\(\int (a+i a \tan (e+f x))^{1+m} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{-1-m} \, dx\) [852]

Optimal result

Integrand size = 47, antiderivative size = 151 \[ \int (a+i a \tan (e+f x))^{1+m} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{-1-m} \, dx=-\frac {(i A+B) (a+i a \tan (e+f x))^{1+m} (c-i c \tan (e+f x))^{-1-m}}{2 f (1+m)}+\frac {2^{-1-m} B \operatorname {Hypergeometric2F1}\left (1+m,1+m,2+m,\frac {1}{2} (1+i \tan (e+f x))\right ) (1-i \tan (e+f x))^{1+m} (a+i a \tan (e+f x))^{1+m} (c-i c \tan (e+f x))^{-1-m}}{f (1+m)} \] Output:

-1/2*(I*A+B)*(a+I*a*tan(f*x+e))^(1+m)*(c-I*c*tan(f*x+e))^(-1-m)/f/(1+m)+2^ 
(-1-m)*B*hypergeom([1+m, 1+m],[2+m],1/2+1/2*I*tan(f*x+e))*(1-I*tan(f*x+e)) 
^(1+m)*(a+I*a*tan(f*x+e))^(1+m)*(c-I*c*tan(f*x+e))^(-1-m)/f/(1+m)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 5.26 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.88 \[ \int (a+i a \tan (e+f x))^{1+m} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{-1-m} \, dx=\frac {a (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^{-m} \left (\frac {2^{1+m} B \operatorname {Hypergeometric2F1}\left (-m,-m,1-m,-\frac {1}{2} i (i+\tan (e+f x))\right ) (1+i \tan (e+f x))^{-m}}{m}+\frac {(i A+B) (-i+\tan (e+f x))}{(1+m) (i+\tan (e+f x))}\right )}{2 c f} \] Input:

Integrate[(a + I*a*Tan[e + f*x])^(1 + m)*(A + B*Tan[e + f*x])*(c - I*c*Tan 
[e + f*x])^(-1 - m),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.106, Rules used = {3042, 4071, 88, 80, 79}

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.

\(\displaystyle \int (a+i a \tan (e+f x))^{m+1} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{-m-1} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (a+i a \tan (e+f x))^{m+1} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{-m-1}dx\)

\(\Big \downarrow \) 4071

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

\(\Big \downarrow \) 88

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

\(\Big \downarrow \) 80

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

\(\Big \downarrow \) 79

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

Input:

Int[(a + I*a*Tan[e + f*x])^(1 + m)*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f 
*x])^(-1 - m),x]
 

Output:

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

Defintions of rubi rules used

rule 79

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( 
a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 
, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] 
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] 
 ||  !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
 

rule 80

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c 
 + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) 
^FracPart[n])   Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) 
), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Integ 
erQ[n] && (RationalQ[m] ||  !SimplerQ[n + 1, m + 1])
 

rule 88

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^Simplify[p + 1], 
 x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] &&  !RationalQ[p] && SumSimpl 
erQ[p, 1]
 

rule 3042

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

rule 4071

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + 
 (f_.)*(x_)])*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Si 
mp[a*(c/f)   Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^(n - 1)*(A + B*x), x], x 
, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && EqQ[b*c 
+ a*d, 0] && EqQ[a^2 + b^2, 0]
 

Maple [F]

\[\int \left (a +i a \tan \left (f x +e \right )\right )^{1+m} \left (A +B \tan \left (f x +e \right )\right ) \left (c -i c \tan \left (f x +e \right )\right )^{-1-m}d x\]

Input:

int((a+I*a*tan(f*x+e))^(1+m)*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(-1-m),x)
 

Output:

int((a+I*a*tan(f*x+e))^(1+m)*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(-1-m),x)
 

Fricas [F]

\[ \int (a+i a \tan (e+f x))^{1+m} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{-1-m} \, dx=\int { {\left (B \tan \left (f x + e\right ) + A\right )} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m + 1} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{-m - 1} \,d x } \] Input:

integrate((a+I*a*tan(f*x+e))^(1+m)*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(-1 
-m),x, algorithm="fricas")
 

