Integrand size = 21, antiderivative size = 278 \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^4} \, dx=-\frac {\left (a^4-6 a^2 b^2+b^4\right ) x}{\left (a^2+b^2\right )^4}-\frac {4 b \log (\sin (c+d x))}{a^5 d}+\frac {4 b^3 \left (5 a^6+6 a^4 b^2+4 a^2 b^4+b^6\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^5 \left (a^2+b^2\right )^4 d}-\frac {b \left (3 a^2+4 b^2\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^3}-\frac {b \left (a^4+4 a^2 b^2+2 b^4\right )}{a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {b \left (a^6+13 a^4 b^2+12 a^2 b^4+4 b^6\right )}{a^4 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))} \] Output:
-(a^4-6*a^2*b^2+b^4)*x/(a^2+b^2)^4-4*b*ln(sin(d*x+c))/a^5/d+4*b^3*(5*a^6+6 *a^4*b^2+4*a^2*b^4+b^6)*ln(a*cos(d*x+c)+b*sin(d*x+c))/a^5/(a^2+b^2)^4/d-1/ 3*b*(3*a^2+4*b^2)/a^2/(a^2+b^2)/d/(a+b*tan(d*x+c))^3-cot(d*x+c)/a/d/(a+b*t an(d*x+c))^3-b*(a^4+4*a^2*b^2+2*b^4)/a^3/(a^2+b^2)^2/d/(a+b*tan(d*x+c))^2- b*(a^6+13*a^4*b^2+12*a^2*b^4+4*b^6)/a^4/(a^2+b^2)^3/d/(a+b*tan(d*x+c))
Result contains complex when optimal does not.
Time = 2.45 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.87 \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^4} \, dx=-\frac {\frac {\cot (c+d x)}{a^4}-\frac {b^6}{3 a^5 \left (a^2+b^2\right ) (b+a \cot (c+d x))^3}+\frac {b^5 \left (3 a^2+2 b^2\right )}{a^5 \left (a^2+b^2\right )^2 (b+a \cot (c+d x))^2}-\frac {b^4 \left (15 a^4+17 a^2 b^2+6 b^4\right )}{a^5 \left (a^2+b^2\right )^3 (b+a \cot (c+d x))}+\frac {i \log (i-\cot (c+d x))}{2 (a-i b)^4}-\frac {i \log (i+\cot (c+d x))}{2 (a+i b)^4}-\frac {4 b^3 \left (5 a^6+6 a^4 b^2+4 a^2 b^4+b^6\right ) \log (b+a \cot (c+d x))}{a^5 \left (a^2+b^2\right )^4}}{d} \] Input:
Integrate[Cot[c + d*x]^2/(a + b*Tan[c + d*x])^4,x]
Output:
-((Cot[c + d*x]/a^4 - b^6/(3*a^5*(a^2 + b^2)*(b + a*Cot[c + d*x])^3) + (b^ 5*(3*a^2 + 2*b^2))/(a^5*(a^2 + b^2)^2*(b + a*Cot[c + d*x])^2) - (b^4*(15*a ^4 + 17*a^2*b^2 + 6*b^4))/(a^5*(a^2 + b^2)^3*(b + a*Cot[c + d*x])) + ((I/2 )*Log[I - Cot[c + d*x]])/(a - I*b)^4 - ((I/2)*Log[I + Cot[c + d*x]])/(a + I*b)^4 - (4*b^3*(5*a^6 + 6*a^4*b^2 + 4*a^2*b^4 + b^6)*Log[b + a*Cot[c + d* x]])/(a^5*(a^2 + b^2)^4))/d)
Time = 2.02 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.21, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.762, Rules used = {3042, 4052, 3042, 4132, 27, 3042, 4132, 27, 3042, 4132, 3042, 4134, 3042, 25, 3956, 4013}
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.
