Integrand size = 40, antiderivative size = 215 \[ \int \frac {\cot ^2(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx=-\frac {\left (3 a^2 b B-b^3 B-a^3 C+3 a b^2 C\right ) x}{\left (a^2+b^2\right )^3}+\frac {B \log (\sin (c+d x))}{a^3 d}-\frac {b \left (6 a^4 b B+3 a^2 b^3 B+b^5 B-3 a^5 C+a^3 b^2 C\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 \left (a^2+b^2\right )^3 d}+\frac {b (b B-a C)}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {b \left (3 a^2 b B+b^3 B-2 a^3 C\right )}{a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \]
-(3*B*a^2*b-B*b^3-C*a^3+3*C*a*b^2)*x/(a^2+b^2)^3+B*ln(sin(d*x+c))/a^3/d-b* (6*B*a^4*b+3*B*a^2*b^3+B*b^5-3*C*a^5+C*a^3*b^2)*ln(a*cos(d*x+c)+b*sin(d*x+ c))/a^3/(a^2+b^2)^3/d+1/2*b*(B*b-C*a)/a/(a^2+b^2)/d/(a+b*tan(d*x+c))^2+b*( 3*B*a^2*b+B*b^3-2*C*a^3)/a^2/(a^2+b^2)^2/d/(a+b*tan(d*x+c))
Result contains complex when optimal does not.
Time = 3.31 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.04 \[ \int \frac {\cot ^2(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx=\frac {-\frac {(B+i C) \log (i-\tan (c+d x))}{(a+i b)^3}+\frac {2 B \log (\tan (c+d x))}{a^3}-\frac {(B-i C) \log (i+\tan (c+d x))}{(a-i b)^3}-\frac {2 b \left (6 a^4 b B+3 a^2 b^3 B+b^5 B-3 a^5 C+a^3 b^2 C\right ) \log (a+b \tan (c+d x))}{a^3 \left (a^2+b^2\right )^3}+\frac {b (b B-a C)}{a \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {2 b \left (3 a^2 b B+b^3 B-2 a^3 C\right )}{a^2 \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}}{2 d} \]
(-(((B + I*C)*Log[I - Tan[c + d*x]])/(a + I*b)^3) + (2*B*Log[Tan[c + d*x]] )/a^3 - ((B - I*C)*Log[I + Tan[c + d*x]])/(a - I*b)^3 - (2*b*(6*a^4*b*B + 3*a^2*b^3*B + b^5*B - 3*a^5*C + a^3*b^2*C)*Log[a + b*Tan[c + d*x]])/(a^3*( a^2 + b^2)^3) + (b*(b*B - a*C))/(a*(a^2 + b^2)*(a + b*Tan[c + d*x])^2) + ( 2*b*(3*a^2*b*B + b^3*B - 2*a^3*C))/(a^2*(a^2 + b^2)^2*(a + b*Tan[c + d*x]) ))/(2*d)
Time = 1.47 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.20, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.325, Rules used = {3042, 4115, 3042, 4092, 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.