Output:

integral(((A - I*B)*e^(2*I*f*x + 2*I*e) + A + I*B)*(2*c/(e^(2*I*f*x + 2*I* 
e) + 1))^(-m - 1)*e^(2*I*e*m - 2*(-I*f*m - I*f)*x + (m + 1)*log(a/c) + (m 
+ 1)*log(2*c/(e^(2*I*f*x + 2*I*e) + 1)) + 2*I*e)/(e^(2*I*f*x + 2*I*e) + 1) 
, x)
 

Sympy [F]

\[ \int (a+i a \tan (e+f x))^{1+m} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{-1-m} \, dx=\int \left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{m + 1} \left (- i c \left (\tan {\left (e + f x \right )} + i\right )\right )^{- m - 1} \left (A + B \tan {\left (e + f x \right )}\right )\, dx \] Input:

integrate((a+I*a*tan(f*x+e))**(1+m)*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))**( 
-1-m),x)
 

Output:

Integral((I*a*(tan(e + f*x) - I))**(m + 1)*(-I*c*(tan(e + f*x) + I))**(-m 
- 1)*(A + B*tan(e + f*x)), x)
 

Maxima [F]

\[ \int (a+i a \tan (e+f x))^{1+m} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{-1-m} \, dx=\int { {\left (B \tan \left (f x + e\right ) + A\right )} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m + 1} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{-m - 1} \,d x } \] Input:

integrate((a+I*a*tan(f*x+e))^(1+m)*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(-1 
-m),x, algorithm="maxima")
 

Output:

-(2*(-I*B*a^(m + 1)*m - I*B*a^(m + 1))*cos(2*f*m*x + 2*e*m) + ((A - I*B)*a 
^(m + 1)*m^2 - (A - I*B)*a^(m + 1)*m)*cos(2*e*m + 2*(f*m + 2*f)*x + 4*e) + 
 ((A + I*B)*a^(m + 1)*m^2 - (A - I*B)*a^(m + 1)*m - 2*I*B*a^(m + 1))*cos(2 
*e*m + 2*(f*m + f)*x + 2*e) - 4*(B*a^(m + 1)*c^(m + 1)*f*m^3 - B*a^(m + 1) 
*c^(m + 1)*f*m + (B*a^(m + 1)*c^(m + 1)*f*m^3 - B*a^(m + 1)*c^(m + 1)*f*m) 
*cos(2*f*x + 2*e) - (-I*B*a^(m + 1)*c^(m + 1)*f*m^3 + I*B*a^(m + 1)*c^(m + 
 1)*f*m)*sin(2*f*x + 2*e))*integrate(((cos(4*f*x + 4*e) + 2*cos(2*f*x + 2* 
e) + 1)*cos(2*f*m*x + 2*e*m) + (sin(4*f*x + 4*e) + 2*sin(2*f*x + 2*e))*sin 
(2*f*m*x + 2*e*m))/((c^(m + 1)*m - c^(m + 1))*cos(4*f*x + 4*e)^2 + 4*(c^(m 
 + 1)*m - c^(m + 1))*cos(2*f*x + 2*e)^2 + (c^(m + 1)*m - c^(m + 1))*sin(4* 
f*x + 4*e)^2 + 4*(c^(m + 1)*m - c^(m + 1))*sin(4*f*x + 4*e)*sin(2*f*x + 2* 
e) + 4*(c^(m + 1)*m - c^(m + 1))*sin(2*f*x + 2*e)^2 + c^(m + 1)*m + 2*(c^( 
m + 1)*m + 2*(c^(m + 1)*m - c^(m + 1))*cos(2*f*x + 2*e) - c^(m + 1))*cos(4 
*f*x + 4*e) + 4*(c^(m + 1)*m - c^(m + 1))*cos(2*f*x + 2*e) - c^(m + 1)), x 
) + 4*(-I*B*a^(m + 1)*c^(m + 1)*f*m^3 + I*B*a^(m + 1)*c^(m + 1)*f*m + (-I* 
B*a^(m + 1)*c^(m + 1)*f*m^3 + I*B*a^(m + 1)*c^(m + 1)*f*m)*cos(2*f*x + 2*e 
) + (B*a^(m + 1)*c^(m + 1)*f*m^3 - B*a^(m + 1)*c^(m + 1)*f*m)*sin(2*f*x + 
2*e))*integrate(-((sin(4*f*x + 4*e) + 2*sin(2*f*x + 2*e))*cos(2*f*m*x + 2* 
e*m) - (cos(4*f*x + 4*e) + 2*cos(2*f*x + 2*e) + 1)*sin(2*f*m*x + 2*e*m))/( 
(c^(m + 1)*m - c^(m + 1))*cos(4*f*x + 4*e)^2 + 4*(c^(m + 1)*m - c^(m + ...
 

