Integrand size = 31, antiderivative size = 192 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=-\frac {\left (a^2 A-A b^2+2 a b B\right ) x}{\left (a^2+b^2\right )^2}-\frac {(2 A b-a B) \log (\sin (c+d x))}{a^3 d}+\frac {b^2 \left (4 a^2 A b+2 A b^3-3 a^3 B-a b^2 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 \left (a^2+b^2\right )^2 d}-\frac {b \left (a^2 A+2 A b^2-a b B\right )}{a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))} \] Output:
-(A*a^2-A*b^2+2*B*a*b)*x/(a^2+b^2)^2-(2*A*b-B*a)*ln(sin(d*x+c))/a^3/d+b^2* (4*A*a^2*b+2*A*b^3-3*B*a^3-B*a*b^2)*ln(a*cos(d*x+c)+b*sin(d*x+c))/a^3/(a^2 +b^2)^2/d-b*(A*a^2+2*A*b^2-B*a*b)/a^2/(a^2+b^2)/d/(a+b*tan(d*x+c))-A*cot(d *x+c)/a/d/(a+b*tan(d*x+c))
Result contains complex when optimal does not.
Time = 2.53 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.01 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\frac {-\frac {2 A \cot (c+d x)}{a^2}+\frac {i (A+i B) \log (i-\tan (c+d x))}{(a+i b)^2}+\frac {2 (-2 A b+a B) \log (\tan (c+d x))}{a^3}-\frac {(i A+B) \log (i+\tan (c+d x))}{(a-i b)^2}-\frac {2 b^2 \left (-4 a^2 A b-2 A b^3+3 a^3 B+a b^2 B\right ) \log (a+b \tan (c+d x))}{a^3 \left (a^2+b^2\right )^2}+\frac {2 b^2 (-A b+a B)}{a^2 \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 d} \] Input:
Integrate[(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2,x]
Output:
((-2*A*Cot[c + d*x])/a^2 + (I*(A + I*B)*Log[I - Tan[c + d*x]])/(a + I*b)^2 + (2*(-2*A*b + a*B)*Log[Tan[c + d*x]])/a^3 - ((I*A + B)*Log[I + Tan[c + d *x]])/(a - I*b)^2 - (2*b^2*(-4*a^2*A*b - 2*A*b^3 + 3*a^3*B + a*b^2*B)*Log[ a + b*Tan[c + d*x]])/(a^3*(a^2 + b^2)^2) + (2*b^2*(-(A*b) + a*B))/(a^2*(a^ 2 + b^2)*(a + b*Tan[c + d*x])))/(2*d)
Time = 1.22 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.16, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.323, Rules used = {3042, 4092, 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.
| \(\displaystyle \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \tan (c+d x)}{\tan (c+d x)^2 (a+b \tan (c+d x))^2}dx\) |
| \(\Big \downarrow \) 4092 |
| \(\displaystyle -\frac {\int \frac {\cot (c+d x) \left (2 A b \tan ^2(c+d x)+a A \tan (c+d x)+2 A b-a B\right )}{(a+b \tan (c+d x))^2}dx}{a}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {2 A b \tan (c+d x)^2+a A \tan (c+d x)+2 A b-a B}{\tan (c+d x) (a+b \tan (c+d x))^2}dx}{a}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle -\frac {\frac {\int \frac {\cot (c+d x) \left ((a A+b B) \tan (c+d x) a^2+b \left (A a^2-b B a+2 A b^2\right ) \tan ^2(c+d x)+\left (a^2+b^2\right ) (2 A b-a B)\right )}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}+\frac {b \left (a^2 A-a b B+2 A b^2\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\int \frac {(a A+b B) \tan (c+d x) a^2+b \left (A a^2-b B a+2 A b^2\right ) \tan (c+d x)^2+\left (a^2+b^2\right ) (2 A b-a B)}{\tan (c+d x) (a+b \tan (c+d x))}dx}{a \left (a^2+b^2\right )}+\frac {b \left (a^2 A-a b B+2 A b^2\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 4134 |
| \(\displaystyle -\frac {\frac {\frac {\left (a^2+b^2\right ) (2 A b-a B) \int \cot (c+d x)dx}{a}-\frac {b^2 \left (-3 a^3 B+4 a^2 A b-a b^2 B+2 A b^3\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}+\frac {a^2 x \left (a^2 A+2 a b B-A b^2\right )}{a^2+b^2}}{a \left (a^2+b^2\right )}+\frac {b \left (a^2 A-a b B+2 A b^2\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {\left (a^2+b^2\right ) (2 A b-a B) \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx}{a}-\frac {b^2 \left (-3 a^3 B+4 a^2 A b-a b^2 B+2 A b^3\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}+\frac {a^2 x \left (a^2 A+2 a b B-A b^2\right )}{a^2+b^2}}{a \left (a^2+b^2\right )}+\frac {b \left (a^2 A-a b B+2 A b^2\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {-\frac {\left (a^2+b^2\right ) (2 A b-a B) \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx}{a}-\frac {b^2 \left (-3 a^3 B+4 a^2 A b-a b^2 B+2 A b^3\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}+\frac {a^2 x \left (a^2 A+2 a b B-A b^2\right )}{a^2+b^2}}{a \left (a^2+b^2\right )}+\frac {b \left (a^2 A-a b B+2 A b^2\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle -\frac {\frac {-\frac {b^2 \left (-3 a^3 B+4 a^2 A b-a b^2 B+2 A b^3\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}+\frac {\left (a^2+b^2\right ) (2 A b-a B) \log (-\sin (c+d x))}{a d}+\frac {a^2 x \left (a^2 A+2 a b B-A b^2\right )}{a^2+b^2}}{a \left (a^2+b^2\right )}+\frac {b \left (a^2 A-a b B+2 A b^2\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 4013 |
| \(\displaystyle -\frac {\frac {b \left (a^2 A-a b B+2 A b^2\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {\frac {\left (a^2+b^2\right ) (2 A b-a B) \log (-\sin (c+d x))}{a d}+\frac {a^2 x \left (a^2 A+2 a b B-A b^2\right )}{a^2+b^2}-\frac {b^2 \left (-3 a^3 B+4 a^2 A b-a b^2 B+2 A b^3\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a d \left (a^2+b^2\right )}}{a \left (a^2+b^2\right )}}{a}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))}\) |
Input:
Int[(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2,x]
Output:
-((A*Cot[c + d*x])/(a*d*(a + b*Tan[c + d*x]))) - (((a^2*(a^2*A - A*b^2 + 2 *a*b*B)*x)/(a^2 + b^2) + ((a^2 + b^2)*(2*A*b - a*B)*Log[-Sin[c + d*x]])/(a *d) - (b^2*(4*a^2*A*b + 2*A*b^3 - 3*a^3*B - a*b^2*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a*(a^2 + b^2)*d))/(a*(a^2 + b^2)) + (b*(a^2*A + 2*A*b^2 - a*b*B))/(a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])))/a
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_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[b*(A*b - a*B)*(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[b* B*(b*c*(m + 1) + a*d*(n + 1)) + A*(a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2 )) - (A*b - a*B)*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b*d*(A*b - a*B)*(m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) && !(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 = 0.48 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.02
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\left (2 A a b -B \,a^{2}+B \,b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-A \,a^{2}+A \,b^{2}-2 B a b \right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}-\frac {A}{a^{2} \tan \left (d x +c \right )}+\frac {\left (-2 A b +B a \right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{3}}+\frac {b^{2} \left (4 A \,a^{2} b +2 A \,b^{3}-3 B \,a^{3}-B a \,b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2} a^{3}}-\frac {\left (A b -B a \right ) b^{2}}{\left (a^{2}+b^{2}\right ) a^{2} \left (a +b \tan \left (d x +c \right )\right )}}{d}\) | \(196\) |
| default | \(\frac {\frac {\frac {\left (2 A a b -B \,a^{2}+B \,b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-A \,a^{2}+A \,b^{2}-2 B a b \right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}-\frac {A}{a^{2} \tan \left (d x +c \right )}+\frac {\left (-2 A b +B a \right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{3}}+\frac {b^{2} \left (4 A \,a^{2} b +2 A \,b^{3}-3 B \,a^{3}-B a \,b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2} a^{3}}-\frac {\left (A b -B a \right ) b^{2}}{\left (a^{2}+b^{2}\right ) a^{2} \left (a +b \tan \left (d x +c \right )\right )}}{d}\) | \(196\) |
| parallelrisch | \(\frac {4 \left (A \,a^{2} b +\frac {1}{2} A \,b^{3}-\frac {3}{4} B \,a^{3}-\frac {1}{4} B a \,b^{2}\right ) b^{2} \left (a +b \tan \left (d x +c \right )\right ) \ln \left (a +b \tan \left (d x +c \right )\right )+a^{3} \left (A a b -\frac {1}{2} B \,a^{2}+\frac {1}{2} B \,b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right ) \ln \left (\sec \left (d x +c \right )^{2}\right )-2 \left (a^{2}+b^{2}\right )^{2} \left (A b -\frac {B a}{2}\right ) \left (a +b \tan \left (d x +c \right )\right ) \ln \left (\tan \left (d x +c \right )\right )-b \left (-2 A \,b^{5}+B a \,b^{4}-3 A \,a^{2} b^{3}-a^{3} \left (A d x -B \right ) b^{2}-a^{4} \left (-2 B d x +A \right ) b +A x \,a^{5} d \right ) \tan \left (d x +c \right )-a^{2} \left (A \left (a^{2}+b^{2}\right )^{2} \cot \left (d x +c \right )+a^{2} d x \left (A \,a^{2}-A \,b^{2}+2 B a b \right )\right )}{\left (a^{2}+b^{2}\right )^{2} a^{3} d \left (a +b \tan \left (d x +c \right )\right )}\) | \(271\) |
