\(\int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^2 \, dx\) [1322]

Optimal result

Integrand size = 25, antiderivative size = 176 \[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^2 \, dx=\frac {d^2 (a+b \tan (e+f x))^{1+m}}{b f (1+m)}+\frac {(c-i d)^2 \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a-i b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (i a+b) f (1+m)}-\frac {(c+i d)^2 \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a+i b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (i a-b) f (1+m)} \] Output:

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)
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.77 \[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^2 \, dx=\frac {\left (\frac {2 d^2}{b}-\frac {i (c-i d)^2 \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a-i b}\right )}{a-i b}+\frac {i (c+i d)^2 \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a+i b}\right )}{a+i b}\right ) (a+b \tan (e+f x))^{1+m}}{2 f (1+m)} \] Input:

Integrate[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^2,x]
 

Output:

(((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))
 

Rubi [A] (verified)

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

\(\displaystyle -\frac {i (c-i d)^2 (a+b \tan (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {a+b \tan (e+f x)}{a-i b}\right )}{2 f (m+1) (a-i b)}+\frac {i (c+i d)^2 (a+b \tan (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {a+b \tan (e+f x)}{a+i b}\right )}{2 f (m+1) (a+i b)}+\frac {d^2 (a+b \tan (e+f x))^{m+1}}{b f (m+1)}\)

Input:

Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^2,x]
 

Output:

(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))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 78

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4020

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]
 

rule 4022

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]
 

rule 4026

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])
 

Maple [F]

\[\int \left (a +b \tan \left (f x +e \right )\right )^{m} \left (c +d \tan \left (f x +e \right )\right )^{2}d x\]

Input:

int((a+b*tan(f*x+e))^m*(c+d*tan(f*x+e))^2,x)
 

Output:

int((a+b*tan(f*x+e))^m*(c+d*tan(f*x+e))^2,x)
 

Fricas [F]

\[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^2 \, dx=\int { {\left (d \tan \left (f x + e\right ) + c\right )}^{2} {\left (b \tan \left (f x + e\right ) + a\right )}^{m} \,d x } \] Input:

integrate((a+b*tan(f*x+e))^m*(c+d*tan(f*x+e))^2,x, algorithm="fricas")
 

Output:

integral((d^2*tan(f*x + e)^2 + 2*c*d*tan(f*x + e) + c^2)*(b*tan(f*x + e) + 
 a)^m, x)
 

Sympy [F]

\[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^2 \, dx=\int \left (a + b \tan {\left (e + f x \right )}\right )^{m} \left (c + d \tan {\left (e + f x \right )}\right )^{2}\, dx \] Input:

integrate((a+b*tan(f*x+e))**m*(c+d*tan(f*x+e))**2,x)
 

Output:

Integral((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**2, x)
 

Maxima [F]

\[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^2 \, dx=\int { {\left (d \tan \left (f x + e\right ) + c\right )}^{2} {\left (b \tan \left (f x + e\right ) + a\right )}^{m} \,d x } \] Input:

integrate((a+b*tan(f*x+e))^m*(c+d*tan(f*x+e))^2,x, algorithm="maxima")
 

Output:

integrate((d*tan(f*x + e) + c)^2*(b*tan(f*x + e) + a)^m, x)
 

Giac [F]

\[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^2 \, dx=\int { {\left (d \tan \left (f x + e\right ) + c\right )}^{2} {\left (b \tan \left (f x + e\right ) + a\right )}^{m} \,d x } \] Input:

integrate((a+b*tan(f*x+e))^m*(c+d*tan(f*x+e))^2,x, algorithm="giac")
 

Output:

integrate((d*tan(f*x + e) + c)^2*(b*tan(f*x + e) + a)^m, x)
 

Mupad [F(-1)]

Timed out. \[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^2 \, dx=\int {\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^m\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^2 \,d x \] Input:

int((a + b*tan(e + f*x))^m*(c + d*tan(e + f*x))^2,x)
 

Output:

int((a + b*tan(e + f*x))^m*(c + d*tan(e + f*x))^2, x)
 

Reduce [F]

\[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^2 \, dx=\left (\int \left (\tan \left (f x +e \right ) b +a \right )^{m}d x \right ) c^{2}+\left (\int \left (\tan \left (f x +e \right ) b +a \right )^{m} \tan \left (f x +e \right )^{2}d x \right ) d^{2}+2 \left (\int \left (\tan \left (f x +e \right ) b +a \right )^{m} \tan \left (f x +e \right )d x \right ) c d \] Input:

int((a+b*tan(f*x+e))^m*(c+d*tan(f*x+e))^2,x)
                                                                                    
                                                                                    
 

Output:

int((tan(e + f*x)*b + a)**m,x)*c**2 + int((tan(e + f*x)*b + a)**m*tan(e + 
f*x)**2,x)*d**2 + 2*int((tan(e + f*x)*b + a)**m*tan(e + f*x),x)*c*d