Integrand size = 23, antiderivative size = 132 \[ \int \frac {\cot ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=-\frac {\cot ^2(e+f x)}{2 a^2 f}-\frac {\log (\cos (e+f x))}{(a-b)^2 f}-\frac {(a+2 b) \log (\tan (e+f x))}{a^3 f}-\frac {(3 a-2 b) b^2 \log \left (a+b \tan ^2(e+f x)\right )}{2 a^3 (a-b)^2 f}+\frac {b^2}{2 a^2 (a-b) f \left (a+b \tan ^2(e+f x)\right )} \]
-1/2*cot(f*x+e)^2/a^2/f-ln(cos(f*x+e))/(a-b)^2/f-(a+2*b)*ln(tan(f*x+e))/a^ 3/f-1/2*(3*a-2*b)*b^2*ln(a+b*tan(f*x+e)^2)/a^3/(a-b)^2/f+1/2*b^2/a^2/(a-b) /f/(a+b*tan(f*x+e)^2)
Time = 0.88 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.74 \[ \int \frac {\cot ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=-\frac {\frac {\cot ^2(e+f x)}{a^2}+\frac {b^3}{a^3 (a-b) \left (b+a \cot ^2(e+f x)\right )}+\frac {(3 a-2 b) b^2 \log \left (b+a \cot ^2(e+f x)\right )}{a^3 (a-b)^2}+\frac {2 \log (\sin (e+f x))}{(a-b)^2}}{2 f} \]
-1/2*(Cot[e + f*x]^2/a^2 + b^3/(a^3*(a - b)*(b + a*Cot[e + f*x]^2)) + ((3* a - 2*b)*b^2*Log[b + a*Cot[e + f*x]^2])/(a^3*(a - b)^2) + (2*Log[Sin[e + f *x]])/(a - b)^2)/f
Time = 0.36 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.91, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3042, 4153, 354, 99, 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.
| \(\displaystyle \int \frac {\cot ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (e+f x)^3 \left (a+b \tan (e+f x)^2\right )^2}dx\) |
| \(\Big \downarrow \) 4153 |
| \(\displaystyle \frac {\int \frac {\cot ^3(e+f x)}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )^2}d\tan (e+f x)}{f}\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {\int \frac {\cot ^2(e+f x)}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )^2}d\tan ^2(e+f x)}{2 f}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {\int \left (-\frac {(3 a-2 b) b^3}{a^3 (a-b)^2 \left (b \tan ^2(e+f x)+a\right )}-\frac {b^3}{a^2 (a-b) \left (b \tan ^2(e+f x)+a\right )^2}+\frac {\cot ^2(e+f x)}{a^2}+\frac {(-a-2 b) \cot (e+f x)}{a^3}+\frac {1}{(a-b)^2 \left (\tan ^2(e+f x)+1\right )}\right )d\tan ^2(e+f x)}{2 f}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {b^2 (3 a-2 b) \log \left (a+b \tan ^2(e+f x)\right )}{a^3 (a-b)^2}-\frac {(a+2 b) \log \left (\tan ^2(e+f x)\right )}{a^3}+\frac {b^2}{a^2 (a-b) \left (a+b \tan ^2(e+f x)\right )}-\frac {\cot (e+f x)}{a^2}+\frac {\log \left (\tan ^2(e+f x)+1\right )}{(a-b)^2}}{2 f}\) |
(-(Cot[e + f*x]/a^2) - ((a + 2*b)*Log[Tan[e + f*x]^2])/a^3 + Log[1 + Tan[e + f*x]^2]/(a - b)^2 - ((3*a - 2*b)*b^2*Log[a + b*Tan[e + f*x]^2])/(a^3*(a - b)^2) + b^2/(a^2*(a - b)*(a + b*Tan[e + f*x]^2)))/(2*f)
3.3.28.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[c*(ff/f) Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2 + f f^2*x^2)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ[n, 2] || EqQ[n, 4] || (IntegerQ[p] && Ratio nalQ[n]))
Time = 0.32 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.92
| method | result | size |
