\(\int \frac {(a+i a \tan (e+f x))^m}{c+d \tan (e+f x)} \, dx\) [1183]

Optimal result

Integrand size = 28, antiderivative size = 122 \[ \int \frac {(a+i a \tan (e+f x))^m}{c+d \tan (e+f x)} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (1,m,1+m,\frac {1}{2} (1+i \tan (e+f x))\right ) (a+i a \tan (e+f x))^m}{2 (i c+d) f m}-\frac {d \operatorname {Hypergeometric2F1}\left (1,m,1+m,-\frac {d (1+i \tan (e+f x))}{i c-d}\right ) (a+i a \tan (e+f x))^m}{\left (c^2+d^2\right ) f m} \] Output:

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

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.89 \[ \int \frac {(a+i a \tan (e+f x))^m}{c+d \tan (e+f x)} \, dx=\frac {\left ((c+i d) \operatorname {Hypergeometric2F1}\left (1,m,1+m,\frac {1}{2} (1+i \tan (e+f x))\right )-2 i d \operatorname {Hypergeometric2F1}\left (1,m,1+m,\frac {d (1+i \tan (e+f x))}{-i c+d}\right )\right ) (a+i a \tan (e+f x))^m}{2 (c+i d) (i c+d) f m} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.16, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3042, 4045, 3042, 3962, 78, 4082, 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.

\(\displaystyle \int \frac {(a+i a \tan (e+f x))^m}{c+d \tan (e+f x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a+i a \tan (e+f x))^m}{c+d \tan (e+f x)}dx\)

\(\Big \downarrow \) 4045

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3962

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

\(\Big \downarrow \) 78

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

\(\Big \downarrow \) 4082

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

\(\Big \downarrow \) 78

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

Input:

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

Output:

((-I)*d*Hypergeometric2F1[1, m, 1 + m, -((d*(1 + I*Tan[e + f*x]))/(I*c - d 
))]*(a + I*a*Tan[e + f*x])^m)/((I*c - d)*(c - I*d)*f*m) - ((I/2)*Hypergeom 
etric2F1[1, m, 1 + m, (a + I*a*Tan[e + f*x])/(2*a)]*(a + I*a*Tan[e + f*x]) 
^m)/((c - I*d)*f*m)
 

Defintions of rubi rules used

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 3962

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[-b/d   S 
ubst[Int[(a + x)^(n - 1)/(a - x), x], x, b*Tan[c + d*x]], x] /; FreeQ[{a, b 
, c, d, n}, x] && EqQ[a^2 + b^2, 0]
 

rule 4045

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)/((c_.) + (d_.)*tan[(e_.) + 
(f_.)*(x_)]), x_Symbol] :> Simp[a/(a*c - b*d)   Int[(a + b*Tan[e + f*x])^m, 
 x], x] - Simp[d/(a*c - b*d)   Int[(a + b*Tan[e + f*x])^m*((b + a*Tan[e + f 
*x])/(c + d*Tan[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && Ne 
Q[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
 

rule 4082

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral((I*a*(tan(e + f*x) - I))**m/(c + d*tan(e + f*x)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(a+i a \tan (e+f x))^m}{c+d \tan (e+f x)} \, dx =\text {Too large to display} \] Input:

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

Output:

( - (tan(e + f*x)*a*i + a)**m*i - int(((tan(e + f*x)*a*i + a)**m*tan(e + f 
*x)**2)/(tan(e + f*x)*c*d*i*m + tan(e + f*x)*c*d*i - tan(e + f*x)*d**2*m + 
 c**2*i*m + c**2*i - c*d*m),x)*c*d*f*m**2 - int(((tan(e + f*x)*a*i + a)**m 
*tan(e + f*x)**2)/(tan(e + f*x)*c*d*i*m + tan(e + f*x)*c*d*i - tan(e + f*x 
)*d**2*m + c**2*i*m + c**2*i - c*d*m),x)*c*d*f*m - int(((tan(e + f*x)*a*i 
+ a)**m*tan(e + f*x)**2)/(tan(e + f*x)*c*d*i*m + tan(e + f*x)*c*d*i - tan( 
e + f*x)*d**2*m + c**2*i*m + c**2*i - c*d*m),x)*d**2*f*i*m**2 - int(((tan( 
e + f*x)*a*i + a)**m*tan(e + f*x))/(tan(e + f*x)*c*d*i*m + tan(e + f*x)*c* 
d*i - tan(e + f*x)*d**2*m + c**2*i*m + c**2*i - c*d*m),x)*c**2*f*m**2 - in 
t(((tan(e + f*x)*a*i + a)**m*tan(e + f*x))/(tan(e + f*x)*c*d*i*m + tan(e + 
 f*x)*c*d*i - tan(e + f*x)*d**2*m + c**2*i*m + c**2*i - c*d*m),x)*c**2*f*m 
 - 2*int(((tan(e + f*x)*a*i + a)**m*tan(e + f*x))/(tan(e + f*x)*c*d*i*m + 
tan(e + f*x)*c*d*i - tan(e + f*x)*d**2*m + c**2*i*m + c**2*i - c*d*m),x)*c 
*d*f*i*m**2 - int(((tan(e + f*x)*a*i + a)**m*tan(e + f*x))/(tan(e + f*x)*c 
*d*i*m + tan(e + f*x)*c*d*i - tan(e + f*x)*d**2*m + c**2*i*m + c**2*i - c* 
d*m),x)*c*d*f*i*m + int(((tan(e + f*x)*a*i + a)**m*tan(e + f*x))/(tan(e + 
f*x)*c*d*i*m + tan(e + f*x)*c*d*i - tan(e + f*x)*d**2*m + c**2*i*m + c**2* 
i - c*d*m),x)*d**2*f*m**2)/(c*f*m)