\(\int (a+b \tan (e+f x))^m (c+d \tan (e+f x)) \, dx\) [1323]

Optimal result

Integrand size = 23, antiderivative size = 143 \[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x)) \, dx=\frac {(c-i d) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a-i b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (i a+b) f (1+m)}+\frac {(i c-d) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a+i b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (a+i b) f (1+m)} \] Output:

1/2*(c-I*d)*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*(I*c-d)*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)/f/(1+m)
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.84 \[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x)) \, dx=\frac {i \left (-\frac {(c-i d) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a-i b}\right )}{a-i b}+\frac {(c+i d) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a+i b}\right )}{a+i b}\right ) (a+b \tan (e+f x))^{1+m}}{2 f (1+m)} \] Input:

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

Output:

((I/2)*(-(((c - I*d)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x 
])/(a - I*b)])/(a - I*b)) + ((c + I*d)*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))/ 
(f*(1 + m))
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {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.

\(\displaystyle \int (c+d \tan (e+f x)) (a+b \tan (e+f x))^m \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (c+d \tan (e+f x)) (a+b \tan (e+f x))^mdx\)

\(\Big \downarrow \) 4022

\(\displaystyle \frac {1}{2} (c+i d) \int (1-i \tan (e+f x)) (a+b \tan (e+f x))^mdx+\frac {1}{2} (c-i d) \int (i \tan (e+f x)+1) (a+b \tan (e+f x))^mdx\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} (c+i d) \int (1-i \tan (e+f x)) (a+b \tan (e+f x))^mdx+\frac {1}{2} (c-i d) \int (i \tan (e+f x)+1) (a+b \tan (e+f x))^mdx\)

\(\Big \downarrow \) 4020

\(\displaystyle \frac {i (c-i d) \int -\frac {(a+b \tan (e+f x))^m}{1-i \tan (e+f x)}d(i \tan (e+f x))}{2 f}-\frac {i (c+i d) \int -\frac {(a+b \tan (e+f x))^m}{i \tan (e+f x)+1}d(-i \tan (e+f x))}{2 f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {i (c+i d) \int \frac {(a+b \tan (e+f x))^m}{i \tan (e+f x)+1}d(-i \tan (e+f x))}{2 f}-\frac {i (c-i d) \int \frac {(a+b \tan (e+f x))^m}{1-i \tan (e+f x)}d(i \tan (e+f x))}{2 f}\)

\(\Big \downarrow \) 78

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

Input:

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

Output:

((-1/2*I)*(c - I*d)*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)*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))
 

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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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