Integrand size = 25, antiderivative size = 215 \[ \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^2 \, dx=-\left (\left (6 a^2 b c d-2 b^3 c d-a^3 \left (c^2-d^2\right )+3 a b^2 \left (c^2-d^2\right )\right ) x\right )-\frac {\left (2 a^3 c d-6 a b^2 c d+3 a^2 b \left (c^2-d^2\right )-b^3 \left (c^2-d^2\right )\right ) \log (\cos (e+f x))}{f}+\frac {2 b (b c+a d) (a c-b d) \tan (e+f x)}{f}+\frac {\left (2 a c d+b \left (c^2-d^2\right )\right ) (a+b \tan (e+f x))^2}{2 f}+\frac {2 c d (a+b \tan (e+f x))^3}{3 f}+\frac {d^2 (a+b \tan (e+f x))^4}{4 b f} \] Output:
-(6*a^2*b*c*d-2*b^3*c*d-a^3*(c^2-d^2)+3*a*b^2*(c^2-d^2))*x-(2*a^3*c*d-6*a* b^2*c*d+3*a^2*b*(c^2-d^2)-b^3*(c^2-d^2))*ln(cos(f*x+e))/f+2*b*(a*d+b*c)*(a *c-b*d)*tan(f*x+e)/f+1/2*(2*a*c*d+b*(c^2-d^2))*(a+b*tan(f*x+e))^2/f+2/3*c* d*(a+b*tan(f*x+e))^3/f+1/4*d^2*(a+b*tan(f*x+e))^4/b/f
Result contains complex when optimal does not.
Time = 1.70 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.03 \[ \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^2 \, dx=\frac {3 d^2 (a+b \tan (e+f x))^4-6 \left (2 a c d+b \left (-c^2+d^2\right )\right ) \left ((i a-b)^3 \log (i-\tan (e+f x))-(i a+b)^3 \log (i+\tan (e+f x))+6 a b^2 \tan (e+f x)+b^3 \tan ^2(e+f x)\right )-4 c d \left (3 i (a+i b)^4 \log (i-\tan (e+f x))-3 i (a-i b)^4 \log (i+\tan (e+f x))+6 b^2 \left (-6 a^2+b^2\right ) \tan (e+f x)-12 a b^3 \tan ^2(e+f x)-2 b^4 \tan ^3(e+f x)\right )}{12 b f} \] Input:
Integrate[(a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^2,x]
Output:
(3*d^2*(a + b*Tan[e + f*x])^4 - 6*(2*a*c*d + b*(-c^2 + d^2))*((I*a - b)^3* Log[I - Tan[e + f*x]] - (I*a + b)^3*Log[I + Tan[e + f*x]] + 6*a*b^2*Tan[e + f*x] + b^3*Tan[e + f*x]^2) - 4*c*d*((3*I)*(a + I*b)^4*Log[I - Tan[e + f* x]] - (3*I)*(a - I*b)^4*Log[I + Tan[e + f*x]] + 6*b^2*(-6*a^2 + b^2)*Tan[e + f*x] - 12*a*b^3*Tan[e + f*x]^2 - 2*b^4*Tan[e + f*x]^3))/(12*b*f)
Time = 0.90 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 4026, 3042, 4011, 3042, 4011, 3042, 4008, 3042, 3956}
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.
| \(\displaystyle \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^2dx\) |
| \(\Big \downarrow \) 4026 |
| \(\displaystyle \int (a+b \tan (e+f x))^3 \left (c^2+2 d \tan (e+f x) c-d^2\right )dx+\frac {d^2 (a+b \tan (e+f x))^4}{4 b f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+b \tan (e+f x))^3 \left (c^2+2 d \tan (e+f x) c-d^2\right )dx+\frac {d^2 (a+b \tan (e+f x))^4}{4 b f}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \int (a+b \tan (e+f x))^2 \left (-2 b c d+a \left (c^2-d^2\right )+\left (2 a c d+b \left (c^2-d^2\right )\right ) \tan (e+f x)\right )dx+\frac {2 c d (a+b \tan (e+f x))^3}{3 f}+\frac {d^2 (a+b \tan (e+f x))^4}{4 b f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+b \tan (e+f x))^2 \left (-2 b c d+a \left (c^2-d^2\right )+\left (2 a c d+b \left (c^2-d^2\right )\right ) \tan (e+f x)\right )dx+\frac {2 c d (a+b \tan (e+f x))^3}{3 f}+\frac {d^2 (a+b \tan (e+f x))^4}{4 b f}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \int (a+b \tan (e+f x)) ((a c-b c-a d-b d) (a c+b c+a d-b d)+2 (b c+a d) (a c-b d) \tan (e+f x))dx+\frac {\left (2 a c d+b \left (c^2-d^2\right )\right ) (a+b \tan (e+f x))^2}{2 f}+\frac {2 c d (a+b \tan (e+f x))^3}{3 f}+\frac {d^2 (a+b \tan (e+f x))^4}{4 b f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+b \tan (e+f x)) ((a c-b c-a d-b d) (a c+b c+a d-b d)+2 (b c+a d) (a c-b d) \tan (e+f x))dx+\frac {\left (2 a c d+b \left (c^2-d^2\right )\right ) (a+b \tan (e+f x))^2}{2 f}+\frac {2 c d (a+b \tan (e+f x))^3}{3 f}+\frac {d^2 (a+b \tan (e+f x))^4}{4 b f}\) |
| \(\Big \downarrow \) 4008 |
| \(\displaystyle \left (2 a^3 c d+3 a^2 b \left (c^2-d^2\right )-6 a b^2 c d-b^3 \left (c^2-d^2\right )\right ) \int \tan (e+f x)dx-x \left (-\left (a^3 \left (c^2-d^2\right )\right )+6 a^2 b c d+3 a b^2 \left (c^2-d^2\right )-2 b^3 c d\right )+\frac {\left (2 a c d+b \left (c^2-d^2\right )\right ) (a+b \tan (e+f x))^2}{2 f}+\frac {2 c d (a+b \tan (e+f x))^3}{3 f}+\frac {2 b (a d+b c) (a c-b d) \tan (e+f x)}{f}+\frac {d^2 (a+b \tan (e+f x))^4}{4 b f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \left (2 a^3 c d+3 a^2 b \left (c^2-d^2\right )-6 a b^2 c d-b^3 \left (c^2-d^2\right )\right ) \int \tan (e+f x)dx-x \left (-\left (a^3 \left (c^2-d^2\right )\right )+6 a^2 b c d+3 a b^2 \left (c^2-d^2\right )-2 b^3 c d\right )+\frac {\left (2 a c d+b \left (c^2-d^2\right )\right ) (a+b \tan (e+f x))^2}{2 f}+\frac {2 c d (a+b \tan (e+f x))^3}{3 f}+\frac {2 b (a d+b c) (a c-b d) \tan (e+f x)}{f}+\frac {d^2 (a+b \tan (e+f x))^4}{4 b f}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle -\frac {\left (2 a^3 c d+3 a^2 b \left (c^2-d^2\right )-6 a b^2 c d-b^3 \left (c^2-d^2\right )\right ) \log (\cos (e+f x))}{f}-x \left (-\left (a^3 \left (c^2-d^2\right )\right )+6 a^2 b c d+3 a b^2 \left (c^2-d^2\right )-2 b^3 c d\right )+\frac {\left (2 a c d+b \left (c^2-d^2\right )\right ) (a+b \tan (e+f x))^2}{2 f}+\frac {2 c d (a+b \tan (e+f x))^3}{3 f}+\frac {2 b (a d+b c) (a c-b d) \tan (e+f x)}{f}+\frac {d^2 (a+b \tan (e+f x))^4}{4 b f}\) |
Input:
Int[(a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^2,x]
Output:
-((6*a^2*b*c*d - 2*b^3*c*d - a^3*(c^2 - d^2) + 3*a*b^2*(c^2 - d^2))*x) - ( (2*a^3*c*d - 6*a*b^2*c*d + 3*a^2*b*(c^2 - d^2) - b^3*(c^2 - d^2))*Log[Cos[ e + f*x]])/f + (2*b*(b*c + a*d)*(a*c - b*d)*Tan[e + f*x])/f + ((2*a*c*d + b*(c^2 - d^2))*(a + b*Tan[e + f*x])^2)/(2*f) + (2*c*d*(a + b*Tan[e + f*x]) ^3)/(3*f) + (d^2*(a + b*Tan[e + f*x])^4)/(4*b*f)
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.) *(x_)]), x_Symbol] :> Simp[(a*c - b*d)*x, x] + (Simp[b*d*(Tan[e + f*x]/f), x] + Simp[(b*c + a*d) Int[Tan[e + f*x], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[b*c + a*d, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*((a + b*Tan[e + f*x])^m/(f*m)), x] + Int [(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[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] && GtQ[m, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[d^2*((a + b*Tan[e + f*x])^(m + 1)/(b*f*( m + 1))), x] + Int[(a + b*Tan[e + f*x])^m*Simp[c^2 - d^2 + 2*c*d*Tan[e + f* x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && !LeQ [m, -1] && !(EqQ[m, 2] && EqQ[a, 0])
