\(\int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx\) [517]

Optimal result

Integrand size = 27, antiderivative size = 164 \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\frac {11 c \sqrt {1-\frac {1}{a^2 x^2}} x}{8 a^2 \sqrt {c-\frac {c}{a x}}}-\frac {11 c \sqrt {1-\frac {1}{a^2 x^2}} x^2}{12 a \sqrt {c-\frac {c}{a x}}}+\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^3}{3 \sqrt {c-\frac {c}{a x}}}-\frac {11 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{8 a^3} \]

[Out]

-11/8*arctanh(c^(1/2)*(1-1/a^2/x^2)^(1/2)/(c-c/a/x)^(1/2))*c^(1/2)/a^3+11/8*c*x*(1-1/a^2/x^2)^(1/2)/a^2/(c-c/a
/x)^(1/2)-11/12*c*x^2*(1-1/a^2/x^2)^(1/2)/a/(c-c/a/x)^(1/2)+1/3*c*x^3*(1-1/a^2/x^2)^(1/2)/(c-c/a/x)^(1/2)

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6313, 893, 887, 889, 214} \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=-\frac {11 c x^2 \sqrt {1-\frac {1}{a^2 x^2}}}{12 a \sqrt {c-\frac {c}{a x}}}+\frac {11 c x \sqrt {1-\frac {1}{a^2 x^2}}}{8 a^2 \sqrt {c-\frac {c}{a x}}}+\frac {c x^3 \sqrt {1-\frac {1}{a^2 x^2}}}{3 \sqrt {c-\frac {c}{a x}}}-\frac {11 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{8 a^3} \]

[In]

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

[Out]

(11*c*Sqrt[1 - 1/(a^2*x^2)]*x)/(8*a^2*Sqrt[c - c/(a*x)]) - (11*c*Sqrt[1 - 1/(a^2*x^2)]*x^2)/(12*a*Sqrt[c - c/(
a*x)]) + (c*Sqrt[1 - 1/(a^2*x^2)]*x^3)/(3*Sqrt[c - c/(a*x)]) - (11*Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[1 - 1/(a^2*x^
2)])/Sqrt[c - c/(a*x)]])/(8*a^3)

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 887

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

Rule 889

Int[Sqrt[(d_) + (e_.)*(x_)]/(((f_.) + (g_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[2*e^2, Subst[I
nt[1/(c*(e*f + d*g) + e^2*g*x^2), x], x, Sqrt[a + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x] &&
NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0]

Rule 893

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

Rule 6313

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[-c^n, Subst[Int[(c +
d*x)^(p - n)*((1 - x^2/a^2)^(n/2)/x^(m + 2)), x], x, 1/x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] &&
IntegerQ[(n - 1)/2] && IntegerQ[m] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p, n/2 + 1] || LtQ[-5, m, -1]) && In
tegerQ[2*p]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{3/2}}{x^4 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{c} \\ & = \frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^3}{3 \sqrt {c-\frac {c}{a x}}}+\frac {11 \text {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}}}{x^3 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{6 a} \\ & = -\frac {11 c \sqrt {1-\frac {1}{a^2 x^2}} x^2}{12 a \sqrt {c-\frac {c}{a x}}}+\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^3}{3 \sqrt {c-\frac {c}{a x}}}-\frac {11 \text {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{8 a^2} \\ & = \frac {11 c \sqrt {1-\frac {1}{a^2 x^2}} x}{8 a^2 \sqrt {c-\frac {c}{a x}}}-\frac {11 c \sqrt {1-\frac {1}{a^2 x^2}} x^2}{12 a \sqrt {c-\frac {c}{a x}}}+\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^3}{3 \sqrt {c-\frac {c}{a x}}}+\frac {11 \text {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}}}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{16 a^3} \\ & = \frac {11 c \sqrt {1-\frac {1}{a^2 x^2}} x}{8 a^2 \sqrt {c-\frac {c}{a x}}}-\frac {11 c \sqrt {1-\frac {1}{a^2 x^2}} x^2}{12 a \sqrt {c-\frac {c}{a x}}}+\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^3}{3 \sqrt {c-\frac {c}{a x}}}+\frac {\left (11 c^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {c}{a^2}+\frac {c^2 x^2}{a^2}} \, dx,x,\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{8 a^5} \\ & = \frac {11 c \sqrt {1-\frac {1}{a^2 x^2}} x}{8 a^2 \sqrt {c-\frac {c}{a x}}}-\frac {11 c \sqrt {1-\frac {1}{a^2 x^2}} x^2}{12 a \sqrt {c-\frac {c}{a x}}}+\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^3}{3 \sqrt {c-\frac {c}{a x}}}-\frac {11 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{8 a^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.90 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.90 \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\frac {\frac {2 a^2 \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} x^2 \left (33-22 a x+8 a^2 x^2\right )}{-1+a x}+33 \sqrt {c} \log (1-a x)-33 \sqrt {c} \log \left (2 a^2 \sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} x^2+c \left (-1-a x+2 a^2 x^2\right )\right )}{48 a^3} \]

[In]

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

[Out]

((2*a^2*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2*(33 - 22*a*x + 8*a^2*x^2))/(-1 + a*x) + 33*Sqrt[c]*Log[1 -
a*x] - 33*Sqrt[c]*Log[2*a^2*Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2 + c*(-1 - a*x + 2*a^2*x^2)])/
(48*a^3)

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.81

method	result	size
default	\(\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (16 a^{\frac {5}{2}} x^{2} \sqrt {\left (a x +1\right ) x}-44 a^{\frac {3}{2}} x \sqrt {\left (a x +1\right ) x}+66 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}-33 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right )\right )}{48 a^{\frac {5}{2}} \left (a x -1\right ) \sqrt {\left (a x +1\right ) x}}\)	\(133\)
risch	\(\frac {\left (8 a^{2} x^{2}-22 a x +33\right ) x \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}{24 a^{2} \left (a x -1\right )}-\frac {11 \ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\left (a x +1\right ) a c x}}{16 a^{2} \sqrt {a^{2} c}\, \left (a x -1\right )}\)	\(160\)

[In]

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

[Out]

1/48*((a*x-1)/(a*x+1))^(1/2)*(a*x+1)*(c*(a*x-1)/a/x)^(1/2)*x*(16*a^(5/2)*x^2*((a*x+1)*x)^(1/2)-44*a^(3/2)*x*((
a*x+1)*x)^(1/2)+66*((a*x+1)*x)^(1/2)*a^(1/2)-33*ln(1/2*(2*((a*x+1)*x)^(1/2)*a^(1/2)+2*a*x+1)/a^(1/2)))/a^(5/2)
/(a*x-1)/((a*x+1)*x)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 337, normalized size of antiderivative = 2.05 \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\left [\frac {33 \, {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (8 \, a^{4} x^{4} - 14 \, a^{3} x^{3} + 11 \, a^{2} x^{2} + 33 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{96 \, {\left (a^{4} x - a^{3}\right )}}, \frac {33 \, {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (8 \, a^{4} x^{4} - 14 \, a^{3} x^{3} + 11 \, a^{2} x^{2} + 33 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{48 \, {\left (a^{4} x - a^{3}\right )}}\right ] \]

[In]

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

[Out]

[1/96*(33*(a*x - 1)*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*(8*a^4*x^4 - 14*a^3*x^3 + 11*a^2*x^2 + 33*a*x)*sqrt(
(a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^4*x - a^3), 1/48*(33*(a*x - 1)*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*(8*a^4*x^4 - 14*
a^3*x^3 + 11*a^2*x^2 + 33*a*x)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^4*x - a^3)]

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[In]

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

[Out]

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