| \(\displaystyle \int \frac {\cot ^2(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {B \tan (c+d x)+C \tan (c+d x)^2}{\tan (c+d x)^2 (a+b \tan (c+d x))^3}dx\) |
| \(\Big \downarrow \) 4115 |
| \(\displaystyle \int \frac {\cot (c+d x) (B+C \tan (c+d x))}{(a+b \tan (c+d x))^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {B+C \tan (c+d x)}{\tan (c+d x) (a+b \tan (c+d x))^3}dx\) |
| \(\Big \downarrow \) 4092 |
| \(\displaystyle \frac {\int \frac {2 \cot (c+d x) \left (b (b B-a C) \tan ^2(c+d x)-a (b B-a C) \tan (c+d x)+\left (a^2+b^2\right ) B\right )}{(a+b \tan (c+d x))^2}dx}{2 a \left (a^2+b^2\right )}+\frac {b (b B-a C)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\cot (c+d x) \left (b (b B-a C) \tan ^2(c+d x)-a (b B-a C) \tan (c+d x)+\left (a^2+b^2\right ) B\right )}{(a+b \tan (c+d x))^2}dx}{a \left (a^2+b^2\right )}+\frac {b (b B-a C)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {b (b B-a C) \tan (c+d x)^2-a (b B-a C) \tan (c+d x)+\left (a^2+b^2\right ) B}{\tan (c+d x) (a+b \tan (c+d x))^2}dx}{a \left (a^2+b^2\right )}+\frac {b (b B-a C)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \frac {\frac {\int \frac {\cot (c+d x) \left (-\left (\left (-C a^2+2 b B a+b^2 C\right ) \tan (c+d x) a^2\right )+b \left (-2 C a^3+3 b B a^2+b^3 B\right ) \tan ^2(c+d x)+\left (a^2+b^2\right )^2 B\right )}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}+\frac {b \left (-2 a^3 C+3 a^2 b B+b^3 B\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}+\frac {b (b B-a C)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {-\left (\left (-C a^2+2 b B a+b^2 C\right ) \tan (c+d x) a^2\right )+b \left (-2 C a^3+3 b B a^2+b^3 B\right ) \tan (c+d x)^2+\left (a^2+b^2\right )^2 B}{\tan (c+d x) (a+b \tan (c+d x))}dx}{a \left (a^2+b^2\right )}+\frac {b \left (-2 a^3 C+3 a^2 b B+b^3 B\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}+\frac {b (b B-a C)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4134 |
| \(\displaystyle \frac {\frac {\frac {B \left (a^2+b^2\right )^2 \int \cot (c+d x)dx}{a}-\frac {b \left (-3 a^5 C+6 a^4 b B+a^3 b^2 C+3 a^2 b^3 B+b^5 B\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}-\frac {a^2 x \left (a^3 (-C)+3 a^2 b B+3 a b^2 C-b^3 B\right )}{a^2+b^2}}{a \left (a^2+b^2\right )}+\frac {b \left (-2 a^3 C+3 a^2 b B+b^3 B\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}+\frac {b (b B-a C)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {B \left (a^2+b^2\right )^2 \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx}{a}-\frac {b \left (-3 a^5 C+6 a^4 b B+a^3 b^2 C+3 a^2 b^3 B+b^5 B\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}-\frac {a^2 x \left (a^3 (-C)+3 a^2 b B+3 a b^2 C-b^3 B\right )}{a^2+b^2}}{a \left (a^2+b^2\right )}+\frac {b \left (-2 a^3 C+3 a^2 b B+b^3 B\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}+\frac {b (b B-a C)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {-\frac {B \left (a^2+b^2\right )^2 \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx}{a}-\frac {b \left (-3 a^5 C+6 a^4 b B+a^3 b^2 C+3 a^2 b^3 B+b^5 B\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}-\frac {a^2 x \left (a^3 (-C)+3 a^2 b B+3 a b^2 C-b^3 B\right )}{a^2+b^2}}{a \left (a^2+b^2\right )}+\frac {b \left (-2 a^3 C+3 a^2 b B+b^3 B\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}+\frac {b (b B-a C)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {\frac {-\frac {b \left (-3 a^5 C+6 a^4 b B+a^3 b^2 C+3 a^2 b^3 B+b^5 B\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}+\frac {B \left (a^2+b^2\right )^2 \log (-\sin (c+d x))}{a d}-\frac {a^2 x \left (a^3 (-C)+3 a^2 b B+3 a b^2 C-b^3 B\right )}{a^2+b^2}}{a \left (a^2+b^2\right )}+\frac {b \left (-2 a^3 C+3 a^2 b B+b^3 B\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}+\frac {b (b B-a C)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4013 |
| \(\displaystyle \frac {b (b B-a C)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {\frac {b \left (-2 a^3 C+3 a^2 b B+b^3 B\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {\frac {B \left (a^2+b^2\right )^2 \log (-\sin (c+d x))}{a d}-\frac {a^2 x \left (a^3 (-C)+3 a^2 b B+3 a b^2 C-b^3 B\right )}{a^2+b^2}-\frac {b \left (-3 a^5 C+6 a^4 b B+a^3 b^2 C+3 a^2 b^3 B+b^5 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a d \left (a^2+b^2\right )}}{a \left (a^2+b^2\right )}}{a \left (a^2+b^2\right )}\) |
(b*(b*B - a*C))/(2*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + ((-((a^2*(3*a ^2*b*B - b^3*B - a^3*C + 3*a*b^2*C)*x)/(a^2 + b^2)) + ((a^2 + b^2)^2*B*Log [-Sin[c + d*x]])/(a*d) - (b*(6*a^4*b*B + 3*a^2*b^3*B + b^5*B - 3*a^5*C + a ^3*b^2*C)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a*(a^2 + b^2)*d))/(a*(a^2 + b^2)) + (b*(3*a^2*b*B + b^3*B - 2*a^3*C))/(a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])))/(a*(a^2 + b^2))
3.1.43.3.1 Defintions of rubi rules used
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_)*((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[1/b^2 Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*(b*B - a*C + b*C*Tan[e + f*x]), x], x] /; FreeQ [{a, b, c, d, e, f, A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[A*b^2 - a*b*B + a^2*C, 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 = 0.66 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.13
| method | result | size |
| derivativedivides | \(\frac {\frac {B \ln \left (\tan \left (d x +c \right )\right )}{a^{3}}+\frac {\frac {\left (-B \,a^{3}+3 B a \,b^{2}-3 C \,a^{2} b +C \,b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-3 B \,a^{2} b +B \,b^{3}+C \,a^{3}-3 C a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}+\frac {b \left (3 B \,a^{2} b +B \,b^{3}-2 C \,a^{3}\right )}{\left (a^{2}+b^{2}\right )^{2} a^{2} \left (a +b \tan \left (d x +c \right )\right )}-\frac {b \left (6 B \,a^{4} b +3 B \,a^{2} b^{3}+B \,b^{5}-3 C \,a^{5}+C \,a^{3} b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3} a^{3}}+\frac {\left (B b -C a \right ) b}{2 \left (a^{2}+b^{2}\right ) a \left (a +b \tan \left (d x +c \right )\right )^{2}}}{d}\) | \(243\) |