Time = 0.18 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.08
| method | result | size |
| parts | \(a^{3} c^{2} x +\frac {\left (3 a \,b^{2} d^{2}+2 b^{3} c d \right ) \left (\frac {\tan \left (f x +e \right )^{3}}{3}-\tan \left (f x +e \right )+\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}+\frac {\left (2 a^{3} c d +3 a^{2} b \,c^{2}\right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2 f}+\frac {\left (3 a^{2} b \,d^{2}+6 a \,b^{2} c d +b^{3} c^{2}\right ) \left (\frac {\tan \left (f x +e \right )^{2}}{2}-\frac {\ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}\right )}{f}+\frac {\left (a^{3} d^{2}+6 a^{2} b c d +3 a \,b^{2} c^{2}\right ) \left (\tan \left (f x +e \right )-\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}+\frac {b^{3} d^{2} \left (\frac {\tan \left (f x +e \right )^{4}}{4}-\frac {\tan \left (f x +e \right )^{2}}{2}+\frac {\ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}\right )}{f}\) | \(232\) |
| norman | \(\left (a^{3} c^{2}-a^{3} d^{2}-6 a^{2} b c d -3 a \,b^{2} c^{2}+3 a \,b^{2} d^{2}+2 b^{3} c d \right ) x +\frac {\left (a^{3} d^{2}+6 a^{2} b c d +3 a \,b^{2} c^{2}-3 a \,b^{2} d^{2}-2 b^{3} c d \right ) \tan \left (f x +e \right )}{f}+\frac {b \left (3 a^{2} d^{2}+6 a b c d +b^{2} c^{2}-b^{2} d^{2}\right ) \tan \left (f x +e \right )^{2}}{2 f}+\frac {b^{3} d^{2} \tan \left (f x +e \right )^{4}}{4 f}+\frac {b^{2} d \left (3 a d +2 b c \right ) \tan \left (f x +e \right )^{3}}{3 f}+\frac {\left (2 a^{3} c d +3 a^{2} b \,c^{2}-3 a^{2} b \,d^{2}-6 a \,b^{2} c d -b^{3} c^{2}+b^{3} d^{2}\right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2 f}\) | \(258\) |
| derivativedivides | \(\frac {\frac {b^{3} d^{2} \tan \left (f x +e \right )^{4}}{4}+a \,b^{2} d^{2} \tan \left (f x +e \right )^{3}+\frac {2 b^{3} c d \tan \left (f x +e \right )^{3}}{3}+\frac {3 a^{2} b \,d^{2} \tan \left (f x +e \right )^{2}}{2}+3 a \,b^{2} c d \tan \left (f x +e \right )^{2}+\frac {b^{3} c^{2} \tan \left (f x +e \right )^{2}}{2}-\frac {b^{3} d^{2} \tan \left (f x +e \right )^{2}}{2}+\tan \left (f x +e \right ) a^{3} d^{2}+6 \tan \left (f x +e \right ) a^{2} b c d +3 \tan \left (f x +e \right ) a \,b^{2} c^{2}-3 \tan \left (f x +e \right ) a \,b^{2} d^{2}-2 \tan \left (f x +e \right ) b^{3} c d +\frac {\left (2 a^{3} c d +3 a^{2} b \,c^{2}-3 a^{2} b \,d^{2}-6 a \,b^{2} c d -b^{3} c^{2}+b^{3} d^{2}\right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}+\left (a^{3} c^{2}-a^{3} d^{2}-6 a^{2} b c d -3 a \,b^{2} c^{2}+3 a \,b^{2} d^{2}+2 b^{3} c d \right ) \arctan \left (\tan \left (f x +e \right )\right )}{f}\) | \(307\) |
| default | \(\frac {\frac {b^{3} d^{2} \tan \left (f x +e \right )^{4}}{4}+a \,b^{2} d^{2} \tan \left (f x +e \right )^{3}+\frac {2 b^{3} c d \tan \left (f x +e \right )^{3}}{3}+\frac {3 a^{2} b \,d^{2} \tan \left (f x +e \right )^{2}}{2}+3 a \,b^{2} c d \tan \left (f x +e \right )^{2}+\frac {b^{3} c^{2} \tan \left (f x +e \right )^{2}}{2}-\frac {b^{3} d^{2} \tan \left (f x +e \right )^{2}}{2}+\tan \left (f x +e \right ) a^{3} d^{2}+6 \tan \left (f x +e \right ) a^{2} b c d +3 \tan \left (f x +e \right ) a \,b^{2} c^{2}-3 \tan \left (f x +e \right ) a \,b^{2} d^{2}-2 \tan \left (f x +e \right ) b^{3} c d +\frac {\left (2 a^{3} c d +3 a^{2} b \,c^{2}-3 a^{2} b \,d^{2}-6 a \,b^{2} c d -b^{3} c^{2}+b^{3} d^{2}\right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}+\left (a^{3} c^{2}-a^{3} d^{2}-6 a^{2} b c d -3 a \,b^{2} c^{2}+3 a \,b^{2} d^{2}+2 b^{3} c d \right ) \arctan \left (\tan \left (f x +e \right )\right )}{f}\) | \(307\) |
