Integrand size = 33, antiderivative size = 280 \[ \int \frac {\cot (d+e x)}{\left (a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right )^{3/2}} \, dx=-\frac {\text {arctanh}\left (\frac {2 a+b \tan ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 a^{3/2} e}+\frac {\text {arctanh}\left (\frac {2 a-b+(b-2 c) \tan ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 (a-b+c)^{3/2} e}+\frac {b^2-2 a c+b c \tan ^2(d+e x)}{a \left (b^2-4 a c\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {b^2-2 a c-b c+(b-2 c) c \tan ^2(d+e x)}{(a-b+c) \left (b^2-4 a c\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \]
-1/2*arctanh(1/2*(2*a+b*tan(e*x+d)^2)/a^(1/2)/(a+b*tan(e*x+d)^2+c*tan(e*x+ d)^4)^(1/2))/a^(3/2)/e+1/2*arctanh(1/2*(2*a-b+(b-2*c)*tan(e*x+d)^2)/(a-b+c )^(1/2)/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2))/(a-b+c)^(3/2)/e+(b^2-2*a* c+b*c*tan(e*x+d)^2)/a/(-4*a*c+b^2)/e/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/ 2)+(-b^2+2*a*c+b*c-(b-2*c)*c*tan(e*x+d)^2)/(a-b+c)/(-4*a*c+b^2)/e/(a+b*tan (e*x+d)^2+c*tan(e*x+d)^4)^(1/2)
Time = 3.89 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.99 \[ \int \frac {\cot (d+e x)}{\left (a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right )^{3/2}} \, dx=\frac {\frac {\left (-\frac {b^2}{2}+2 a c\right ) \text {arctanh}\left (\frac {2 a+b \tan ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{a^{3/2}}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {-2 a+b-(b-2 c) \tan ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 (a-b+c)^{3/2}}+\frac {b^2-2 a c+b c \tan ^2(d+e x)}{a \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {b^2-2 a c-b c+(b-2 c) c \tan ^2(d+e x)}{(a-b+c) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}}{\left (b^2-4 a c\right ) e} \]
(((-1/2*b^2 + 2*a*c)*ArcTanh[(2*a + b*Tan[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])])/a^(3/2) - ((b^2 - 4*a*c)*ArcTanh[( -2*a + b - (b - 2*c)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])])/(2*(a - b + c)^(3/2)) + (b^2 - 2*a*c + b*c* Tan[d + e*x]^2)/(a*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - (b^2 - 2*a*c - b*c + (b - 2*c)*c*Tan[d + e*x]^2)/((a - b + c)*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]))/((b^2 - 4*a*c)*e)
Time = 0.56 (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, 4183, 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.
\(\displaystyle \int \frac {\cot (d+e x)}{\left (a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\tan (d+e x) \left (a+b \tan (d+e x)^2+c \tan (d+e x)^4\right )^{3/2}}dx\) |
\(\Big \downarrow \) 4183 |
\(\displaystyle \frac {\int \frac {\cot (d+e x)}{\left (\tan ^2(d+e x)+1\right ) \left (c \tan ^4(d+e x)+b \tan ^2(d+e x)+a\right )^{3/2}}d\tan (d+e x)}{e}\) |
\(\Big \downarrow \) 1578 |
\(\displaystyle \frac {\int \frac {\cot (d+e x)}{\left (\tan ^2(d+e x)+1\right ) \left (c \tan ^4(d+e x)+b \tan ^2(d+e x)+a\right )^{3/2}}d\tan ^2(d+e x)}{2 e}\) |
\(\Big \downarrow \) 1289 |
\(\displaystyle \frac {\int \left (\frac {\cot (d+e x)}{\left (c \tan ^4(d+e x)+b \tan ^2(d+e x)+a\right )^{3/2}}+\frac {1}{\left (-\tan ^2(d+e x)-1\right ) \left (c \tan ^4(d+e x)+b \tan ^2(d+e x)+a\right )^{3/2}}\right )d\tan ^2(d+e x)}{2 e}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {\text {arctanh}\left (\frac {2 a+b \tan ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{a^{3/2}}+\frac {\text {arctanh}\left (\frac {2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt {a-b+c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{(a-b+c)^{3/2}}+\frac {2 \left (-2 a c+b^2+b c \tan ^2(d+e x)\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {2 \left (-2 a c+b^2+c (b-2 c) \tan ^2(d+e x)-b c\right )}{(a-b+c) \left (b^2-4 a c\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}}{2 e}\) |
