Integrand size = 24, antiderivative size = 37 \[ \int (d \cot (e+f x))^n (a+i a \tan (e+f x)) \, dx=-\frac {i a (d \cot (e+f x))^n \operatorname {Hypergeometric2F1}(1,n,1+n,-i \cot (e+f x))}{f n} \] Output:
-I*a*(d*cot(f*x+e))^n*hypergeom([1, n],[1+n],-I*cot(f*x+e))/f/n
Time = 0.14 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00 \[ \int (d \cot (e+f x))^n (a+i a \tan (e+f x)) \, dx=-\frac {i a (d \cot (e+f x))^n \operatorname {Hypergeometric2F1}(1,n,1+n,-i \cot (e+f x))}{f n} \] Input:
Integrate[(d*Cot[e + f*x])^n*(a + I*a*Tan[e + f*x]),x]
Output:
((-I)*a*(d*Cot[e + f*x])^n*Hypergeometric2F1[1, n, 1 + n, (-I)*Cot[e + f*x ]])/(f*n)
Time = 0.33 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3042, 4156, 3042, 4020, 27, 74}
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.
\(\displaystyle \int (a+i a \tan (e+f x)) (d \cot (e+f x))^n \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (a+i a \tan (e+f x)) (d \cot (e+f x))^ndx\) |
\(\Big \downarrow \) 4156 |
\(\displaystyle d \int (d \cot (e+f x))^{n-1} (\cot (e+f x) a+i a)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle d \int \left (-d \tan \left (e+f x+\frac {\pi }{2}\right )\right )^{n-1} \left (i a-a \tan \left (e+f x+\frac {\pi }{2}\right )\right )dx\) |
\(\Big \downarrow \) 4020 |
\(\displaystyle -\frac {i a^2 d \int \frac {(d \cot (e+f x))^{n-1}}{a (i \cot (e+f x) a+a)}d(a \cot (e+f x))}{f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {i a d \int \frac {(d \cot (e+f x))^{n-1}}{i \cot (e+f x) a+a}d(a \cot (e+f x))}{f}\) |
\(\Big \downarrow \) 74 |
\(\displaystyle -\frac {i a (d \cot (e+f x))^n \operatorname {Hypergeometric2F1}(1,n,n+1,-i \cot (e+f x))}{f n}\) |
Input:
Int[(d*Cot[e + f*x])^n*(a + I*a*Tan[e + f*x]),x]
Output:
((-I)*a*(d*Cot[e + f*x])^n*Hypergeometric2F1[1, n, 1 + n, (-I)*Cot[e + f*x ]])/(f*n)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x )^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] && !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[c*(d/f) Subst[Int[(a + (b/d)*x)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[ b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]
Int[(cot[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x _)]^(n_.))^(p_.), x_Symbol] :> Simp[d^(n*p) Int[(d*Cot[e + f*x])^(m - n*p )*(b + a*Cot[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x] && !IntegerQ[m] && IntegersQ[n, p]
\[\int \left (d \cot \left (f x +e \right )\right )^{n} \left (a +i a \tan \left (f x +e \right )\right )d x\]
Input:
int((d*cot(f*x+e))^n*(a+I*a*tan(f*x+e)),x)
Output:
int((d*cot(f*x+e))^n*(a+I*a*tan(f*x+e)),x)
\[ \int (d \cot (e+f x))^n (a+i a \tan (e+f x)) \, dx=\int { {\left (i \, a \tan \left (f x + e\right ) + a\right )} \left (d \cot \left (f x + e\right )\right )^{n} \,d x } \] Input:
integrate((d*cot(f*x+e))^n*(a+I*a*tan(f*x+e)),x, algorithm="fricas")
Output:
integral(2*a*((I*d*e^(2*I*f*x + 2*I*e) + I*d)/(e^(2*I*f*x + 2*I*e) - 1))^n *e^(2*I*f*x + 2*I*e)/(e^(2*I*f*x + 2*I*e) + 1), x)
\[ \int (d \cot (e+f x))^n (a+i a \tan (e+f x)) \, dx=i a \left (\int \left (- i \left (d \cot {\left (e + f x \right )}\right )^{n}\right )\, dx + \int \left (d \cot {\left (e + f x \right )}\right )^{n} \tan {\left (e + f x \right )}\, dx\right ) \] Input:
integrate((d*cot(f*x+e))**n*(a+I*a*tan(f*x+e)),x)
Output:
I*a*(Integral(-I*(d*cot(e + f*x))**n, x) + Integral((d*cot(e + f*x))**n*ta n(e + f*x), x))
\[ \int (d \cot (e+f x))^n (a+i a \tan (e+f x)) \, dx=\int { {\left (i \, a \tan \left (f x + e\right ) + a\right )} \left (d \cot \left (f x + e\right )\right )^{n} \,d x } \] Input:
integrate((d*cot(f*x+e))^n*(a+I*a*tan(f*x+e)),x, algorithm="maxima")
Output:
integrate((I*a*tan(f*x + e) + a)*(d*cot(f*x + e))^n, x)
\[ \int (d \cot (e+f x))^n (a+i a \tan (e+f x)) \, dx=\int { {\left (i \, a \tan \left (f x + e\right ) + a\right )} \left (d \cot \left (f x + e\right )\right )^{n} \,d x } \] Input:
integrate((d*cot(f*x+e))^n*(a+I*a*tan(f*x+e)),x, algorithm="giac")
Output:
integrate((I*a*tan(f*x + e) + a)*(d*cot(f*x + e))^n, x)
Timed out. \[ \int (d \cot (e+f x))^n (a+i a \tan (e+f x)) \, dx=\int {\left (d\,\mathrm {cot}\left (e+f\,x\right )\right )}^n\,\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right ) \,d x \] Input:
int((d*cot(e + f*x))^n*(a + a*tan(e + f*x)*1i),x)
Output:
int((d*cot(e + f*x))^n*(a + a*tan(e + f*x)*1i), x)
\[ \int (d \cot (e+f x))^n (a+i a \tan (e+f x)) \, dx=d^{n} a \left (\int \cot \left (f x +e \right )^{n}d x +\left (\int \cot \left (f x +e \right )^{n} \tan \left (f x +e \right )d x \right ) i \right ) \] Input:
int((d*cot(f*x+e))^n*(a+I*a*tan(f*x+e)),x)
Output:
d**n*a*(int(cot(e + f*x)**n,x) + int(cot(e + f*x)**n*tan(e + f*x),x)*i)