| parallelrisch | \(\frac {-36 \ln \left (1+\tan \left (f x +e \right )^{2}\right ) a \,b^{2} c d +12 a^{3} c^{2} f x -12 a^{3} d^{2} f x -72 a^{2} b c d f x +12 a \,b^{2} d^{2} \tan \left (f x +e \right )^{3}+8 b^{3} c d \tan \left (f x +e \right )^{3}+18 a^{2} b \,d^{2} \tan \left (f x +e \right )^{2}+36 \tan \left (f x +e \right ) a \,b^{2} c^{2}-36 \tan \left (f x +e \right ) a \,b^{2} d^{2}-24 \tan \left (f x +e \right ) b^{3} c d +3 b^{3} d^{2} \tan \left (f x +e \right )^{4}+6 b^{3} c^{2} \tan \left (f x +e \right )^{2}-6 b^{3} d^{2} \tan \left (f x +e \right )^{2}+12 \tan \left (f x +e \right ) a^{3} d^{2}+36 a \,b^{2} c d \tan \left (f x +e \right )^{2}+72 \tan \left (f x +e \right ) a^{2} b c d -36 a \,b^{2} c^{2} f x +36 a \,b^{2} d^{2} f x +24 b^{3} c d f x +12 \ln \left (1+\tan \left (f x +e \right )^{2}\right ) a^{3} c d +18 \ln \left (1+\tan \left (f x +e \right )^{2}\right ) a^{2} b \,c^{2}-18 \ln \left (1+\tan \left (f x +e \right )^{2}\right ) a^{2} b \,d^{2}+6 \ln \left (1+\tan \left (f x +e \right )^{2}\right ) b^{3} d^{2}-6 \ln \left (1+\tan \left (f x +e \right )^{2}\right ) b^{3} c^{2}}{12 f}\) | \(367\) |
| risch | \(-i b^{3} c^{2} x +i b^{3} d^{2} x -6 a^{2} b c d x -3 a \,b^{2} c^{2} x +3 a \,b^{2} d^{2} x +2 b^{3} c d x +\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) b^{3} c^{2}}{f}-\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) b^{3} d^{2}}{f}+\frac {2 i \left (-3 i b^{3} c^{2} {\mathrm e}^{2 i \left (f x +e \right )}+6 i b^{3} d^{2} {\mathrm e}^{2 i \left (f x +e \right )}+6 i b^{3} d^{2} {\mathrm e}^{4 i \left (f x +e \right )}+6 i b^{3} d^{2} {\mathrm e}^{6 i \left (f x +e \right )}+18 a^{2} b c d -8 b^{3} c d +9 a \,b^{2} c^{2}-12 a \,b^{2} d^{2}+3 a^{3} d^{2}-6 i b^{3} c^{2} {\mathrm e}^{4 i \left (f x +e \right )}-3 i b^{3} c^{2} {\mathrm e}^{6 i \left (f x +e \right )}-24 b^{3} c d \,{\mathrm e}^{4 i \left (f x +e \right )}+27 a \,b^{2} c^{2} {\mathrm e}^{2 i \left (f x +e \right )}-12 b^{3} c d \,{\mathrm e}^{6 i \left (f x +e \right )}+27 a \,b^{2} c^{2} {\mathrm e}^{4 i \left (f x +e \right )}-36 a \,b^{2} d^{2} {\mathrm e}^{4 i \left (f x +e \right )}+9 a \,b^{2} c^{2} {\mathrm e}^{6 i \left (f x +e \right )}-18 a \,b^{2} d^{2} {\mathrm e}^{6 i \left (f x +e \right )}-20 b^{3} c d \,{\mathrm e}^{2 i \left (f x +e \right )}-30 a \,b^{2} d^{2} {\mathrm e}^{2 i \left (f x +e \right )}+54 a^{2} b c d \,{\mathrm e}^{2 i \left (f x +e \right )}+18 a^{2} b c d \,{\mathrm e}^{6 i \left (f x +e \right )}+54 a^{2} b c d \,{\mathrm e}^{4 i \left (f x +e \right )}-9 i a^{2} b \,d^{2} {\mathrm e}^{6 i \left (f x +e \right )}-18 i a^{2} b \,d^{2} {\mathrm e}^{4 i \left (f x +e \right )}-9 i a^{2} b \,d^{2} {\mathrm e}^{2 i \left (f x +e \right )}+9 a^{3} d^{2} {\mathrm e}^{2 i \left (f x +e \right )}+9 a^{3} d^{2} {\mathrm e}^{4 i \left (f x +e \right )}+3 a^{3} d^{2} {\mathrm e}^{6 i \left (f x +e \right )}-18 i a \,b^{2} c d \,{\mathrm e}^{2 i \left (f x +e \right )}-18 i a \,b^{2} c d \,{\mathrm e}^{6 i \left (f x +e \right )}-36 i a \,b^{2} c d \,{\mathrm e}^{4 i \left (f x +e \right )}\right )}{3 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{4}}-3 i a^{2} b \,d^{2} x -\frac {2 \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) a^{3} c d}{f}-\frac {3 \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) a^{2} b \,c^{2}}{f}+\frac {3 \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) a^{2} b \,d^{2}}{f}+2 i a^{3} c d x +3 i a^{2} b \,c^{2} x -\frac {2 i b^{3} c^{2} e}{f}+\frac {2 i b^{3} d^{2} e}{f}+a^{3} c^{2} x -a^{3} d^{2} x +\frac {6 \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) a \,b^{2} c d}{f}-6 i a \,b^{2} c d x +\frac {4 i a^{3} c d e}{f}+\frac {6 i a^{2} b \,c^{2} e}{f}-\frac {6 i a^{2} b \,d^{2} e}{f}-\frac {12 i a \,b^{2} c d e}{f}\) | \(872\) |
Input:
int((a+b*tan(f*x+e))^3*(c+d*tan(f*x+e))^2,x,method=_RETURNVERBOSE)
Output:
a^3*c^2*x+(3*a*b^2*d^2+2*b^3*c*d)/f*(1/3*tan(f*x+e)^3-tan(f*x+e)+arctan(ta n(f*x+e)))+1/2*(2*a^3*c*d+3*a^2*b*c^2)/f*ln(1+tan(f*x+e)^2)+(3*a^2*b*d^2+6 *a*b^2*c*d+b^3*c^2)/f*(1/2*tan(f*x+e)^2-1/2*ln(1+tan(f*x+e)^2))+(a^3*d^2+6 *a^2*b*c*d+3*a*b^2*c^2)/f*(tan(f*x+e)-arctan(tan(f*x+e)))+b^3*d^2/f*(1/4*t an(f*x+e)^4-1/2*tan(f*x+e)^2+1/2*ln(1+tan(f*x+e)^2))
Time = 0.09 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.17 \[ \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^2 \, dx=\frac {3 \, b^{3} d^{2} \tan \left (f x + e\right )^{4} + 4 \, {\left (2 \, b^{3} c d + 3 \, a b^{2} d^{2}\right )} \tan \left (f x + e\right )^{3} + 12 \, {\left ({\left (a^{3} - 3 \, a b^{2}\right )} c^{2} - 2 \, {\left (3 \, a^{2} b - b^{3}\right )} c d - {\left (a^{3} - 3 \, a b^{2}\right )} d^{2}\right )} f x + 6 \, {\left (b^{3} c^{2} + 6 \, a b^{2} c d + {\left (3 \, a^{2} b - b^{3}\right )} d^{2}\right )} \tan \left (f x + e\right )^{2} - 6 \, {\left ({\left (3 \, a^{2} b - b^{3}\right )} c^{2} + 2 \, {\left (a^{3} - 3 \, a b^{2}\right )} c d - {\left (3 \, a^{2} b - b^{3}\right )} d^{2}\right )} \log \left (\frac {1}{\tan \left (f x + e\right )^{2} + 1}\right ) + 12 \, {\left (3 \, a b^{2} c^{2} + 2 \, {\left (3 \, a^{2} b - b^{3}\right )} c d + {\left (a^{3} - 3 \, a b^{2}\right )} d^{2}\right )} \tan \left (f x + e\right )}{12 \, f} \] Input:
integrate((a+b*tan(f*x+e))^3*(c+d*tan(f*x+e))^2,x, algorithm="fricas")
Output:
1/12*(3*b^3*d^2*tan(f*x + e)^4 + 4*(2*b^3*c*d + 3*a*b^2*d^2)*tan(f*x + e)^ 3 + 12*((a^3 - 3*a*b^2)*c^2 - 2*(3*a^2*b - b^3)*c*d - (a^3 - 3*a*b^2)*d^2) *f*x + 6*(b^3*c^2 + 6*a*b^2*c*d + (3*a^2*b - b^3)*d^2)*tan(f*x + e)^2 - 6* ((3*a^2*b - b^3)*c^2 + 2*(a^3 - 3*a*b^2)*c*d - (3*a^2*b - b^3)*d^2)*log(1/ (tan(f*x + e)^2 + 1)) + 12*(3*a*b^2*c^2 + 2*(3*a^2*b - b^3)*c*d + (a^3 - 3 *a*b^2)*d^2)*tan(f*x + e))/f
Leaf count of result is larger than twice the leaf count of optimal. 445 vs. \(2 (190) = 380\).