| default | \(\frac {\frac {B \ln \left (\tan \left (d x +c \right )\right )}{a^{3}}+\frac {\frac {\left (-B \,a^{3}+3 B a \,b^{2}-3 C \,a^{2} b +C \,b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-3 B \,a^{2} b +B \,b^{3}+C \,a^{3}-3 C a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}+\frac {b \left (3 B \,a^{2} b +B \,b^{3}-2 C \,a^{3}\right )}{\left (a^{2}+b^{2}\right )^{2} a^{2} \left (a +b \tan \left (d x +c \right )\right )}-\frac {b \left (6 B \,a^{4} b +3 B \,a^{2} b^{3}+B \,b^{5}-3 C \,a^{5}+C \,a^{3} b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3} a^{3}}+\frac {\left (B b -C a \right ) b}{2 \left (a^{2}+b^{2}\right ) a \left (a +b \tan \left (d x +c \right )\right )^{2}}}{d}\) | \(243\) |
| parallelrisch | \(\frac {-12 b \left (B \,a^{4} b +\frac {1}{2} B \,a^{2} b^{3}+\frac {1}{6} B \,b^{5}-\frac {1}{2} C \,a^{5}+\frac {1}{6} C \,a^{3} b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2} \ln \left (a +b \tan \left (d x +c \right )\right )-a^{3} \left (a +b \tan \left (d x +c \right )\right )^{2} \left (B \,a^{3}-3 B a \,b^{2}+3 C \,a^{2} b -C \,b^{3}\right ) \ln \left (\sec \left (d x +c \right )^{2}\right )+2 B \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )^{2} \ln \left (\tan \left (d x +c \right )\right )-6 \left (-\frac {C \,a^{6} d x}{3}+b \left (B d x -\frac {C}{3}\right ) a^{5}+\frac {b^{2} \left (2 C d x +B \right ) a^{4}}{2}-\frac {b^{3} \left (B d x +C \right ) a^{3}}{3}+\frac {2 B \,a^{2} b^{4}}{3}+\frac {B \,b^{6}}{6}\right ) b^{2} \tan \left (d x +c \right )^{2}-12 a^{4} d b \left (B \,a^{2} b -\frac {1}{3} B \,b^{3}-\frac {1}{3} C \,a^{3}+C a \,b^{2}\right ) x \tan \left (d x +c \right )-6 a^{2} \left (-\frac {C \,a^{6} d x}{3}+b \left (B d x +\frac {C}{2}\right ) a^{5}-\frac {2 b^{2} \left (-\frac {3 C d x}{2}+B \right ) a^{4}}{3}-\frac {b^{3} \left (B d x -2 C \right ) a^{3}}{3}-B \,a^{2} b^{4}+\frac {C a \,b^{5}}{6}-\frac {B \,b^{6}}{3}\right )}{2 \left (a^{2}+b^{2}\right )^{3} a^{3} d \left (a +b \tan \left (d x +c \right )\right )^{2}}\) | \(375\) |
| norman | \(\frac {-\frac {b^{2} \left (3 B \,a^{2} b -B \,b^{3}-C \,a^{3}+3 C a \,b^{2}\right ) x \tan \left (d x +c \right )^{3}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}-\frac {b \left (4 B \,a^{2} b^{2}+2 B \,b^{4}-3 C \,a^{3} b -C a \,b^{3}\right ) \tan \left (d x +c \right )^{2}}{d \,a^{2} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {b^{2} \left (7 B \,a^{2} b^{2}+3 B \,b^{4}-5 C \,a^{3} b -C a \,b^{3}\right ) \tan \left (d x +c \right )^{3}}{2 d \,a^{3} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (3 B \,a^{2} b -B \,b^{3}-C \,a^{3}+3 C a \,b^{2}\right ) a^{2} x \tan \left (d x +c \right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}-\frac {2 b \left (3 B \,a^{2} b -B \,b^{3}-C \,a^{3}+3 C a \,b^{2}\right ) a x \tan \left (d x +c \right )^{2}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}}{\tan \left (d x +c \right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {B \ln \left (\tan \left (d x +c \right )\right )}{a^{3} d}-\frac {\left (B \,a^{3}-3 B a \,b^{2}+3 C \,a^{2} b -C \,b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {b \left (6 B \,a^{4} b +3 B \,a^{2} b^{3}+B \,b^{5}-3 C \,a^{5}+C \,a^{3} b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) a^{3} d}\) | \(515\) |