| derivativedivides | \(\frac {-\frac {1}{2 a^{2} \tan \left (f x +e \right )^{2}}+\frac {\left (-2 b -a \right ) \ln \left (\tan \left (f x +e \right )\right )}{a^{3}}+\frac {\ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2 \left (a -b \right )^{2}}-\frac {b^{3} \left (\frac {\left (3 a -2 b \right ) \ln \left (a +b \tan \left (f x +e \right )^{2}\right )}{b}-\frac {a \left (a -b \right )}{b \left (a +b \tan \left (f x +e \right )^{2}\right )}\right )}{2 a^{3} \left (a -b \right )^{2}}}{f}\) | \(122\) |
| default | \(\frac {-\frac {1}{2 a^{2} \tan \left (f x +e \right )^{2}}+\frac {\left (-2 b -a \right ) \ln \left (\tan \left (f x +e \right )\right )}{a^{3}}+\frac {\ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2 \left (a -b \right )^{2}}-\frac {b^{3} \left (\frac {\left (3 a -2 b \right ) \ln \left (a +b \tan \left (f x +e \right )^{2}\right )}{b}-\frac {a \left (a -b \right )}{b \left (a +b \tan \left (f x +e \right )^{2}\right )}\right )}{2 a^{3} \left (a -b \right )^{2}}}{f}\) | \(122\) |
| parallelrisch | \(\frac {-3 \left (a -\frac {2 b}{3}\right ) b^{2} \left (a +b \tan \left (f x +e \right )^{2}\right ) \ln \left (a +b \tan \left (f x +e \right )^{2}\right )+a^{3} \left (a +b \tan \left (f x +e \right )^{2}\right ) \ln \left (\sec \left (f x +e \right )^{2}\right )-\left (a -b \right ) \left (2 \left (a +2 b \right ) \left (a -b \right ) \left (a +b \tan \left (f x +e \right )^{2}\right ) \ln \left (\tan \left (f x +e \right )\right )+a \left (\cot \left (f x +e \right )^{2} a \left (a -b \right )+b \left (a -2 b \right )\right )\right )}{2 \left (a -b \right )^{2} a^{3} f \left (a +b \tan \left (f x +e \right )^{2}\right )}\) | \(155\) |
| norman | \(\frac {-\frac {1}{2 a f}+\frac {\left (-a \,b^{2}+2 b^{3}\right ) \tan \left (f x +e \right )^{2}}{2 a^{2} f b \left (a -b \right )}}{\tan \left (f x +e \right )^{2} \left (a +b \tan \left (f x +e \right )^{2}\right )}+\frac {\ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2 f \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (a +2 b \right ) \ln \left (\tan \left (f x +e \right )\right )}{a^{3} f}-\frac {b^{2} \left (3 a -2 b \right ) \ln \left (a +b \tan \left (f x +e \right )^{2}\right )}{2 a^{3} f \left (a^{2}-2 a b +b^{2}\right )}\) | \(165\) |
| risch | \(-\frac {i x}{a^{2}-2 a b +b^{2}}+\frac {2 i x}{a^{2}}+\frac {2 i e}{a^{2} f}+\frac {4 i b x}{a^{3}}+\frac {4 i b e}{a^{3} f}+\frac {6 i b^{2} x}{a^{2} \left (a^{2}-2 a b +b^{2}\right )}+\frac {6 i b^{2} e}{a^{2} f \left (a^{2}-2 a b +b^{2}\right )}-\frac {4 i b^{3} x}{a^{3} \left (a^{2}-2 a b +b^{2}\right )}-\frac {4 i b^{3} e}{a^{3} f \left (a^{2}-2 a b +b^{2}\right )}+\frac {2 a^{3} {\mathrm e}^{6 i \left (f x +e \right )}-6 a^{2} b \,{\mathrm e}^{6 i \left (f x +e \right )}+6 a \,b^{2} {\mathrm e}^{6 i \left (f x +e \right )}-4 b^{3} {\mathrm e}^{6 i \left (f x +e \right )}+4 a^{3} {\mathrm e}^{4 i \left (f x +e \right )}-4 a^{2} b \,{\mathrm e}^{4 i \left (f x +e \right )}-4 a \,b^{2} {\mathrm e}^{4 i \left (f x +e \right )}+8 b^{3} {\mathrm e}^{4 i \left (f x +e \right )}+2 a^{3} {\mathrm e}^{2 i \left (f x +e \right )}-6 