| \(\displaystyle \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (c+d x)^2 (a+b \tan (c+d x))^4}dx\) |
| \(\Big \downarrow \) 4052 |
| \(\displaystyle -\frac {\int \frac {\cot (c+d x) \left (4 b \tan ^2(c+d x)+a \tan (c+d x)+4 b\right )}{(a+b \tan (c+d x))^4}dx}{a}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {4 b \tan (c+d x)^2+a \tan (c+d x)+4 b}{\tan (c+d x) (a+b \tan (c+d x))^4}dx}{a}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle -\frac {\frac {\int \frac {3 \cot (c+d x) \left (\tan (c+d x) a^3+b \left (3 a^2+4 b^2\right ) \tan ^2(c+d x)+4 b \left (a^2+b^2\right )\right )}{(a+b \tan (c+d x))^3}dx}{3 a \left (a^2+b^2\right )}+\frac {b \left (3 a^2+4 b^2\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}}{a}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\int \frac {\cot (c+d x) \left (\tan (c+d x) a^3+b \left (3 a^2+4 b^2\right ) \tan ^2(c+d x)+4 b \left (a^2+b^2\right )\right )}{(a+b \tan (c+d x))^3}dx}{a \left (a^2+b^2\right )}+\frac {b \left (3 a^2+4 b^2\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}}{a}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\int \frac {\tan (c+d x) a^3+b \left (3 a^2+4 b^2\right ) \tan (c+d x)^2+4 b \left (a^2+b^2\right )}{\tan (c+d x) (a+b \tan (c+d x))^3}dx}{a \left (a^2+b^2\right )}+\frac {b \left (3 a^2+4 b^2\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}}{a}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {2 \cot (c+d x) \left (\left (a^2-b^2\right ) \tan (c+d x) a^3+4 b \left (a^2+b^2\right )^2+2 b \left (a^4+4 b^2 a^2+2 b^4\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2}dx}{2 a \left (a^2+b^2\right )}+\frac {b \left (a^4+4 a^2 b^2+2 b^4\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a \left (a^2+b^2\right )}+\frac {b \left (3 a^2+4 b^2\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}}{a}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {\cot (c+d x) \left (\left (a^2-b^2\right ) \tan (c+d x) a^3+4 b \left (a^2+b^2\right )^2+2 b \left (a^4+4 b^2 a^2+2 b^4\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2}dx}{a \left (a^2+b^2\right )}+\frac {b \left (a^4+4 a^2 b^2+2 b^4\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a \left (a^2+b^2\right )}+\frac {b \left (3 a^2+4 b^2\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}}{a}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {\left (a^2-b^2\right ) \tan (c+d x) a^3+4 b \left (a^2+b^2\right )^2+2 b \left (a^4+4 b^2 a^2+2 b^4\right ) \tan (c+d x)^2}{\tan (c+d x) (a+b \tan (c+d x))^2}dx}{a \left (a^2+b^2\right )}+\frac {b \left (a^4+4 a^2 b^2+2 b^4\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a \left (a^2+b^2\right )}+\frac {b \left (3 a^2+4 b^2\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}}{a}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\int \frac {\cot (c+d x) \left (\left (a^2-3 b^2\right ) \tan (c+d x) a^5+4 b \left (a^2+b^2\right )^3+b \left (a^6+13 b^2 a^4+12 b^4 a^2+4 b^6\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}+\frac {b \left (a^6+13 a^4 b^2+12 a^2 b^4+4 b^6\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}+\frac {b \left (a^4+4 a^2 b^2+2 b^4\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a \left (a^2+b^2\right )}+\frac {b \left (3 a^2+4 b^2\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}}{a}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\int \frac {\left (a^2-3 b^2\right ) \tan (c+d x) a^5+4 b \left (a^2+b^2\right )^3+b \left (a^6+13 b^2 a^4+12 b^4 a^2+4 b^6\right ) \tan (c+d x)^2}{\tan (c+d x) (a+b \tan (c+d x))}dx}{a \left (a^2+b^2\right )}+\frac {b \left (a^6+13 a^4 b^2+12 a^2 b^4+4 b^6\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}+\frac {b \left (a^4+4 a^2 b^2+2 b^4\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a \left (a^2+b^2\right )}+\frac {b \left (3 a^2+4 b^2\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}}{a}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 4134 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\frac {4 b \left (a^2+b^2\right )^3 \int \cot (c+d x)dx}{a}-\frac {4 b^3 \left (5 a^6+6 a^4 b^2+4 a^2 b^4+b^6\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}+\frac {a^4 x \left (a^4-6 a^2 b^2+b^4\right )}{a^2+b^2}}{a \left (a^2+b^2\right )}+\frac {b \left (a^6+13 a^4 b^2+12 a^2 b^4+4 b^6\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}+\frac {b \left (a^4+4 a^2 b^2+2 b^4\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a \left (a^2+b^2\right )}+\frac {b \left (3 a^2+4 b^2\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}}{a}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\frac {4 b \left (a^2+b^2\right )^3 \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx}{a}-\frac {4 b^3 \left (5 a^6+6 a^4 b^2+4 a^2 b^4+b^6\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}+\frac {a^4 x \left (a^4-6 a^2 b^2+b^4\right )}{a^2+b^2}}{a \left (a^2+b^2\right )}+\frac {b \left (a^6+13 a^4 b^2+12 a^2 b^4+4 b^6\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}+\frac {b \left (a^4+4 a^2 b^2+2 b^4\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a \left (a^2+b^2\right )}+\frac {b \left (3 a^2+4 b^2\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}}{a}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\frac {\frac {-\frac {4 b \left (a^2+b^2\right )^3 \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx}{a}-\frac {4 b^3 \left (5 a^6+6 a^4 b^2+4 a^2 b^4+b^6\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}+\frac {a^4 x \left (a^4-6 a^2 b^2+b^4\right )}{a^2+b^2}}{a \left (a^2+b^2\right )}+\frac {b \left (a^6+13 a^4 b^2+12 a^2 b^4+4 b^6\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}+\frac {b \left (a^4+4 a^2 b^2+2 b^4\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a \left (a^2+b^2\right )}+\frac {b \left (3 a^2+4 b^2\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}}{a}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle -\frac {\frac {\frac {\frac {-\frac {4 b^3 \left (5 a^6+6 a^4 b^2+4 a^2 b^4+b^6\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}+\frac {4 b \left (a^2+b^2\right )^3 \log (-\sin (c+d x))}{a d}+\frac {a^4 x \left (a^4-6 a^2 b^2+b^4\right )}{a^2+b^2}}{a \left (a^2+b^2\right )}+\frac {b \left (a^6+13 a^4 b^2+12 a^2 b^4+4 b^6\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}+\frac {b \left (a^4+4 a^2 b^2+2 b^4\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a \left (a^2+b^2\right )}+\frac {b \left (3 a^2+4 b^2\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}}{a}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 4013 |
| \(\displaystyle -\frac {\frac {b \left (3 a^2+4 b^2\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac {\frac {b \left (a^4+4 a^2 b^2+2 b^4\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {\frac {b \left (a^6+13 a^4 b^2+12 a^2 b^4+4 b^6\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {\frac {4 b \left (a^2+b^2\right )^3 \log (-\sin (c+d x))}{a d}+\frac {a^4 x \left (a^4-6 a^2 b^2+b^4\right )}{a^2+b^2}-\frac {4 b^3 \left (5 a^6+6 a^4 b^2+4 a^2 b^4+b^6\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a d \left (a^2+b^2\right )}}{a \left (a^2+b^2\right )}}{a \left (a^2+b^2\right )}}{a \left (a^2+b^2\right )}}{a}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^3}\) |
Input:
Int[Cot[c + d*x]^2/(a + b*Tan[c + d*x])^4,x]
Output:
-(Cot[c + d*x]/(a*d*(a + b*Tan[c + d*x])^3)) - ((b*(3*a^2 + 4*b^2))/(3*a*( a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) + ((b*(a^4 + 4*a^2*b^2 + 2*b^4))/(a*( a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (((a^4*(a^4 - 6*a^2*b^2 + b^4)*x)/( a^2 + b^2) + (4*b*(a^2 + b^2)^3*Log[-Sin[c + d*x]])/(a*d) - (4*b^3*(5*a^6 + 6*a^4*b^2 + 4*a^2*b^4 + b^6)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a*(a ^2 + b^2)*d))/(a*(a^2 + b^2)) + (b*(a^6 + 13*a^4*b^2 + 12*a^2*b^4 + 4*b^6) )/(a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])))/(a*(a^2 + b^2)))/(a*(a^2 + b^2)) )/a
