\(\int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^3} \, dx\) [347]

Optimal result

Integrand size = 23, antiderivative size = 106 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^3} \, dx=\frac {\sqrt {c-a c x}}{2 x^2}-\frac {9 a \sqrt {c-a c x}}{4 x}+\frac {23}{4} a^2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )-4 \sqrt {2} a^2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right ) \]

[Out]

23/4*a^2*arctanh((-a*c*x+c)^(1/2)/c^(1/2))*c^(1/2)-4*a^2*arctanh(1/2*(-a*c*x+c)^(1/2)*2^(1/2)/c^(1/2))*2^(1/2)
*c^(1/2)+1/2*(-a*c*x+c)^(1/2)/x^2-9/4*a*(-a*c*x+c)^(1/2)/x

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {6302, 6265, 21, 100, 156, 162, 65, 214, 212} \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^3} \, dx=\frac {23}{4} a^2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )-4 \sqrt {2} a^2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )+\frac {\sqrt {c-a c x}}{2 x^2}-\frac {9 a \sqrt {c-a c x}}{4 x} \]

[In]

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

[Out]

Sqrt[c - a*c*x]/(2*x^2) - (9*a*Sqrt[c - a*c*x])/(4*x) + (23*a^2*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]])/4 -
4*Sqrt[2]*a^2*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])]

Rule 21

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

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 100

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

Rule 156

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -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 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 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 6265

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, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] &&  !(IntegerQ[p] || 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-a c x}}{x^3} \, dx \\ & = -\int \frac {(1-a x) \sqrt {c-a c x}}{x^3 (1+a x)} \, dx \\ & = -\frac {\int \frac {(c-a c x)^{3/2}}{x^3 (1+a x)} \, dx}{c} \\ & = \frac {\sqrt {c-a c x}}{2 x^2}+\frac {\int \frac {\frac {9 a c^2}{2}-\frac {7}{2} a^2 c^2 x}{x^2 (1+a x) \sqrt {c-a c x}} \, dx}{2 c} \\ & = \frac {\sqrt {c-a c x}}{2 x^2}-\frac {9 a \sqrt {c-a c x}}{4 x}-\frac {\int \frac {\frac {23 a^2 c^3}{4}-\frac {9}{4} a^3 c^3 x}{x (1+a x) \sqrt {c-a c x}} \, dx}{2 c^2} \\ & = \frac {\sqrt {c-a c x}}{2 x^2}-\frac {9 a \sqrt {c-a c x}}{4 x}-\frac {1}{8} \left (23 a^2 c\right ) \int \frac {1}{x \sqrt {c-a c x}} \, dx+\left (4 a^3 c\right ) \int \frac {1}{(1+a x) \sqrt {c-a c x}} \, dx \\ & = \frac {\sqrt {c-a c x}}{2 x^2}-\frac {9 a \sqrt {c-a c x}}{4 x}+\frac {1}{4} (23 a) \text {Subst}\left (\int \frac {1}{\frac {1}{a}-\frac {x^2}{a c}} \, dx,x,\sqrt {c-a c x}\right )-\left (8 a^2\right ) \text {Subst}\left (\int \frac {1}{2-\frac {x^2}{c}} \, dx,x,\sqrt {c-a c x}\right ) \\ & = \frac {\sqrt {c-a c x}}{2 x^2}-\frac {9 a \sqrt {c-a c x}}{4 x}+\frac {23}{4} a^2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )-4 \sqrt {2} a^2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.88 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^3} \, dx=\frac {(2-9 a x) \sqrt {c-a c x}}{4 x^2}+\frac {23}{4} a^2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )-4 \sqrt {2} a^2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right ) \]

[In]

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

[Out]

((2 - 9*a*x)*Sqrt[c - a*c*x])/(4*x^2) + (23*a^2*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]])/4 - 4*Sqrt[2]*a^2*Sq
rt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])]

Maple [A] (verified)

Time = 0.62 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.75

method	result	size
pseudoelliptic	\(-\frac {\sqrt {-c \left (a x -1\right )}\, \left (9 a x -2\right ) \sqrt {c}+a^{2} c \,x^{2} \left (16 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-c \left (a x -1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )-23 \,\operatorname {arctanh}\left (\frac {\sqrt {-c \left (a x -1\right )}}{\sqrt {c}}\right )\right )}{4 \sqrt {c}\, x^{2}}\)	\(80\)
risch	\(\frac {\left (9 a^{2} x^{2}-11 a x +2\right ) c}{4 x^{2} \sqrt {-c \left (a x -1\right )}}-\frac {a^{2} \left (-\frac {46 \,\operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right )}{\sqrt {c}}+\frac {32 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{\sqrt {c}}\right ) c}{8}\)	\(84\)
derivativedivides	\(2 c^{2} a^{2} \left (\frac {\frac {\frac {9 \left (-a c x +c \right )^{\frac {3}{2}}}{8}-\frac {7 c \sqrt {-a c x +c}}{8}}{a^{2} c^{2} x^{2}}+\frac {23 \,\operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right )}{8 \sqrt {c}}}{c}-\frac {2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{c^{\frac {3}{2}}}\right )\)	\(94\)
default	\(2 c^{2} a^{2} \left (\frac {\frac {\frac {9 \left (-a c x +c \right )^{\frac {3}{2}}}{8}-\frac {7 c \sqrt {-a c x +c}}{8}}{a^{2} c^{2} x^{2}}+\frac {23 \,\operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right )}{8 \sqrt {c}}}{c}-\frac {2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{c^{\frac {3}{2}}}\right )\)	\(94\)

