Integrand size = 26, antiderivative size = 157 \[ \int \frac {(d \cot (e+f x))^n}{a+i a \tan (e+f x)} \, dx=-\frac {(d \cot (e+f x))^{2+n}}{2 d^2 f (i a+a \cot (e+f x))}-\frac {i n (d \cot (e+f x))^{2+n} \operatorname {Hypergeometric2F1}\left (1,\frac {2+n}{2},\frac {4+n}{2},-\cot ^2(e+f x)\right )}{2 a d^2 f (2+n)}+\frac {(1+n) (d \cot (e+f x))^{3+n} \operatorname {Hypergeometric2F1}\left (1,\frac {3+n}{2},\frac {5+n}{2},-\cot ^2(e+f x)\right )}{2 a d^3 f (3+n)} \]
-1/2*(d*cot(f*x+e))^(2+n)/d^2/f/(I*a+a*cot(f*x+e))-1/2*I*n*(d*cot(f*x+e))^ (2+n)*hypergeom([1, 1+1/2*n],[2+1/2*n],-cot(f*x+e)^2)/a/d^2/f/(2+n)+1/2*(1 +n)*(d*cot(f*x+e))^(3+n)*hypergeom([1, 3/2+1/2*n],[5/2+1/2*n],-cot(f*x+e)^ 2)/a/d^3/f/(3+n)
Time = 1.09 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.83 \[ \int \frac {(d \cot (e+f x))^n}{a+i a \tan (e+f x)} \, dx=\frac {\cot ^2(e+f x) (d \cot (e+f x))^n \left (-\frac {1}{i+\cot (e+f x)}+\frac {-i n (3+n) \operatorname {Hypergeometric2F1}\left (1,\frac {2+n}{2},\frac {4+n}{2},-\cot ^2(e+f x)\right )+\left (2+3 n+n^2\right ) \cot (e+f x) \operatorname {Hypergeometric2F1}\left (1,\frac {3+n}{2},\frac {5+n}{2},-\cot ^2(e+f x)\right )}{(2+n) (3+n)}\right )}{2 a f} \]
(Cot[e + f*x]^2*(d*Cot[e + f*x])^n*(-(I + Cot[e + f*x])^(-1) + ((-I)*n*(3 + n)*Hypergeometric2F1[1, (2 + n)/2, (4 + n)/2, -Cot[e + f*x]^2] + (2 + 3* n + n^2)*Cot[e + f*x]*Hypergeometric2F1[1, (3 + n)/2, (5 + n)/2, -Cot[e + f*x]^2])/((2 + n)*(3 + n))))/(2*a*f)
Time = 0.67 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.02, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3042, 4156, 3042, 4035, 25, 3042, 4021, 3042, 3957, 278}
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.
\(\displaystyle \int \frac {(d \cot (e+f x))^n}{a+i a \tan (e+f x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(d \cot (e+f x))^n}{a+i a \tan (e+f x)}dx\) |
\(\Big \downarrow \) 4156 |
\(\displaystyle \frac {\int \frac {(d \cot (e+f x))^{n+1}}{\cot (e+f x) a+i a}dx}{d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {\left (-d \tan \left (e+f x+\frac {\pi }{2}\right )\right )^{n+1}}{i a-a \tan \left (e+f x+\frac {\pi }{2}\right )}dx}{d}\) |
\(\Big \downarrow \) 4035 |
\(\displaystyle \frac {-\frac {\int -(d \cot (e+f x))^{n+1} (i a d n-a d (n+1) \cot (e+f x))dx}{2 a^2 d}-\frac {(d \cot (e+f x))^{n+2}}{2 d f (a \cot (e+f x)+i a)}}{d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {\int (d \cot (e+f x))^{n+1} (i a d n-a d (n+1) \cot (e+f x))dx}{2 a^2 d}-\frac {(d \cot (e+f x))^{n+2}}{2 d f (a \cot (e+f x)+i a)}}{d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\int \left (-d \tan \left (e+f x+\frac {\pi }{2}\right )\right )^{n+1} \left (i a d n+a d (n+1) \tan \left (e+f x+\frac {\pi }{2}\right )\right )dx}{2 a^2 d}-\frac {(d \cot (e+f x))^{n+2}}{2 d f (a \cot (e+f x)+i a)}}{d}\) |
\(\Big \downarrow \) 4021 |
\(\displaystyle \frac {\frac {-a (n+1) \int (d \cot (e+f x))^{n+2}dx+i a d n \int (d \cot (e+f x))^{n+1}dx}{2 a^2 d}-\frac {(d \cot (e+f x))^{n+2}}{2 d f (a \cot (e+f x)+i a)}}{d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {-a (n+1) \int \left (-d \tan \left (e+f x+\frac {\pi }{2}\right )\right )^{n+2}dx+i a d n \int \left (-d \tan \left (e+f x+\frac {\pi }{2}\right )\right )^{n+1}dx}{2 a^2 d}-\frac {(d \cot (e+f x))^{n+2}}{2 d f (a \cot (e+f x)+i a)}}{d}\) |
