\(\int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^n \, dx\) [886]

Optimal result

Integrand size = 23, antiderivative size = 157 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^n \, dx=-\frac {\operatorname {AppellF1}\left (-\frac {1}{2},-n,1,\frac {1}{2},-\frac {b \tan (c+d x)}{a},-i \tan (c+d x)\right ) \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^n \left (\frac {a+b \tan (c+d x)}{a}\right )^{-n}}{d}-\frac {\operatorname {AppellF1}\left (-\frac {1}{2},-n,1,\frac {1}{2},-\frac {b \tan (c+d x)}{a},i \tan (c+d x)\right ) \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^n \left (\frac {a+b \tan (c+d x)}{a}\right )^{-n}}{d} \] Output:

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.10, 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.

\(\displaystyle \int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^n \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cot (c+d x)^{3/2} (a+b \tan (c+d x))^ndx\)

\(\Big \downarrow \) 4729

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {(a+b \tan (c+d x))^n}{\tan ^{\frac {3}{2}}(c+d x)}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {(a+b \tan (c+d x))^n}{\tan (c+d x)^{3/2}}dx\)

\(\Big \downarrow \) 4058

\(\displaystyle \frac {\sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {(a+b \tan (c+d x))^n}{\tan ^{\frac {3}{2}}(c+d x) \left (\tan ^2(c+d x)+1\right )}d\tan (c+d x)}{d}\)

\(\Big \downarrow \) 615

\(\displaystyle \frac {\sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \left (\frac {i (a+b \tan (c+d x))^n}{2 \tan ^{\frac {3}{2}}(c+d x) (\tan (c+d x)+i)}+\frac {i (a+b \tan (c+d x))^n}{2 (i-\tan (c+d x)) \tan ^{\frac {3}{2}}(c+d x)}\right )d\tan (c+d x)}{d}\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-((AppellF1[-1/2, 1, -n, 1/2, (-I) 
*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*(a + b*Tan[c + d*x])^n)/(Sqrt[Tan[c 
+ d*x]]*(1 + (b*Tan[c + d*x])/a)^n)) - (AppellF1[-1/2, 1, -n, 1/2, I*Tan[c 
 + d*x], -((b*Tan[c + d*x])/a)]*(a + b*Tan[c + d*x])^n)/(Sqrt[Tan[c + d*x] 
]*(1 + (b*Tan[c + d*x])/a)^n)))/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]
 

rule 4729

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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((b*tan(d*x + c) + a)^n*cot(d*x + c)^(3/2), x)
 

Sympy [F(-1)]

Timed out. \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^n \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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