| norman | \(\frac {\frac {\left (A \,a^{2} b +2 A \,b^{3}-B a \,b^{2}\right ) b \tan \left (d x +c \right )^{2}}{d \,a^{3} \left (a^{2}+b^{2}\right )}-\frac {A}{a d}-\frac {a \left (A \,a^{2}-A \,b^{2}+2 B a b \right ) x \tan \left (d x +c \right )}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {b \left (A \,a^{2}-A \,b^{2}+2 B a b \right ) x \tan \left (d x +c \right )^{2}}{a^{4}+2 a^{2} b^{2}+b^{4}}}{\tan \left (d x +c \right ) \left (a +b \tan \left (d x +c \right )\right )}+\frac {b^{2} \left (4 A \,a^{2} b +2 A \,b^{3}-3 B \,a^{3}-B a \,b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) a^{3} d}-\frac {\left (2 A b -B a \right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{3} d}+\frac {\left (2 A a b -B \,a^{2}+B \,b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\) | \(307\) |
| risch | \(\frac {4 i A b c}{a^{3} d}+\frac {x A}{2 i a b -a^{2}+b^{2}}+\frac {6 i B \,b^{2} x}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {8 i b^{3} A c}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) a d}+\frac {2 i b^{4} B x}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) a^{2}}+\frac {6 i B \,b^{2} c}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {4 i A b x}{a^{3}}-\frac {4 i b^{5} A x}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) a^{3}}-\frac {2 i x B}{a^{2}}-\frac {8 i b^{3} A x}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) a}-\frac {4 i b^{5} A c}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) a^{3} d}-\frac {2 i \left (A \,a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-2 A \,b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+B a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-2 i A \,a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}-2 i A a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+A \,a^{4}+2 A \,a^{2} b^{2}+2 A \,b^{4}-B a \,b^{3}\right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) \left (i b +a \right ) \left (-i b +a \right )^{2} \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+a \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +a \right ) a^{2} d}-\frac {i x B}{2 i a b -a^{2}+b^{2}}+\frac {2 i b^{4} B c}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) a^{2} d}-\frac {2 i B c}{a^{2} d}-\frac {2 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) A b}{a^{3} d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B}{a^{2} d}+\frac {4 b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) A}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) a d}+\frac {2 b^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) A}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) a^{3} d}-\frac {3 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) B \,b^{2}}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {b^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) B}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) a^{2} d}\) | \(759\) |
Input:
int(cot(d*x+c)^2*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^2,x,method=_RETURNVERBO SE)
Output:
1/d*(1/(a^2+b^2)^2*(1/2*(2*A*a*b-B*a^2+B*b^2)*ln(1+tan(d*x+c)^2)+(-A*a^2+A *b^2-2*B*a*b)*arctan(tan(d*x+c)))-1/a^2*A/tan(d*x+c)+(-2*A*b+B*a)/a^3*ln(t an(d*x+c))+b^2*(4*A*a^2*b+2*A*b^3-3*B*a^3-B*a*b^2)/(a^2+b^2)^2/a^3*ln(a+b* tan(d*x+c))-(A*b-B*a)*b^2/(a^2+b^2)/a^2/(a+b*tan(d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 465 vs. \(2 (190) = 380\).