\(\Big \downarrow \) 3957 |
\(\displaystyle \frac {\frac {\frac {a d (n+1) \int \frac {(d \cot (e+f x))^{n+2}}{\cot ^2(e+f x) d^2+d^2}d(d \cot (e+f x))}{f}-\frac {i a d^2 n \int \frac {(d \cot (e+f x))^{n+1}}{\cot ^2(e+f x) d^2+d^2}d(d \cot (e+f x))}{f}}{2 a^2 d}-\frac {(d \cot (e+f x))^{n+2}}{2 d f (a \cot (e+f x)+i a)}}{d}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle \frac {\frac {\frac {a (n+1) (d \cot (e+f x))^{n+3} \operatorname {Hypergeometric2F1}\left (1,\frac {n+3}{2},\frac {n+5}{2},-\cot ^2(e+f x)\right )}{d f (n+3)}-\frac {i a n (d \cot (e+f x))^{n+2} \operatorname {Hypergeometric2F1}\left (1,\frac {n+2}{2},\frac {n+4}{2},-\cot ^2(e+f x)\right )}{f (n+2)}}{2 a^2 d}-\frac {(d \cot (e+f x))^{n+2}}{2 d f (a \cot (e+f x)+i a)}}{d}\) |
(-1/2*(d*Cot[e + f*x])^(2 + n)/(d*f*(I*a + a*Cot[e + f*x])) + (((-I)*a*n*( d*Cot[e + f*x])^(2 + n)*Hypergeometric2F1[1, (2 + n)/2, (4 + n)/2, -Cot[e + f*x]^2])/(f*(2 + n)) + (a*(1 + n)*(d*Cot[e + f*x])^(3 + n)*Hypergeometri c2F1[1, (3 + n)/2, (5 + n)/2, -Cot[e + f*x]^2])/(d*f*(3 + n)))/(2*a^2*d))/ d
3.8.91.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( -b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && (ILtQ[p, 0 ] || GtQ[a, 0])
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d Subst[Int [x^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[n]
Int[((b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Tan[e + f*x])^m, x], x] + Simp[d/b Int [(b*Tan[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x] && NeQ[c^ 2 + d^2, 0] && !IntegerQ[2*m]
Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-a)*((c + d*Tan[e + f*x])^(n + 1)/(2*f*(b* c - a*d)*(a + b*Tan[e + f*x]))), x] + Simp[1/(2*a*(b*c - a*d)) Int[(c + d *Tan[e + f*x])^n*Simp[b*c + a*d*(n - 1) - b*d*n*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && !GtQ[n, 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]
\[\int \frac {\left (d \cot \left (f x +e \right )\right )^{n}}{a +i a \tan \left (f x +e \right )}d x\]
\[ \int \frac {(d \cot (e+f x))^n}{a+i a \tan (e+f x)} \, dx=\int { \frac {\left (d \cot \left (f x + e\right )\right )^{n}}{i \, a \tan \left (f x + e\right ) + a} \,d x } \]
integral(1/2*((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) + 1)*e^(-2*I*f*x - 2*I*e)/a, x)
\[ \int \frac {(d \cot (e+f x))^n}{a+i a \tan (e+f x)} \, dx=- \frac {i \int \frac {\left (d \cot {\left (e + f x \right )}\right )^{n}}{\tan {\left (e + f x \right )} - i}\, dx}{a} \]
Exception generated. \[ \int \frac {(d \cot (e+f x))^n}{a+i a \tan (e+f x)} \, dx=\text {Exception raised: RuntimeError} \]
\[ \int \frac {(d \cot (e+f x))^n}{a+i a \tan (e+f x)} \, dx=\int { \frac {\left (d \cot \left (f x + e\right )\right )^{n}}{i \, a \tan \left (f x + e\right ) + a} \,d x } \]
Timed out. \[ \int \frac {(d \cot (e+f x))^n}{a+i a \tan (e+f x)} \, dx=\int \frac {{\left (d\,\mathrm {cot}\left (e+f\,x\right )\right )}^n}{a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}} \,d x \]