\(\int \frac {\tan ^m(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx\) [705]

Optimal result

Integrand size = 23, antiderivative size = 196 \[ \int \frac {\tan ^m(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=-\frac {\operatorname {AppellF1}\left (-\frac {1}{2},-m,1,\frac {1}{2},\frac {a+b \tan (c+d x)}{a},\frac {a+b \tan (c+d x)}{a-i b}\right ) \tan ^m(c+d x) \left (-\frac {b \tan (c+d x)}{a}\right )^{-m}}{(i a+b) d \sqrt {a+b \tan (c+d x)}}+\frac {\operatorname {AppellF1}\left (-\frac {1}{2},-m,1,\frac {1}{2},\frac {a+b \tan (c+d x)}{a},\frac {a+b \tan (c+d x)}{a+i b}\right ) \tan ^m(c+d x) \left (-\frac {b \tan (c+d x)}{a}\right )^{-m}}{(i a-b) d \sqrt {a+b \tan (c+d x)}} \] Output:

-AppellF1(-1/2,-m,1,1/2,(a+b*tan(d*x+c))/a,(a+b*tan(d*x+c))/(a-I*b))*tan(d 
*x+c)^m/(I*a+b)/d/((-b*tan(d*x+c)/a)^m)/(a+b*tan(d*x+c))^(1/2)+AppellF1(-1 
/2,-m,1,1/2,(a+b*tan(d*x+c))/a,(a+b*tan(d*x+c))/(a+I*b))*tan(d*x+c)^m/(I*a 
-b)/d/((-b*tan(d*x+c)/a)^m)/(a+b*tan(d*x+c))^(1/2)
 

Mathematica [F]

\[ \int \frac {\tan ^m(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\int \frac {\tan ^m(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx \] Input:

Integrate[Tan[c + d*x]^m/(a + b*Tan[c + d*x])^(3/2),x]
 

Output:

Integrate[Tan[c + d*x]^m/(a + b*Tan[c + d*x])^(3/2), x]
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.90, 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

\(\displaystyle \frac {\frac {\tan ^{m+1}(c+d x) \sqrt {\frac {b \tan (c+d x)}{a}+1} \operatorname {AppellF1}\left (m+1,\frac {3}{2},1,m+2,-\frac {b \tan (c+d x)}{a},-i \tan (c+d x)\right )}{2 a (m+1) \sqrt {a+b \tan (c+d x)}}+\frac {\tan ^{m+1}(c+d x) \sqrt {\frac {b \tan (c+d x)}{a}+1} \operatorname {AppellF1}\left (m+1,\frac {3}{2},1,m+2,-\frac {b \tan (c+d x)}{a},i \tan (c+d x)\right )}{2 a (m+1) \sqrt {a+b \tan (c+d x)}}}{d}\)

Input:

Int[Tan[c + d*x]^m/(a + b*Tan[c + d*x])^(3/2),x]
 

Output:

((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
 

Defintions of rubi rules used

rule 615

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]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4058

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]
 

Maple [F]

\[\int \frac {\tan \left (d x +c \right )^{m}}{\left (a +b \tan \left (d x +c \right )\right )^{\frac {3}{2}}}d x\]

Input:

int(tan(d*x+c)^m/(a+b*tan(d*x+c))^(3/2),x)
 

Output:

int(tan(d*x+c)^m/(a+b*tan(d*x+c))^(3/2),x)
 

Fricas [F]

\[ \int \frac {\tan ^m(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\int { \frac {\tan \left (d x + c\right )^{m}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(tan(d*x+c)^m/(a+b*tan(d*x+c))^(3/2),x, algorithm="fricas")
 

Output:

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)
 

Sympy [F]

\[ \int \frac {\tan ^m(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\int \frac {\tan ^{m}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(tan(d*x+c)**m/(a+b*tan(d*x+c))**(3/2),x)
 

Output:

Integral(tan(c + d*x)**m/(a + b*tan(c + d*x))**(3/2), x)
 

Maxima [F]

\[ \int \frac {\tan ^m(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\int { \frac {\tan \left (d x + c\right )^{m}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(tan(d*x+c)^m/(a+b*tan(d*x+c))^(3/2),x, algorithm="maxima")
 

Output:

integrate(tan(d*x + c)^m/(b*tan(d*x + c) + a)^(3/2), x)
 

Giac [F]

\[ \int \frac {\tan ^m(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\int { \frac {\tan \left (d x + c\right )^{m}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(tan(d*x+c)^m/(a+b*tan(d*x+c))^(3/2),x, algorithm="giac")
 

Output:

integrate(tan(d*x + c)^m/(b*tan(d*x + c) + a)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\tan ^m(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\int \frac {{\mathrm {tan}\left (c+d\,x\right )}^m}{{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{3/2}} \,d x \] Input:

int(tan(c + d*x)^m/(a + b*tan(c + d*x))^(3/2),x)
 

Output:

int(tan(c + d*x)^m/(a + b*tan(c + d*x))^(3/2), x)
 

Reduce [F]

\[ \int \frac {\tan ^m(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\left (\int \frac {\tan \left (d x +c \right )^{m} \sqrt {a +\tan \left (d x +c \right ) b}}{2 \tan \left (d x +c \right )^{2} b^{2} m -\tan \left (d x +c \right )^{2} b^{2}+4 \tan \left (d x +c \right ) a b m -2 \tan \left (d x +c \right ) a b +2 a^{2} m -a^{2}}d x \right ) \left (2 m -1\right ) \] Input:

int(tan(d*x+c)^m/(a+b*tan(d*x+c))^(3/2),x)
 

Output:

int((tan(c + d*x)**m*sqrt(tan(c + d*x)*b + a))/(2*tan(c + d*x)**2*b**2*m - 
 tan(c + d*x)**2*b**2 + 4*tan(c + d*x)*a*b*m - 2*tan(c + d*x)*a*b + 2*a**2 
*m - a**2),x)*(2*m - 1)