Integrand size = 23, antiderivative size = 179 \[ \int \frac {\tan ^m(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\frac {\operatorname {AppellF1}\left (1+m,\frac {3}{2},1,2+m,-\frac {b \tan (c+d x)}{a},-i \tan (c+d x)\right ) \tan ^{1+m}(c+d x) \sqrt {1+\frac {b \tan (c+d x)}{a}}}{2 a d (1+m) \sqrt {a+b \tan (c+d x)}}+\frac {\operatorname {AppellF1}\left (1+m,\frac {3}{2},1,2+m,-\frac {b \tan (c+d x)}{a},i \tan (c+d x)\right ) \tan ^{1+m}(c+d x) \sqrt {1+\frac {b \tan (c+d x)}{a}}}{2 a d (1+m) \sqrt {a+b \tan (c+d x)}} \]
1/2*AppellF1(1+m,1,3/2,2+m,-I*tan(d*x+c),-b*tan(d*x+c)/a)*(1+b*tan(d*x+c)/ a)^(1/2)*tan(d*x+c)^(1+m)/a/d/(1+m)/(a+b*tan(d*x+c))^(1/2)+1/2*AppellF1(1+ m,1,3/2,2+m,I*tan(d*x+c),-b*tan(d*x+c)/a)*(1+b*tan(d*x+c)/a)^(1/2)*tan(d*x +c)^(1+m)/a/d/(1+m)/(a+b*tan(d*x+c))^(1/2)
\[ \int \frac {\tan ^m(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\int \frac {\tan ^m(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx \]
Time = 0.37 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {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.
\(\displaystyle \int \frac {\tan ^m(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\tan (c+d x)^m}{(a+b \tan (c+d x))^{3/2}}dx\) |
\(\Big \downarrow \) 4058 |
\(\displaystyle \frac {\int \frac {\tan ^m(c+d x)}{(a+b \tan (c+d x))^{3/2} \left (\tan ^2(c+d x)+1\right )}d\tan (c+d x)}{d}\) |
\(\Big \downarrow \) 615 |
\(\displaystyle \frac {\int \left (\frac {i \tan ^m(c+d x)}{2 (i-\tan (c+d x)) (a+b \tan (c+d x))^{3/2}}+\frac {i \tan ^m(c+d x)}{2 (\tan (c+d x)+i) (a+b \tan (c+d x))^{3/2}}\right )d\tan (c+d x)}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {\tan ^{m+1}(c+d x) \sqrt {\frac {b \tan (c+d x)}{a}+1} \operatorname {AppellF1}\left (m+1,\frac {3}{2},1,m+2,-\frac {b \tan (c+d x)}{a},-i \tan (c+d x)\right )}{2 a (m+1) \sqrt {a+b \tan (c+d x)}}+\frac {\tan ^{m+1}(c+d x) \sqrt {\frac {b \tan (c+d x)}{a}+1} \operatorname {AppellF1}\left (m+1,\frac {3}{2},1,m+2,-\frac {b \tan (c+d x)}{a},i \tan (c+d x)\right )}{2 a (m+1) \sqrt {a+b \tan (c+d x)}}}{d}\) |
((AppellF1[1 + m, 3/2, 1, 2 + m, -((b*Tan[c + d*x])/a), (-I)*Tan[c + d*x]] *Tan[c + d*x]^(1 + m)*Sqrt[1 + (b*Tan[c + d*x])/a])/(2*a*(1 + m)*Sqrt[a + b*Tan[c + d*x]]) + (AppellF1[1 + m, 3/2, 1, 2 + m, -((b*Tan[c + d*x])/a), I*Tan[c + d*x]]*Tan[c + d*x]^(1 + m)*Sqrt[1 + (b*Tan[c + d*x])/a])/(2*a*(1 + m)*Sqrt[a + b*Tan[c + d*x]]))/d
3.8.5.3.1 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]
\[\int \frac {\tan ^{m}\left (d x +c \right )}{\left (a +b \tan \left (d x +c \right )\right )^{\frac {3}{2}}}d x\]
\[ \int \frac {\tan ^m(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\int { \frac {\tan \left (d x + c\right )^{m}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
integral(sqrt(b*tan(d*x + c) + a)*tan(d*x + c)^m/(b^2*tan(d*x + c)^2 + 2*a *b*tan(d*x + c) + a^2), x)
\[ \int \frac {\tan ^m(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\int \frac {\tan ^{m}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\tan ^m(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\int { \frac {\tan \left (d x + c\right )^{m}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\tan ^m(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\tan ^m(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\int \frac {{\mathrm {tan}\left (c+d\,x\right )}^m}{{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{3/2}} \,d x \]