Integrand size = 26, antiderivative size = 81 \[ \int \frac {(a+i a \tan (c+d x))^n}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2 \operatorname {AppellF1}\left (\frac {5}{2},1-n,1,\frac {7}{2},-i \tan (c+d x),i \tan (c+d x)\right ) (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n}{5 d \cot ^{\frac {5}{2}}(c+d x)} \]
2/5*AppellF1(5/2,1-n,1,7/2,-I*tan(d*x+c),I*tan(d*x+c))*(a+I*a*tan(d*x+c))^ n/d/cot(d*x+c)^(5/2)/((1+I*tan(d*x+c))^n)
\[ \int \frac {(a+i a \tan (c+d x))^n}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {(a+i a \tan (c+d x))^n}{\cot ^{\frac {3}{2}}(c+d x)} \, dx \]
Time = 0.44 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.42, 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, 395, 394}
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.
\(\displaystyle \int \frac {(a+i a \tan (c+d x))^n}{\cot ^{\frac {3}{2}}(c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a+i a \tan (c+d x))^n}{\cot (c+d x)^{3/2}}dx\) |
\(\Big \downarrow \) 4729 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \tan ^{\frac {3}{2}}(c+d x) (i \tan (c+d x) a+a)^ndx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \tan (c+d x)^{3/2} (i \tan (c+d x) a+a)^ndx\) |
\(\Big \downarrow \) 4047 |
\(\displaystyle \frac {i a^2 \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int -\frac {\tan ^{\frac {3}{2}}(c+d x) (i \tan (c+d x) a+a)^{n-1}}{a (a-i a \tan (c+d x))}d(i a \tan (c+d x))}{d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {i a^2 \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {\tan ^{\frac {3}{2}}(c+d x) (i \tan (c+d x) a+a)^{n-1}}{a (a-i a \tan (c+d x))}d(i a \tan (c+d x))}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {i a \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {\tan ^{\frac {3}{2}}(c+d x) (i \tan (c+d x) a+a)^{n-1}}{a-i a \tan (c+d x)}d(i a \tan (c+d x))}{d}\) |
\(\Big \downarrow \) 148 |
\(\displaystyle \frac {2 a^2 \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {a^3 \tan ^4(c+d x) \left (a-i a^3 \tan ^2(c+d x)\right )^{n-1}}{i a^2 \tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 a \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {a^4 \tan ^4(c+d x) \left (a-i a^3 \tan ^2(c+d x)\right )^{n-1}}{i a^2 \tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d}\) |
\(\Big \downarrow \) 395 |
\(\displaystyle \frac {2 \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (1-i a^2 \tan ^2(c+d x)\right )^{-n} \left (a-i a^3 \tan ^2(c+d x)\right )^n \int \frac {a^4 \tan ^4(c+d x) \left (1-i a^2 \tan ^2(c+d x)\right )^{n-1}}{i a^2 \tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d}\) |
\(\Big \downarrow \) 394 |
\(\displaystyle \frac {2 i a^5 \tan ^{\frac {11}{2}}(c+d x) \sqrt {\cot (c+d x)} \left (1-i a^2 \tan ^2(c+d x)\right )^{-n} \left (a-i a^3 \tan ^2(c+d x)\right )^n \operatorname {AppellF1}\left (\frac {5}{2},1,1-n,\frac {7}{2},-i a^2 \tan ^2(c+d x),i a^2 \tan ^2(c+d x)\right )}{5 d}\) |
(((2*I)/5)*a^5*AppellF1[5/2, 1, 1 - n, 7/2, (-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]^(11/2)*(a - I*a^3*Tan[c + d*x]^2)^n)/(d*(1 - I*a^2*Tan[c + d*x]^2)^n)
3.8.97.3.1 Defintions of rubi rules used
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_.)*((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]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/2 , -p, -q, 1 + (m + 1)/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; FreeQ[{a, b, c, d, e, m, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, 1] && (Int egerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^ FracPart[p]) Int[(e*x)^m*(1 + b*(x^2/a))^p*(c + d*x^2)^q, x], x] /; FreeQ [{a, b, c, d, e, m, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, 1] && !(IntegerQ[p] || GtQ[a, 0])
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]
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]
\[\int \frac {\left (a +i a \tan \left (d x +c \right )\right )^{n}}{\cot \left (d x +c \right )^{\frac {3}{2}}}d x\]
\[ \int \frac {(a+i a \tan (c+d x))^n}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n}}{\cot \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
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))*(e^(4*I*d*x + 4*I*c) - 2 *e^(2*I*d*x + 2*I*c) + 1)/(e^(4*I*d*x + 4*I*c) + 2*e^(2*I*d*x + 2*I*c) + 1 ), x)
\[ \int \frac {(a+i a \tan (c+d x))^n}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{n}}{\cot ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \]
\[ \int \frac {(a+i a \tan (c+d x))^n}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n}}{\cot \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(a+i a \tan (c+d x))^n}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n}}{\cot \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(a+i a \tan (c+d x))^n}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^n}{{\mathrm {cot}\left (c+d\,x\right )}^{3/2}} \,d x \]