[In]

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

[Out]

-1/4/c^(1/2)*((-c*(a*x-1))^(1/2)*(9*a*x-2)*c^(1/2)+a^2*c*x^2*(16*2^(1/2)*arctanh(1/2*(-c*(a*x-1))^(1/2)*2^(1/2
)/c^(1/2))-23*arctanh((-c*(a*x-1))^(1/2)/c^(1/2))))/x^2

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.92 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^3} \, dx=\left [\frac {16 \, \sqrt {2} a^{2} \sqrt {c} x^{2} \log \left (\frac {a c x + 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) + 23 \, a^{2} \sqrt {c} x^{2} \log \left (\frac {a c x - 2 \, \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{x}\right ) - 2 \, \sqrt {-a c x + c} {\left (9 \, a x - 2\right )}}{8 \, x^{2}}, \frac {16 \, \sqrt {2} a^{2} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{2 \, c}\right ) - 23 \, a^{2} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {-c}}{c}\right ) - \sqrt {-a c x + c} {\left (9 \, a x - 2\right )}}{4 \, x^{2}}\right ] \]

[In]

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

[Out]

[1/8*(16*sqrt(2)*a^2*sqrt(c)*x^2*log((a*c*x + 2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a*x + 1)) + 23*a^2*sq
rt(c)*x^2*log((a*c*x - 2*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/x) - 2*sqrt(-a*c*x + c)*(9*a*x - 2))/x^2, 1/4*(16*sqr
t(2)*a^2*sqrt(-c)*x^2*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(-c)/c) - 23*a^2*sqrt(-c)*x^2*arctan(sqrt(-a*c*x
+ c)*sqrt(-c)/c) - sqrt(-a*c*x + c)*(9*a*x - 2))/x^2]

Sympy [F]

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

[In]

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

[Out]

Integral(sqrt(-c*(a*x - 1))*(a*x - 1)/(x**3*(a*x + 1)), x)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.43 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^3} \, dx=\frac {1}{8} \, a^{2} c^{2} {\left (\frac {2 \, {\left (9 \, {\left (-a c x + c\right )}^{\frac {3}{2}} - 7 \, \sqrt {-a c x + c} c\right )}}{{\left (a c x - c\right )}^{2} c + 2 \, {\left (a c x - c\right )} c^{2} + c^{3}} + \frac {16 \, \sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right )}{c^{\frac {3}{2}}} - \frac {23 \, \log \left (\frac {\sqrt {-a c x + c} - \sqrt {c}}{\sqrt {-a c x + c} + \sqrt {c}}\right )}{c^{\frac {3}{2}}}\right )} \]

[In]

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

[Out]

1/8*a^2*c^2*(2*(9*(-a*c*x + c)^(3/2) - 7*sqrt(-a*c*x + c)*c)/((a*c*x - c)^2*c + 2*(a*c*x - c)*c^2 + c^3) + 16*
sqrt(2)*log(-(sqrt(2)*sqrt(c) - sqrt(-a*c*x + c))/(sqrt(2)*sqrt(c) + sqrt(-a*c*x + c)))/c^(3/2) - 23*log((sqrt
(-a*c*x + c) - sqrt(c))/(sqrt(-a*c*x + c) + sqrt(c)))/c^(3/2))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^3} \, dx=\frac {4 \, \sqrt {2} a^{2} c \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{\sqrt {-c}} - \frac {23 \, a^{2} c \arctan \left (\frac {\sqrt {-a c x + c}}{\sqrt {-c}}\right )}{4 \, \sqrt {-c}} + \frac {9 \, {\left (-a c x + c\right )}^{\frac {3}{2}} a^{2} c - 7 \, \sqrt {-a c x + c} a^{2} c^{2}}{4 \, a^{2} c^{2} x^{2}} \]

[In]

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

[Out]

4*sqrt(2)*a^2*c*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)/sqrt(-c))/sqrt(-c) - 23/4*a^2*c*arctan(sqrt(-a*c*x + c)/sq
rt(-c))/sqrt(-c) + 1/4*(9*(-a*c*x + c)^(3/2)*a^2*c - 7*sqrt(-a*c*x + c)*a^2*c^2)/(a^2*c^2*x^2)

Mupad [B] (verification not implemented)

Time = 4.02 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.83 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^3} \, dx=\frac {9\,{\left (c-a\,c\,x\right )}^{3/2}}{4\,c\,x^2}-\frac {a^2\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{\sqrt {c}}\right )\,23{}\mathrm {i}}{4}-\frac {7\,\sqrt {c-a\,c\,x}}{4\,x^2}+\sqrt {2}\,a^2\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{2\,\sqrt {c}}\right )\,4{}\mathrm {i} \]

[In]

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

[Out]

(9*(c - a*c*x)^(3/2))/(4*c*x^2) - (a^2*c^(1/2)*atan(((c - a*c*x)^(1/2)*1i)/c^(1/2))*23i)/4 - (7*(c - a*c*x)^(1
/2))/(4*x^2) + 2^(1/2)*a^2*c^(1/2)*atan((2^(1/2)*(c - a*c*x)^(1/2)*1i)/(2*c^(1/2)))*4i