(-(ArcTanh[(2*a + b*Tan[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])]/a^(3/2)) + ArcTanh[(2*a - b + (b - 2*c)*Tan[d + e*x]^ 2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])]/(a - b + c)^(3/2) + (2*(b^2 - 2*a*c + b*c*Tan[d + e*x]^2))/(a*(b^2 - 4*a*c)*Sq rt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - (2*(b^2 - 2*a*c - b*c + (b - 2*c)*c*Tan[d + e*x]^2))/((a - b + c)*(b^2 - 4*a*c)*Sqrt[a + b*Tan[d + e* x]^2 + c*Tan[d + e*x]^4]))/(2*e)
3.1.49.3.1 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[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]
\[\int \frac {\cot \left (e x +d \right )}{\left (a +b \tan \left (e x +d \right )^{2}+c \tan \left (e x +d \right )^{4}\right )^{\frac {3}{2}}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 963 vs. \(2 (256) = 512\).
Time = 2.62 (sec) , antiderivative size = 3951, normalized size of antiderivative = 14.11 \[ \int \frac {\cot (d+e x)}{\left (a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right )^{3/2}} \, dx=\text {Too large to display} \]
[-1/4*((a^3*b^2 - 4*a^4*c + (a^2*b^2*c - 4*a^3*c^2)*tan(e*x + d)^4 + (a^2* b^3 - 4*a^3*b*c)*tan(e*x + d)^2)*sqrt(a - b + c)*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*tan(e*x + d)^4 + 2*(4*a*b - 3*b^2 - 4*(a - b)*c)*tan(e*x + d)^2 + 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^ 2 + 2*a - b)*sqrt(a - b + c) + 8*a^2 - 8*a*b + b^2 + 4*a*c)/(tan(e*x + d)^ 4 + 2*tan(e*x + d)^2 + 1)) + (a^3*b^2 - 2*a^2*b^3 + a*b^4 - 4*a^2*c^3 - (4 *a*c^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^2 - 2*a*b^3 + b^4)*c)*tan(e*x + d)^4 - (8*a^3 - 8*a^2*b - a*b^2) *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 - 2*(2*a^4 - 4 *a^3*b + a^2*b^2 + a*b^3)*c)*sqrt(a)*log(((b^2 + 4*a*c)*tan(e*x + d)^4 + 8 *a*b*tan(e*x + d)^2 - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(b*t an(e*x + d)^2 + 2*a)*sqrt(a) + 8*a^2)/tan(e*x + d)^4) - 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 - ((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)*tan(e*x + d)^2 - (3*a^3*b - 2*a^2*b^2 - 2*a*b^3)*c)*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a))/(( 4*a^3*c^4 + (8*a^4 - 8*a^3*b - a^2*b^2)*c^3 + 2*(2*a^5 - 4*a^4*b + a^3*b^2 + a^2*b^3)*c^2 - (a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*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^2*b^ 3)*c^2 - 2*(2*a^5*b - 4*a^4*b^2 + a^3*b^3 + a^2*b^4)*c)*e*tan(e*x + d)^...
\[ \int \frac {\cot (d+e x)}{\left (a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right )^{3/2}} \, dx=\int \frac {\cot {\left (d + e x \right )}}{\left (a + b \tan ^{2}{\left (d + e x \right )} + c \tan ^{4}{\left (d + e x \right )}\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {\cot (d+e x)}{\left (a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right )^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]
Timed out. \[ \int \frac {\cot (d+e x)}{\left (a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right )^{3/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\cot (d+e x)}{\left (a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right )^{3/2}} \, dx=\int \frac {\mathrm {cot}\left (d+e\,x\right )}{{\left (c\,{\mathrm {tan}\left (d+e\,x\right )}^4+b\,{\mathrm {tan}\left (d+e\,x\right )}^2+a\right )}^{3/2}} \,d x \]