Integrand size = 45, antiderivative size = 560 \[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\frac {\left (b c (2+m) \left (b^2 d (B c+(A-C) d) (3+m) (4+m)-2 (b c-a d) (3 a C d-b (3 c C+B d (4+m)))\right )+d \left (b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) (2+m) (3+m) (4+m)-a \left (b^2 d (B c+(A-C) d) (3+m) (4+m)-2 (b c-a d) (3 a C d-b (3 c C+B d (4+m)))\right )\right )\right ) (a+b \tan (e+f x))^{1+m}}{b^4 f (1+m) (2+m) (3+m) (4+m)}+\frac {(A-i B-C) (c-i d)^3 \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a-i b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (i a+b) f (1+m)}-\frac {(A+i B-C) (c+i d)^3 \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a+i b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (i a-b) f (1+m)}+\frac {d \left (b^2 d (B c+(A-C) d) (3+m) (4+m)-2 (b c-a d) (3 a C d-b (3 c C+B d (4+m)))\right ) \tan (e+f x) (a+b \tan (e+f x))^{1+m}}{b^3 f (2+m) (3+m) (4+m)}-\frac {(3 a C d-b (3 c C+B d (4+m))) (a+b \tan (e+f x))^{1+m} (c+d \tan (e+f x))^2}{b^2 f (3+m) (4+m)}+\frac {C (a+b \tan (e+f x))^{1+m} (c+d \tan (e+f x))^3}{b f (4+m)} \]
(b*c*(2+m)*(b^2*d*(B*c+(A-C)*d)*(3+m)*(4+m)-2*(-a*d+b*c)*(3*C*a*d-b*(3*C*c +B*d*(4+m))))+d*(b^3*(2*c*(A-C)*d+B*(c^2-d^2))*(2+m)*(3+m)*(4+m)-a*(b^2*d* (B*c+(A-C)*d)*(3+m)*(4+m)-2*(-a*d+b*c)*(3*C*a*d-b*(3*C*c+B*d*(4+m))))))*(a +b*tan(f*x+e))^(1+m)/b^4/f/(1+m)/(2+m)/(3+m)/(4+m)+1/2*(A-I*B-C)*(c-I*d)^3 *hypergeom([1, 1+m],[2+m],(a+b*tan(f*x+e))/(a-I*b))*(a+b*tan(f*x+e))^(1+m) /(I*a+b)/f/(1+m)-1/2*(A+I*B-C)*(c+I*d)^3*hypergeom([1, 1+m],[2+m],(a+b*tan (f*x+e))/(a+I*b))*(a+b*tan(f*x+e))^(1+m)/(I*a-b)/f/(1+m)+d*(b^2*d*(B*c+(A- C)*d)*(3+m)*(4+m)-2*(-a*d+b*c)*(3*C*a*d-b*(3*C*c+B*d*(4+m))))*tan(f*x+e)*( a+b*tan(f*x+e))^(1+m)/b^3/f/(2+m)/(3+m)/(4+m)-(3*C*a*d-b*(3*C*c+B*d*(4+m)) )*(a+b*tan(f*x+e))^(1+m)*(c+d*tan(f*x+e))^2/b^2/f/(3+m)/(4+m)+C*(a+b*tan(f *x+e))^(1+m)*(c+d*tan(f*x+e))^3/b/f/(4+m)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1390\) vs. \(2(560)=1120\).
