\(\int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx\) [259]

Optimal result

Integrand size = 20, antiderivative size = 136 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=-\frac {a^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x^2}{2 \left (a-\frac {1}{x}\right ) (c-a c x)^{5/2}}+\frac {a^{3/2} \left (1-\frac {1}{a x}\right )^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )}{2 \sqrt {2} \left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}} \]

[Out]

1/4*a^(3/2)*(1-1/a/x)^(5/2)*arctanh(2^(1/2)*(1/x)^(1/2)/a^(1/2)/(1+1/a/x)^(1/2))/(1/x)^(5/2)/(-a*c*x+c)^(5/2)*
2^(1/2)-1/2*a^2*(1-1/a/x)^(5/2)*x^2*(1+1/a/x)^(1/2)/(a-1/x)/(-a*c*x+c)^(5/2)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6311, 6316, 96, 95, 212} \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\frac {a^{3/2} \left (1-\frac {1}{a x}\right )^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )}{2 \sqrt {2} \left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}}-\frac {a^2 x^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}{2 \left (a-\frac {1}{x}\right ) (c-a c x)^{5/2}} \]

[In]

Int[1/(E^ArcCoth[a*x]*(c - a*c*x)^(5/2)),x]

[Out]

-1/2*(a^2*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)]*x^2)/((a - x^(-1))*(c - a*c*x)^(5/2)) + (a^(3/2)*(1 - 1/(a*x))
^(5/2)*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/(2*Sqrt[2]*(x^(-1))^(5/2)*(c - a*c*x)^(5/2
))

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*
x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Dist[n*((d*e - c*f)/((m + 1)*(b*e - a*f
))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 6311

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

Rule 6316

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_), x_Symbol] :> Dist[(-c^p)*x^m*(1/x)^m, S
ubst[Int[(1 + d*(x/c))^p*((1 + x/a)^(n/2)/(x^(m + 2)*(1 - x/a)^(n/2))), x], x, 1/x], x] /; FreeQ[{a, c, d, m,
n, p}, x] && EqQ[c^2 - a^2*d^2, 0] &&  !IntegerQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) &&  !IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\left (1-\frac {1}{a x}\right )^{5/2} x^{5/2}\right ) \int \frac {e^{-\coth ^{-1}(a x)}}{\left (1-\frac {1}{a x}\right )^{5/2} x^{5/2}} \, dx}{(c-a c x)^{5/2}} \\ & = -\frac {\left (1-\frac {1}{a x}\right )^{5/2} \text {Subst}\left (\int \frac {\sqrt {x}}{\left (1-\frac {x}{a}\right )^2 \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}} \\ & = -\frac {a^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x^2}{2 \left (a-\frac {1}{x}\right ) (c-a c x)^{5/2}}+\frac {\left (a \left (1-\frac {1}{a x}\right )^{5/2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{4 \left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}} \\ & = -\frac {a^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x^2}{2 \left (a-\frac {1}{x}\right ) (c-a c x)^{5/2}}+\frac {\left (a \left (1-\frac {1}{a x}\right )^{5/2}\right ) \text {Subst}\left (\int \frac {1}{1-\frac {2 x^2}{a}} \, dx,x,\frac {\sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{a x}}}\right )}{2 \left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}} \\ & = -\frac {a^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x^2}{2 \left (a-\frac {1}{x}\right ) (c-a c x)^{5/2}}+\frac {a^{3/2} \left (1-\frac {1}{a x}\right )^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )}{2 \sqrt {2} \left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.85 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\frac {\sqrt {1-\frac {1}{a x}} x \left (-2 \sqrt {a} \sqrt {1+\frac {1}{a x}}+\sqrt {2} \sqrt {\frac {1}{x}} (-1+a x) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )\right )}{4 \sqrt {a} c^2 (-1+a x) \sqrt {c-a c x}} \]

[In]

Integrate[1/(E^ArcCoth[a*x]*(c - a*c*x)^(5/2)),x]

[Out]

(Sqrt[1 - 1/(a*x)]*x*(-2*Sqrt[a]*Sqrt[1 + 1/(a*x)] + Sqrt[2]*Sqrt[x^(-1)]*(-1 + a*x)*ArcTanh[(Sqrt[2]*Sqrt[x^(
-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])]))/(4*Sqrt[a]*c^2*(-1 + a*x)*Sqrt[c - a*c*x])

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.90

method	result	size
default	\(\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, \left (-\sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) a c x +\sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c +2 \sqrt {-c \left (a x +1\right )}\, \sqrt {c}\right )}{4 c^{\frac {7}{2}} \left (a x -1\right )^{2} \sqrt {-c \left (a x +1\right )}\, a}\)	\(123\)

[In]

int(((a*x-1)/(a*x+1))^(1/2)/(-a*c*x+c)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/4*((a*x-1)/(a*x+1))^(1/2)*(a*x+1)*(-c*(a*x-1))^(1/2)*(-2^(1/2)*arctan(1/2*(-c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2)
)*a*c*x+2^(1/2)*arctan(1/2*(-c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2))*c+2*(-c*(a*x+1))^(1/2)*c^(1/2))/c^(7/2)/(a*x-1)
^2/(-c*(a*x+1))^(1/2)/a

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 281, normalized size of antiderivative = 2.07 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\left [-\frac {\sqrt {2} {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) - 4 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{8 \, {\left (a^{3} c^{3} x^{2} - 2 \, a^{2} c^{3} x + a c^{3}\right )}}, -\frac {\sqrt {2} {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) - 2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{4 \, {\left (a^{3} c^{3} x^{2} - 2 \, a^{2} c^{3} x + a c^{3}\right )}}\right ] \]

[In]

integrate(((a*x-1)/(a*x+1))^(1/2)/(-a*c*x+c)^(5/2),x, algorithm="fricas")

[Out]

[-1/8*(sqrt(2)*(a^2*x^2 - 2*a*x + 1)*sqrt(-c)*log(-(a^2*c*x^2 + 2*a*c*x - 2*sqrt(2)*sqrt(-a*c*x + c)*(a*x + 1)
*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1)) - 3*c)/(a^2*x^2 - 2*a*x + 1)) - 4*sqrt(-a*c*x + c)*(a*x + 1)*sqrt((a*x - 1
)/(a*x + 1)))/(a^3*c^3*x^2 - 2*a^2*c^3*x + a*c^3), -1/4*(sqrt(2)*(a^2*x^2 - 2*a*x + 1)*sqrt(c)*arctan(sqrt(2)*
sqrt(-a*c*x + c)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))/(a*c*x - c)) - 2*sqrt(-a*c*x + c)*(a*x + 1)*sqrt((a*x - 1)/
(a*x + 1)))/(a^3*c^3*x^2 - 2*a^2*c^3*x + a*c^3)]

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate(((a*x-1)/(a*x+1))**(1/2)/(-a*c*x+c)**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\int { \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{{\left (-a c x + c\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(((a*x-1)/(a*x+1))^(1/2)/(-a*c*x+c)^(5/2),x, algorithm="maxima")

[Out]

integrate(sqrt((a*x - 1)/(a*x + 1))/(-a*c*x + c)^(5/2), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(((a*x-1)/(a*x+1))^(1/2)/(-a*c*x+c)^(5/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\int \frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{{\left (c-a\,c\,x\right )}^{5/2}} \,d x \]

[In]

int(((a*x - 1)/(a*x + 1))^(1/2)/(c - a*c*x)^(5/2),x)

[Out]

int(((a*x - 1)/(a*x + 1))^(1/2)/(c - a*c*x)^(5/2), x)