a^{2} b \,{\mathrm e}^{2 i \left (f x +e \right )}+6 a \,b^{2} {\mathrm e}^{2 i \left (f x +e \right )}-4 b^{3} {\mathrm e}^{2 i \left (f x +e \right )}}{a^{2} f \left (a -b \right )^{2} \left (a \,{\mathrm e}^{4 i \left (f x +e \right )}-b \,{\mathrm e}^{4 i \left (f x +e \right )}+2 a \,{\mathrm e}^{2 i \left (f x +e \right )}+2 b \,{\mathrm e}^{2 i \left (f x +e \right )}+a -b \right ) \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{2}}-\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )}{a^{2} f}-\frac {2 \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right ) b}{a^{3} f}-\frac {3 b^{2} \ln \left ({\mathrm e}^{4 i \left (f x +e \right )}+\frac {2 \left (a +b \right ) {\mathrm e}^{2 i \left (f x +e \right )}}{a -b}+1\right )}{2 a^{2} f \left (a^{2}-2 a b +b^{2}\right )}+\frac {b^{3} \ln \left ({\mathrm e}^{4 i \left (f x +e \right )}+\frac {2 \left (a +b \right ) {\mathrm e}^{2 i \left (f x +e \right )}}{a -b}+1\right )}{a^{3} f \left (a^{2}-2 a b +b^{2}\right )}\) | \(562\) |
1/f*(-1/2/a^2/tan(f*x+e)^2+(-2*b-a)/a^3*ln(tan(f*x+e))+1/2/(a-b)^2*ln(1+ta n(f*x+e)^2)-1/2*b^3/a^3/(a-b)^2*((3*a-2*b)/b*ln(a+b*tan(f*x+e)^2)-a*(a-b)/ b/(a+b*tan(f*x+e)^2)))
Leaf count of result is larger than twice the leaf count of optimal. 292 vs. \(2 (126) = 252\).
Time = 0.30 (sec) , antiderivative size = 292, normalized size of antiderivative = 2.21 \[ \int \frac {\cot ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=-\frac {{\left (a^{3} b - 2 \, a^{2} b^{2} + 2 \, a b^{3}\right )} \tan \left (f x + e\right )^{4} + a^{4} - 2 \, a^{3} b + a^{2} b^{2} + {\left (a^{4} - a^{3} b - a^{2} b^{2} + 2 \, a b^{3}\right )} \tan \left (f x + e\right )^{2} + {\left ({\left (a^{3} b - 3 \, a b^{3} + 2 \, b^{4}\right )} \tan \left (f x + e\right )^{4} + {\left (a^{4} - 3 \, a^{2} b^{2} + 2 \, a b^{3}\right )} \tan \left (f x + e\right )^{2}\right )} \log \left (\frac {\tan \left (f x + e\right )^{2}}{\tan \left (f x + e\right )^{2} + 1}\right ) + {\left ({\left (3 \, a b^{3} - 2 \, b^{4}\right )} \tan \left (f x + e\right )^{4} + {\left (3 \, a^{2} b^{2} - 2 \, a b^{3}\right )} \tan \left (f x + e\right )^{2}\right )} \log \left (\frac {b \tan \left (f x + e\right )^{2} + a}{\tan \left (f x + e\right )^{2} + 1}\right )}{2 \, {\left ({\left (a^{5} b - 2 \, a^{4} b^{2} + a^{3} b^{3}\right )} f \tan \left (f x + e\right )^{4} + {\left (a^{6} - 2 \, a^{5} b + a^{4} b^{2}\right )} f \tan \left (f x + e\right )^{2}\right )}} \]
-1/2*((a^3*b - 2*a^2*b^2 + 2*a*b^3)*tan(f*x + e)^4 + a^4 - 2*a^3*b + a^2*b ^2 + (a^4 - a^3*b - a^2*b^2 + 2*a*b^3)*tan(f*x + e)^2 + ((a^3*b - 3*a*b^3 + 2*b^4)*tan(f*x + e)^4 + (a^4 - 3*a^2*b^2 + 2*a*b^3)*tan(f*x + e)^2)*log( tan(f*x + e)^2/(tan(f*x + e)^2 + 1)) + ((3*a*b^3 - 2*b^4)*tan(f*x + e)^4 + (3*a^2*b^2 - 2*a*b^3)*tan(f*x + e)^2)*log((b*tan(f*x + e)^2 + a)/(tan(f*x + e)^2 + 1)))/((a^5*b - 2*a^4*b^2 + a^3*b^3)*f*tan(f*x + e)^4 + (a^6 - 2* a^5*b + a^4*b^2)*f*tan(f*x + e)^2)
Leaf count of result is larger than twice the leaf count of optimal. 3568 vs. \(2 (107) = 214\).
