\(\int e^{2 \coth ^{-1}(a x)} (c-\frac {c}{a x})^p \, dx\) [566]

Optimal result

Integrand size = 22, antiderivative size = 57 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\left (c-\frac {c}{a x}\right )^p x+\frac {(2-p) \left (c-\frac {c}{a x}\right )^p \operatorname {Hypergeometric2F1}\left (1,p,1+p,1-\frac {1}{a x}\right )}{a p} \] Output:

(c-c/a/x)^p*x+(2-p)*(c-c/a/x)^p*hypergeom([1, p],[p+1],1-1/a/x)/a/p
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.81 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\frac {\left (c-\frac {c}{a x}\right )^p \left (a p x-(-2+p) \operatorname {Hypergeometric2F1}\left (1,p,1+p,1-\frac {1}{a x}\right )\right )}{a p} \] Input:

Integrate[E^(2*ArcCoth[a*x])*(c - c/(a*x))^p,x]
 

Output:

((c - c/(a*x))^p*(a*p*x - (-2 + p)*Hypergeometric2F1[1, p, 1 + p, 1 - 1/(a 
*x)]))/(a*p)
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.21, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6717, 6683, 1035, 281, 899, 87, 75}

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.

\(\displaystyle \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx\)

\(\Big \downarrow \) 6717

\(\displaystyle -\int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^pdx\)

\(\Big \downarrow \) 6683

\(\displaystyle -\int \frac {\left (c-\frac {c}{a x}\right )^p (a x+1)}{1-a x}dx\)

\(\Big \downarrow \) 1035

\(\displaystyle -\int \frac {\left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^p}{\frac {1}{x}-a}dx\)

\(\Big \downarrow \) 281

\(\displaystyle \frac {c \int \left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{p-1}dx}{a}\)

\(\Big \downarrow \) 899

\(\displaystyle -\frac {c \int \left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{p-1} x^2d\frac {1}{x}}{a}\)

\(\Big \downarrow \) 87

\(\displaystyle -\frac {c \left ((2-p) \int \left (c-\frac {c}{a x}\right )^{p-1} xd\frac {1}{x}-\frac {a x \left (c-\frac {c}{a x}\right )^p}{c}\right )}{a}\)

\(\Big \downarrow \) 75

\(\displaystyle -\frac {c \left (-\frac {(2-p) \left (c-\frac {c}{a x}\right )^p \operatorname {Hypergeometric2F1}\left (1,p,p+1,1-\frac {1}{a x}\right )}{c p}-\frac {a x \left (c-\frac {c}{a x}\right )^p}{c}\right )}{a}\)

Input:

Int[E^(2*ArcCoth[a*x])*(c - c/(a*x))^p,x]
 

Output:

-((c*(-((a*(c - c/(a*x))^p*x)/c) - ((2 - p)*(c - c/(a*x))^p*Hypergeometric 
2F1[1, p, 1 + p, 1 - 1/(a*x)])/(c*p)))/a)
 

Defintions of rubi rules used

rule 75

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x 
)^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))*Hypergeometric2F1[-m, n + 1, n + 2, 1 + 
 d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (IntegerQ[m] 
 || GtQ[-d/(b*c), 0])
 

rule 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

rule 281

Int[(u_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_ 
Symbol] :> Simp[(b/d)^p   Int[u*(c + d*x^n)^(p + q), x], x] /; FreeQ[{a, b, 
 c, d, n, p, q}, x] && EqQ[b*c - a*d, 0] && IntegerQ[p] &&  !(IntegerQ[q] & 
& SimplerQ[a + b*x^n, c + d*x^n])
 

rule 899

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol 
] :> -Subst[Int[(a + b/x^n)^p*((c + d/x^n)^q/x^2), x], x, 1/x] /; FreeQ[{a, 
 b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0]
 

rule 1035

Int[((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_.) + (b_.)*(x_)^(n_.))^(p_.)*((e_) 
 + (f_.)*(x_)^(n_.))^(r_.), x_Symbol] :> Int[x^(n*(p + r))*(b + a/x^n)^p*(c 
 + d/x^n)^q*(f + e/x^n)^r, x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && EqQ[ 
mn, -n] && IntegerQ[p] && IntegerQ[r]
 

rule 6683

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(u_.)*((c_) + (d_.)/(x_))^(p_), x_Symbol] 
:> Int[u*(c + d/x)^p*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] /; FreeQ[{a, c, 
d, p}, x] && EqQ[c^2 - a^2*d^2, 0] &&  !IntegerQ[p] && IntegerQ[n/2] &&  !G 
tQ[c, 0]
 

rule 6717

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Simp[(-1)^(n/2)   Int[ 
u*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[a, x] && IntegerQ[n/2]
 

Maple [F]

\[\int \frac {\left (a x +1\right ) \left (c -\frac {c}{a x}\right )^{p}}{a x -1}d x\]

Input:

int(1/(a*x-1)*(a*x+1)*(c-c/a/x)^p,x)
 

Output:

int(1/(a*x-1)*(a*x+1)*(c-c/a/x)^p,x)
 

Fricas [F]

\[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\int { \frac {{\left (a x + 1\right )} {\left (c - \frac {c}{a x}\right )}^{p}}{a x - 1} \,d x } \] Input:

integrate(1/(a*x-1)*(a*x+1)*(c-c/a/x)^p,x, algorithm="fricas")
 