| risch | \(\frac {2 i C \,b^{3} x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {x C}{3 i a^{2} b -i b^{3}-a^{3}+3 a \,b^{2}}+\frac {6 i b^{4} B x}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) a}+\frac {2 i C \,b^{3} c}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {6 i b^{4} B c}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) a d}+\frac {2 i b^{6} B x}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) a^{3}}-\frac {2 i B x}{a^{3}}+\frac {12 i a \,b^{2} B c}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d}-\frac {6 i a^{2} b C c}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d}+\frac {2 i \left (3 i B \,a^{2} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-2 i C \,a^{3} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-2 B a \,b^{5} {\mathrm e}^{2 i \left (d x +c \right )}+C \,a^{2} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-4 i B \,a^{2} b^{4}+3 i C \,a^{3} b^{3}-B a \,b^{5}+i B \,b^{6} {\mathrm e}^{2 i \left (d x +c \right )}-4 B \,a^{3} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+3 C \,a^{4} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-i B \,b^{6}-4 B \,a^{3} b^{3}+3 C \,a^{4} b^{2}\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 )^{2} a^{2} d \left (-i b +a \right )^{3}}+\frac {12 i a \,b^{2} B x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {6 i a^{2} b C x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {i x B}{3 i a^{2} b -i b^{3}-a^{3}+3 a \,b^{2}}+\frac {2 i b^{6} B c}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) a^{3} d}-\frac {2 i B c}{a^{3} d}-\frac {6 a \,b^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) B}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d}-\frac {3 b^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) B}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) a d}-\frac {b^{6} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) B}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) a^{3} d}+\frac {3 a^{2} b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) C}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) C \,b^{3}}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {B \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{3} d}\) | \(1021\) |
1/d*(1/a^3*B*ln(tan(d*x+c))+1/(a^2+b^2)^3*(1/2*(-B*a^3+3*B*a*b^2-3*C*a^2*b +C*b^3)*ln(1+tan(d*x+c)^2)+(-3*B*a^2*b+B*b^3+C*a^3-3*C*a*b^2)*arctan(tan(d *x+c)))+b*(3*B*a^2*b+B*b^3-2*C*a^3)/(a^2+b^2)^2/a^2/(a+b*tan(d*x+c))-b*(6* B*a^4*b+3*B*a^2*b^3+B*b^5-3*C*a^5+C*a^3*b^2)/(a^2+b^2)^3/a^3*ln(a+b*tan(d* x+c))+1/2*(B*b-C*a)*b/(a^2+b^2)/a/(a+b*tan(d*x+c))^2)