Time = 6.54 (sec) , antiderivative size = 1390, normalized size of antiderivative = 2.48 \[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\frac {C (a+b \tan (e+f x))^{1+m} (c+d \tan (e+f x))^3}{b f (4+m)}+\frac {\frac {(3 b c C-3 a C d+b B d (4+m)) (a+b \tan (e+f x))^{1+m} (c+d \tan (e+f x))^2}{b f (3+m)}+\frac {\frac {d \left (b^2 d (B c+(A-C) d) (3+m) (4+m)+2 (b c-a d) (3 b c C-3 a C d+b B d (4+m))\right ) \tan (e+f x) (a+b \tan (e+f x))^{1+m}}{b f (2+m)}-\frac {\frac {\left (-b c (2+m) \left (b^2 d (B c+(A-C) d) (3+m) (4+m)+2 (b c-a d) (3 b c C-3 a C d+b B d (4+m))\right )+d \left (-b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) (2+m) (3+m) (4+m)+a \left (b^2 d (B c+(A-C) d) (3+m) (4+m)+2 (b c-a d) (3 b c C-3 a C d+b B d (4+m))\right )\right )\right ) (a+b \tan (e+f x))^{1+m}}{b f (1+m)}+\frac {i \left (a d \left (b^2 d (B c+(A-C) d) (3+m) (4+m)+2 (b c-a d) (3 b c C-3 a C d+b B d (4+m))\right )+b c (2+m) \left (b^2 d (B c+(A-C) d) (3+m) (4+m)+2 (b c-a d) (3 b c C-3 a C d+b B d (4+m))\right )-b c (2+m) (-((2 a d+b c (1+m)) (3 b c C-3 a C d+b B d (4+m)))+b c (3+m) (A b c (4+m)-C (3 a d+b c (1+m))))-d \left (-b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) (2+m) (3+m) (4+m)+a \left (b^2 d (B c+(A-C) d) (3+m) (4+m)+2 (b c-a d) (3 b c C-3 a C d+b B d (4+m))\right )\right )-i b (2+m) \left (b^2 c \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) (3+m) (4+m)-d \left (b^2 d (B c+(A-C) d) (3+m) (4+m)+2 (b c-a d) (3 b c C-3 a C d+b B d (4+m))\right )+d (-((2 a d+b c (1+m)) (3 b c C-3 a C d+b B d (4+m)))+b c (3+m) (A b c (4+m)-C (3 a d+b c (1+m))))\right )\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {-i a-i b \tan (e+f x)}{-i a+b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (a+i b) f (1+m)}-\frac {i \left (a d \left (b^2 d (B c+(A-C) d) (3+m) (4+m)+2 (b c-a d) (3 b c C-3 a C d+b B d (4+m))\right )+b c (2+m) \left (b^2 d (B c+(A-C) d) (3+m) (4+m)+2 (b c-a d) (3 b c C-3 a C d+b B d (4+m))\right )-b c (2+m) (-((2 a d+b c (1+m)) (3 b c C-3 a C d+b B d (4+m)))+b c (3+m) (A b c (4+m)-C (3 a d+b c (1+m))))-d \left (-b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) (2+m) (3+m) (4+m)+a \left (b^2 d (B c+(A-C) d) (3+m) (4+m)+2 (b c-a d) (3 b c C-3 a C d+b B d (4+m))\right )\right )+i b (2+m) \left (b^2 c \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) (3+m) (4+m)-d \left (b^2 d (B c+(A-C) d) (3+m) (4+m)+2 (b c-a d) (3 b c C-3 a C d+b B d (4+m))\right )+d (-((2 a d+b c (1+m)) (3 b c C-3 a C d+b B d (4+m)))+b c (3+m) (A b c (4+m)-C (3 a d+b c (1+m))))\right )\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {i a+i b \tan (e+f x)}{-i a-b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (a-i b) f (1+m)}}{b (2+m)}}{b (3+m)}}{b (4+m)} \]
Integrate[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^3*(A + B*Tan[e + f*x ] + C*Tan[e + f*x]^2),x]
(C*(a + b*Tan[e + f*x])^(1 + m)*(c + d*Tan[e + f*x])^3)/(b*f*(4 + m)) + (( (3*b*c*C - 3*a*C*d + b*B*d*(4 + m))*(a + b*Tan[e + f*x])^(1 + m)*(c + d*Ta n[e + f*x])^2)/(b*f*(3 + m)) + ((d*(b^2*d*(B*c + (A - C)*d)*(3 + m)*(4 + m ) + 2*(b*c - a*d)*(3*b*c*C - 3*a*C*d + b*B*d*(4 + m)))*Tan[e + f*x]*(a + b *Tan[e + f*x])^(1 + m))/(b*f*(2 + m)) - (((-(b*c*(2 + m)*(b^2*d*(B*c + (A - C)*d)*(3 + m)*(4 + m) + 2*(b*c - a*d)*(3*b*c*C - 3*a*C*d + b*B*d*(4 + m) ))) + d*(-(b^3*(2*c*(A - C)*d + B*(c^2 - d^2))*(2 + m)*(3 + m)*(4 + m)) + a*(b^2*d*(B*c + (A - C)*d)*(3 + m)*(4 + m) + 2*(b*c - a*d)*(3*b*c*C - 3*a* C*d + b*B*d*(4 + m)))))*(a + b*Tan[e + f*x])^(1 + m))/(b*f*(1 + m)) + ((I/ 2)*(a*d*(b^2*d*(B*c + (A - C)*d)*(3 + m)*(4 + m) + 2*(b*c - a*d)*(3*b*c*C - 3*a*C*d + b*B*d*(4 + m))) + b*c*(2 + m)*(b^2*d*(B*c + (A - C)*d)*(3 + m) *(4 + m) + 2*(b*c - a*d)*(3*b*c*C - 3*a*C*d + b*B*d*(4 + m))) - b*c*(2 + m )*(-((2*a*d + b*c*(1 + m))*(3*b*c*C - 3*a*C*d + b*B*d*(4 + m))) + b*c*(3 + m)*(A*b*c*(4 + m) - C*(3*a*d + b*c*(1 + m)))) - d*(-(b^3*(2*c*(A - C)*d + B*(c^2 - d^2))*(2 + m)*(3 + m)*(4 + m)) + a*(b^2*d*(B*c + (A - C)*d)*(3 + m)*(4 + m) + 2*(b*c - a*d)*(3*b*c*C - 3*a*C*d + b*B*d*(4 + m)))) - I*b*(2 + m)*(b^2*c*(2*c*(A - C)*d + B*(c^2 - d^2))*(3 + m)*(4 + m) - d*(b^2*d*(B *c + (A - C)*d)*(3 + m)*(4 + m) + 2*(b*c - a*d)*(3*b*c*C - 3*a*C*d + b*B*d *(4 + m))) + d*(-((2*a*d + b*c*(1 + m))*(3*b*c*C - 3*a*C*d + b*B*d*(4 + m) )) + b*c*(3 + m)*(A*b*c*(4 + m) - C*(3*a*d + b*c*(1 + m))))))*Hypergeom...