Output:

integral((a*x + 1)*((a*c*x - c)/(a*x))^p/(a*x - 1), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 5.10 (sec) , antiderivative size = 269, normalized size of antiderivative = 4.72 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=a \left (\begin {cases} \frac {0^{p} x}{a} + \frac {0^{p} \log {\left (a x - 1 \right )}}{a^{2}} - \frac {a^{- p} c^{p} p x^{2 - p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (2 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, 2 - p \\ 3 - p \end {matrix}\middle | {a x} \right )}}{\Gamma \left (3 - p\right ) \Gamma \left (p + 1\right )} & \text {for}\: \left |{a x}\right | > 1 \\\frac {0^{p} x}{a} + \frac {0^{p} \log {\left (- a x + 1 \right )}}{a^{2}} - \frac {a^{- p} c^{p} p x^{2 - p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (2 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, 2 - p \\ 3 - p \end {matrix}\middle | {a x} \right )}}{\Gamma \left (3 - p\right ) \Gamma \left (p + 1\right )} & \text {otherwise} \end {cases}\right ) + \begin {cases} \frac {0^{p} \log {\left (a x - 1 \right )}}{a} - \frac {a^{- p} c^{p} p x^{1 - p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (1 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, 1 - p \\ 2 - p \end {matrix}\middle | {a x} \right )}}{\Gamma \left (2 - p\right ) \Gamma \left (p + 1\right )} & \text {for}\: \left |{a x}\right | > 1 \\\frac {0^{p} \log {\left (- a x + 1 \right )}}{a} - \frac {a^{- p} c^{p} p x^{1 - p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (1 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, 1 - p \\ 2 - p \end {matrix}\middle | {a x} \right )}}{\Gamma \left (2 - p\right ) \Gamma \left (p + 1\right )} & \text {otherwise} \end {cases} \] Input:

integrate(1/(a*x-1)*(a*x+1)*(c-c/a/x)**p,x)
 

Output:

a*Piecewise((0**p*x/a + 0**p*log(a*x - 1)/a**2 - c**p*p*x**(2 - p)*exp(I*p 
i*p)*gamma(p)*gamma(2 - p)*hyper((1 - p, 2 - p), (3 - p,), a*x)/(a**p*gamm 
a(3 - p)*gamma(p + 1)), Abs(a*x) > 1), (0**p*x/a + 0**p*log(-a*x + 1)/a**2 
 - c**p*p*x**(2 - p)*exp(I*pi*p)*gamma(p)*gamma(2 - p)*hyper((1 - p, 2 - p 
), (3 - p,), a*x)/(a**p*gamma(3 - p)*gamma(p + 1)), True)) + Piecewise((0* 
*p*log(a*x - 1)/a - c**p*p*x**(1 - p)*exp(I*pi*p)*gamma(p)*gamma(1 - p)*hy 
per((1 - p, 1 - p), (2 - p,), a*x)/(a**p*gamma(2 - p)*gamma(p + 1)), Abs(a 
*x) > 1), (0**p*log(-a*x + 1)/a - c**p*p*x**(1 - p)*exp(I*pi*p)*gamma(p)*g 
amma(1 - p)*hyper((1 - p, 1 - p), (2 - p,), a*x)/(a**p*gamma(2 - p)*gamma( 
p + 1)), True))
 

Maxima [F]

\[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\int { \frac {{\left (a x + 1\right )} {\left (c - \frac {c}{a x}\right )}^{p}}{a x - 1} \,d x } \] Input:

integrate(1/(a*x-1)*(a*x+1)*(c-c/a/x)^p,x, algorithm="maxima")
 

Output:

integrate((a*x + 1)*(c - c/(a*x))^p/(a*x - 1), x)
 

Giac [F]

\[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\int { \frac {{\left (a x + 1\right )} {\left (c - \frac {c}{a x}\right )}^{p}}{a x - 1} \,d x } \] Input:

integrate(1/(a*x-1)*(a*x+1)*(c-c/a/x)^p,x, algorithm="giac")
 

Output:

integrate((a*x + 1)*(c - c/(a*x))^p/(a*x - 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\int \frac {{\left (c-\frac {c}{a\,x}\right )}^p\,\left (a\,x+1\right )}{a\,x-1} \,d x \] Input:

int(((c - c/(a*x))^p*(a*x + 1))/(a*x - 1),x)
 

Output:

int(((c - c/(a*x))^p*(a*x + 1))/(a*x - 1), x)
 

Reduce [F]

\[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\frac {\left (a c x -c \right )^{p} x -x^{p} \left (\int \frac {\left (a c x -c \right )^{p}}{x^{p} a x -x^{p}}d x \right ) p +2 x^{p} \left (\int \frac {\left (a c x -c \right )^{p}}{x^{p} a x -x^{p}}d x \right )}{x^{p} a^{p}} \] Input:

int(1/(a*x-1)*(a*x+1)*(c-c/a/x)^p,x)
 

Output:

((a*c*x - c)**p*x - x**p*int((a*c*x - c)**p/(x**p*a*x - x**p),x)*p + 2*x** 
p*int((a*c*x - c)**p/(x**p*a*x - x**p),x))/(x**p*a**p)