\(\int \cot (c+d x) (a+i a \tan (c+d x))^m \, dx\) [330]

Optimal result

Integrand size = 22, antiderivative size = 89 \[ \int \cot (c+d x) (a+i a \tan (c+d x))^m \, dx=\frac {\operatorname {Hypergeometric2F1}\left (1,m,1+m,\frac {1}{2} (1+i \tan (c+d x))\right ) (a+i a \tan (c+d x))^m}{2 d m}-\frac {\operatorname {Hypergeometric2F1}(1,m,1+m,1+i \tan (c+d x)) (a+i a \tan (c+d x))^m}{d m} \] Output:

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.76 \[ \int \cot (c+d x) (a+i a \tan (c+d x))^m \, dx=\frac {\left (\operatorname {Hypergeometric2F1}\left (1,m,1+m,\frac {1}{2} (1+i \tan (c+d x))\right )-2 \operatorname {Hypergeometric2F1}(1,m,1+m,1+i \tan (c+d x))\right ) (a+i a \tan (c+d x))^m}{2 d m} \] Input:

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

Output:

((Hypergeometric2F1[1, m, 1 + m, (1 + I*Tan[c + d*x])/2] - 2*Hypergeometri 
c2F1[1, m, 1 + m, 1 + I*Tan[c + d*x]])*(a + I*a*Tan[c + d*x])^m)/(2*d*m)
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {3042, 4045, 3042, 3962, 78, 4082, 75}

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.

\(\displaystyle \int \cot (c+d x) (a+i a \tan (c+d x))^m \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4045

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3962

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

\(\Big \downarrow \) 78

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

\(\Big \downarrow \) 4082

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

\(\Big \downarrow \) 75

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 75

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

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]

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

Input:

int(cot(d*x+c)*(a+I*a*tan(d*x+c))^m,x)
 

Output:

int(cot(d*x+c)*(a+I*a*tan(d*x+c))^m,x)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

integrate(cot(d*x+c)*(a+I*a*tan(d*x+c))**m,x)
 

Output:

Integral((I*a*(tan(c + d*x) - I))**m*cot(c + d*x), x)
 

Maxima [F]

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

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

Output:

integrate((I*a*tan(d*x + c) + a)^m*cot(d*x + c), x)
 

Giac [F]

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

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

Output:

integrate((I*a*tan(d*x + c) + a)^m*cot(d*x + c), x)
 

Mupad [F(-1)]

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

int(cot(c + d*x)*(a + a*tan(c + d*x)*1i)^m,x)
 

Output:

int(cot(c + d*x)*(a + a*tan(c + d*x)*1i)^m, x)
 

Reduce [F]

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

int(cot(d*x+c)*(a+I*a*tan(d*x+c))^m,x)
                                                                                    
                                                                                    
 

Output:

int((tan(c + d*x)*a*i + a)**m*cot(c + d*x),x)