Time = 3.41 (sec) , antiderivative size = 580, normalized size of antiderivative = 1.04, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.356, Rules used = {3042, 4130, 3042, 4130, 25, 3042, 4120, 25, 3042, 4113, 3042, 4022, 3042, 4020, 25, 78}
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.
| \(\displaystyle \int (c+d \tan (e+f x))^3 (a+b \tan (e+f x))^m \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (c+d \tan (e+f x))^3 (a+b \tan (e+f x))^m \left (A+B \tan (e+f x)+C \tan (e+f x)^2\right )dx\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {\int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^2 \left ((3 b c C-3 a d C+b B d (m+4)) \tan ^2(e+f x)+b (B c+(A-C) d) (m+4) \tan (e+f x)+A b c (m+4)-C (3 a d+b c (m+1))\right )dx}{b (m+4)}+\frac {C (c+d \tan (e+f x))^3 (a+b \tan (e+f x))^{m+1}}{b f (m+4)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^2 \left ((3 b c C-3 a d C+b B d (m+4)) \tan (e+f x)^2+b (B c+(A-C) d) (m+4) \tan (e+f x)+A b c (m+4)-C (3 a d+b c (m+1))\right )dx}{b (m+4)}+\frac {C (c+d \tan (e+f x))^3 (a+b \tan (e+f x))^{m+1}}{b f (m+4)}\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {\frac {\int -(a+b \tan (e+f x))^m (c+d \tan (e+f x)) \left (-\left (\left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) (m+3) (m+4) \tan (e+f x) b^2\right )-c (m+3) (A b c (m+4)-C (3 a d+b c (m+1))) b-\left (d (B c+(A-C) d) (m+3) (m+4) b^2+2 (b c-a d) (3 b c C-3 a d C+b B d (m+4))\right ) \tan ^2(e+f x)+(2 a d+b c (m+1)) (3 b c C-3 a d C+b B d (m+4))\right )dx}{b (m+3)}+\frac {(c+d \tan (e+f x))^2 (-3 a C d+b B d (m+4)+3 b c C) (a+b \tan (e+f x))^{m+1}}{b f (m+3)}}{b (m+4)}+\frac {C (c+d \tan (e+f x))^3 (a+b \tan (e+f x))^{m+1}}{b f (m+4)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {(c+d \tan (e+f x))^2 (-3 a C d+b B d (m+4)+3 b c C) (a+b \tan (e+f x))^{m+1}}{b f (m+3)}-\frac {\int (a+b \tan (e+f x))^m (c+d \tan (e+f x)) \left (-\left (\left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) (m+3) (m+4) \tan (e+f x) b^2\right )-c (m+3) (A b c (m+4)-C (3 a d+b c (m+1))) b-\left (d (B c+(A-C) d) (m+3) (m+4) b^2+2 (b c-a d) (3 b c C-3 a d C+b B d (m+4))\right ) \tan ^2(e+f x)+(2 a d+b c (m+1)) (3 b c C-3 a d C+b B d (m+4))\right )dx}{b (m+3)}}{b (m+4)}+\frac {C (c+d \tan (e+f x))^3 (a+b \tan (e+f x))^{m+1}}{b f (m+4)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {(c+d \tan (e+f x))^2 (-3 a C d+b B d (m+4)+3 b c C) (a+b \tan (e+f x))^{m+1}}{b f (m+3)}-\frac {\int (a+b \tan (e+f x))^m (c+d \tan (e+f x)) \left (-\left (\left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) (m+3) (m+4) \tan (e+f x) b^2\right )-c (m+3) (A b c (m+4)-C (3 a d+b c (m+1))) b-\left (d (B c+(A-C) d) (m+3) (m+4) b^2+2 (b c-a d) (3 b c C-3 a d C+b B d (m+4))\right ) \tan (e+f x)^2+(2 a d+b c (m+1)) (3 b c C-3 a d C+b B d (m+4))\right )dx}{b (m+3)}}{b (m+4)}+\frac {C (c+d \tan (e+f x))^3 (a+b \tan (e+f x))^{m+1}}{b f (m+4)}\) |
| \(\Big \downarrow \) 4120 |