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)* (x_)]), x_Symbol] :> Simp[(c/(b*f))*Log[RemoveContent[a*Cos[e + f*x] + b*Si n[e + f*x], x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d))), x] + Simp[1 /((m + 1)*(a^2 + b^2)*(b*c - a*d)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*Tan[e + f*x]^2, x], x], x] / ; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || Integ erQ[m]) && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 + b^2)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1) - b^2*d* (m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d )*(A*b - a*B - b*C)*Tan[e + f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Ta n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ [b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Int[((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^ 2)/(((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.) *(x_)])), x_Symbol] :> Simp[(a*(A*c - c*C + B*d) + b*(B*c - A*d + C*d))*(x/ ((a^2 + b^2)*(c^2 + d^2))), x] + (Simp[(A*b^2 - a*b*B + a^2*C)/((b*c - a*d) *(a^2 + b^2)) Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x] - Sim p[(c^2*C - B*c*d + A*d^2)/((b*c - a*d)*(c^2 + d^2)) Int[(d - c*Tan[e + f* x])/(c + d*Tan[e + f*x]), x], x]) /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
Time = 2.90 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.95
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\left (4 a^{3} b -4 a \,b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-a^{4}+6 b^{2} a^{2}-b^{4}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}-\frac {b^{3}}{3 a^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {b^{3} \left (10 a^{4}+9 b^{2} a^{2}+3 b^{4}\right )}{a^{4} \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {b^{3} \left (2 a^{2}+b^{2}\right )}{a^{3} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {4 b^{3} \left (5 a^{6}+6 a^{4} b^{2}+4 a^{2} b^{4}+b^{6}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{5} \left (a^{2}+b^{2}\right )^{4}}-\frac {1}{a^{4} \tan \left (d x +c \right )}-\frac {4 b \ln \left (\tan \left (d x +c \right )\right )}{a^{5}}}{d}\) | \(264\) |
| default | \(\frac {\frac {\frac {\left (4 a^{3} b -4 a \,b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-a^{4}+6 b^{2} a^{2}-b^{4}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}-\frac {b^{3}}{3 a^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {b^{3} \left (10 a^{4}+9 b^{2} a^{2}+3 b^{4}\right )}{a^{4} \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {b^{3} \left (2 a^{2}+b^{2}\right )}{a^{3} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {4 b^{3} \left (5 a^{6}+6 a^{4} b^{2}+4 a^{2} b^{4}+b^{6}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{5} \left (a^{2}+b^{2}\right )^{4}}-\frac {1}{a^{4} \tan \left (d x +c \right )}-\frac {4 b \ln \left (\tan \left (d x +c \right )\right )}{a^{5}}}{d}\) | \(264\) |
| parallelrisch | \(\frac {20 \left (a^{6}+\frac {6}{5} a^{4} b^{2}+\frac {4}{5} a^{2} b^{4}+\frac {1}{5} b^{6}\right ) b^{3} \left (a +b \tan \left (d x +c \right )\right )^{3} \ln \left (a +b \tan \left (d x +c \right )\right )+2 b \,a^{6} \left (a -b \right ) \left (a +b \right ) \left (a +b \tan \left (d x +c \right )\right )^{3} \ln \left (\sec \left (d x +c \right )^{2}\right )-4 b \left (a^{2}+b^{2}\right )^{4} \left (a +b \tan \left (d x +c \right )\right )^{3} \ln \left (\tan \left (d x +c \right )\right )-b^{3} \left (a^{9} d x -6 x \,a^{7} b^{2} d +a^{5} b^{4} d x -3 a^{8} b -\frac {73}{3} a^{6} b^{3}-43 a^{4} b^{5}-29 a^{2} b^{7}-\frac {22}{3} b^{9}\right ) \tan \left (d x +c \right )^{3}-3 b^{2} a \left (a^{9} d x -6 x \,a^{7} b^{2} d +a^{5} b^{4} d x -\frac {8}{3} a^{8} b -\frac {59}{3} a^{6} b^{3}-\frac {104}{3} a^{4} b^{5}-\frac {71}{3} a^{2} b^{7}-6 b^{9}\right ) \tan \left (d x +c \right )^{2}-3 b \left (a^{9} d x -6 x \,a^{7} b^{2} d +a^{5} b^{4} d x -2 a^{8} b -13 a^{6} b^{3}-\frac {68}{3} a^{4} b^{5}-\frac {47}{3} a^{2} b^{7}-4 b^{9}\right ) a^{2} \tan \left (d x +c \right )-\left (\left (a^{2}+b^{2}\right )^{4} \cot \left (d x +c \right )+a^{8} d x -6 a^{6} b^{2} d x +a^{4} b^{4} d x \right ) a^{4}}{d \,a^{5} \left (a^{2}+b^{2}\right )^{4} \left (a +b \tan \left (d x +c \right )\right )^{3}}\) | \(418\) |