Time = 111.58 (sec) , antiderivative size = 3568, normalized size of antiderivative = 27.03 \[ \int \frac {\cot ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=\text {Too large to display} \]
Piecewise(((log(tan(e + f*x)**2 + 1)/(2*f) - log(tan(e + f*x))/f - 1/(2*f* tan(e + f*x)**2))/a**2, Eq(b, 0)), ((log(tan(e + f*x)**2 + 1)/(2*f) - log( tan(e + f*x))/f - 1/(2*f*tan(e + f*x)**2) + 1/(4*f*tan(e + f*x)**4) - 1/(6 *f*tan(e + f*x)**6))/b**2, Eq(a, 0)), (6*log(tan(e + f*x)**2 + 1)*tan(e + f*x)**6/(4*a**2*f*tan(e + f*x)**6 + 8*a**2*f*tan(e + f*x)**4 + 4*a**2*f*ta n(e + f*x)**2) + 12*log(tan(e + f*x)**2 + 1)*tan(e + f*x)**4/(4*a**2*f*tan (e + f*x)**6 + 8*a**2*f*tan(e + f*x)**4 + 4*a**2*f*tan(e + f*x)**2) + 6*lo g(tan(e + f*x)**2 + 1)*tan(e + f*x)**2/(4*a**2*f*tan(e + f*x)**6 + 8*a**2* f*tan(e + f*x)**4 + 4*a**2*f*tan(e + f*x)**2) - 12*log(tan(e + f*x))*tan(e + f*x)**6/(4*a**2*f*tan(e + f*x)**6 + 8*a**2*f*tan(e + f*x)**4 + 4*a**2*f *tan(e + f*x)**2) - 24*log(tan(e + f*x))*tan(e + f*x)**4/(4*a**2*f*tan(e + f*x)**6 + 8*a**2*f*tan(e + f*x)**4 + 4*a**2*f*tan(e + f*x)**2) - 12*log(t an(e + f*x))*tan(e + f*x)**2/(4*a**2*f*tan(e + f*x)**6 + 8*a**2*f*tan(e + f*x)**4 + 4*a**2*f*tan(e + f*x)**2) - 6*tan(e + f*x)**4/(4*a**2*f*tan(e + f*x)**6 + 8*a**2*f*tan(e + f*x)**4 + 4*a**2*f*tan(e + f*x)**2) - 9*tan(e + f*x)**2/(4*a**2*f*tan(e + f*x)**6 + 8*a**2*f*tan(e + f*x)**4 + 4*a**2*f*t an(e + f*x)**2) - 2/(4*a**2*f*tan(e + f*x)**6 + 8*a**2*f*tan(e + f*x)**4 + 4*a**2*f*tan(e + f*x)**2), Eq(a, b)), (zoo*(log(tan(e + f*x)**2 + 1)/(2*f ) - log(tan(e + f*x))/f - 1/(2*f*tan(e + f*x)**2)), Eq(b, -a/tan(e + f*x)* *2)), (zoo*x/a**2, Eq(e, -f*x)), (x*cot(e)**3/(a + b*tan(e)**2)**2, Eq(...
Time = 0.27 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.42 \[ \int \frac {\cot ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=-\frac {\frac {{\left (3 \, a b^{2} - 2 \, b^{3}\right )} \log \left (-{\left (a - b\right )} \sin \left (f x + e\right )^{2} + a\right )}{a^{5} - 2 \, a^{4} b + a^{3} b^{2}} - \frac {a^{3} - 2 \, a^{2} b + a b^{2} - {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - 2 \, b^{3}\right )} \sin \left (f x + e\right )^{2}}{{\left (a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right )} \sin \left (f x + e\right )^{4} - {\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} \sin \left (f x + e\right )^{2}} + \frac {{\left (a + 2 \, b\right )} \log \left (\sin \left (f x + e\right )^{2}\right )}{a^{3}}}{2 \, f} \]
-1/2*((3*a*b^2 - 2*b^3)*log(-(a - b)*sin(f*x + e)^2 + a)/(a^5 - 2*a^4*b + a^3*b^2) - (a^3 - 2*a^2*b + a*b^2 - (a^3 - 3*a^2*b + 3*a*b^2 - 2*b^3)*sin( f*x + e)^2)/((a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*sin(f*x + e)^4 - (a^5 - 2*a^4*b + a^3*b^2)*sin(f*x + e)^2) + (a + 2*b)*log(sin(f*x + e)^2)/a^3)/f
Leaf count of result is larger than twice the leaf count of optimal. 640 vs. \(2 (126) = 252\).
