\(\int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx\) [528]

Optimal result

Integrand size = 27, antiderivative size = 86 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=2 \sqrt {c-\frac {c}{a x}}+2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )-4 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6302, 6268, 25, 445, 457, 86, 162, 65, 214} \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )-4 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )+2 \sqrt {c-\frac {c}{a x}} \]

[In]

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

[Out]

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

Rule 25

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

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 86

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

Rule 162

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

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 445

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

Rule 457

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

Rule 6268

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] &
&  !GtQ[c, 0]

Rule 6302

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

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=2 \sqrt {c-\frac {c}{a x}}+2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )-4 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right ) \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(164\) vs. \(2(69)=138\).

Time = 0.52 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.92

method	result	size
risch	\(2 \sqrt {\frac {c \left (a x -1\right )}{a x}}+\frac {\left (\frac {a \ln \left (\frac {-\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-a c x}\right )}{\sqrt {a^{2} c}}+\frac {2 \sqrt {2}\, \ln \left (\frac {4 c -3 \left (x +\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-3 \left (x +\frac {1}{a}\right ) a c +2 c}}{x +\frac {1}{a}}\right )}{\sqrt {c}}\right ) \sqrt {c \left (a x -1\right ) a x}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}{a x -1}\)	\(165\)
default	\(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (2 \sqrt {\left (a x -1\right ) x}\, \sqrt {\frac {1}{a}}\, a^{\frac {3}{2}} x^{2}-4 \sqrt {\frac {1}{a}}\, \sqrt {a \,x^{2}-x}\, a^{\frac {3}{2}} x^{2}-3 \sqrt {\frac {1}{a}}\, \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a \,x^{2}-2 \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, a -3 a x +1}{a x +1}\right ) \sqrt {a}\, \sqrt {2}\, x^{2}+2 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}\, \sqrt {\frac {1}{a}}+2 \sqrt {\frac {1}{a}}\, \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a \,x^{2}\right )}{x \sqrt {\left (a x -1\right ) x}\, \sqrt {a}\, \sqrt {\frac {1}{a}}}\)	\(228\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 203, normalized size of antiderivative = 2.36 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=\left [2 \, \sqrt {2} \sqrt {c} \log \left (\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} - 3 \, a c x + c}{a x + 1}\right ) + \sqrt {c} \log \left (-2 \, a c x - 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) + 2 \, \sqrt {\frac {a c x - c}{a x}}, 4 \, \sqrt {2} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{2 \, c}\right ) - 2 \, \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right ) + 2 \, \sqrt {\frac {a c x - c}{a x}}\right ] \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (70) = 140\).

Time = 4.28 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.64 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=\begin {cases} - \frac {2 a \left (\frac {c^{2} \operatorname {atan}{\left (\frac {\sqrt {c - \frac {c}{a x}}}{\sqrt {- c}} \right )}}{a \sqrt {- c}} - \frac {2 \sqrt {2} c^{2} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {c - \frac {c}{a x}}}{2 \sqrt {- c}} \right )}}{a \sqrt {- c}} - \frac {c \sqrt {c - \frac {c}{a x}}}{a}\right )}{c} & \text {for}\: \frac {c}{a} \neq 0 \\- \frac {3 a \sqrt {c} \left (\frac {\log {\left (- \frac {2}{x} \right )}}{a} - \frac {\log {\left (2 a + \frac {2}{x} \right )}}{a}\right )}{2} + \frac {\sqrt {c} \log {\left (\frac {a}{x} + \frac {1}{x^{2}} \right )}}{2} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-2*a*(c**2*atan(sqrt(c - c/(a*x))/sqrt(-c))/(a*sqrt(-c)) - 2*sqrt(2)*c**2*atan(sqrt(2)*sqrt(c - c/(
a*x))/(2*sqrt(-c)))/(a*sqrt(-c)) - c*sqrt(c - c/(a*x))/a)/c, Ne(c/a, 0)), (-3*a*sqrt(c)*(log(-2/x)/a - log(2*a
+ 2/x)/a)/2 + sqrt(c)*log(a/x + x**(-2))/2, True))

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[In]

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

[Out]

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