| norman | \(\frac {\frac {b \left (6 a^{6} b +33 a^{4} b^{3}+35 b^{5} a^{2}+12 b^{7}\right ) \tan \left (d x +c \right )^{2}}{d \,a^{3} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {b^{2} \left (8 a^{6} b +51 a^{4} b^{3}+53 b^{5} a^{2}+18 b^{7}\right ) \tan \left (d x +c \right )^{3}}{d \,a^{4} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {1}{a d}-\frac {b^{3} \left (a^{4}-6 b^{2} a^{2}+b^{4}\right ) x \tan \left (d x +c \right )^{4}}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}+\frac {b^{3} \left (9 a^{6} b +64 a^{4} b^{3}+65 b^{5} a^{2}+22 b^{7}\right ) \tan \left (d x +c \right )^{4}}{3 d \,a^{5} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {\left (a^{4}-6 b^{2} a^{2}+b^{4}\right ) a^{3} x \tan \left (d x +c \right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}-\frac {3 b \left (a^{4}-6 b^{2} a^{2}+b^{4}\right ) a^{2} x \tan \left (d x +c \right )^{2}}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}-\frac {3 b^{2} \left (a^{4}-6 b^{2} a^{2}+b^{4}\right ) a x \tan \left (d x +c \right )^{3}}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}}{\tan \left (d x +c \right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {4 b \ln \left (\tan \left (d x +c \right )\right )}{a^{5} d}+\frac {2 a b \left (a^{2}-b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}+\frac {4 b^{3} \left (5 a^{6}+6 a^{4} b^{2}+4 a^{2} b^{4}+b^{6}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{d \,a^{5} \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}\) | \(652\) |
| risch | \(\text {Expression too large to display}\) | \(1370\) |
Input:
int(cot(d*x+c)^2/(a+b*tan(d*x+c))^4,x,method=_RETURNVERBOSE)
Output:
1/d*(1/(a^2+b^2)^4*(1/2*(4*a^3*b-4*a*b^3)*ln(1+tan(d*x+c)^2)+(-a^4+6*a^2*b ^2-b^4)*arctan(tan(d*x+c)))-1/3*b^3/a^2/(a^2+b^2)/(a+b*tan(d*x+c))^3-b^3*( 10*a^4+9*a^2*b^2+3*b^4)/a^4/(a^2+b^2)^3/(a+b*tan(d*x+c))-b^3*(2*a^2+b^2)/a ^3/(a^2+b^2)^2/(a+b*tan(d*x+c))^2+4*b^3*(5*a^6+6*a^4*b^2+4*a^2*b^4+b^6)/a^ 5/(a^2+b^2)^4*ln(a+b*tan(d*x+c))-1/a^4/tan(d*x+c)-4/a^5*b*ln(tan(d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 925 vs. \(2 (276) = 552\).
Time = 0.19 (sec) , antiderivative size = 925, normalized size of antiderivative = 3.33 \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^4} \, dx =\text {Too large to display} \] Input:
integrate(cot(d*x+c)^2/(a+b*tan(d*x+c))^4,x, algorithm="fricas")
Output:
-1/3*(3*a^12 + 12*a^10*b^2 + 18*a^8*b^4 + 12*a^6*b^6 + 3*a^4*b^8 - (37*a^6 *b^6 + 21*a^4*b^8 + 6*a^2*b^10 - 3*(a^9*b^3 - 6*a^7*b^5 + a^5*b^7)*d*x)*ta n(d*x + c)^4 + 3*(a^9*b^3 - 23*a^7*b^5 + 4*a^5*b^7 + 10*a^3*b^9 + 4*a*b^11 + 3*(a^10*b^2 - 6*a^8*b^4 + a^6*b^6)*d*x)*tan(d*x + c)^3 + 3*(3*a^10*b^2 - 3*a^8*b^4 + 40*a^6*b^6 + 34*a^4*b^8 + 10*a^2*b^10 + 3*(a^11*b - 6*a^9*b^ 3 + a^7*b^5)*d*x)*tan(d*x + c)^2 + 6*((a^8*b^4 + 4*a^6*b^6 + 6*a^4*b^8 + 4 *a^2*b^10 + b^12)*tan(d*x + c)^4 + 3*(a^9*b^3 + 4*a^7*b^5 + 6*a^5*b^7 + 4* a^3*b^9 + a*b^11)*tan(d*x + c)^3 + 3*(a^10*b^2 + 4*a^8*b^4 + 6*a^6*b^6 + 4 *a^4*b^8 + a^2*b^10)*tan(d*x + c)^2 + (a^11*b + 4*a^9*b^3 + 6*a^7*b^5 + 4* a^5*b^7 + a^3*b^9)*tan(d*x + c))*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1)) - 6*((5*a^6*b^6 + 6*a^4*b^8 + 4*a^2*b^10 + b^12)*tan(d*x + c)^4 + 3*(5*a^7 *b^5 + 6*a^5*b^7 + 4*a^3*b^9 + a*b^11)*tan(d*x + c)^3 + 3*(5*a^8*b^4 + 6*a ^6*b^6 + 4*a^4*b^8 + a^2*b^10)*tan(d*x + c)^2 + (5*a^9*b^3 + 6*a^7*b^5 + 4 *a^5*b^7 + a^3*b^9)*tan(d*x + c))*log((b^2*tan(d*x + c)^2 + 2*a*b*tan(d*x + c) + a^2)/(tan(d*x + c)^2 + 1)) + (9*a^11*b + 36*a^9*b^3 + 108*a^7*b^5 + 81*a^5*b^7 + 22*a^3*b^9 + 3*(a^12 - 6*a^10*b^2 + a^8*b^4)*d*x)*tan(d*x + c))/((a^13*b^3 + 4*a^11*b^5 + 6*a^9*b^7 + 4*a^7*b^9 + a^5*b^11)*d*tan(d*x + c)^4 + 3*(a^14*b^2 + 4*a^12*b^4 + 6*a^10*b^6 + 4*a^8*b^8 + a^6*b^10)*d*t an(d*x + c)^3 + 3*(a^15*b + 4*a^13*b^3 + 6*a^11*b^5 + 4*a^9*b^7 + a^7*b^9) *d*tan(d*x + c)^2 + (a^16 + 4*a^14*b^2 + 6*a^12*b^4 + 4*a^10*b^6 + a^8*...
