Integrand size = 25, antiderivative size = 455 \[ \int \frac {(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^3} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a-i b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (i a+b) (c-i d)^3 f (1+m)}+\frac {\operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a+i b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (a+i b) (i c-d)^3 f (1+m)}+\frac {d^2 \left (2 a^2 d^2 \left (3 c^2-d^2\right )-4 a b c d \left (c^2 (3-m)-d^2 (1+m)\right )-b^2 \left (d^4 (1-m) m+2 c^2 d^2 \left (1+3 m-m^2\right )-c^4 \left (6-5 m+m^2\right )\right )\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {d (a+b \tan (e+f x))}{b c-a d}\right ) (a+b \tan (e+f x))^{1+m}}{2 (b c-a d)^3 \left (c^2+d^2\right )^3 f (1+m)}+\frac {d^2 (a+b \tan (e+f x))^{1+m}}{2 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {d^2 \left (4 a c d-b \left (d^2 (1-m)+c^2 (5-m)\right )\right ) (a+b \tan (e+f x))^{1+m}}{2 (b c-a d)^2 \left (c^2+d^2\right )^2 f (c+d \tan (e+f x))} \]
1/2*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)/(c-I*d)^3/f/(1+m)+1/2*hypergeom([1, 1+m],[2+m],(a+b*tan(f*x+e) )/(a+I*b))*(a+b*tan(f*x+e))^(1+m)/(a+I*b)/(I*c-d)^3/f/(1+m)+1/2*d^2*(2*a^2 *d^2*(3*c^2-d^2)-4*a*b*c*d*(c^2*(3-m)-d^2*(1+m))-b^2*(d^4*(1-m)*m+2*c^2*d^ 2*(-m^2+3*m+1)-c^4*(m^2-5*m+6)))*hypergeom([1, 1+m],[2+m],-d*(a+b*tan(f*x+ e))/(-a*d+b*c))*(a+b*tan(f*x+e))^(1+m)/(-a*d+b*c)^3/(c^2+d^2)^3/f/(1+m)+1/ 2*d^2*(a+b*tan(f*x+e))^(1+m)/(-a*d+b*c)/(c^2+d^2)/f/(c+d*tan(f*x+e))^2-1/2 *d^2*(4*a*c*d-b*(d^2*(1-m)+c^2*(5-m)))*(a+b*tan(f*x+e))^(1+m)/(-a*d+b*c)^2 /(c^2+d^2)^2/f/(c+d*tan(f*x+e))
Time = 6.31 (sec) , antiderivative size = 670, normalized size of antiderivative = 1.47 \[ \int \frac {(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^3} \, dx=-\frac {d^2 (a+b \tan (e+f x))^{1+m}}{2 (-b c+a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {-\frac {\left (d^2 \left (2 c (b c-a d)+b d^2 (1-m)\right )-c \left (-2 d^2 (b c-a d)-b c d^2 (1-m)\right )\right ) (a+b \tan (e+f x))^{1+m}}{(-b c+a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))}-\frac {-\frac {\left (4 c^2 d^2 (b c-a d)^2+b c^2 d^2 \left (4 a c d-b d^2 (1-m)-b c^2 (5-m)\right ) m+d^2 \left (-d^2 (2 a d-b c (3-m)) (a d-b c (1+m))-\left (2 b c^2-2 a c d+b d^2 (1-m)\right ) \left (a c d-b \left (c^2-d^2 m\right )\right )\right )\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {d (a+b \tan (e+f x))}{-b c+a d}\right ) (a+b \tan (e+f x))^{1+m}}{(-b c+a d) \left (c^2+d^2\right ) f (1+m)}+\frac {\frac {i \left (2 c (b c-a d)^2 \left (c^2-3 d^2\right )-2 i d (b c-a d)^2 \left (3 c^2-d^2\right )\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {-i a-i b \tan (e+f x)}{-i a+b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (a+i b) f (1+m)}-\frac {i \left (2 c (b c-a d)^2 \left (c^2-3 d^2\right )+2 i d (b c-a d)^2 \left (3 c^2-d^2\right )\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {i a+i b \tan (e+f x)}{-i a-b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (a-i b) f (1+m)}}{c^2+d^2}}{(-b c+a d) \left (c^2+d^2\right )}}{2 (-b c+a d) \left (c^2+d^2\right )} \]
