3.9.51 \(\int (a+i a \tan (e+f x))^m (A+B \tan (e+f x)) (c-i c \tan (e+f x))^n \, dx\) [851]

3.9.51.1 Optimal result

Integrand size = 41, antiderivative size = 150 \[ \int (a+i a \tan (e+f x))^m (A+B \tan (e+f x)) (c-i c \tan (e+f x))^n \, dx=\frac {(i A+B) (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^n}{2 f n}-\frac {2^{-1+n} (B (m-n)+i A (m+n)) \operatorname {Hypergeometric2F1}\left (m,-n,1+m,\frac {1}{2} (1+i \tan (e+f x))\right ) (1-i \tan (e+f x))^{-n} (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^n}{f m n} \]

output

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

3.9.51.2 Mathematica [A] (verified)

Time = 1.13 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.81 \[ \int (a+i a \tan (e+f x))^m (A+B \tan (e+f x)) (c-i c \tan (e+f x))^n \, dx=\frac {2^{-1+n} \left ((-i A-B) \operatorname {Hypergeometric2F1}\left (m,1-n,1+m,\frac {1}{2} (1+i \tan (e+f x))\right )+2 B \operatorname {Hypergeometric2F1}\left (m,-n,1+m,\frac {1}{2} (1+i \tan (e+f x))\right )\right ) (1-i \tan (e+f x))^{-n} (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^n}{f m} \]

input

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

output

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

3.9.51.3 Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, 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.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int (a+i a \tan (e+f x))^m (A+B \tan (e+f x)) (c-i c \tan (e+f x))^ndx\)

\(\Big \downarrow \) 4071

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

\(\Big \downarrow \) 88

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

\(\Big \downarrow \) 80

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

\(\Big \downarrow \) 79

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

input

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

output

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

3.9.51.3.1 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]
 

3.9.51.4 Maple [F]

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

input

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

output

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

3.9.51.5 Fricas [F]

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

input

integrate((a+I*a*tan(f*x+e))^m*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^n,x, al 
gorithm="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))^n*e^(2*I*f*m*x + 2*I*e*m + m*log(a/c) + m*log(2*c/(e^(2*I*f*x + 2 
*I*e) + 1)))/(e^(2*I*f*x + 2*I*e) + 1), x)
 

3.9.51.6 Sympy [F]

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

input

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

output

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

3.9.51.7 Maxima [F]

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

input

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

output

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

3.9.51.8 Giac [F]

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

input

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

output

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

3.9.51.9 Mupad [F(-1)]

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

input

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

output

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