Exception generated. \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^4} \, dx=\text {Exception raised: AttributeError} \] Input:
integrate(cot(d*x+c)**2/(a+b*tan(d*x+c))**4,x)
Output:
Exception raised: AttributeError >> 'NoneType' object has no attribute 'pr imitive'
Time = 0.13 (sec) , antiderivative size = 516, normalized size of antiderivative = 1.86 \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^4} \, dx=-\frac {\frac {3 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {12 \, {\left (5 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{13} + 4 \, a^{11} b^{2} + 6 \, a^{9} b^{4} + 4 \, a^{7} b^{6} + a^{5} b^{8}} - \frac {6 \, {\left (a^{3} b - a b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {3 \, a^{9} + 9 \, a^{7} b^{2} + 9 \, a^{5} b^{4} + 3 \, a^{3} b^{6} + 3 \, {\left (a^{6} b^{3} + 13 \, a^{4} b^{5} + 12 \, a^{2} b^{7} + 4 \, b^{9}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (3 \, a^{7} b^{2} + 31 \, a^{5} b^{4} + 30 \, a^{3} b^{6} + 10 \, a b^{8}\right )} \tan \left (d x + c\right )^{2} + {\left (9 \, a^{8} b + 64 \, a^{6} b^{3} + 65 \, a^{4} b^{5} + 22 \, a^{2} b^{7}\right )} \tan \left (d x + c\right )}{{\left (a^{10} b^{3} + 3 \, a^{8} b^{5} + 3 \, a^{6} b^{7} + a^{4} b^{9}\right )} \tan \left (d x + c\right )^{4} + 3 \, {\left (a^{11} b^{2} + 3 \, a^{9} b^{4} + 3 \, a^{7} b^{6} + a^{5} b^{8}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{12} b + 3 \, a^{10} b^{3} + 3 \, a^{8} b^{5} + a^{6} b^{7}\right )} \tan \left (d x + c\right )^{2} + {\left (a^{13} + 3 \, a^{11} b^{2} + 3 \, a^{9} b^{4} + a^{7} b^{6}\right )} \tan \left (d x + c\right )} + \frac {12 \, b \log \left (\tan \left (d x + c\right )\right )}{a^{5}}}{3 \, d} \] Input:
integrate(cot(d*x+c)^2/(a+b*tan(d*x+c))^4,x, algorithm="maxima")
Output:
-1/3*(3*(a^4 - 6*a^2*b^2 + b^4)*(d*x + c)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4 *a^2*b^6 + b^8) - 12*(5*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*log(b*tan(d *x + c) + a)/(a^13 + 4*a^11*b^2 + 6*a^9*b^4 + 4*a^7*b^6 + a^5*b^8) - 6*(a^ 3*b - a*b^3)*log(tan(d*x + c)^2 + 1)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2* b^6 + b^8) + (3*a^9 + 9*a^7*b^2 + 9*a^5*b^4 + 3*a^3*b^6 + 3*(a^6*b^3 + 13* a^4*b^5 + 12*a^2*b^7 + 4*b^9)*tan(d*x + c)^3 + 3*(3*a^7*b^2 + 31*a^5*b^4 + 30*a^3*b^6 + 10*a*b^8)*tan(d*x + c)^2 + (9*a^8*b + 64*a^6*b^3 + 65*a^4*b^ 5 + 22*a^2*b^7)*tan(d*x + c))/((a^10*b^3 + 3*a^8*b^5 + 3*a^6*b^7 + a^4*b^9 )*tan(d*x + c)^4 + 3*(a^11*b^2 + 3*a^9*b^4 + 3*a^7*b^6 + a^5*b^8)*tan(d*x + c)^3 + 3*(a^12*b + 3*a^10*b^3 + 3*a^8*b^5 + a^6*b^7)*tan(d*x + c)^2 + (a ^13 + 3*a^11*b^2 + 3*a^9*b^4 + a^7*b^6)*tan(d*x + c)) + 12*b*log(tan(d*x + c))/a^5)/d
