\(\int \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^n \, dx\) [795]

Optimal result

Integrand size = 26, antiderivative size = 79 \[ \int \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^n \, dx=\frac {2 \operatorname {AppellF1}\left (\frac {1}{2},1-n,1,\frac {3}{2},-i \tan (c+d x),i \tan (c+d x)\right ) (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n}{d \sqrt {\cot (c+d x)}} \] Output:

2*AppellF1(1/2,1-n,1,3/2,-I*tan(d*x+c),I*tan(d*x+c))*(a+I*a*tan(d*x+c))^n/ 
d/cot(d*x+c)^(1/2)/((1+I*tan(d*x+c))^n)
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.39 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.41, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3042, 4729, 3042, 4047, 25, 27, 148, 27, 334, 333}

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.

\(\displaystyle \int \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^n \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^ndx\)

\(\Big \downarrow \) 4729

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4047

\(\displaystyle \frac {i a^2 \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int -\frac {(i \tan (c+d x) a+a)^{n-1}}{a \sqrt {\tan (c+d x)} (a-i a \tan (c+d x))}d(i a \tan (c+d x))}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {i a^2 \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {(i \tan (c+d x) a+a)^{n-1}}{a \sqrt {\tan (c+d x)} (a-i a \tan (c+d x))}d(i a \tan (c+d x))}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {i a \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {(i \tan (c+d x) a+a)^{n-1}}{\sqrt {\tan (c+d x)} (a-i a \tan (c+d x))}d(i a \tan (c+d x))}{d}\)

\(\Big \downarrow \) 148

\(\displaystyle \frac {2 a^2 \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {\left (a-i a^3 \tan ^2(c+d x)\right )^{n-1}}{a \left (i a^2 \tan ^2(c+d x)+1\right )}d\sqrt {\tan (c+d x)}}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 a \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {\left (a-i a^3 \tan ^2(c+d x)\right )^{n-1}}{i a^2 \tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d}\)

\(\Big \downarrow \) 334

\(\displaystyle \frac {2 \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (1-i a^2 \tan ^2(c+d x)\right )^{-n} \left (a-i a^3 \tan ^2(c+d x)\right )^n \int \frac {\left (1-i a^2 \tan ^2(c+d x)\right )^{n-1}}{i a^2 \tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d}\)

\(\Big \downarrow \) 333

\(\displaystyle \frac {2 i a \tan ^{\frac {3}{2}}(c+d x) \sqrt {\cot (c+d x)} \left (1-i a^2 \tan ^2(c+d x)\right )^{-n} \left (a-i a^3 \tan ^2(c+d x)\right )^n \operatorname {AppellF1}\left (\frac {1}{2},1,1-n,\frac {3}{2},-i a^2 \tan ^2(c+d x),i a^2 \tan ^2(c+d x)\right )}{d}\)

Input:

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

Output:

((2*I)*a*AppellF1[1/2, 1, 1 - n, 3/2, (-I)*a^2*Tan[c + d*x]^2, I*a^2*Tan[c 
 + d*x]^2]*Sqrt[Cot[c + d*x]]*Tan[c + d*x]^(3/2)*(a - I*a^3*Tan[c + d*x]^2 
)^n)/(d*(1 - I*a^2*Tan[c + d*x]^2)^n)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 148

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))^(p_.), 
x_] :> With[{k = Denominator[m]}, Simp[k/b   Subst[Int[x^(k*(m + 1) - 1)*(c 
 + d*(x^k/b))^n*(e + f*(x^k/b))^p, x], x, (b*x)^(1/k)], x]] /; FreeQ[{b, c, 
 d, e, f, n, p}, x] && FractionQ[m] && IntegerQ[p]
 

rule 333

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[a^p*c^q*x*AppellF1[1/2, -p, -q, 3/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; F 
reeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[p] || GtQ[a, 
0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 334

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[a^IntPart[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p])   Int[ 
(1 + b*(x^2/a))^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, p, q}, x] && 
NeQ[b*c - a*d, 0] &&  !(IntegerQ[p] || GtQ[a, 0])
 

rule 3042

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

rule 4047

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

rule 4729

Int[(cot[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Simp[(c*Cot[a 
+ b*x])^m*(c*Tan[a + b*x])^m   Int[ActivateTrig[u]/(c*Tan[a + b*x])^m, x], 
x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownTangentIntegrandQ[u, 
x]
 

Maple [F]

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

Input:

int(cot(d*x+c)^(1/2)*(a+I*a*tan(d*x+c))^n,x)
 

Output:

int(cot(d*x+c)^(1/2)*(a+I*a*tan(d*x+c))^n,x)
 

Fricas [F]

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

integrate(cot(d*x+c)^(1/2)*(a+I*a*tan(d*x+c))^n,x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate(cot(d*x+c)**(1/2)*(a+I*a*tan(d*x+c))**n,x)
 

Output:

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

Maxima [F]

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

integrate(cot(d*x+c)^(1/2)*(a+I*a*tan(d*x+c))^n,x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate(cot(d*x+c)^(1/2)*(a+I*a*tan(d*x+c))^n,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int(cot(c + d*x)^(1/2)*(a + a*tan(c + d*x)*1i)^n,x)
 

Output:

int(cot(c + d*x)^(1/2)*(a + a*tan(c + d*x)*1i)^n, x)
 

Reduce [F]

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

int(cot(d*x+c)^(1/2)*(a+I*a*tan(d*x+c))^n,x)
 

Output:

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