| \(\displaystyle \frac {\frac {(c+d \tan (e+f x))^2 (-3 a C d+b B d (m+4)+3 b c C) (a+b \tan (e+f x))^{m+1}}{b f (m+3)}-\frac {-\frac {\int -(a+b \tan (e+f x))^m \left (-\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) (m+2) \left (m^2+7 m+12\right ) \tan (e+f x) b^3\right )+c (m+2) ((2 a d+b c (m+1)) (3 b c C-3 a d C+b B d (m+4))-b c (m+3) (A b c (m+4)-C (3 a d+b c (m+1)))) b-\left (b c (m+2) \left (d (B c+(A-C) d) (m+3) (m+4) b^2+2 (b c-a d) (3 b c C-3 a d C+b B d (m+4))\right )+d \left (b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) (m+2) (m+3) (m+4)-a \left (d (B c+(A-C) d) (m+3) (m+4) b^2+2 (b c-a d) (3 b c C-3 a d C+b B d (m+4))\right )\right )\right ) \tan ^2(e+f x)+a d \left (d (B c+(A-C) d) (m+3) (m+4) b^2+2 (b c-a d) (3 b c C-3 a d C+b B d (m+4))\right )\right )dx}{b (m+2)}-\frac {d \tan (e+f x) (a+b \tan (e+f x))^{m+1} \left (2 (b c-a d) (-3 a C d+b B d (m+4)+3 b c C)+b^2 d (m+3) (m+4) (d (A-C)+B c)\right )}{b f (m+2)}}{b (m+3)}}{b (m+4)}+\frac {C (c+d \tan (e+f x))^3 (a+b \tan (e+f x))^{m+1}}{b f (m+4)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {(c+d \tan (e+f x))^2 (-3 a C d+b B d (m+4)+3 b c C) (a+b \tan (e+f x))^{m+1}}{b f (m+3)}-\frac {\frac {\int (a+b \tan (e+f x))^m \left (-\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) (m+2) \left (m^2+7 m+12\right ) \tan (e+f x) b^3\right )+c (m+2) ((2 a d+b c (m+1)) (3 b c C-3 a d C+b B d (m+4))-b c (m+3) (A b c (m+4)-C (3 a d+b c (m+1)))) b-\left (b c (m+2) \left (d (B c+(A-C) d) (m+3) (m+4) b^2+2 (b c-a d) (3 b c C-3 a d C+b B d (m+4))\right )+d \left (b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) (m+2) (m+3) (m+4)-a \left (d (B c+(A-C) d) (m+3) (m+4) b^2+2 (b c-a d) (3 b c C-3 a d C+b B d (m+4))\right )\right )\right ) \tan ^2(e+f x)+a d \left (d (B c+(A-C) d) (m+3) (m+4) b^2+2 (b c-a d) (3 b c C-3 a d C+b B d (m+4))\right )\right )dx}{b (m+2)}-\frac {d \tan (e+f x) (a+b \tan (e+f x))^{m+1} \left (2 (b c-a d) (-3 a C d+b B d (m+4)+3 b c C)+b^2 d (m+3) (m+4) (d (A-C)+B c)\right )}{b f (m+2)}}{b (m+3)}}{b (m+4)}+\frac {C (c+d \tan (e+f x))^3 (a+b \tan (e+f x))^{m+1}}{b f (m+4)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {(c+d \tan (e+f x))^2 (-3 a C d+b B d (m+4)+3 b c C) (a+b \tan (e+f x))^{m+1}}{b f (m+3)}-\frac {\frac {\int (a+b \tan (e+f x))^m \left (-\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) (m+2) \left (m^2+7 m+12\right ) \tan (e+f x) b^3\right )+c (m+2) ((2 a d+b c (m+1)) (3 b c C-3 a d C+b B d (m+4))-b c (m+3) (A b c (m+4)-C (3 a d+b c (m+1)))) b-\left (b c (m+2) \left (d (B c+(A-C) d) (m+3) (m+4) b^2+2 (b c-a d) (3 b c C-3 a d C+b B d (m+4))\right )+d \left (b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) (m+2) (m+3) (m+4)-a \left (d (B c+(A-C) d) (m+3) (m+4) b^2+2 (b c-a d) (3 b c C-3 a d C+b B d (m+4))\right )\right )\right ) \tan (e+f x)^2+a d \left (d (B c+(A-C) d) (m+3) (m+4) b^2+2 (b c-a d) (3 b c C-3 a d C+b B d (m+4))\right )\right )dx}{b (m+2)}-\frac {d \tan (e+f x) (a+b \tan (e+f x))^{m+1} \left (2 (b c-a d) (-3 a C d+b B d (m+4)+3 b c C)+b^2 d (m+3) (m+4) (d (A-C)+B c)\right )}{b f (m+2)}}{b (m+3)}}{b (m+4)}+\frac {C (c+d \tan (e+f x))^3 (a+b \tan (e+f x))^{m+1}}{b f (m+4)}\) |