Time = 0.25 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.62 \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^4} \, dx=-\frac {{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} {\left (d x + c\right )}}{a^{8} d + 4 \, a^{6} b^{2} d + 6 \, a^{4} b^{4} d + 4 \, a^{2} b^{6} d + b^{8} d} + \frac {2 \, {\left (a^{3} b - a b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} d + 4 \, a^{6} b^{2} d + 6 \, a^{4} b^{4} d + 4 \, a^{2} b^{6} d + b^{8} d} + \frac {4 \, {\left (5 \, a^{6} b^{4} + 6 \, a^{4} b^{6} + 4 \, a^{2} b^{8} + b^{10}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{13} b d + 4 \, a^{11} b^{3} d + 6 \, a^{9} b^{5} d + 4 \, a^{7} b^{7} d + a^{5} b^{9} d} - \frac {4 \, b \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{5} d} - \frac {3 \, a^{12} + 12 \, a^{10} b^{2} + 18 \, a^{8} b^{4} + 12 \, a^{6} b^{6} + 3 \, a^{4} b^{8} + 3 \, {\left (a^{9} b^{3} + 14 \, a^{7} b^{5} + 25 \, a^{5} b^{7} + 16 \, a^{3} b^{9} + 4 \, a b^{11}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (3 \, a^{10} b^{2} + 34 \, a^{8} b^{4} + 61 \, a^{6} b^{6} + 40 \, a^{4} b^{8} + 10 \, a^{2} b^{10}\right )} \tan \left (d x + c\right )^{2} + {\left (9 \, a^{11} b + 73 \, a^{9} b^{3} + 129 \, a^{7} b^{5} + 87 \, a^{5} b^{7} + 22 \, a^{3} b^{9}\right )} \tan \left (d x + c\right )}{3 \, {\left (a^{2} + b^{2}\right )}^{4} {\left (b \tan \left (d x + c\right ) + a\right )}^{3} a^{5} d \tan \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^2/(a+b*tan(d*x+c))^4,x, algorithm="giac")
Output:
-(a^4 - 6*a^2*b^2 + b^4)*(d*x + c)/(a^8*d + 4*a^6*b^2*d + 6*a^4*b^4*d + 4* a^2*b^6*d + b^8*d) + 2*(a^3*b - a*b^3)*log(tan(d*x + c)^2 + 1)/(a^8*d + 4* a^6*b^2*d + 6*a^4*b^4*d + 4*a^2*b^6*d + b^8*d) + 4*(5*a^6*b^4 + 6*a^4*b^6 + 4*a^2*b^8 + b^10)*log(abs(b*tan(d*x + c) + a))/(a^13*b*d + 4*a^11*b^3*d + 6*a^9*b^5*d + 4*a^7*b^7*d + a^5*b^9*d) - 4*b*log(abs(tan(d*x + c)))/(a^5 *d) - 1/3*(3*a^12 + 12*a^10*b^2 + 18*a^8*b^4 + 12*a^6*b^6 + 3*a^4*b^8 + 3* (a^9*b^3 + 14*a^7*b^5 + 25*a^5*b^7 + 16*a^3*b^9 + 4*a*b^11)*tan(d*x + c)^3 + 3*(3*a^10*b^2 + 34*a^8*b^4 + 61*a^6*b^6 + 40*a^4*b^8 + 10*a^2*b^10)*tan (d*x + c)^2 + (9*a^11*b + 73*a^9*b^3 + 129*a^7*b^5 + 87*a^5*b^7 + 22*a^3*b ^9)*tan(d*x + c))/((a^2 + b^2)^4*(b*tan(d*x + c) + a)^3*a^5*d*tan(d*x + c) )
Time = 2.56 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.55 \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {4\,b^3\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (5\,a^6+6\,a^4\,b^2+4\,a^2\,b^4+b^6\right )}{a^5\,d\,{\left (a^2+b^2\right )}^4}-\frac {\frac {1}{a}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (a^6\,b^3+13\,a^4\,b^5+12\,a^2\,b^7+4\,b^9\right )}{a^4\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (3\,a^6\,b^2+31\,a^4\,b^4+30\,a^2\,b^6+10\,b^8\right )}{a^3\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (9\,a^6\,b+64\,a^4\,b^3+65\,a^2\,b^5+22\,b^7\right )}{3\,a^2\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}}{d\,\left (a^3\,\mathrm {tan}\left (c+d\,x\right )+3\,a^2\,b\,{\mathrm {tan}\left (c+d\,x\right )}^2+3\,a\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^3+b^3\,{\mathrm {tan}\left (c+d\,x\right )}^4\right )}-\frac {4\,b\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{a^5\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,d\,\left (a^4\,1{}\mathrm {i}-4\,a^3\,b-a^2\,b^2\,6{}\mathrm {i}+4\,a\,b^3+b^4\,1{}\mathrm {i}\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (a^4-a^3\,b\,4{}\mathrm {i}-6\,a^2\,b^2+a\,b^3\,4{}\mathrm {i}+b^4\right )} \] Input:
