Integrand size = 33, antiderivative size = 280 \[ \int \frac {\tan (d+e x)}{\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}} \, dx=\frac {\text {arctanh}\left (\frac {2 a+b \cot ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 a^{3/2} e}-\frac {\text {arctanh}\left (\frac {2 a-b+(b-2 c) \cot ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 (a-b+c)^{3/2} e}-\frac {b^2-2 a c+b c \cot ^2(d+e x)}{a \left (b^2-4 a c\right ) e \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}+\frac {b^2-2 a c-b c+(b-2 c) c \cot ^2(d+e x)}{(a-b+c) \left (b^2-4 a c\right ) e \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \]
1/2*arctanh(1/2*(2*a+b*cot(e*x+d)^2)/a^(1/2)/(a+b*cot(e*x+d)^2+c*cot(e*x+d )^4)^(1/2))/a^(3/2)/e-1/2*arctanh(1/2*(2*a-b+(b-2*c)*cot(e*x+d)^2)/(a-b+c) ^(1/2)/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2))/(a-b+c)^(3/2)/e+(-b^2+2*a* c-b*c*cot(e*x+d)^2)/a/(-4*a*c+b^2)/e/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/ 2)+(b^2-2*a*c-b*c+(b-2*c)*c*cot(e*x+d)^2)/(a-b+c)/(-4*a*c+b^2)/e/(a+b*cot( e*x+d)^2+c*cot(e*x+d)^4)^(1/2)
Time = 13.75 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.25 \[ \int \frac {\tan (d+e x)}{\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}} \, dx=\frac {\frac {2 \sqrt {2} \sqrt {a} \left (-b^3+b c (3 a+c)+\left (b^3-2 b^2 c+4 a c^2+b c (-3 a+c)\right ) \cos (2 (d+e x))\right ) \csc ^2(d+e x)}{(a-b+c) \left (-b^2+4 a c\right ) \sqrt {(3 a+b+3 c-4 (a-c) \cos (2 (d+e x))+(a-b+c) \cos (4 (d+e x))) \csc ^4(d+e x)}}+\frac {\left ((a-b+c)^{3/2} \text {arctanh}\left (\frac {b+2 a \tan ^2(d+e x)}{2 \sqrt {a} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}\right )+a^{3/2} \text {arctanh}\left (\frac {-b+2 c+(-2 a+b) \tan ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}\right )\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \tan ^2(d+e x)}{(a-b+c)^{3/2} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}}{2 a^{3/2} e} \]
((2*Sqrt[2]*Sqrt[a]*(-b^3 + b*c*(3*a + c) + (b^3 - 2*b^2*c + 4*a*c^2 + b*c *(-3*a + c))*Cos[2*(d + e*x)])*Csc[d + e*x]^2)/((a - b + c)*(-b^2 + 4*a*c) *Sqrt[(3*a + b + 3*c - 4*(a - c)*Cos[2*(d + e*x)] + (a - b + c)*Cos[4*(d + e*x)])*Csc[d + e*x]^4]) + (((a - b + c)^(3/2)*ArcTanh[(b + 2*a*Tan[d + e* x]^2)/(2*Sqrt[a]*Sqrt[c + b*Tan[d + e*x]^2 + a*Tan[d + e*x]^4])] + a^(3/2) *ArcTanh[(-b + 2*c + (-2*a + b)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[c + b*Tan[d + e*x]^2 + a*Tan[d + e*x]^4])])*Sqrt[a + b*Cot[d + e*x]^2 + c*Co t[d + e*x]^4]*Tan[d + e*x]^2)/((a - b + c)^(3/2)*Sqrt[c + b*Tan[d + e*x]^2 + a*Tan[d + e*x]^4]))/(2*a^(3/2)*e)
Time = 0.55 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {3042, 4184, 1578, 1289, 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.
| \(\displaystyle \int \frac {\tan (d+e x)}{\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cot (d+e x) \left (a+b \cot (d+e x)^2+c \cot (d+e x)^4\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4184 |
| \(\displaystyle -\frac {\int \frac {\tan (d+e x)}{\left (\cot ^2(d+e x)+1\right ) \left (c \cot ^4(d+e x)+b \cot ^2(d+e x)+a\right )^{3/2}}d\cot (d+e x)}{e}\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle -\frac {\int \frac {\tan (d+e x)}{\left (\cot ^2(d+e x)+1\right ) \left (c \cot ^4(d+e x)+b \cot ^2(d+e x)+a\right )^{3/2}}d\cot ^2(d+e x)}{2 e}\) |
| \(\Big \downarrow \) 1289 |
| \(\displaystyle -\frac {\int \left (\frac {\tan (d+e x)}{\left (c \cot ^4(d+e x)+b \cot ^2(d+e x)+a\right )^{3/2}}+\frac {1}{\left (-\cot ^2(d+e x)-1\right ) \left (c \cot ^4(d+e x)+b \cot ^2(d+e x)+a\right )^{3/2}}\right )d\cot ^2(d+e x)}{2 e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-\frac {\text {arctanh}\left (\frac {2 a+b \cot ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{a^{3/2}}+\frac {\text {arctanh}\left (\frac {2 a+(b-2 c) \cot ^2(d+e x)-b}{2 \sqrt {a-b+c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{(a-b+c)^{3/2}}+\frac {2 \left (-2 a c+b^2+b c \cot ^2(d+e x)\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}-\frac {2 \left (-2 a c+b^2+c (b-2 c) \cot ^2(d+e x)-b c\right )}{(a-b+c) \left (b^2-4 a c\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}}{2 e}\) |
-1/2*(-(ArcTanh[(2*a + b*Cot[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x ]^2 + c*Cot[d + e*x]^4])]/a^(3/2)) + ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])] /(a - b + c)^(3/2) + (2*(b^2 - 2*a*c + b*c*Cot[d + e*x]^2))/(a*(b^2 - 4*a* c)*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]) - (2*(b^2 - 2*a*c - b*c + (b - 2*c)*c*Cot[d + e*x]^2))/((a - b + c)*(b^2 - 4*a*c)*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]))/e
3.1.31.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && ( IntegerQ[p] || (ILtQ[m, 0] && ILtQ[n, 0]))
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int egerQ[(m - 1)/2]
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]
\[\int \frac {\tan \left (e x +d \right )}{\left (a +b \cot \left (e x +d \right )^{2}+c \cot \left (e x +d \right )^{4}\right )^{\frac {3}{2}}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 1020 vs. \(2 (256) = 512\).