| \(\Big \downarrow \) 4113 |
| \(\displaystyle \frac {\frac {(c+d \tan (e+f x))^2 (-3 a C d+b B d (m+4)+3 b c C) (a+b \tan (e+f x))^{m+1}}{b f (m+3)}-\frac {\frac {\int (a+b \tan (e+f x))^m \left (-\left (\left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) (m+2) (m+3) (m+4) b^3\right )-\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) (m+2) \left (m^2+7 m+12\right ) \tan (e+f x) b^3\right )dx-\frac {(a+b \tan (e+f x))^{m+1} \left (b c (m+2) \left (2 (b c-a d) (-3 a C d+b B d (m+4)+3 b c C)+b^2 d (m+3) (m+4) (d (A-C)+B c)\right )+d \left (b^3 (m+2) (m+3) (m+4) \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )-a \left (2 (b c-a d) (-3 a C d+b B d (m+4)+3 b c C)+b^2 d (m+3) (m+4) (d (A-C)+B c)\right )\right )\right )}{b f (m+1)}}{b (m+2)}-\frac {d \tan (e+f x) (a+b \tan (e+f x))^{m+1} \left (2 (b c-a d) (-3 a C d+b B d (m+4)+3 b c C)+b^2 d (m+3) (m+4) (d (A-C)+B c)\right )}{b f (m+2)}}{b (m+3)}}{b (m+4)}+\frac {C (c+d \tan (e+f x))^3 (a+b \tan (e+f x))^{m+1}}{b f (m+4)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {(c+d \tan (e+f x))^2 (-3 a C d+b B d (m+4)+3 b c C) (a+b \tan (e+f x))^{m+1}}{b f (m+3)}-\frac {\frac {\int (a+b \tan (e+f x))^m \left (-\left (\left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) (m+2) (m+3) (m+4) b^3\right )-\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) (m+2) \left (m^2+7 m+12\right ) \tan (e+f x) b^3\right )dx-\frac {(a+b \tan (e+f x))^{m+1} \left (b c (m+2) \left (2 (b c-a d) (-3 a C d+b B d (m+4)+3 b c C)+b^2 d (m+3) (m+4) (d (A-C)+B c)\right )+d \left (b^3 (m+2) (m+3) (m+4) \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )-a \left (2 (b c-a d) (-3 a C d+b B d (m+4)+3 b c C)+b^2 d (m+3) (m+4) (d (A-C)+B c)\right )\right )\right )}{b f (m+1)}}{b (m+2)}-\frac {d \tan (e+f x) (a+b \tan (e+f x))^{m+1} \left (2 (b c-a d) (-3 a C d+b B d (m+4)+3 b c C)+b^2 d (m+3) (m+4) (d (A-C)+B c)\right )}{b f (m+2)}}{b (m+3)}}{b (m+4)}+\frac {C (c+d \tan (e+f x))^3 (a+b \tan (e+f x))^{m+1}}{b f (m+4)}\) |
| \(\Big \downarrow \) 4022 |
| \(\displaystyle \frac {C (c+d \tan (e+f x))^3 (a+b \tan (e+f x))^{m+1}}{b f (m+4)}+\frac {\frac {(c+d \tan (e+f x))^2 (-3 a C d+b B d (m+4)+3 b c C) (a+b \tan (e+f x))^{m+1}}{b f (m+3)}-\frac {-\frac {d \tan (e+f x) (a+b \tan (e+f x))^{m+1} \left (2 (b c-a d) (-3 a C d+b B d (m+4)+3 b c C)+b^2 d (m+3) (m+4) (d (A-C)+B c)\right )}{b f (m+2)}+\frac {-\frac {1}{2} b^3 (m+2) (m+3) (m+4) (c+i d)^3 (A+i B-C) \int (1-i \tan (e+f x)) (a+b \tan (e+f x))^mdx-\frac {1}{2} b^3 (m+2) (m+3) (m+4) (c-i d)^3 (A-i B-C) \int (i \tan (e+f x)+1) (a+b \tan (e+f x))^mdx-\frac {(a+b \tan (e+f x))^{m+1} \left (b c (m+2) \left (2 (b c-a d) (-3 a C d+b B d (m+4)+3 b c C)+b^2 d (m+3) (m+4) (d (A-C)+B c)\right )+d \left (b^3 (m+2) (m+3) (m+4) \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )-a \left (2 (b c-a d) (-3 a C d+b B d (m+4)+3 b c C)+b^2 d (m+3) (m+4) (d (A-C)+B c)\right )\right )\right )}{b f (m+1)}}{b (m+2)}}{b (m+3)}}{b (m+4)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {C (c+d \tan (e+f x))^3 (a+b \tan (e+f x))^{m+1}}{b f (m+4)}+\frac {\frac {(c+d \tan (e+f x))^2 (-3 a C d+b B d (m+4)+3 b c C) (a+b \tan (e+f x))^{m+1}}{b f (m+3)}-\frac {-\frac {d \tan (e+f x) (a+b \tan (e+f x))^{m+1} \left (2 (b c-a d) (-3 a C d+b B d (m+4)+3 b c C)+b^2 d (m+3) (m+4) (d (A-C)+B c)\right )}{b f (m+2)}+\frac {-\frac {1}{2} b^3 (m+2) (m+3) (m+4) (c+i d)^3 (A+i B-C) \int (1-i \tan (e+f x)) (a+b \tan (e+f x))^mdx-\frac {1}{2} b^3 (m+2) (m+3) (m+4) (c-i d)^3 (A-i B-C) \int (i \tan (e+f x)+1) (a+b \tan (e+f x))^mdx-\frac {(a+b \tan (e+f x))^{m+1} \left (b c (m+2) \left (2 (b c-a d) (-3 a C d+b B d (m+4)+3 b c C)+b^2 d (m+3) (m+4) (d (A-C)+B c)\right )+d \left (b^3 (m+2) (m+3) (m+4) \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )-a \left (2 (b c-a d) (-3 a C d+b B d (m+4)+3 b c C)+b^2 d (m+3) (m+4) (d (A-C)+B c)\right )\right )\right )}{b f (m+1)}}{b (m+2)}}{b (m+3)}}{b (m+4)}\) |
| \(\Big \downarrow \) 4020 |
| \(\displaystyle \frac {C (c+d \tan (e+f x))^3 (a+b \tan (e+f x))^{m+1}}{b f (m+4)}+\frac {\frac {(c+d \tan (e+f x))^2 (-3 a C d+b B d (m+4)+3 b c C) (a+b \tan (e+f x))^{m+1}}{b f (m+3)}-\frac {-\frac {d \tan (e+f x) (a+b \tan (e+f x))^{m+1} \left (2 (b c-a d) (-3 a C d+b B d (m+4)+3 b c C)+b^2 d (m+3) (m+4) (d (A-C)+B c)\right )}{b f (m+2)}+\frac {-\frac {i b^3 (m+2) (m+3) (m+4) (c-i d)^3 (A-i B-C) \int -\frac {(a+b \tan (e+f x))^m}{1-i \tan (e+f x)}d(i \tan (e+f x))}{2 f}+\frac {i b^3 (m+2) (m+3) (m+4) (c+i d)^3 (A+i B-C) \int -\frac {(a+b \tan (e+f x))^m}{i \tan (e+f x)+1}d(-i \tan (e+f x))}{2 f}-\frac {(a+b \tan (e+f x))^{m+1} \left (b c (m+2) \left (2 (b c-a d) (-3 a C d+b B d (m+4)+3 b c C)+b^2 d (m+3) (m+4) (d (A-C)+B c)\right )+d \left (b^3 (m+2) (m+3) (m+4) \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )-a \left (2 (b c-a d) (-3 a C d+b B d (m+4)+3 b c C)+b^2 d (m+3) (m+4) (d (A-C)+B c)\right )\right )\right )}{b f (m+1)}}{b (m+2)}}{b (m+3)}}{b (m+4)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {C (c+d \tan (e+f x))^3 (a+b \tan (e+f x))^{m+1}}{b f (m+4)}+\frac {\frac {(c+d \tan (e+f x))^2 (-3 a C d+b B d (m+4)+3 b c C) (a+b \tan (e+f x))^{m+1}}{b f (m+3)}-\frac {-\frac {d \tan (e+f x) (a+b \tan (e+f x))^{m+1} \left (2 (b c-a d) (-3 a C d+b B d (m+4)+3 b c C)+b^2 d (m+3) (m+4) (d (A-C)+B c)\right )}{b f (m+2)}+\frac {\frac {i b^3 (m+2) (m+3) (m+4) (c-i d)^3 (A-i B-C) \int \frac {(a+b \tan (e+f x))^m}{1-i \tan (e+f x)}d(i \tan (e+f x))}{2 f}-\frac {i b^3 (m+2) (m+3) (m+4) (c+i d)^3 (A+i B-C) \int \frac {(a+b \tan (e+f x))^m}{i \tan (e+f x)+1}d(-i \tan (e+f x))}{2 f}-\frac {(a+b \tan (e+f x))^{m+1} \left (b c (m+2) \left (2 (b c-a d) (-3 a C d+b B d (m+4)+3 b c C)+b^2 d (m+3) (m+4) (d (A-C)+B c)\right )+d \left (b^3 (m+2) (m+3) (m+4) \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )-a \left (2 (b c-a d) (-3 a C d+b B d (m+4)+3 b c C)+b^2 d (m+3) (m+4) (d (A-C)+B c)\right )\right )\right )}{b f (m+1)}}{b (m+2)}}{b (m+3)}}{b (m+4)}\) |