int(cot(c + d*x)^2/(a + b*tan(c + d*x))^4,x)
Output:
(4*b^3*log(a + b*tan(c + d*x))*(5*a^6 + b^6 + 4*a^2*b^4 + 6*a^4*b^2))/(a^5 *d*(a^2 + b^2)^4) - (log(tan(c + d*x) + 1i)*1i)/(2*d*(a*b^3*4i - a^3*b*4i + a^4 + b^4 - 6*a^2*b^2)) - (1/a + (tan(c + d*x)^3*(4*b^9 + 12*a^2*b^7 + 1 3*a^4*b^5 + a^6*b^3))/(a^4*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (tan(c + d*x)^2*(10*b^8 + 30*a^2*b^6 + 31*a^4*b^4 + 3*a^6*b^2))/(a^3*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (tan(c + d*x)*(9*a^6*b + 22*b^7 + 65*a^2*b^5 + 6 4*a^4*b^3))/(3*a^2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)))/(d*(a^3*tan(c + d *x) + b^3*tan(c + d*x)^4 + 3*a^2*b*tan(c + d*x)^2 + 3*a*b^2*tan(c + d*x)^3 )) - (4*b*log(tan(c + d*x)))/(a^5*d) - log(tan(c + d*x) - 1i)/(2*d*(4*a*b^ 3 - 4*a^3*b + a^4*1i + b^4*1i - a^2*b^2*6i))
Time = 0.29 (sec) , antiderivative size = 2948, normalized size of antiderivative = 10.60 \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^4} \, dx =\text {Too large to display} \] Input:
int(cot(d*x+c)^2/(a+b*tan(d*x+c))^4,x)
Output:
(12*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**3*a**11*b**2 - 48*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**3*a**9*b**4 + 36*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**3*a**7*b**6 - 1 2*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)*a**11*b**2 + 12*c os(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)*a**9*b**4 + 60*cos(c + d*x)*log(tan((c + d*x)/2)**2*a - 2*tan((c + d*x)/2)*b - a)*sin(c + d*x) **3*a**9*b**4 - 108*cos(c + d*x)*log(tan((c + d*x)/2)**2*a - 2*tan((c + d* x)/2)*b - a)*sin(c + d*x)**3*a**7*b**6 - 168*cos(c + d*x)*log(tan((c + d*x )/2)**2*a - 2*tan((c + d*x)/2)*b - a)*sin(c + d*x)**3*a**5*b**8 - 132*cos( c + d*x)*log(tan((c + d*x)/2)**2*a - 2*tan((c + d*x)/2)*b - a)*sin(c + d*x )**3*a**3*b**10 - 36*cos(c + d*x)*log(tan((c + d*x)/2)**2*a - 2*tan((c + d *x)/2)*b - a)*sin(c + d*x)**3*a*b**12 - 60*cos(c + d*x)*log(tan((c + d*x)/ 2)**2*a - 2*tan((c + d*x)/2)*b - a)*sin(c + d*x)*a**9*b**4 - 72*cos(c + d* x)*log(tan((c + d*x)/2)**2*a - 2*tan((c + d*x)/2)*b - a)*sin(c + d*x)*a**7 *b**6 - 48*cos(c + d*x)*log(tan((c + d*x)/2)**2*a - 2*tan((c + d*x)/2)*b - a)*sin(c + d*x)*a**5*b**8 - 12*cos(c + d*x)*log(tan((c + d*x)/2)**2*a - 2 *tan((c + d*x)/2)*b - a)*sin(c + d*x)*a**3*b**10 - 12*cos(c + d*x)*log(tan ((c + d*x)/2))*sin(c + d*x)**3*a**11*b**2 - 12*cos(c + d*x)*log(tan((c + d *x)/2))*sin(c + d*x)**3*a**9*b**4 + 72*cos(c + d*x)*log(tan((c + d*x)/2))* sin(c + d*x)**3*a**7*b**6 + 168*cos(c + d*x)*log(tan((c + d*x)/2))*sin(...