Integrand size = 25, antiderivative size = 176 \[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^2 \, dx=\frac {d^2 (a+b \tan (e+f x))^{1+m}}{b f (1+m)}+\frac {(c-i d)^2 \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a-i b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (i a+b) f (1+m)}-\frac {(c+i d)^2 \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a+i b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (i a-b) f (1+m)} \]
d^2*(a+b*tan(f*x+e))^(1+m)/b/f/(1+m)+1/2*(c-I*d)^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)/f/(1+m)-1/2*(c+ I*d)^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)/f/(1+m)
Time = 0.27 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.77 \[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^2 \, dx=\frac {\left (\frac {2 d^2}{b}-\frac {i (c-i d)^2 \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a-i b}\right )}{a-i b}+\frac {i (c+i d)^2 \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a+i b}\right )}{a+i b}\right ) (a+b \tan (e+f x))^{1+m}}{2 f (1+m)} \]
(((2*d^2)/b - (I*(c - I*d)^2*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan [e + f*x])/(a - I*b)])/(a - I*b) + (I*(c + I*d)^2*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a + I*b)])/(a + I*b))*(a + b*Tan[e + f*x] )^(1 + m))/(2*f*(1 + m))
Time = 0.56 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3042, 4026, 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.
\(\displaystyle \int (c+d \tan (e+f x))^2 (a+b \tan (e+f x))^m \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (c+d \tan (e+f x))^2 (a+b \tan (e+f x))^mdx\) |
\(\Big \downarrow \) 4026 |
\(\displaystyle \int (a+b \tan (e+f x))^m \left (c^2+2 d \tan (e+f x) c-d^2\right )dx+\frac {d^2 (a+b \tan (e+f x))^{m+1}}{b f (m+1)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (a+b \tan (e+f x))^m \left (c^2+2 d \tan (e+f x) c-d^2\right )dx+\frac {d^2 (a+b \tan (e+f x))^{m+1}}{b f (m+1)}\) |
\(\Big \downarrow \) 4022 |
\(\displaystyle \frac {1}{2} (c+i d)^2 \int (1-i \tan (e+f x)) (a+b \tan (e+f x))^mdx+\frac {1}{2} (c-i d)^2 \int (i \tan (e+f x)+1) (a+b \tan (e+f x))^mdx+\frac {d^2 (a+b \tan (e+f x))^{m+1}}{b f (m+1)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} (c+i d)^2 \int (1-i \tan (e+f x)) (a+b \tan (e+f x))^mdx+\frac {1}{2} (c-i d)^2 \int (i \tan (e+f x)+1) (a+b \tan (e+f x))^mdx+\frac {d^2 (a+b \tan (e+f x))^{m+1}}{b f (m+1)}\) |
\(\Big \downarrow \) 4020 |
\(\displaystyle \frac {i (c-i d)^2 \int -\frac {(a+b \tan (e+f x))^m}{1-i \tan (e+f x)}d(i \tan (e+f x))}{2 f}-\frac {i (c+i d)^2 \int -\frac {(a+b \tan (e+f x))^m}{i \tan (e+f x)+1}d(-i \tan (e+f x))}{2 f}+\frac {d^2 (a+b \tan (e+f x))^{m+1}}{b f (m+1)}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {i (c-i d)^2 \int \frac {(a+b \tan (e+f x))^m}{1-i \tan (e+f x)}d(i \tan (e+f x))}{2 f}+\frac {i (c+i d)^2 \int \frac {(a+b \tan (e+f x))^m}{i \tan (e+f x)+1}d(-i \tan (e+f x))}{2 f}+\frac {d^2 (a+b \tan (e+f x))^{m+1}}{b f (m+1)}\) |
\(\Big \downarrow \) 78 |
\(\displaystyle -\frac {i (c-i d)^2 (a+b \tan (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {a+b \tan (e+f x)}{a-i b}\right )}{2 f (m+1) (a-i b)}+\frac {i (c+i d)^2 (a+b \tan (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {a+b \tan (e+f x)}{a+i b}\right )}{2 f (m+1) (a+i b)}+\frac {d^2 (a+b \tan (e+f x))^{m+1}}{b f (m+1)}\) |
(d^2*(a + b*Tan[e + f*x])^(1 + m))/(b*f*(1 + m)) - ((I/2)*(c - I*d)^2*Hype rgeometric2F1[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)^2*Hypergeometr ic2F1[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))
3.14.6.3.1 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_)*((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])
\[\int \left (a +b \tan \left (f x +e \right )\right )^{m} \left (c +d \tan \left (f x +e \right )\right )^{2}d x\]
\[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^2 \, dx=\int { {\left (d \tan \left (f x + e\right ) + c\right )}^{2} {\left (b \tan \left (f x + e\right ) + a\right )}^{m} \,d x } \]
\[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^2 \, dx=\int \left (a + b \tan {\left (e + f x \right )}\right )^{m} \left (c + d \tan {\left (e + f x \right )}\right )^{2}\, dx \]
\[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^2 \, dx=\int { {\left (d \tan \left (f x + e\right ) + c\right )}^{2} {\left (b \tan \left (f x + e\right ) + a\right )}^{m} \,d x } \]
\[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^2 \, dx=\int { {\left (d \tan \left (f x + e\right ) + c\right )}^{2} {\left (b \tan \left (f x + e\right ) + a\right )}^{m} \,d x } \]
Timed out. \[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^2 \, dx=\int {\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^m\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^2 \,d x \]