-1/2*(d^2*(a + b*Tan[e + f*x])^(1 + m))/((-(b*c) + a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^2) - (-(((d^2*(2*c*(b*c - a*d) + b*d^2*(1 - m)) - c*(-2*d ^2*(b*c - a*d) - b*c*d^2*(1 - m)))*(a + b*Tan[e + f*x])^(1 + m))/((-(b*c) + a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x]))) - (-(((4*c^2*d^2*(b*c - a*d)^2 + b*c^2*d^2*(4*a*c*d - b*d^2*(1 - m) - b*c^2*(5 - m))*m + d^2*(-(d^2*(2*a *d - b*c*(3 - m))*(a*d - b*c*(1 + m))) - (2*b*c^2 - 2*a*c*d + b*d^2*(1 - m ))*(a*c*d - b*(c^2 - d^2*m))))*Hypergeometric2F1[1, 1 + m, 2 + m, (d*(a + b*Tan[e + f*x]))/(-(b*c) + a*d)]*(a + b*Tan[e + f*x])^(1 + m))/((-(b*c) + a*d)*(c^2 + d^2)*f*(1 + m))) + (((I/2)*(2*c*(b*c - a*d)^2*(c^2 - 3*d^2) - (2*I)*d*(b*c - a*d)^2*(3*c^2 - d^2))*Hypergeometric2F1[1, 1 + m, 2 + m, (( -I)*a - I*b*Tan[e + f*x])/((-I)*a + b)]*(a + b*Tan[e + f*x])^(1 + m))/((a + I*b)*f*(1 + m)) - ((I/2)*(2*c*(b*c - a*d)^2*(c^2 - 3*d^2) + (2*I)*d*(b*c - a*d)^2*(3*c^2 - d^2))*Hypergeometric2F1[1, 1 + m, 2 + m, -((I*a + I*b*T an[e + f*x])/((-I)*a - b))]*(a + b*Tan[e + f*x])^(1 + m))/((a - I*b)*f*(1 + m)))/(c^2 + d^2))/((-(b*c) + a*d)*(c^2 + d^2)))/(2*(-(b*c) + a*d)*(c^2 + d^2))
Time = 2.63 (sec) , antiderivative size = 529, normalized size of antiderivative = 1.16, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 4052, 3042, 4132, 3042, 4136, 27, 3042, 4022, 3042, 4020, 25, 78, 4117, 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.
| \(\displaystyle \int \frac {(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^3}dx\) |
| \(\Big \downarrow \) 4052 |
| \(\displaystyle \frac {\int \frac {(a+b \tan (e+f x))^m \left (b (1-m) \tan ^2(e+f x) d^2+b (1-m) d^2-2 (b c-a d) \tan (e+f x) d+2 c (b c-a d)\right )}{(c+d \tan (e+f x))^2}dx}{2 \left (c^2+d^2\right ) (b c-a d)}+\frac {d^2 (a+b \tan (e+f x))^{m+1}}{2 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(a+b \tan (e+f x))^m \left (b (1-m) \tan (e+f x)^2 d^2+b (1-m) d^2-2 (b c-a d) \tan (e+f x) d+2 c (b c-a d)\right )}{(c+d \tan (e+f x))^2}dx}{2 \left (c^2+d^2\right ) (b c-a d)}+\frac {d^2 (a+b \tan (e+f x))^{m+1}}{2 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^2}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \frac {\frac {\int \frac {(a+b \tan (e+f x))^m \left (\left (2 c^4-d^2 \left (-m^2+5 m+2\right ) c^2-d^4 (1-m) m\right ) b^2+d^2 \left (-b (5-m) c^2+4 a d c-b d^2 (1-m)\right ) m \tan ^2(e+f x) b-4 a c d \left (c^2-d^2 (m+1)\right ) b+2 a^2 d^2 \left (c^2-d^2\right )-4 c d (b c-a d)^2 \tan (e+f x)\right )}{c+d \tan (e+f x)}dx}{\left (c^2+d^2\right ) (b c-a d)}-\frac {d^2 \left (4 a c d-b c^2 (5-m)-b d^2 (1-m)\right ) (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}}{2 \left (c^2+d^2\right ) (b c-a d)}+\frac {d^2 (a+b \tan (e+f x))^{m+1}}{2 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {(a+b \tan (e+f x))^m \left (\left (2 c^4-d^2 \left (-m^2+5 m+2\right ) c^2-d^4 (1-m) m\right ) b^2+d^2 \left (-b (5-m) c^2+4 a d c-b d^2 (1-m)\right ) m \tan (e+f x)^2 b-4 a c d \left (c^2-d^2 (m+1)\right ) b+2 a^2 d^2 \left (c^2-d^2\right )-4 c d (b c-a d)^2 \tan (e+f x)\right )}{c+d \tan (e+f x)}dx}{\left (c^2+d^2\right ) (b c-a d)}-\frac {d^2 \left (4 a c d-b c^2 (5-m)-b d^2 (1-m)\right ) (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}}{2 \left (c^2+d^2\right ) (b c-a d)}+\frac {d^2 (a+b \tan (e+f x))^{m+1}}{2 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^2}\) |