Time = 0.23 (sec) , antiderivative size = 445, normalized size of antiderivative = 2.07 \[ \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^2 \, dx=\begin {cases} a^{3} c^{2} x + \frac {a^{3} c d \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{f} - a^{3} d^{2} x + \frac {a^{3} d^{2} \tan {\left (e + f x \right )}}{f} + \frac {3 a^{2} b c^{2} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} - 6 a^{2} b c d x + \frac {6 a^{2} b c d \tan {\left (e + f x \right )}}{f} - \frac {3 a^{2} b d^{2} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {3 a^{2} b d^{2} \tan ^{2}{\left (e + f x \right )}}{2 f} - 3 a b^{2} c^{2} x + \frac {3 a b^{2} c^{2} \tan {\left (e + f x \right )}}{f} - \frac {3 a b^{2} c d \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{f} + \frac {3 a b^{2} c d \tan ^{2}{\left (e + f x \right )}}{f} + 3 a b^{2} d^{2} x + \frac {a b^{2} d^{2} \tan ^{3}{\left (e + f x \right )}}{f} - \frac {3 a b^{2} d^{2} \tan {\left (e + f x \right )}}{f} - \frac {b^{3} c^{2} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {b^{3} c^{2} \tan ^{2}{\left (e + f x \right )}}{2 f} + 2 b^{3} c d x + \frac {2 b^{3} c d \tan ^{3}{\left (e + f x \right )}}{3 f} - \frac {2 b^{3} c d \tan {\left (e + f x \right )}}{f} + \frac {b^{3} d^{2} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {b^{3} d^{2} \tan ^{4}{\left (e + f x \right )}}{4 f} - \frac {b^{3} d^{2} \tan ^{2}{\left (e + f x \right )}}{2 f} & \text {for}\: f \neq 0 \\x \left (a + b \tan {\left (e \right )}\right )^{3} \left (c + d \tan {\left (e \right )}\right )^{2} & \text {otherwise} \end {cases} \] Input:
integrate((a+b*tan(f*x+e))**3*(c+d*tan(f*x+e))**2,x)
Output:
Piecewise((a**3*c**2*x + a**3*c*d*log(tan(e + f*x)**2 + 1)/f - a**3*d**2*x + a**3*d**2*tan(e + f*x)/f + 3*a**2*b*c**2*log(tan(e + f*x)**2 + 1)/(2*f) - 6*a**2*b*c*d*x + 6*a**2*b*c*d*tan(e + f*x)/f - 3*a**2*b*d**2*log(tan(e + f*x)**2 + 1)/(2*f) + 3*a**2*b*d**2*tan(e + f*x)**2/(2*f) - 3*a*b**2*c**2 *x + 3*a*b**2*c**2*tan(e + f*x)/f - 3*a*b**2*c*d*log(tan(e + f*x)**2 + 1)/ f + 3*a*b**2*c*d*tan(e + f*x)**2/f + 3*a*b**2*d**2*x + a*b**2*d**2*tan(e + f*x)**3/f - 3*a*b**2*d**2*tan(e + f*x)/f - b**3*c**2*log(tan(e + f*x)**2 + 1)/(2*f) + b**3*c**2*tan(e + f*x)**2/(2*f) + 2*b**3*c*d*x + 2*b**3*c*d*t an(e + f*x)**3/(3*f) - 2*b**3*c*d*tan(e + f*x)/f + b**3*d**2*log(tan(e + f *x)**2 + 1)/(2*f) + b**3*d**2*tan(e + f*x)**4/(4*f) - b**3*d**2*tan(e + f* x)**2/(2*f), Ne(f, 0)), (x*(a + b*tan(e))**3*(c + d*tan(e))**2, True))
Time = 0.12 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.18 \[ \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^2 \, dx=\frac {3 \, b^{3} d^{2} \tan \left (f x + e\right )^{4} + 4 \, {\left (2 \, b^{3} c d + 3 \, a b^{2} d^{2}\right )} \tan \left (f x + e\right )^{3} + 6 \, {\left (b^{3} c^{2} + 6 \, a b^{2} c d + {\left (3 \, a^{2} b - b^{3}\right )} d^{2}\right )} \tan \left (f x + e\right )^{2} + 12 \, {\left ({\left (a^{3} - 3 \, a b^{2}\right )} c^{2} - 2 \, {\left (3 \, a^{2} b - b^{3}\right )} c d - {\left (a^{3} - 3 \, a b^{2}\right )} d^{2}\right )} {\left (f x + e\right )} + 6 \, {\left ({\left (3 \, a^{2} b - b^{3}\right )} c^{2} + 2 \, {\left (a^{3} - 3 \, a b^{2}\right )} c d - {\left (3 \, a^{2} b - b^{3}\right )} d^{2}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right ) + 12 \, {\left (3 \, a b^{2} c^{2} + 2 \, {\left (3 \, a^{2} b - b^{3}\right )} c d + {\left (a^{3} - 3 \, a b^{2}\right )} d^{2}\right )} \tan \left (f x + e\right )}{12 \, f} \] Input:
integrate((a+b*tan(f*x+e))^3*(c+d*tan(f*x+e))^2,x, algorithm="maxima")
Output:
1/12*(3*b^3*d^2*tan(f*x + e)^4 + 4*(2*b^3*c*d + 3*a*b^2*d^2)*tan(f*x + e)^ 3 + 6*(b^3*c^2 + 6*a*b^2*c*d + (3*a^2*b - b^3)*d^2)*tan(f*x + e)^2 + 12*(( a^3 - 3*a*b^2)*c^2 - 2*(3*a^2*b - b^3)*c*d - (a^3 - 3*a*b^2)*d^2)*(f*x + e ) + 6*((3*a^2*b - b^3)*c^2 + 2*(a^3 - 3*a*b^2)*c*d - (3*a^2*b - b^3)*d^2)* log(tan(f*x + e)^2 + 1) + 12*(3*a*b^2*c^2 + 2*(3*a^2*b - b^3)*c*d + (a^3 - 3*a*b^2)*d^2)*tan(f*x + e))/f
Time = 0.34 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.63 \[ \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^2 \, dx=\frac {{\left (a^{3} c^{2} - 3 \, a b^{2} c^{2} - 6 \, a^{2} b c d + 2 \, b^{3} c d - a^{3} d^{2} + 3 \, a b^{2} d^{2}\right )} {\left (f x + e\right )}}{f} + \frac {{\left (3 \, a^{2} b c^{2} - b^{3} c^{2} + 2 \, a^{3} c d - 6 \, a b^{2} c d - 3 \, a^{2} b d^{2} + b^{3} d^{2}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{2 \, f} + \frac {3 \, b^{3} d^{2} f^{3} \tan \left (f x + e\right )^{4} + 8 \, b^{3} c d f^{3} \tan \left (f x + e\right )^{3} + 12 \, a b^{2} d^{2} f^{3} \tan \left (f x + e\right )^{3} + 6 \, b^{3} c^{2} f^{3} \tan \left (f x + e\right )^{2} + 36 \, a b^{2} c d f^{3} \tan \left (f x + e\right )^{2} + 18 \, a^{2} b d^{2} f^{3} \tan \left (f x + e\right )^{2} - 6 \, b^{3} d^{2} f^{3} \tan \left (f x + e\right )^{2} + 36 \, a b^{2} c^{2} f^{3} \tan \left (f x + e\right ) + 72 \, a^{2} b c d f^{3} \tan \left (f x + e\right ) - 24 \, b^{3} c d f^{3} \tan \left (f x + e\right ) + 12 \, a^{3} d^{2} f^{3} \tan \left (f x + e\right ) - 36 \, a b^{2} d^{2} f^{3} \tan \left (f x + e\right )}{12 \, f^{4}} \] Input:
integrate((a+b*tan(f*x+e))^3*(c+d*tan(f*x+e))^2,x, algorithm="giac")
Output:
(a^3*c^2 - 3*a*b^2*c^2 - 6*a^2*b*c*d + 2*b^3*c*d - a^3*d^2 + 3*a*b^2*d^2)* (f*x + e)/f + 1/2*(3*a^2*b*c^2 - b^3*c^2 + 2*a^3*c*d - 6*a*b^2*c*d - 3*a^2 *b*d^2 + b^3*d^2)*log(tan(f*x + e)^2 + 1)/f + 1/12*(3*b^3*d^2*f^3*tan(f*x + e)^4 + 8*b^3*c*d*f^3*tan(f*x + e)^3 + 12*a*b^2*d^2*f^3*tan(f*x + e)^3 + 6*b^3*c^2*f^3*tan(f*x + e)^2 + 36*a*b^2*c*d*f^3*tan(f*x + e)^2 + 18*a^2*b* d^2*f^3*tan(f*x + e)^2 - 6*b^3*d^2*f^3*tan(f*x + e)^2 + 36*a*b^2*c^2*f^3*t an(f*x + e) + 72*a^2*b*c*d*f^3*tan(f*x + e) - 24*b^3*c*d*f^3*tan(f*x + e) + 12*a^3*d^2*f^3*tan(f*x + e) - 36*a*b^2*d^2*f^3*tan(f*x + e))/f^4