Leaf count of result is larger than twice the leaf count of optimal. 683 vs. \(2 (213) = 426\).
Time = 0.33 (sec) , antiderivative size = 683, normalized size of antiderivative = 3.18 \[ \int \frac {\cot ^2(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx=-\frac {7 \, C a^{5} b^{3} - 9 \, B a^{4} b^{4} + C a^{3} b^{5} - 3 \, B a^{2} b^{6} - 2 \, {\left (C a^{8} - 3 \, B a^{7} b - 3 \, C a^{6} b^{2} + B a^{5} b^{3}\right )} d x - {\left (5 \, C a^{5} b^{3} - 7 \, B a^{4} b^{4} - C a^{3} b^{5} - B a^{2} b^{6} + 2 \, {\left (C a^{6} b^{2} - 3 \, B a^{5} b^{3} - 3 \, C a^{4} b^{4} + B a^{3} b^{5}\right )} d x\right )} \tan \left (d x + c\right )^{2} - {\left (B a^{8} + 3 \, B a^{6} b^{2} + 3 \, B a^{4} b^{4} + B a^{2} b^{6} + {\left (B a^{6} b^{2} + 3 \, B a^{4} b^{4} + 3 \, B a^{2} b^{6} + B b^{8}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (B a^{7} b + 3 \, B a^{5} b^{3} + 3 \, B a^{3} b^{5} + B a b^{7}\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 (3 \, C a^{7} b - 6 \, B a^{6} b^{2} - C a^{5} b^{3} - 3 \, B a^{4} b^{4} - B a^{2} b^{6} + {\left (3 \, C a^{5} b^{3} - 6 \, B a^{4} b^{4} - C a^{3} b^{5} - 3 \, B a^{2} b^{6} - B b^{8}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (3 \, C a^{6} b^{2} - 6 \, B a^{5} b^{3} - C a^{4} b^{4} - 3 \, B a^{3} b^{5} - B a b^{7}\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 (3 \, C a^{6} b^{2} - 4 \, B a^{5} b^{3} - 3 \, C a^{4} b^{4} + 3 \, B a^{3} b^{5} + B a b^{7} + 2 \, {\left (C a^{7} b - 3 \, B a^{6} b^{2} - 3 \, C a^{5} b^{3} + B a^{4} b^{4}\right )} d x\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{9} b^{2} + 3 \, a^{7} b^{4} + 3 \, a^{5} b^{6} + a^{3} b^{8}\right )} d \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{10} b + 3 \, a^{8} b^{3} + 3 \, a^{6} b^{5} + a^{4} b^{7}\right )} d \tan \left (d x + c\right ) + {\left (a^{11} + 3 \, a^{9} b^{2} + 3 \, a^{7} b^{4} + a^{5} b^{6}\right )} d\right )}} \]
integrate(cot(d*x+c)^2*(B*tan(d*x+c)+C*tan(d*x+c)^2)/(a+b*tan(d*x+c))^3,x, algorithm="fricas")
-1/2*(7*C*a^5*b^3 - 9*B*a^4*b^4 + C*a^3*b^5 - 3*B*a^2*b^6 - 2*(C*a^8 - 3*B *a^7*b - 3*C*a^6*b^2 + B*a^5*b^3)*d*x - (5*C*a^5*b^3 - 7*B*a^4*b^4 - C*a^3 *b^5 - B*a^2*b^6 + 2*(C*a^6*b^2 - 3*B*a^5*b^3 - 3*C*a^4*b^4 + B*a^3*b^5)*d *x)*tan(d*x + c)^2 - (B*a^8 + 3*B*a^6*b^2 + 3*B*a^4*b^4 + B*a^2*b^6 + (B*a ^6*b^2 + 3*B*a^4*b^4 + 3*B*a^2*b^6 + B*b^8)*tan(d*x + c)^2 + 2*(B*a^7*b + 3*B*a^5*b^3 + 3*B*a^3*b^5 + B*a*b^7)*tan(d*x + c))*log(tan(d*x + c)^2/(tan (d*x + c)^2 + 1)) - (3*C*a^7*b - 6*B*a^6*b^2 - C*a^5*b^3 - 3*B*a^4*b^4 - B *a^2*b^6 + (3*C*a^5*b^3 - 6*B*a^4*b^4 - C*a^3*b^5 - 3*B*a^2*b^6 - B*b^8)*t an(d*x + c)^2 + 2*(3*C*a^6*b^2 - 6*B*a^5*b^3 - C*a^4*b^4 - 3*B*a^3*b^5 - B *a*b^7)*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*(3*C*a^6*b^2 - 4*B*a^5*b^3 - 3*C*a^4*b^4 + 3*B*a ^3*b^5 + B*a*b^7 + 2*(C*a^7*b - 3*B*a^6*b^2 - 3*C*a^5*b^3 + B*a^4*b^4)*d*x )*tan(d*x + c))/((a^9*b^2 + 3*a^7*b^4 + 3*a^5*b^6 + a^3*b^8)*d*tan(d*x + c )^2 + 2*(a^10*b + 3*a^8*b^3 + 3*a^6*b^5 + a^4*b^7)*d*tan(d*x + c) + (a^11 + 3*a^9*b^2 + 3*a^7*b^4 + a^5*b^6)*d)