| \(\Big \downarrow \) 4136 |
| \(\displaystyle \frac {\frac {\frac {d^2 \left (2 a^2 d^2 \left (3 c^2-d^2\right )-4 a b c d \left (c^2 (3-m)-d^2 (m+1)\right )-\left (b^2 \left (-\left (c^4 \left (m^2-5 m+6\right )\right )+2 c^2 d^2 \left (-m^2+3 m+1\right )+d^4 (1-m) m\right )\right )\right ) \int \frac {(a+b \tan (e+f x))^m \left (\tan ^2(e+f x)+1\right )}{c+d \tan (e+f x)}dx}{c^2+d^2}+\frac {\int 2 (a+b \tan (e+f x))^m \left (c (b c-a d)^2 \left (c^2-3 d^2\right )-d (b c-a d)^2 \left (3 c^2-d^2\right ) \tan (e+f x)\right )dx}{c^2+d^2}}{\left (c^2+d^2\right ) (b c-a d)}-\frac {d^2 \left (4 a c d-b c^2 (5-m)-b d^2 (1-m)\right ) (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}}{2 \left (c^2+d^2\right ) (b c-a d)}+\frac {d^2 (a+b \tan (e+f x))^{m+1}}{2 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {d^2 \left (2 a^2 d^2 \left (3 c^2-d^2\right )-4 a b c d \left (c^2 (3-m)-d^2 (m+1)\right )-\left (b^2 \left (-\left (c^4 \left (m^2-5 m+6\right )\right )+2 c^2 d^2 \left (-m^2+3 m+1\right )+d^4 (1-m) m\right )\right )\right ) \int \frac {(a+b \tan (e+f x))^m \left (\tan ^2(e+f x)+1\right )}{c+d \tan (e+f x)}dx}{c^2+d^2}+\frac {2 \int (a+b \tan (e+f x))^m \left (c (b c-a d)^2 \left (c^2-3 d^2\right )-d (b c-a d)^2 \left (3 c^2-d^2\right ) \tan (e+f x)\right )dx}{c^2+d^2}}{\left (c^2+d^2\right ) (b c-a d)}-\frac {d^2 \left (4 a c d-b c^2 (5-m)-b d^2 (1-m)\right ) (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}}{2 \left (c^2+d^2\right ) (b c-a d)}+\frac {d^2 (a+b \tan (e+f x))^{m+1}}{2 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {d^2 \left (2 a^2 d^2 \left (3 c^2-d^2\right )-4 a b c d \left (c^2 (3-m)-d^2 (m+1)\right )-\left (b^2 \left (-\left (c^4 \left (m^2-5 m+6\right )\right )+2 c^2 d^2 \left (-m^2+3 m+1\right )+d^4 (1-m) m\right )\right )\right ) \int \frac {(a+b \tan (e+f x))^m \left (\tan (e+f x)^2+1\right )}{c+d \tan (e+f x)}dx}{c^2+d^2}+\frac {2 \int (a+b \tan (e+f x))^m \left (c (b c-a d)^2 \left (c^2-3 d^2\right )-d (b c-a d)^2 \left (3 c^2-d^2\right ) \tan (e+f x)\right )dx}{c^2+d^2}}{\left (c^2+d^2\right ) (b c-a d)}-\frac {d^2 \left (4 a c d-b c^2 (5-m)-b d^2 (1-m)\right ) (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}}{2 \left (c^2+d^2\right ) (b c-a d)}+\frac {d^2 (a+b \tan (e+f x))^{m+1}}{2 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^2}\) |
| \(\Big \downarrow \) 4022 |