Time = 0.13 (sec) , antiderivative size = 465, normalized size of antiderivative = 2.42 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=-\frac {2 \, A a^{6} + 4 \, A a^{4} b^{2} + 2 \, A a^{2} b^{4} + 2 \, {\left (B a^{3} b^{3} - A a^{2} b^{4} + {\left (A a^{5} b + 2 \, B a^{4} b^{2} - A a^{3} b^{3}\right )} d x\right )} \tan \left (d x + c\right )^{2} - {\left ({\left (B a^{5} b - 2 \, A a^{4} b^{2} + 2 \, B a^{3} b^{3} - 4 \, A a^{2} b^{4} + B a b^{5} - 2 \, A b^{6}\right )} \tan \left (d x + c\right )^{2} + {\left (B a^{6} - 2 \, A a^{5} b + 2 \, B a^{4} b^{2} - 4 \, A a^{3} b^{3} + B a^{2} b^{4} - 2 \, A a b^{5}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + {\left ({\left (3 \, B a^{3} b^{3} - 4 \, A a^{2} b^{4} + B a b^{5} - 2 \, A b^{6}\right )} \tan \left (d x + c\right )^{2} + {\left (3 \, B a^{4} b^{2} - 4 \, A a^{3} b^{3} + B a^{2} b^{4} - 2 \, A a b^{5}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + 2 \, {\left (A a^{5} b + 2 \, A a^{3} b^{3} - B a^{2} b^{4} + 2 \, A a b^{5} + {\left (A a^{6} + 2 \, B a^{5} b - A a^{4} b^{2}\right )} d x\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{7} b + 2 \, a^{5} b^{3} + a^{3} b^{5}\right )} d \tan \left (d x + c\right )^{2} + {\left (a^{8} + 2 \, a^{6} b^{2} + a^{4} b^{4}\right )} d \tan \left (d x + c\right )\right )}} \] Input:
integrate(cot(d*x+c)^2*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^2,x, algorithm="f ricas")
Output:
-1/2*(2*A*a^6 + 4*A*a^4*b^2 + 2*A*a^2*b^4 + 2*(B*a^3*b^3 - A*a^2*b^4 + (A* a^5*b + 2*B*a^4*b^2 - A*a^3*b^3)*d*x)*tan(d*x + c)^2 - ((B*a^5*b - 2*A*a^4 *b^2 + 2*B*a^3*b^3 - 4*A*a^2*b^4 + B*a*b^5 - 2*A*b^6)*tan(d*x + c)^2 + (B* a^6 - 2*A*a^5*b + 2*B*a^4*b^2 - 4*A*a^3*b^3 + B*a^2*b^4 - 2*A*a*b^5)*tan(d *x + c))*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1)) + ((3*B*a^3*b^3 - 4*A*a^ 2*b^4 + B*a*b^5 - 2*A*b^6)*tan(d*x + c)^2 + (3*B*a^4*b^2 - 4*A*a^3*b^3 + B *a^2*b^4 - 2*A*a*b^5)*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)) + 2*(A*a^5*b + 2*A*a^3*b^3 - B*a^2*b^4 + 2*A*a*b^5 + (A*a^6 + 2*B*a^5*b - A*a^4*b^2)*d*x)*tan(d*x + c))/((a^7*b + 2*a^5*b^3 + a^3*b^5)*d*tan(d*x + c)^2 + (a^8 + 2*a^6*b^2 + a^4*b^4)*d*ta n(d*x + c))
Result contains complex when optimal does not.
