Integrand size = 23, antiderivative size = 159 \[ \int \frac {(a+b \tan (c+d x))^n}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\frac {\operatorname {AppellF1}\left (\frac {5}{2},1,-n,\frac {7}{2},-i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right ) (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {\operatorname {AppellF1}\left (\frac {5}{2},1,-n,\frac {7}{2},i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right ) (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}}{5 d \cot ^{\frac {5}{2}}(c+d x)} \]
1/5*AppellF1(5/2,1,-n,7/2,-I*tan(d*x+c),-b*tan(d*x+c)/a)*(a+b*tan(d*x+c))^ n/d/cot(d*x+c)^(5/2)/((1+b*tan(d*x+c)/a)^n)+1/5*AppellF1(5/2,1,-n,7/2,I*ta n(d*x+c),-b*tan(d*x+c)/a)*(a+b*tan(d*x+c))^n/d/cot(d*x+c)^(5/2)/((1+b*tan( d*x+c)/a)^n)
\[ \int \frac {(a+b \tan (c+d x))^n}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {(a+b \tan (c+d x))^n}{\cot ^{\frac {3}{2}}(c+d x)} \, dx \]
Time = 0.49 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 4729, 3042, 4058, 615, 2009}
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.
\(\displaystyle \int \frac {(a+b \tan (c+d x))^n}{\cot ^{\frac {3}{2}}(c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a+b \tan (c+d x))^n}{\cot (c+d x)^{3/2}}dx\) |
\(\Big \downarrow \) 4729 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^ndx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \tan (c+d x)^{3/2} (a+b \tan (c+d x))^ndx\) |
\(\Big \downarrow \) 4058 |
\(\displaystyle \frac {\sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^n}{\tan ^2(c+d x)+1}d\tan (c+d x)}{d}\) |
\(\Big \downarrow \) 615 |
\(\displaystyle \frac {\sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \left (\frac {i \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^n}{2 (i-\tan (c+d x))}+\frac {i \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^n}{2 (\tan (c+d x)+i)}\right )d\tan (c+d x)}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{5} \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^n \left (\frac {b \tan (c+d x)}{a}+1\right )^{-n} \operatorname {AppellF1}\left (\frac {5}{2},1,-n,\frac {7}{2},-i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right )+\frac {1}{5} \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^n \left (\frac {b \tan (c+d x)}{a}+1\right )^{-n} \operatorname {AppellF1}\left (\frac {5}{2},1,-n,\frac {7}{2},i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right )\right )}{d}\) |
(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*((AppellF1[5/2, 1, -n, 7/2, (-I)*Ta n[c + d*x], -((b*Tan[c + d*x])/a)]*Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x]) ^n)/(5*(1 + (b*Tan[c + d*x])/a)^n) + (AppellF1[5/2, 1, -n, 7/2, I*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^n)/(5 *(1 + (b*Tan[c + d*x])/a)^n)))/d
3.9.89.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, S imp[ff/f Subst[Int[(a + b*ff*x)^m*((c + d*ff*x)^n/(1 + ff^2*x^2)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a *d, 0] && NeQ[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]
\[\int \frac {\left (a +b \tan \left (d x +c \right )\right )^{n}}{\cot \left (d x +c \right )^{\frac {3}{2}}}d x\]
\[ \int \frac {(a+b \tan (c+d x))^n}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (b \tan \left (d x + c\right ) + a\right )}^{n}}{\cot \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(a+b \tan (c+d x))^n}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\left (a + b \tan {\left (c + d x \right )}\right )^{n}}{\cot ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \]
\[ \int \frac {(a+b \tan (c+d x))^n}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (b \tan \left (d x + c\right ) + a\right )}^{n}}{\cot \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(a+b \tan (c+d x))^n}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (b \tan \left (d x + c\right ) + a\right )}^{n}}{\cot \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(a+b \tan (c+d x))^n}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^n}{{\mathrm {cot}\left (c+d\,x\right )}^{3/2}} \,d x \]