| \(\displaystyle \frac {d^2 (a+b \tan (e+f x))^{m+1}}{2 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^2}+\frac {-\frac {d^2 \left (4 a c d-b c^2 (5-m)-b d^2 (1-m)\right ) (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}+\frac {\frac {d^2 \left (2 a^2 d^2 \left (3 c^2-d^2\right )-4 a b c d \left (c^2 (3-m)-d^2 (m+1)\right )-\left (b^2 \left (-\left (c^4 \left (m^2-5 m+6\right )\right )+2 c^2 d^2 \left (-m^2+3 m+1\right )+d^4 (1-m) m\right )\right )\right ) \int \frac {(a+b \tan (e+f x))^m \left (\tan (e+f x)^2+1\right )}{c+d \tan (e+f x)}dx}{c^2+d^2}+\frac {2 \left (\frac {1}{2} (c-i d)^3 (b c-a d)^2 \int (1-i \tan (e+f x)) (a+b \tan (e+f x))^mdx+\frac {1}{2} (c+i d)^3 (b c-a d)^2 \int (i \tan (e+f x)+1) (a+b \tan (e+f x))^mdx\right )}{c^2+d^2}}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (c^2+d^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {d^2 (a+b \tan (e+f x))^{m+1}}{2 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^2}+\frac {-\frac {d^2 \left (4 a c d-b c^2 (5-m)-b d^2 (1-m)\right ) (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}+\frac {\frac {d^2 \left (2 a^2 d^2 \left (3 c^2-d^2\right )-4 a b c d \left (c^2 (3-m)-d^2 (m+1)\right )-\left (b^2 \left (-\left (c^4 \left (m^2-5 m+6\right )\right )+2 c^2 d^2 \left (-m^2+3 m+1\right )+d^4 (1-m) m\right )\right )\right ) \int \frac {(a+b \tan (e+f x))^m \left (\tan (e+f x)^2+1\right )}{c+d \tan (e+f x)}dx}{c^2+d^2}+\frac {2 \left (\frac {1}{2} (c-i d)^3 (b c-a d)^2 \int (1-i \tan (e+f x)) (a+b \tan (e+f x))^mdx+\frac {1}{2} (c+i d)^3 (b c-a d)^2 \int (i \tan (e+f x)+1) (a+b \tan (e+f x))^mdx\right )}{c^2+d^2}}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (c^2+d^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 4020 |
| \(\displaystyle \frac {d^2 (a+b \tan (e+f x))^{m+1}}{2 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^2}+\frac {-\frac {d^2 \left (4 a c d-b c^2 (5-m)-b d^2 (1-m)\right ) (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}+\frac {\frac {d^2 \left (2 a^2 d^2 \left (3 c^2-d^2\right )-4 a b c d \left (c^2 (3-m)-d^2 (m+1)\right )-\left (b^2 \left (-\left (c^4 \left (m^2-5 m+6\right )\right )+2 c^2 d^2 \left (-m^2+3 m+1\right )+d^4 (1-m) m\right )\right )\right ) \int \frac {(a+b \tan (e+f x))^m \left (\tan (e+f x)^2+1\right )}{c+d \tan (e+f x)}dx}{c^2+d^2}+\frac {2 \left (\frac {i (c+i d)^3 (b c-a d)^2 \int -\frac {(a+b \tan (e+f x))^m}{1-i \tan (e+f x)}d(i \tan (e+f x))}{2 f}-\frac {i (c-i d)^3 (b c-a d)^2 \int -\frac {(a+b \tan (e+f x))^m}{i \tan (e+f x)+1}d(-i \tan (e+f x))}{2 f}\right )}{c^2+d^2}}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (c^2+d^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d^2 (a+b \tan (e+f x))^{m+1}}{2 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^2}+\frac {-\frac {d^2 \left (4 a c d-b c^2 (5-m)-b d^2 (1-m)\right ) (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}+\frac {\frac {d^2 \left (2 a^2 d^2 \left (3 c^2-d^2\right )-4 a b c d \left (c^2 (3-m)-d^2 (m+1)\right )-\left (b^2 \left (-\left (c^4 \left (m^2-5 m+6\right )\right )+2 c^2 d^2 \left (-m^2+3 m+1\right )+d^4 (1-m) m\right )\right )\right ) \int \frac {(a+b \tan (e+f x))^m \left (\tan (e+f x)^2+1\right )}{c+d \tan (e+f x)}dx}{c^2+d^2}+\frac {2 \left (\frac {i (c-i d)^3 (b c-a d)^2 \int \frac {(a+b \tan (e+f x))^m}{i \tan (e+f x)+1}d(-i \tan (e+f x))}{2 f}-\frac {i (c+i d)^3 (b c-a d)^2 \int \frac {(a+b \tan (e+f x))^m}{1-i \tan (e+f x)}d(i \tan (e+f x))}{2 f}\right )}{c^2+d^2}}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (c^2+d^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 78 |