| \(\Big \downarrow \) 78 |
| \(\displaystyle \frac {C (c+d \tan (e+f x))^3 (a+b \tan (e+f x))^{m+1}}{b f (m+4)}+\frac {\frac {(c+d \tan (e+f x))^2 (-3 a C d+b B d (m+4)+3 b c C) (a+b \tan (e+f x))^{m+1}}{b f (m+3)}-\frac {-\frac {d \tan (e+f x) (a+b \tan (e+f x))^{m+1} \left (2 (b c-a d) (-3 a C d+b B d (m+4)+3 b c C)+b^2 d (m+3) (m+4) (d (A-C)+B c)\right )}{b f (m+2)}+\frac {\frac {i b^3 (m+2) (m+3) (m+4) (c-i d)^3 (A-i B-C) (a+b \tan (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {a+b \tan (e+f x)}{a-i b}\right )}{2 f (m+1) (a-i b)}-\frac {i b^3 (m+2) (m+3) (m+4) (c+i d)^3 (A+i B-C) (a+b \tan (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {a+b \tan (e+f x)}{a+i b}\right )}{2 f (m+1) (a+i b)}-\frac {(a+b \tan (e+f x))^{m+1} \left (b c (m+2) \left (2 (b c-a d) (-3 a C d+b B d (m+4)+3 b c C)+b^2 d (m+3) (m+4) (d (A-C)+B c)\right )+d \left (b^3 (m+2) (m+3) (m+4) \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )-a \left (2 (b c-a d) (-3 a C d+b B d (m+4)+3 b c C)+b^2 d (m+3) (m+4) (d (A-C)+B c)\right )\right )\right )}{b f (m+1)}}{b (m+2)}}{b (m+3)}}{b (m+4)}\) |
(C*(a + b*Tan[e + f*x])^(1 + m)*(c + d*Tan[e + f*x])^3)/(b*f*(4 + m)) + (( (3*b*c*C - 3*a*C*d + b*B*d*(4 + m))*(a + b*Tan[e + f*x])^(1 + m)*(c + d*Ta n[e + f*x])^2)/(b*f*(3 + m)) - (-((d*(b^2*d*(B*c + (A - C)*d)*(3 + m)*(4 + m) + 2*(b*c - a*d)*(3*b*c*C - 3*a*C*d + b*B*d*(4 + m)))*Tan[e + f*x]*(a + b*Tan[e + f*x])^(1 + m))/(b*f*(2 + m))) + (-(((b*c*(2 + m)*(b^2*d*(B*c + (A - C)*d)*(3 + m)*(4 + m) + 2*(b*c - a*d)*(3*b*c*C - 3*a*C*d + b*B*d*(4 + m))) + d*(b^3*(2*c*(A - C)*d + B*(c^2 - d^2))*(2 + m)*(3 + m)*(4 + m) - a *(b^2*d*(B*c + (A - C)*d)*(3 + m)*(4 + m) + 2*(b*c - a*d)*(3*b*c*C - 3*a*C *d + b*B*d*(4 + m)))))*(a + b*Tan[e + f*x])^(1 + m))/(b*f*(1 + m))) + ((I/ 2)*b^3*(A - I*B - C)*(c - I*d)^3*(2 + m)*(3 + m)*(4 + m)*Hypergeometric2F1 [1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a - I*b)]*(a + b*Tan[e + f*x])^(1 + m))/((a - I*b)*f*(1 + m)) - ((I/2)*b^3*(A + I*B - C)*(c + I*d)^3*(2 + m) *(3 + m)*(4 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/( a + I*b)]*(a + b*Tan[e + f*x])^(1 + m))/((a + I*b)*f*(1 + m)))/(b*(2 + m)) )/(b*(3 + m)))/(b*(4 + m))
3.2.65.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b *c - a*d)^n*((a + b*x)^(m + 1)/(b^(n + 1)*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m}, x] && !IntegerQ[m] && IntegerQ[n]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[c*(d/f) Subst[Int[(a + (b/d)*x)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[ b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c + I*d)/2 Int[(a + b*Tan[e + f*x])^m*( 1 - I*Tan[e + f*x]), x], x] + Simp[(c - I*d)/2 Int[(a + b*Tan[e + f*x])^m *(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && !IntegerQ[m]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])^m*Si mp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && !LeQ[m, -1]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.) *(x_)])^(n_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[b*C*Tan[e + f*x]*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 2))), x] - Simp[1/(d*(n + 2)) Int[(c + d*Tan[e + f*x])^n*Si mp[b*c*C - a*A*d*(n + 2) - (A*b + a*B - b*C)*d*(n + 2)*Tan[e + f*x] - (a*C* d*(n + 2) - b*(c*C - B*d*(n + 2)))*Tan[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[c^2 + d^2, 0] && !LtQ[n, -1]