Time = 2.67 (sec) , antiderivative size = 4153, normalized size of antiderivative = 14.83 \[ \int \frac {\tan (d+e x)}{\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}} \, dx=\text {Too large to display} \]
[-1/4*((4*a*c^4 - (a^3*b^2 - 2*a^2*b^3 + a*b^4 - 4*a^2*c^3 - (8*a^3 - 8*a^ 2*b - a*b^2)*c^2 - 2*(2*a^4 - 4*a^3*b + a^2*b^2 + a*b^3)*c)*tan(e*x + d)^4 + (8*a^2 - 8*a*b - b^2)*c^3 + 2*(2*a^3 - 4*a^2*b + a*b^2 + b^3)*c^2 - (a^ 2*b^3 - 2*a*b^4 + b^5 - 4*a*b*c^3 - (8*a^2*b - 8*a*b^2 - b^3)*c^2 - 2*(2*a ^3*b - 4*a^2*b^2 + a*b^3 + b^4)*c)*tan(e*x + d)^2 - (a^2*b^2 - 2*a*b^3 + b ^4)*c)*sqrt(a)*log(8*a^2*tan(e*x + d)^4 + 8*a*b*tan(e*x + d)^2 + b^2 + 4*a *c + 4*(2*a*tan(e*x + d)^4 + b*tan(e*x + d)^2)*sqrt(a)*sqrt((a*tan(e*x + d )^4 + b*tan(e*x + d)^2 + c)/tan(e*x + d)^4)) - (a^2*b^2*c - 4*a^3*c^2 + (a ^3*b^2 - 4*a^4*c)*tan(e*x + d)^4 + (a^2*b^3 - 4*a^3*b*c)*tan(e*x + d)^2)*s qrt(a - b + c)*log(((8*a^2 - 8*a*b + b^2 + 4*a*c)*tan(e*x + d)^4 + 2*(4*a* b - 3*b^2 - 4*(a - b)*c)*tan(e*x + d)^2 + b^2 + 4*(a - 2*b)*c + 8*c^2 - 4* ((2*a - b)*tan(e*x + d)^4 + (b - 2*c)*tan(e*x + d)^2)*sqrt(a - b + c)*sqrt ((a*tan(e*x + d)^4 + b*tan(e*x + d)^2 + c)/tan(e*x + d)^4))/(tan(e*x + d)^ 4 + 2*tan(e*x + d)^2 + 1)) - 4*((a^2*b^3 - a*b^4 + 2*a^2*c^3 + (2*a^3 - 5* a^2*b - a*b^2)*c^2 - (3*a^3*b - 2*a^2*b^2 - 2*a*b^3)*c)*tan(e*x + d)^4 - ( (2*a^2 + a*b)*c^3 + (2*a^3 - a^2*b - 2*a*b^2)*c^2 - (a^2*b^2 - a*b^3)*c)*t an(e*x + d)^2)*sqrt((a*tan(e*x + d)^4 + b*tan(e*x + d)^2 + c)/tan(e*x + d) ^4))/((a^5*b^2 - 2*a^4*b^3 + a^3*b^4 - 4*a^4*c^3 - (8*a^5 - 8*a^4*b - a^3* b^2)*c^2 - 2*(2*a^6 - 4*a^5*b + a^4*b^2 + a^3*b^3)*c)*e*tan(e*x + d)^4 + ( a^4*b^3 - 2*a^3*b^4 + a^2*b^5 - 4*a^3*b*c^3 - (8*a^4*b - 8*a^3*b^2 - a^...
\[ \int \frac {\tan (d+e x)}{\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}} \, dx=\int \frac {\tan {\left (d + e x \right )}}{\left (a + b \cot ^{2}{\left (d + e x \right )} + c \cot ^{4}{\left (d + e x \right )}\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {\tan (d+e x)}{\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]
Timed out. \[ \int \frac {\tan (d+e x)}{\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\tan (d+e x)}{\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}} \, dx=\int \frac {\mathrm {tan}\left (d+e\,x\right )}{{\left (c\,{\mathrm {cot}\left (d+e\,x\right )}^4+b\,{\mathrm {cot}\left (d+e\,x\right )}^2+a\right )}^{3/2}} \,d x \]