Time = 2.72 (sec) , antiderivative size = 8145, normalized size of antiderivative = 42.42 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \] Input:
integrate(cot(d*x+c)**2*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))**2,x)
Output:
Piecewise(((-A*x - A/(d*tan(c + d*x)) - B*log(tan(c + d*x)**2 + 1)/(2*d) + B*log(tan(c + d*x))/d)/a**2, Eq(b, 0)), ((A*x + A/(d*tan(c + d*x)) - A/(3 *d*tan(c + d*x)**3) + B*log(tan(c + d*x)**2 + 1)/(2*d) - B*log(tan(c + d*x ))/d - B/(2*d*tan(c + d*x)**2))/b**2, Eq(a, 0)), (-9*A*d*x*tan(c + d*x)**3 /(4*a**2*d*tan(c + d*x)**3 + 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) - 18*I*A*d*x*tan(c + d*x)**2/(4*a**2*d*tan(c + d*x)**3 + 8*I*a**2*d *tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) + 9*A*d*x*tan(c + d*x)/(4*a**2*d *tan(c + d*x)**3 + 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) - 4 *I*A*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**3/(4*a**2*d*tan(c + d*x)**3 + 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) + 8*A*log(tan(c + d*x) **2 + 1)*tan(c + d*x)**2/(4*a**2*d*tan(c + d*x)**3 + 8*I*a**2*d*tan(c + d* x)**2 - 4*a**2*d*tan(c + d*x)) + 4*I*A*log(tan(c + d*x)**2 + 1)*tan(c + d* x)/(4*a**2*d*tan(c + d*x)**3 + 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) + 8*I*A*log(tan(c + d*x))*tan(c + d*x)**3/(4*a**2*d*tan(c + d*x)* *3 + 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) - 16*A*log(tan(c + d*x))*tan(c + d*x)**2/(4*a**2*d*tan(c + d*x)**3 + 8*I*a**2*d*tan(c + d*x )**2 - 4*a**2*d*tan(c + d*x)) - 8*I*A*log(tan(c + d*x))*tan(c + d*x)/(4*a* *2*d*tan(c + d*x)**3 + 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) - 9*A*tan(c + d*x)**2/(4*a**2*d*tan(c + d*x)**3 + 8*I*a**2*d*tan(c + d*x) **2 - 4*a**2*d*tan(c + d*x)) - 14*I*A*tan(c + d*x)/(4*a**2*d*tan(c + d*...
Time = 0.11 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.36 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=-\frac {\frac {2 \, {\left (A a^{2} + 2 \, B a b - A b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (3 \, B a^{3} b^{2} - 4 \, A a^{2} b^{3} + B a b^{4} - 2 \, A b^{5}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}} + \frac {{\left (B a^{2} - 2 \, A a b - B b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (A a^{3} + A a b^{2} + {\left (A a^{2} b - B a b^{2} + 2 \, A b^{3}\right )} \tan \left (d x + c\right )\right )}}{{\left (a^{4} b + a^{2} b^{3}\right )} \tan \left (d x + c\right )^{2} + {\left (a^{5} + a^{3} b^{2}\right )} \tan \left (d x + c\right )} - \frac {2 \, {\left (B a - 2 \, A b\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{3}}}{2 \, d} \] Input:
integrate(cot(d*x+c)^2*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^2,x, algorithm="m axima")
Output:
-1/2*(2*(A*a^2 + 2*B*a*b - A*b^2)*(d*x + c)/(a^4 + 2*a^2*b^2 + b^4) + 2*(3 *B*a^3*b^2 - 4*A*a^2*b^3 + B*a*b^4 - 2*A*b^5)*log(b*tan(d*x + c) + a)/(a^7 + 2*a^5*b^2 + a^3*b^4) + (B*a^2 - 2*A*a*b - B*b^2)*log(tan(d*x + c)^2 + 1 )/(a^4 + 2*a^2*b^2 + b^4) + 2*(A*a^3 + A*a*b^2 + (A*a^2*b - B*a*b^2 + 2*A* b^3)*tan(d*x + c))/((a^4*b + a^2*b^3)*tan(d*x + c)^2 + (a^5 + a^3*b^2)*tan (d*x + c)) - 2*(B*a - 2*A*b)*log(tan(d*x + c))/a^3)/d