Time = 0.96 (sec) , antiderivative size = 640, normalized size of antiderivative = 4.85 \[ \int \frac {\cot ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=-\frac {\frac {12 \, {\left (3 \, a b^{2} - 2 \, b^{3}\right )} \log \left (a + \frac {2 \, a {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {4 \, b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {a {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )}{a^{5} - 2 \, a^{4} b + a^{3} b^{2}} - \frac {24 \, \log \left ({\left | -\frac {\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1} + 1 \right |}\right )}{a^{2} - 2 \, a b + b^{2}} - \frac {3 \, a^{4} - 6 \, a^{3} b + 3 \, a^{2} b^{2} + \frac {10 \, a^{4} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {24 \, a^{3} b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {42 \, a^{2} b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {20 \, a b^{3} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {11 \, a^{4} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {22 \, a^{3} b {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {27 \, a^{2} b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {16 \, a b^{3} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {16 \, b^{4} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {4 \, a^{4} {\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {12 \, a^{2} b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {8 \, a b^{3} {\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} {\left (\frac {a {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {2 \, a {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {4 \, b {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a {\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}} + \frac {12 \, {\left (a + 2 \, b\right )} \log \left (\frac {{\left | -\cos \left (f x + e\right ) + 1 \right |}}{{\left | \cos \left (f x + e\right ) + 1 \right |}}\right )}{a^{3}} - \frac {3 \, {\left (\cos \left (f x + e\right ) - 1\right )}}{a^{2} {\left (\cos \left (f x + e\right ) + 1\right )}}}{24 \, f} \]
-1/24*(12*(3*a*b^2 - 2*b^3)*log(a + 2*a*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) - 4*b*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) + a*(cos(f*x + e) - 1)^2/( cos(f*x + e) + 1)^2)/(a^5 - 2*a^4*b + a^3*b^2) - 24*log(abs(-(cos(f*x + e) - 1)/(cos(f*x + e) + 1) + 1))/(a^2 - 2*a*b + b^2) - (3*a^4 - 6*a^3*b + 3* a^2*b^2 + 10*a^4*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) - 24*a^3*b*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) + 42*a^2*b^2*(cos(f*x + e) - 1)/(cos(f*x + e ) + 1) - 20*a*b^3*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) + 11*a^4*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2 - 22*a^3*b*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2 + 27*a^2*b^2*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2 - 16*a* b^3*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2 - 16*b^4*(cos(f*x + e) - 1)^ 2/(cos(f*x + e) + 1)^2 + 4*a^4*(cos(f*x + e) - 1)^3/(cos(f*x + e) + 1)^3 + 12*a^2*b^2*(cos(f*x + e) - 1)^3/(cos(f*x + e) + 1)^3 - 8*a*b^3*(cos(f*x + e) - 1)^3/(cos(f*x + e) + 1)^3)/((a^5 - 2*a^4*b + a^3*b^2)*(a*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) + 2*a*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2 - 4*b*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2 + a*(cos(f*x + e) - 1)^3/ (cos(f*x + e) + 1)^3)) + 12*(a + 2*b)*log(abs(-cos(f*x + e) + 1)/abs(cos(f *x + e) + 1))/a^3 - 3*(cos(f*x + e) - 1)/(a^2*(cos(f*x + e) + 1)))/f
Time = 11.54 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.09 \[ \int \frac {\cot ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=\frac {\ln \left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )\,\left (\frac {b}{a^3}+\frac {1}{2\,a^2}-\frac {1}{2\,{\left (a-b\right )}^2}\right )}{f}-\frac {\frac {1}{2\,a}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (a\,b-2\,b^2\right )}{2\,a^2\,\left (a-b\right )}}{f\,\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^4+a\,{\mathrm {tan}\left (e+f\,x\right )}^2\right )}+\frac {\ln \left ({\mathrm {tan}\left (e+f\,x\right )}^2+1\right )}{2\,f\,{\left (a-b\right )}^2}-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )\right )\,\left (a+2\,b\right )}{a^3\,f} \]