (\frac {\cot ^3(d+e x)}{\left (c \tan ^2(d+e x)+b \tan (d+e x)+a\right )^{3/2}}-\frac {\cot (d+e x)}{\left (c \tan ^2(d+e x)+b \tan (d+e x)+a\right )^{3/2}}+\frac {\tan (d+e x)}{\left (\tan ^2(d+e x)+1\right ) \left (c \tan ^2(d+e x)+b \tan (d+e x)+a\right )^{3/2}}\right )d\tan (d+e x)}{e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {\left (5 b^2-12 a c\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \cot ^2(d+e x)}{2 a^2 \left (b^2-4 a c\right )}+\frac {2 \left (b^2+c \tan (d+e x) b-2 a c\right ) \cot ^2(d+e x)}{a \left (b^2-4 a c\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}+\frac {b \left (15 b^2-52 a c\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a} \cot (d+e x)}{4 a^3 \left (b^2-4 a c\right )}-\frac {3 \left (5 b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b \tan (d+e x)}{2 \sqrt {a} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{8 a^{7/2}}+\frac {\text {arctanh}\left (\frac {2 a+b \tan (d+e x)}{2 \sqrt {a} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{a^{3/2}}+\frac {\sqrt {2 a-2 c-\sqrt {a^2-2 c a+b^2+c^2}} \sqrt {a^2-2 c a-b^2+c^2+(a-c) \sqrt {a^2-2 c a+b^2+c^2}} \text {arctanh}\left (\frac {b^2-\left (2 a-2 c-\sqrt {a^2-2 c a+b^2+c^2}\right ) \tan (d+e x) b-(a-c) \left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right )}{\sqrt {2} \sqrt {2 a-2 c-\sqrt {a^2-2 c a+b^2+c^2}} \sqrt {a^2-2 c a-b^2+c^2+(a-c) \sqrt {a^2-2 c a+b^2+c^2}} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{\sqrt {2} \left (a^2-2 c a+b^2+c^2\right )^{3/2}}-\frac {\sqrt {2 a-2 c+\sqrt {a^2-2 c a+b^2+c^2}} \sqrt {a^2-2 c a-b^2+c^2-(a-c) \sqrt {a^2-2 c a+b^2+c^2}} \text {arctanh}\left (\frac {b^2-\left (2 a-2 c+\sqrt {a^2-2 c a+b^2+c^2}\right ) \tan (d+e x) b-(a-c) \left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right )}{\sqrt {2} \sqrt {2 a-2 c+\sqrt {a^2-2 c a+b^2+c^2}} \sqrt {a^2-2 c a-b^2+c^2-(a-c) \sqrt {a^2-2 c a+b^2+c^2}} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{\sqrt {2} \left (a^2-2 c a+b^2+c^2\right )^{3/2}}-\frac {2 \left (b^2+c \tan (d+e x) b-2 a c\right )}{a \left (b^2-4 a c\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}+\frac {2 \left (a \left (b^2-2 (a-c) c\right )+b c (a+c) \tan (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}}{e}\) |
(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) - (3*(5*b^2 - 4*a*c)*ArcTanh[(2*a + b*Tan[d + e*x]) /(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(8*a^(7/2)) + ( 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]]*ArcTanh[(b^2 - (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*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]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*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)) - (2*(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*Cot[d + e *x]^2*(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*(a*(b^2 - 2*(a - c)*c) + b*c*(a + c)*Tan[ d + e*x]))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)*Sqrt[a + b*Tan[d + e*x] + c...
3.1.26.3.1 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. 39674 vs. \(2 (922) = 1844\).
Time = 13.37 (sec) , antiderivative size = 79389, normalized size of antiderivative = 78.84 \[ \int \frac {\cot ^3(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=\text {Too large to display} \]
\[ \int \frac {\cot ^3(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=\int \frac {\cot ^{3}{\left (d + e x \right )}}{\left (a + b \tan {\left (d + e x \right )} + c \tan ^{2}{\left (d + e x \right )}\right )^{\frac {3}{2}}}\, dx \]
Timed out. \[ \int \frac {\cot ^3(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\cot ^3(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\cot ^3(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=\text {Hanged} \]