Time = 0.36 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.54 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=-\frac {{\left (A a^{2} + 2 \, B a b - A b^{2}\right )} {\left (d x + c\right )}}{a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d} - \frac {{\left (B a^{2} - 2 \, A a b - B b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, {\left (a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right )}} - \frac {{\left (3 \, B a^{3} b^{3} - 4 \, A a^{2} b^{4} + B a b^{5} - 2 \, A b^{6}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{7} b d + 2 \, a^{5} b^{3} d + a^{3} b^{5} d} + \frac {{\left (B a - 2 \, A b\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{3} d} - \frac {A a^{5} + 2 \, A a^{3} b^{2} + A a b^{4} + {\left (A a^{4} b - B a^{3} b^{2} + 3 \, A a^{2} b^{3} - B a b^{4} + 2 \, A b^{5}\right )} \tan \left (d x + c\right )}{{\left (a^{2} + b^{2}\right )}^{2} {\left (b \tan \left (d x + c\right ) + a\right )} a^{2} d \tan \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^2*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^2,x, algorithm="g iac")
Output:
-(A*a^2 + 2*B*a*b - A*b^2)*(d*x + c)/(a^4*d + 2*a^2*b^2*d + b^4*d) - 1/2*( B*a^2 - 2*A*a*b - B*b^2)*log(tan(d*x + c)^2 + 1)/(a^4*d + 2*a^2*b^2*d + b^ 4*d) - (3*B*a^3*b^3 - 4*A*a^2*b^4 + B*a*b^5 - 2*A*b^6)*log(abs(b*tan(d*x + c) + a))/(a^7*b*d + 2*a^5*b^3*d + a^3*b^5*d) + (B*a - 2*A*b)*log(abs(tan( d*x + c)))/(a^3*d) - (A*a^5 + 2*A*a^3*b^2 + A*a*b^4 + (A*a^4*b - B*a^3*b^2 + 3*A*a^2*b^3 - B*a*b^4 + 2*A*b^5)*tan(d*x + c))/((a^2 + b^2)^2*(b*tan(d* x + c) + a)*a^2*d*tan(d*x + c))
Time = 6.81 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.20 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\frac {b^2\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (-3\,B\,a^3+4\,A\,a^2\,b-B\,a\,b^2+2\,A\,b^3\right )}{a^3\,d\,{\left (a^2+b^2\right )}^2}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (2\,A\,b-B\,a\right )}{a^3\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B+A\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^2+a\,b\,2{}\mathrm {i}+b^2\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (A+B\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^2\,1{}\mathrm {i}+2\,a\,b+b^2\,1{}\mathrm {i}\right )}-\frac {\frac {A}{a}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (A\,a^2\,b-B\,a\,b^2+2\,A\,b^3\right )}{a^2\,\left (a^2+b^2\right )}}{d\,\left (b\,{\mathrm {tan}\left (c+d\,x\right )}^2+a\,\mathrm {tan}\left (c+d\,x\right )\right )} \] Input:
int((cot(c + d*x)^2*(A + B*tan(c + d*x)))/(a + b*tan(c + d*x))^2,x)
Output:
(log(tan(c + d*x) + 1i)*(A*1i + B))/(2*d*(a*b*2i - a^2 + b^2)) - (log(tan( c + d*x))*(2*A*b - B*a))/(a^3*d) - (A/a + (tan(c + d*x)*(2*A*b^3 + A*a^2*b - B*a*b^2))/(a^2*(a^2 + b^2)))/(d*(a*tan(c + d*x) + b*tan(c + d*x)^2)) + (log(tan(c + d*x) - 1i)*(A + B*1i))/(2*d*(2*a*b - a^2*1i + b^2*1i)) + (b^2 *log(a + b*tan(c + d*x))*(2*A*b^3 - 3*B*a^3 + 4*A*a^2*b - B*a*b^2))/(a^3*d *(a^2 + b^2)^2)
Time = 0.22 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.88 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\frac {-\cos \left (d x +c \right ) a^{3}-\cos \left (d x +c \right ) a \,b^{2}+\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right ) a^{2} b +\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b -a \right ) \sin \left (d x +c \right ) b^{3}-\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right ) a^{2} b -\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right ) b^{3}-\sin \left (d x +c \right ) a^{3} d x}{\sin \left (d x +c \right ) a^{2} d \left (a^{2}+b^{2}\right )} \] Input:
int(cot(d*x+c)^2*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^2,x)
Output:
( - cos(c + d*x)*a**3 - cos(c + d*x)*a*b**2 + log(tan((c + d*x)/2)**2 + 1) *sin(c + d*x)*a**2*b + log(tan((c + d*x)/2)**2*a - 2*tan((c + d*x)/2)*b - a)*sin(c + d*x)*b**3 - log(tan((c + d*x)/2))*sin(c + d*x)*a**2*b - log(tan ((c + d*x)/2))*sin(c + d*x)*b**3 - sin(c + d*x)*a**3*d*x)/(sin(c + d*x)*a* *2*d*(a**2 + b**2))