| \(\displaystyle \frac {d^2 (a+b \tan (e+f x))^{m+1}}{2 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^2}+\frac {-\frac {d^2 \left (4 a c d-b c^2 (5-m)-b d^2 (1-m)\right ) (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}+\frac {\frac {d^2 \left (2 a^2 d^2 \left (3 c^2-d^2\right )-4 a b c d \left (c^2 (3-m)-d^2 (m+1)\right )-\left (b^2 \left (-\left (c^4 \left (m^2-5 m+6\right )\right )+2 c^2 d^2 \left (-m^2+3 m+1\right )+d^4 (1-m) m\right )\right )\right ) \int \frac {(a+b \tan (e+f x))^m \left (\tan (e+f x)^2+1\right )}{c+d \tan (e+f x)}dx}{c^2+d^2}+\frac {2 \left (\frac {i (c-i d)^3 (b c-a d)^2 (a+b \tan (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {a+b \tan (e+f x)}{a+i b}\right )}{2 f (m+1) (a+i b)}-\frac {i (c+i d)^3 (b c-a d)^2 (a+b \tan (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {a+b \tan (e+f x)}{a-i b}\right )}{2 f (m+1) (a-i b)}\right )}{c^2+d^2}}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (c^2+d^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle \frac {d^2 (a+b \tan (e+f x))^{m+1}}{2 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^2}+\frac {-\frac {d^2 \left (4 a c d-b c^2 (5-m)-b d^2 (1-m)\right ) (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}+\frac {\frac {d^2 \left (2 a^2 d^2 \left (3 c^2-d^2\right )-4 a b c d \left (c^2 (3-m)-d^2 (m+1)\right )-\left (b^2 \left (-\left (c^4 \left (m^2-5 m+6\right )\right )+2 c^2 d^2 \left (-m^2+3 m+1\right )+d^4 (1-m) m\right )\right )\right ) \int \frac {(a+b \tan (e+f x))^m}{c+d \tan (e+f x)}d\tan (e+f x)}{f \left (c^2+d^2\right )}+\frac {2 \left (\frac {i (c-i d)^3 (b c-a d)^2 (a+b \tan (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {a+b \tan (e+f x)}{a+i b}\right )}{2 f (m+1) (a+i b)}-\frac {i (c+i d)^3 (b c-a d)^2 (a+b \tan (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {a+b \tan (e+f x)}{a-i b}\right )}{2 f (m+1) (a-i b)}\right )}{c^2+d^2}}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (c^2+d^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 78 |
| \(\displaystyle \frac {d^2 (a+b \tan (e+f x))^{m+1}}{2 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^2}+\frac {-\frac {d^2 \left (4 a c d-b c^2 (5-m)-b d^2 (1-m)\right ) (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}+\frac {\frac {d^2 \left (2 a^2 d^2 \left (3 c^2-d^2\right )-4 a b c d \left (c^2 (3-m)-d^2 (m+1)\right )-\left (b^2 \left (-\left (c^4 \left (m^2-5 m+6\right )\right )+2 c^2 d^2 \left (-m^2+3 m+1\right )+d^4 (1-m) m\right )\right )\right ) (a+b \tan (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,-\frac {d (a+b \tan (e+f x))}{b c-a d}\right )}{f (m+1) \left (c^2+d^2\right ) (b c-a d)}+\frac {2 \left (\frac {i (c-i d)^3 (b c-a d)^2 (a+b \tan (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {a+b \tan (e+f x)}{a+i b}\right )}{2 f (m+1) (a+i b)}-\frac {i (c+i d)^3 (b c-a d)^2 (a+b \tan (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {a+b \tan (e+f x)}{a-i b}\right )}{2 f (m+1) (a-i b)}\right )}{c^2+d^2}}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (c^2+d^2\right ) (b c-a d)}\) |
