Integrand size = 25, antiderivative size = 301 \[ \int \frac {(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^2} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a-i b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (i a+b) (c-i d)^2 f (1+m)}-\frac {\operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a+i b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (i a-b) (c+i d)^2 f (1+m)}-\frac {d^2 \left (2 a c d-b \left (c^2 (2-m)-d^2 m\right )\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {d (a+b \tan (e+f x))}{b c-a d}\right ) (a+b \tan (e+f x))^{1+m}}{(b c-a d)^2 \left (c^2+d^2\right )^2 f (1+m)}+\frac {d^2 (a+b \tan (e+f x))^{1+m}}{(b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))} \]
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)^2/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)/(I*a-b)/(c+I*d)^2/f/(1+m)-d^2*(2*a*c*d-b *(c^2*(2-m)-d^2*m))*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)^2/(c^2+d^2)^2/f/(1+m)+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))
Time = 4.52 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^2} \, dx=\frac {(a+b \tan (e+f x))^{1+m} \left (-\frac {i \left (\frac {(c+i d)^2 (-b c+a d) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a-i b}\right )}{a-i b}+\frac {(c-i d)^2 (b c-a d) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a+i b}\right )}{a+i b}\right )}{\left (c^2+d^2\right ) (1+m)}-\frac {2 d^2 \left (2 a c d+b c^2 (-2+m)+b d^2 m\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {d (a+b \tan (e+f x))}{-b c+a d}\right )}{(-b c+a d) \left (c^2+d^2\right ) (1+m)}-\frac {2 d^2}{c+d \tan (e+f x)}\right )}{2 (-b c+a d) \left (c^2+d^2\right ) f} \]
((a + b*Tan[e + f*x])^(1 + m)*(((-I)*(((c + I*d)^2*(-(b*c) + a*d)*Hypergeo metric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a - I*b)])/(a - I*b) + (( c - I*d)^2*(b*c - a*d)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f *x])/(a + I*b)])/(a + I*b)))/((c^2 + d^2)*(1 + m)) - (2*d^2*(2*a*c*d + b*c ^2*(-2 + m) + b*d^2*m)*Hypergeometric2F1[1, 1 + m, 2 + m, (d*(a + b*Tan[e + f*x]))/(-(b*c) + a*d)])/((-(b*c) + a*d)*(c^2 + d^2)*(1 + m)) - (2*d^2)/( c + d*Tan[e + f*x])))/(2*(-(b*c) + a*d)*(c^2 + d^2)*f)
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.
| \(\displaystyle \int \frac {(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^2}dx\) |
| \(\Big \downarrow \) 4052 |
| \(\displaystyle \frac {\int -\frac {(a+b \tan (e+f x))^m \left (b d^2 m \tan ^2(e+f x)+d (b c-a d) \tan (e+f x)+a c d-b \left (c^2-d^2 m\right )\right )}{c+d \tan (e+f x)}dx}{\left (c^2+d^2\right ) (b c-a d)}+\frac {d^2 (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d^2 (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}-\frac {\int -\frac {(a+b \tan (e+f x))^m \left (b c^2-a d c-b d^2 m \tan ^2(e+f x)-b d^2 m-d (b c-a d) \tan (e+f x)\right )}{c+d \tan (e+f x)}dx}{\left (c^2+d^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b \tan (e+f x))^m \left (b d^2 m \tan ^2(e+f x)+d (b c-a d) \tan (e+f x)+a c d-b \left (c^2-d^2 m\right )\right )}{c+d \tan (e+f x)}dx}{\left (c^2+d^2\right ) (b c-a d)}+\frac {d^2 (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d^2 (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}-\frac {\int -\frac {(a+b \tan (e+f x))^m \left (b c^2-a d c-b d^2 m \tan ^2(e+f x)-b d^2 m-d (b c-a d) \tan (e+f x)\right )}{c+d \tan (e+f x)}dx}{\left (c^2+d^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b \tan (e+f x))^m \left (b d^2 m \tan ^2(e+f x)+d (b c-a d) \tan (e+f x)+a c d-b \left (c^2-d^2 m\right )\right )}{c+d \tan (e+f x)}dx}{\left (c^2+d^2\right ) (b c-a d)}+\frac {d^2 (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d^2 (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}-\frac {\int -\frac {(a+b \tan (e+f x))^m \left (b c^2-a d c-b d^2 m \tan ^2(e+f x)-b d^2 m-d (b c-a d) \tan (e+f x)\right )}{c+d \tan (e+f x)}dx}{\left (c^2+d^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b \tan (e+f x))^m \left (b d^2 m \tan ^2(e+f x)+d (b c-a d) \tan (e+f x)+a c d-b \left (c^2-d^2 m\right )\right )}{c+d \tan (e+f x)}dx}{\left (c^2+d^2\right ) (b c-a d)}+\frac {d^2 (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d^2 (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}-\frac {\int -\frac {(a+b \tan (e+f x))^m \left (b c^2-a d c-b d^2 m \tan ^2(e+f x)-b d^2 m-d (b c-a d) \tan (e+f x)\right )}{c+d \tan (e+f x)}dx}{\left (c^2+d^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b \tan (e+f x))^m \left (b d^2 m \tan ^2(e+f x)+d (b c-a d) \tan (e+f x)+a c d-b \left (c^2-d^2 m\right )\right )}{c+d \tan (e+f x)}dx}{\left (c^2+d^2\right ) (b c-a d)}+\frac {d^2 (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d^2 (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}-\frac {\int -\frac {(a+b \tan (e+f x))^m \left (b c^2-a d c-b d^2 m \tan ^2(e+f x)-b d^2 m-d (b c-a d) \tan (e+f x)\right )}{c+d \tan (e+f x)}dx}{\left (c^2+d^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b \tan (e+f x))^m \left (b d^2 m \tan ^2(e+f x)+d (b c-a d) \tan (e+f x)+a c d-b \left (c^2-d^2 m\right )\right )}{c+d \tan (e+f x)}dx}{\left (c^2+d^2\right ) (b c-a d)}+\frac {d^2 (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d^2 (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}-\frac {\int -\frac {(a+b \tan (e+f x))^m \left (b c^2-a d c-b d^2 m \tan ^2(e+f x)-b d^2 m-d (b c-a d) \tan (e+f x)\right )}{c+d \tan (e+f x)}dx}{\left (c^2+d^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b \tan (e+f x))^m \left (b d^2 m \tan ^2(e+f x)+d (b c-a d) \tan (e+f x)+a c d-b \left (c^2-d^2 m\right )\right )}{c+d \tan (e+f x)}dx}{\left (c^2+d^2\right ) (b c-a d)}+\frac {d^2 (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d^2 (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}-\frac {\int -\frac {(a+b \tan (e+f x))^m \left (b c^2-a d c-b d^2 m \tan ^2(e+f x)-b d^2 m-d (b c-a d) \tan (e+f x)\right )}{c+d \tan (e+f x)}dx}{\left (c^2+d^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b \tan (e+f x))^m \left (b d^2 m \tan ^2(e+f x)+d (b c-a d) \tan (e+f x)+a c d-b \left (c^2-d^2 m\right )\right )}{c+d \tan (e+f x)}dx}{\left (c^2+d^2\right ) (b c-a d)}+\frac {d^2 (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d^2 (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}-\frac {\int -\frac {(a+b \tan (e+f x))^m \left (b c^2-a d c-b d^2 m \tan ^2(e+f x)-b d^2 m-d (b c-a d) \tan (e+f x)\right )}{c+d \tan (e+f x)}dx}{\left (c^2+d^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b \tan (e+f x))^m \left (b d^2 m \tan ^2(e+f x)+d (b c-a d) \tan (e+f x)+a c d-b \left (c^2-d^2 m\right )\right )}{c+d \tan (e+f x)}dx}{\left (c^2+d^2\right ) (b c-a d)}+\frac {d^2 (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d^2 (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}-\frac {\int -\frac {(a+b \tan (e+f x))^m \left (b c^2-a d c-b d^2 m \tan ^2(e+f x)-b d^2 m-d (b c-a d) \tan (e+f x)\right )}{c+d \tan (e+f x)}dx}{\left (c^2+d^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b \tan (e+f x))^m \left (b d^2 m \tan ^2(e+f x)+d (b c-a d) \tan (e+f x)+a c d-b \left (c^2-d^2 m\right )\right )}{c+d \tan (e+f x)}dx}{\left (c^2+d^2\right ) (b c-a d)}+\frac {d^2 (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d^2 (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}-\frac {\int -\frac {(a+b \tan (e+f x))^m \left (b c^2-a d c-b d^2 m \tan ^2(e+f x)-b d^2 m-d (b c-a d) \tan (e+f x)\right )}{c+d \tan (e+f x)}dx}{\left (c^2+d^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b \tan (e+f x))^m \left (b d^2 m \tan ^2(e+f x)+d (b c-a d) \tan (e+f x)+a c d-b \left (c^2-d^2 m\right )\right )}{c+d \tan (e+f x)}dx}{\left (c^2+d^2\right ) (b c-a d)}+\frac {d^2 (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d^2 (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}-\frac {\int -\frac {(a+b \tan (e+f x))^m \left (b c^2-a d c-b d^2 m \tan ^2(e+f x)-b d^2 m-d (b c-a d) \tan (e+f x)\right )}{c+d \tan (e+f x)}dx}{\left (c^2+d^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b \tan (e+f x))^m \left (b d^2 m \tan ^2(e+f x)+d (b c-a d) \tan (e+f x)+a c d-b \left (c^2-d^2 m\right )\right )}{c+d \tan (e+f x)}dx}{\left (c^2+d^2\right ) (b c-a d)}+\frac {d^2 (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d^2 (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}-\frac {\int -\frac {(a+b \tan (e+f x))^m \left (b c^2-a d c-b d^2 m \tan ^2(e+f x)-b d^2 m-d (b c-a d) \tan (e+f x)\right )}{c+d \tan (e+f x)}dx}{\left (c^2+d^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b \tan (e+f x))^m \left (b d^2 m \tan ^2(e+f x)+d (b c-a d) \tan (e+f x)+a c d-b \left (c^2-d^2 m\right )\right )}{c+d \tan (e+f x)}dx}{\left (c^2+d^2\right ) (b c-a d)}+\frac {d^2 (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d^2 (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}-\frac {\int -\frac {(a+b \tan (e+f x))^m \left (b c^2-a d c-b d^2 m \tan ^2(e+f x)-b d^2 m-d (b c-a d) \tan (e+f x)\right )}{c+d \tan (e+f x)}dx}{\left (c^2+d^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b \tan (e+f x))^m \left (b d^2 m \tan ^2(e+f x)+d (b c-a d) \tan (e+f x)+a c d-b \left (c^2-d^2 m\right )\right )}{c+d \tan (e+f x)}dx}{\left (c^2+d^2\right ) (b c-a d)}+\frac {d^2 (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d^2 (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}-\frac {\int -\frac {(a+b \tan (e+f x))^m \left (b c^2-a d c-b d^2 m \tan ^2(e+f x)-b d^2 m-d (b c-a d) \tan (e+f x)\right )}{c+d \tan (e+f x)}dx}{\left (c^2+d^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(a+b \tan (e+f x))^m \left (b d^2 m \tan ^2(e+f x)+d (b c-a d) \tan (e+f x)+a c d-b \left (c^2-d^2 m\right )\right )}{c+d \tan (e+f x)}dx}{\left (c^2+d^2\right ) (b c-a d)}+\frac {d^2 (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}\) |
3.14.10.3.1 Defintions of rubi rules used
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])))
\[\int \frac {\left (a +b \tan \left (f x +e \right )\right )^{m}}{\left (c +d \tan \left (f x +e \right )\right )^{2}}d x\]
\[ \int \frac {(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^2} \, dx=\int { \frac {{\left (b \tan \left (f x + e\right ) + a\right )}^{m}}{{\left (d \tan \left (f x + e\right ) + c\right )}^{2}} \,d x } \]
\[ \int \frac {(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^2} \, dx=\int \frac {\left (a + b \tan {\left (e + f x \right )}\right )^{m}}{\left (c + d \tan {\left (e + f x \right )}\right )^{2}}\, dx \]
\[ \int \frac {(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^2} \, dx=\int { \frac {{\left (b \tan \left (f x + e\right ) + a\right )}^{m}}{{\left (d \tan \left (f x + e\right ) + c\right )}^{2}} \,d x } \]
\[ \int \frac {(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^2} \, dx=\int { \frac {{\left (b \tan \left (f x + e\right ) + a\right )}^{m}}{{\left (d \tan \left (f x + e\right ) + c\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^2} \, dx=\int \frac {{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^m}{{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^2} \,d x \]