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[C*(a + b*Tan[e + f*x])^m*((c + d*Tan[ e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1)) Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C *(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f*x] - (C* m*(b*c - a*d) - b*B*d*(m + n + 1))*Tan[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] && GtQ[m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[ c, 0] && NeQ[a, 0])))
\[\int \left (a +b \tan \left (f x +e \right )\right )^{m} \left (c +d \tan \left (f x +e \right )\right )^{3} \left (A +B \tan \left (f x +e \right )+C \tan \left (f x +e \right )^{2}\right )d x\]
\[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\int { {\left (C \tan \left (f x + e\right )^{2} + B \tan \left (f x + e\right ) + A\right )} {\left (d \tan \left (f x + e\right ) + c\right )}^{3} {\left (b \tan \left (f x + e\right ) + a\right )}^{m} \,d x } \]
integrate((a+b*tan(f*x+e))^m*(c+d*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+ e)^2),x, algorithm="fricas")
integral((C*d^3*tan(f*x + e)^5 + (3*C*c*d^2 + B*d^3)*tan(f*x + e)^4 + A*c^ 3 + (3*C*c^2*d + 3*B*c*d^2 + A*d^3)*tan(f*x + e)^3 + (C*c^3 + 3*B*c^2*d + 3*A*c*d^2)*tan(f*x + e)^2 + (B*c^3 + 3*A*c^2*d)*tan(f*x + e))*(b*tan(f*x + e) + a)^m, x)
\[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\int \left (a + b \tan {\left (e + f x \right )}\right )^{m} \left (c + d \tan {\left (e + f x \right )}\right )^{3} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )\, dx \]
Integral((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**3*(A + B*tan(e + f* x) + C*tan(e + f*x)**2), x)
Timed out. \[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\text {Timed out} \]
integrate((a+b*tan(f*x+e))^m*(c+d*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+ e)^2),x, algorithm="maxima")
\[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\int { {\left (C \tan \left (f x + e\right )^{2} + B \tan \left (f x + e\right ) + A\right )} {\left (d \tan \left (f x + e\right ) + c\right )}^{3} {\left (b \tan \left (f x + e\right ) + a\right )}^{m} \,d x } \]
integrate((a+b*tan(f*x+e))^m*(c+d*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+ e)^2),x, algorithm="giac")
integrate((C*tan(f*x + e)^2 + B*tan(f*x + e) + A)*(d*tan(f*x + e) + c)^3*( b*tan(f*x + e) + a)^m, x)
Timed out. \[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\int {\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^m\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^3\,\left (C\,{\mathrm {tan}\left (e+f\,x\right )}^2+B\,\mathrm {tan}\left (e+f\,x\right )+A\right ) \,d x \]