Time = 2.35 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.20 \[ \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^2 \, dx=x\,\left (a^3\,c^2-a^3\,d^2-6\,a^2\,b\,c\,d-3\,a\,b^2\,c^2+3\,a\,b^2\,d^2+2\,b^3\,c\,d\right )+\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (a^3\,d^2-b^2\,d\,\left (3\,a\,d+2\,b\,c\right )+3\,a\,b^2\,c^2+6\,a^2\,b\,c\,d\right )}{f}-\frac {\ln \left ({\mathrm {tan}\left (e+f\,x\right )}^2+1\right )\,\left (-a^3\,c\,d-\frac {3\,a^2\,b\,c^2}{2}+\frac {3\,a^2\,b\,d^2}{2}+3\,a\,b^2\,c\,d+\frac {b^3\,c^2}{2}-\frac {b^3\,d^2}{2}\right )}{f}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (\frac {3\,a^2\,b\,d^2}{2}+3\,a\,b^2\,c\,d+\frac {b^3\,c^2}{2}-\frac {b^3\,d^2}{2}\right )}{f}+\frac {b^3\,d^2\,{\mathrm {tan}\left (e+f\,x\right )}^4}{4\,f}+\frac {b^2\,d\,{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (3\,a\,d+2\,b\,c\right )}{3\,f} \] Input:
int((a + b*tan(e + f*x))^3*(c + d*tan(e + f*x))^2,x)
Output:
x*(a^3*c^2 - a^3*d^2 - 3*a*b^2*c^2 + 3*a*b^2*d^2 + 2*b^3*c*d - 6*a^2*b*c*d ) + (tan(e + f*x)*(a^3*d^2 - b^2*d*(3*a*d + 2*b*c) + 3*a*b^2*c^2 + 6*a^2*b *c*d))/f - (log(tan(e + f*x)^2 + 1)*((b^3*c^2)/2 - (b^3*d^2)/2 - (3*a^2*b* c^2)/2 + (3*a^2*b*d^2)/2 - a^3*c*d + 3*a*b^2*c*d))/f + (tan(e + f*x)^2*((b ^3*c^2)/2 - (b^3*d^2)/2 + (3*a^2*b*d^2)/2 + 3*a*b^2*c*d))/f + (b^3*d^2*tan (e + f*x)^4)/(4*f) + (b^2*d*tan(e + f*x)^3*(3*a*d + 2*b*c))/(3*f)
Time = 0.23 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.70 \[ \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^2 \, dx=\frac {12 \,\mathrm {log}\left (\tan \left (f x +e \right )^{2}+1\right ) a^{3} c d +18 \,\mathrm {log}\left (\tan \left (f x +e \right )^{2}+1\right ) a^{2} b \,c^{2}-18 \,\mathrm {log}\left (\tan \left (f x +e \right )^{2}+1\right ) a^{2} b \,d^{2}-36 \,\mathrm {log}\left (\tan \left (f x +e \right )^{2}+1\right ) a \,b^{2} c d -6 \,\mathrm {log}\left (\tan \left (f x +e \right )^{2}+1\right ) b^{3} c^{2}+6 \,\mathrm {log}\left (\tan \left (f x +e \right )^{2}+1\right ) b^{3} d^{2}+3 \tan \left (f x +e \right )^{4} b^{3} d^{2}+12 \tan \left (f x +e \right )^{3} a \,b^{2} d^{2}+8 \tan \left (f x +e \right )^{3} b^{3} c d +18 \tan \left (f x +e \right )^{2} a^{2} b \,d^{2}+36 \tan \left (f x +e \right )^{2} a \,b^{2} c d +6 \tan \left (f x +e \right )^{2} b^{3} c^{2}-6 \tan \left (f x +e \right )^{2} b^{3} d^{2}+12 \tan \left (f x +e \right ) a^{3} d^{2}+72 \tan \left (f x +e \right ) a^{2} b c d +36 \tan \left (f x +e \right ) a \,b^{2} c^{2}-36 \tan \left (f x +e \right ) a \,b^{2} d^{2}-24 \tan \left (f x +e \right ) b^{3} c d +12 a^{3} c^{2} f x -12 a^{3} d^{2} f x -72 a^{2} b c d f x -36 a \,b^{2} c^{2} f x +36 a \,b^{2} d^{2} f x +24 b^{3} c d f x}{12 f} \] Input:
int((a+b*tan(f*x+e))^3*(c+d*tan(f*x+e))^2,x)
Output:
(12*log(tan(e + f*x)**2 + 1)*a**3*c*d + 18*log(tan(e + f*x)**2 + 1)*a**2*b *c**2 - 18*log(tan(e + f*x)**2 + 1)*a**2*b*d**2 - 36*log(tan(e + f*x)**2 + 1)*a*b**2*c*d - 6*log(tan(e + f*x)**2 + 1)*b**3*c**2 + 6*log(tan(e + f*x) **2 + 1)*b**3*d**2 + 3*tan(e + f*x)**4*b**3*d**2 + 12*tan(e + f*x)**3*a*b* *2*d**2 + 8*tan(e + f*x)**3*b**3*c*d + 18*tan(e + f*x)**2*a**2*b*d**2 + 36 *tan(e + f*x)**2*a*b**2*c*d + 6*tan(e + f*x)**2*b**3*c**2 - 6*tan(e + f*x) **2*b**3*d**2 + 12*tan(e + f*x)*a**3*d**2 + 72*tan(e + f*x)*a**2*b*c*d + 3 6*tan(e + f*x)*a*b**2*c**2 - 36*tan(e + f*x)*a*b**2*d**2 - 24*tan(e + f*x) *b**3*c*d + 12*a**3*c**2*f*x - 12*a**3*d**2*f*x - 72*a**2*b*c*d*f*x - 36*a *b**2*c**2*f*x + 36*a*b**2*d**2*f*x + 24*b**3*c*d*f*x)/(12*f)