(d^2*(a + b*Tan[e + f*x])^(1 + m))/(2*(b*c - a*d)*(c^2 + d^2)*f*(c + d*Tan [e + f*x])^2) + (-((d^2*(4*a*c*d - b*d^2*(1 - m) - b*c^2*(5 - m))*(a + b*T an[e + f*x])^(1 + m))/((b*c - a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x]))) + ((d^2*(2*a^2*d^2*(3*c^2 - d^2) - 4*a*b*c*d*(c^2*(3 - m) - d^2*(1 + m)) - b ^2*(d^4*(1 - m)*m + 2*c^2*d^2*(1 + 3*m - m^2) - c^4*(6 - 5*m + m^2)))*Hype rgeometric2F1[1, 1 + m, 2 + m, -((d*(a + b*Tan[e + f*x]))/(b*c - a*d))]*(a + b*Tan[e + f*x])^(1 + m))/((b*c - a*d)*(c^2 + d^2)*f*(1 + m)) + (2*(((-1 /2*I)*(c + I*d)^3*(b*c - a*d)^2*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)*(c - I*d)^3*(b*c - a*d)^2*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))))/(c^2 + d^2))/((b*c - a*d)*(c^2 + d^2)))/(2*(b*c - a*d)*(c^2 + d ^2))
3.14.11.3.1 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[((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_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d))), x] + Simp[1 /((m + 1)*(a^2 + b^2)*(b*c - a*d)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*Tan[e + f*x]^2, x], x], x] / ; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || Integ erQ[m]) && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[A/f Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]
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[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[1/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^ n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Simp[ (A*b^2 - a*b*B + a^2*C)/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^n*((1 + Tan[ e + f*x]^2)/(a + b*Tan[e + f*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[n, 0] && !LeQ[n, -1]
\[\int \frac {\left (a +b \tan \left (f x +e \right )\right )^{m}}{\left (c +d \tan \left (f x +e \right )\right )^{3}}d x\]
\[ \int \frac {(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^3} \, dx=\int { \frac {{\left (b \tan \left (f x + e\right ) + a\right )}^{m}}{{\left (d \tan \left (f x + e\right ) + c\right )}^{3}} \,d x } \]
integral((b*tan(f*x + e) + a)^m/(d^3*tan(f*x + e)^3 + 3*c*d^2*tan(f*x + e) ^2 + 3*c^2*d*tan(f*x + e) + c^3), x)
\[ \int \frac {(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^3} \, dx=\int \frac {\left (a + b \tan {\left (e + f x \right )}\right )^{m}}{\left (c + d \tan {\left (e + f x \right )}\right )^{3}}\, dx \]
\[ \int \frac {(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^3} \, dx=\int { \frac {{\left (b \tan \left (f x + e\right ) + a\right )}^{m}}{{\left (d \tan \left (f x + e\right ) + c\right )}^{3}} \,d x } \]
\[ \int \frac {(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^3} \, dx=\int { \frac {{\left (b \tan \left (f x + e\right ) + a\right )}^{m}}{{\left (d \tan \left (f x + e\right ) + c\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^3} \, dx=\int \frac {{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^m}{{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^3} \,d x \]