Exception generated. \[ \int \frac {\cot ^2(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx=\text {Exception raised: AttributeError} \]
Time = 0.34 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.73 \[ \int \frac {\cot ^2(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {2 \, {\left (C a^{3} - 3 \, B a^{2} b - 3 \, C a b^{2} + B b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left (3 \, C a^{5} b - 6 \, B a^{4} b^{2} - C a^{3} b^{3} - 3 \, B a^{2} b^{4} - B b^{6}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{9} + 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} + a^{3} b^{6}} - \frac {{\left (B a^{3} + 3 \, C a^{2} b - 3 \, B a b^{2} - C b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {5 \, C a^{4} b - 7 \, B a^{3} b^{2} + C a^{2} b^{3} - 3 \, B a b^{4} + 2 \, {\left (2 \, C a^{3} b^{2} - 3 \, B a^{2} b^{3} - B b^{5}\right )} \tan \left (d x + c\right )}{a^{8} + 2 \, a^{6} b^{2} + a^{4} b^{4} + {\left (a^{6} b^{2} + 2 \, a^{4} b^{4} + a^{2} b^{6}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b + 2 \, a^{5} b^{3} + a^{3} b^{5}\right )} \tan \left (d x + c\right )} + \frac {2 \, B \log \left (\tan \left (d x + c\right )\right )}{a^{3}}}{2 \, d} \]
integrate(cot(d*x+c)^2*(B*tan(d*x+c)+C*tan(d*x+c)^2)/(a+b*tan(d*x+c))^3,x, algorithm="maxima")
1/2*(2*(C*a^3 - 3*B*a^2*b - 3*C*a*b^2 + B*b^3)*(d*x + c)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 2*(3*C*a^5*b - 6*B*a^4*b^2 - C*a^3*b^3 - 3*B*a^2*b^4 - B*b^6)*log(b*tan(d*x + c) + a)/(a^9 + 3*a^7*b^2 + 3*a^5*b^4 + a^3*b^6) - (B*a^3 + 3*C*a^2*b - 3*B*a*b^2 - C*b^3)*log(tan(d*x + c)^2 + 1)/(a^6 + 3* a^4*b^2 + 3*a^2*b^4 + b^6) - (5*C*a^4*b - 7*B*a^3*b^2 + C*a^2*b^3 - 3*B*a* b^4 + 2*(2*C*a^3*b^2 - 3*B*a^2*b^3 - B*b^5)*tan(d*x + c))/(a^8 + 2*a^6*b^2 + a^4*b^4 + (a^6*b^2 + 2*a^4*b^4 + a^2*b^6)*tan(d*x + c)^2 + 2*(a^7*b + 2 *a^5*b^3 + a^3*b^5)*tan(d*x + c)) + 2*B*log(tan(d*x + c))/a^3)/d
Leaf count of result is larger than twice the leaf count of optimal. 479 vs. \(2 (213) = 426\).
Time = 1.24 (sec) , antiderivative size = 479, normalized size of antiderivative = 2.23 \[ \int \frac {\cot ^2(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {2 \, {\left (C a^{3} - 3 \, B a^{2} b - 3 \, C a b^{2} + B b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (B a^{3} + 3 \, C a^{2} b - 3 \, B a b^{2} - C b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left (3 \, C a^{5} b^{2} - 6 \, B a^{4} b^{3} - C a^{3} b^{4} - 3 \, B a^{2} b^{5} - B b^{7}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{9} b + 3 \, a^{7} b^{3} + 3 \, a^{5} b^{5} + a^{3} b^{7}} + \frac {2 \, B \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{3}} - \frac {9 \, C a^{5} b^{3} \tan \left (d x + c\right )^{2} - 18 \, B a^{4} b^{4} \tan \left (d x + c\right )^{2} - 3 \, C a^{3} b^{5} \tan \left (d x + c\right )^{2} - 9 \, B a^{2} b^{6} \tan \left (d x + c\right )^{2} - 3 \, B b^{8} \tan \left (d x + c\right )^{2} + 22 \, C a^{6} b^{2} \tan \left (d x + c\right ) - 42 \, B a^{5} b^{3} \tan \left (d x + c\right ) - 2 \, C a^{4} b^{4} \tan \left (d x + c\right ) - 26 \, B a^{3} b^{5} \tan \left (d x + c\right ) - 8 \, B a b^{7} \tan \left (d x + c\right ) + 14 \, C a^{7} b - 25 \, B a^{6} b^{2} + 3 \, C a^{5} b^{3} - 19 \, B a^{4} b^{4} + C a^{3} b^{5} - 6 \, B a^{2} b^{6}}{{\left (a^{9} + 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} + a^{3} b^{6}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{2}}}{2 \, d} \]
1/2*(2*(C*a^3 - 3*B*a^2*b - 3*C*a*b^2 + B*b^3)*(d*x + c)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - (B*a^3 + 3*C*a^2*b - 3*B*a*b^2 - C*b^3)*log(tan(d*x + c)^2 + 1)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 2*(3*C*a^5*b^2 - 6*B*a^4* b^3 - C*a^3*b^4 - 3*B*a^2*b^5 - B*b^7)*log(abs(b*tan(d*x + c) + a))/(a^9*b + 3*a^7*b^3 + 3*a^5*b^5 + a^3*b^7) + 2*B*log(abs(tan(d*x + c)))/a^3 - (9* C*a^5*b^3*tan(d*x + c)^2 - 18*B*a^4*b^4*tan(d*x + c)^2 - 3*C*a^3*b^5*tan(d *x + c)^2 - 9*B*a^2*b^6*tan(d*x + c)^2 - 3*B*b^8*tan(d*x + c)^2 + 22*C*a^6 *b^2*tan(d*x + c) - 42*B*a^5*b^3*tan(d*x + c) - 2*C*a^4*b^4*tan(d*x + c) - 26*B*a^3*b^5*tan(d*x + c) - 8*B*a*b^7*tan(d*x + c) + 14*C*a^7*b - 25*B*a^ 6*b^2 + 3*C*a^5*b^3 - 19*B*a^4*b^4 + C*a^3*b^5 - 6*B*a^2*b^6)/((a^9 + 3*a^ 7*b^2 + 3*a^5*b^4 + a^3*b^6)*(b*tan(d*x + c) + a)^2))/d
Time = 10.87 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.47 \[ \int \frac {\cot ^2(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {-5\,C\,a^3\,b+7\,B\,a^2\,b^2-C\,a\,b^3+3\,B\,b^4}{2\,a\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (-2\,C\,a^3\,b^2+3\,B\,a^2\,b^3+B\,b^5\right )}{a^2\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{d\,\left (a^2+2\,a\,b\,\mathrm {tan}\left (c+d\,x\right )+b^2\,{\mathrm {tan}\left (c+d\,x\right )}^2\right )}+\frac {B\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{a^3\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-C+B\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^3\,1{}\mathrm {i}+3\,a^2\,b+a\,b^2\,3{}\mathrm {i}-b^3\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B-C\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^3+a^2\,b\,3{}\mathrm {i}+3\,a\,b^2-b^3\,1{}\mathrm {i}\right )}-\frac {b\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (-3\,C\,a^5+6\,B\,a^4\,b+C\,a^3\,b^2+3\,B\,a^2\,b^3+B\,b^5\right )}{a^3\,d\,{\left (a^2+b^2\right )}^3} \]
((3*B*b^4 + 7*B*a^2*b^2 - C*a*b^3 - 5*C*a^3*b)/(2*a*(a^4 + b^4 + 2*a^2*b^2 )) + (tan(c + d*x)*(B*b^5 + 3*B*a^2*b^3 - 2*C*a^3*b^2))/(a^2*(a^4 + b^4 + 2*a^2*b^2)))/(d*(a^2 + b^2*tan(c + d*x)^2 + 2*a*b*tan(c + d*x))) + (B*log( tan(c + d*x)))/(a^3*d) + (log(tan(c + d*x) - 1i)*(B*1i - C))/(2*d*(a*b^2*3 i + 3*a^2*b - a^3*1i - b^3)) + (log(tan(c + d*x) + 1i)*(B - C*1i))/(2*d*(3 *a*b^2 + a^2*b*3i - a^3 - b^3*1i)) - (b*log(a + b*tan(c + d*x))*(B*b^5 - 3 *C*a^5 + 3*B*a^2*b^3 + C*a^3*b^2 + 6*B*a^4*b))/(a^3*d*(a^2 + b^2)^3)