\(\int e^{-\coth ^{-1}(a x)} (c-\frac {c}{a x})^{3/2} \, dx\) [465]

Optimal result

Integrand size = 24, antiderivative size = 140 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=-\frac {2 \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{3/2}}{a \left (1-\frac {1}{a x}\right )^{3/2}}+\frac {\sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{3/2} x}{\left (1-\frac {1}{a x}\right )^{3/2}}-\frac {5 \left (c-\frac {c}{a x}\right )^{3/2} \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )}{a \left (1-\frac {1}{a x}\right )^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6317, 6314, 91, 81, 65, 214} \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=-\frac {5 \text {arctanh}\left (\sqrt {\frac {1}{a x}+1}\right ) \left (c-\frac {c}{a x}\right )^{3/2}}{a \left (1-\frac {1}{a x}\right )^{3/2}}+\frac {x \sqrt {\frac {1}{a x}+1} \left (c-\frac {c}{a x}\right )^{3/2}}{\left (1-\frac {1}{a x}\right )^{3/2}}-\frac {2 \sqrt {\frac {1}{a x}+1} \left (c-\frac {c}{a x}\right )^{3/2}}{a \left (1-\frac {1}{a x}\right )^{3/2}} \]

[In]

Int[(c - c/(a*x))^(3/2)/E^ArcCoth[a*x],x]

[Out]

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

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 81

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

Rule 91

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c - a*d
)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d*e - c*f)*(n + 1))), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 214

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

Rule 6314

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

Rule 6317

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

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.50 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=\frac {c \sqrt {c-\frac {c}{a x}} \left (\sqrt {1+\frac {1}{a x}} (-2+a x)-5 \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )\right )}{a \sqrt {1-\frac {1}{a x}}} \]

[In]

Integrate[(c - c/(a*x))^(3/2)/E^ArcCoth[a*x],x]

[Out]

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

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.84

method	result	size
default	\(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, c \left (-2 a^{\frac {3}{2}} x \sqrt {\left (a x +1\right ) x}+5 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a x +4 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}\right )}{2 a^{\frac {3}{2}} \left (a x -1\right ) \sqrt {\left (a x +1\right ) x}}\)	\(118\)
risch	\(\frac {\left (a^{2} x^{2}-a x -2\right ) c \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}{a \left (a x -1\right )}-\frac {5 \ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right ) c \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\left (a x +1\right ) a c x}}{2 \sqrt {a^{2} c}\, \left (a x -1\right )}\)	\(151\)

[In]

int((c-c/a/x)^(3/2)*((a*x-1)/(a*x+1))^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 315, normalized size of antiderivative = 2.25 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=\left [\frac {5 \, {\left (a c x - c\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (a^{2} c x^{2} - a c x - 2 \, c\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{4 \, {\left (a^{2} x - a\right )}}, \frac {5 \, {\left (a c x - c\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (a^{2} c x^{2} - a c x - 2 \, c\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a^{2} x - a\right )}}\right ] \]

[In]

integrate((c-c/a/x)^(3/2)*((a*x-1)/(a*x+1))^(1/2),x, algorithm="fricas")

[Out]

[1/4*(5*(a*c*x - c)*sqrt(c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^3*x^3 + 3*a^2*x^2 + a*x)*sqrt(c)*sqrt((a*x -
1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 4*(a^2*c*x^2 - a*c*x - 2*c)*sqrt((a*x - 1)/(a*x + 1))*
sqrt((a*c*x - c)/(a*x)))/(a^2*x - a), 1/2*(5*(a*c*x - c)*sqrt(-c)*arctan(2*(a^2*x^2 + a*x)*sqrt(-c)*sqrt((a*x
- 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) + 2*(a^2*c*x^2 - a*c*x - 2*c)*sqrt((a*x - 1
)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^2*x - a)]

Sympy [F(-1)]

Timed out. \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=\text {Timed out} \]

[In]

integrate((c-c/a/x)**(3/2)*((a*x-1)/(a*x+1))**(1/2),x)

[Out]

Timed out

Maxima [F]

\[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=\int { {\left (c - \frac {c}{a x}\right )}^{\frac {3}{2}} \sqrt {\frac {a x - 1}{a x + 1}} \,d x } \]

[In]

integrate((c-c/a/x)^(3/2)*((a*x-1)/(a*x+1))^(1/2),x, algorithm="maxima")

[Out]

integrate((c - c/(a*x))^(3/2)*sqrt((a*x - 1)/(a*x + 1)), x)

Giac [F(-2)]

Exception generated. \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((c-c/a/x)^(3/2)*((a*x-1)/(a*x+1))^(1/2),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. \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=\int {\left (c-\frac {c}{a\,x}\right )}^{3/2}\,\sqrt {\frac {a\,x-1}{a\,x+1}} \,d x \]

[In]

int((c - c/(a*x))^(3/2)*((a*x - 1)/(a*x + 1))^(1/2),x)

[Out]

int((c - c/(a*x))^(3/2)*((a*x - 1)/(a*x + 1))^(1/2), x)