Giac [F]

\[ \int (a+i a \tan (e+f x))^{1+m} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{-1-m} \, dx=\int { {\left (B \tan \left (f x + e\right ) + A\right )} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m + 1} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{-m - 1} \,d x } \] Input:

integrate((a+I*a*tan(f*x+e))^(1+m)*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(-1 
-m),x, algorithm="giac")
 

Output:

integrate((B*tan(f*x + e) + A)*(I*a*tan(f*x + e) + a)^(m + 1)*(-I*c*tan(f* 
x + e) + c)^(-m - 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int (a+i a \tan (e+f x))^{1+m} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{-1-m} \, dx=\int \frac {\left (A+B\,\mathrm {tan}\left (e+f\,x\right )\right )\,{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{m+1}}{{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{m+1}} \,d x \] Input:

int(((A + B*tan(e + f*x))*(a + a*tan(e + f*x)*1i)^(m + 1))/(c - c*tan(e + 
f*x)*1i)^(m + 1),x)
 

Output:

int(((A + B*tan(e + f*x))*(a + a*tan(e + f*x)*1i)^(m + 1))/(c - c*tan(e + 
f*x)*1i)^(m + 1), x)
 

Reduce [F]

\[ \int (a+i a \tan (e+f x))^{1+m} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{-1-m} \, dx=-\frac {a \left (\left (\int \frac {\left (\tan \left (f x +e \right ) a i +a \right )^{m}}{\left (-\tan \left (f x +e \right ) c i +c \right )^{m} \tan \left (f x +e \right ) i -\left (-\tan \left (f x +e \right ) c i +c \right )^{m}}d x \right ) a +\left (\int \frac {\left (\tan \left (f x +e \right ) a i +a \right )^{m} \tan \left (f x +e \right )^{2}}{\left (-\tan \left (f x +e \right ) c i +c \right )^{m} \tan \left (f x +e \right ) i -\left (-\tan \left (f x +e \right ) c i +c \right )^{m}}d x \right ) b i +\left (\int \frac {\left (\tan \left (f x +e \right ) a i +a \right )^{m} \tan \left (f x +e \right )}{\left (-\tan \left (f x +e \right ) c i +c \right )^{m} \tan \left (f x +e \right ) i -\left (-\tan \left (f x +e \right ) c i +c \right )^{m}}d x \right ) a i +\left (\int \frac {\left (\tan \left (f x +e \right ) a i +a \right )^{m} \tan \left (f x +e \right )}{\left (-\tan \left (f x +e \right ) c i +c \right )^{m} \tan \left (f x +e \right ) i -\left (-\tan \left (f x +e \right ) c i +c \right )^{m}}d x \right ) b \right )}{c} \] Input:

int((a+I*a*tan(f*x+e))^(1+m)*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(-1-m),x)
 

Output:

( - a*(int((tan(e + f*x)*a*i + a)**m/(( - tan(e + f*x)*c*i + c)**m*tan(e + 
 f*x)*i - ( - tan(e + f*x)*c*i + c)**m),x)*a + int(((tan(e + f*x)*a*i + a) 
**m*tan(e + f*x)**2)/(( - tan(e + f*x)*c*i + c)**m*tan(e + f*x)*i - ( - ta 
n(e + f*x)*c*i + c)**m),x)*b*i + int(((tan(e + f*x)*a*i + a)**m*tan(e + f* 
x))/(( - tan(e + f*x)*c*i + c)**m*tan(e + f*x)*i - ( - tan(e + f*x)*c*i + 
c)**m),x)*a*i + int(((tan(e + f*x)*a*i + a)**m*tan(e + f*x))/(( - tan(e + 
f*x)*c*i + c)**m*tan(e + f*x)*i - ( - tan(e + f*x)*c*i + c)**m),x)*b))/c