3.5.17 \(\int \frac {e^{-\tanh ^{-1}(a x)} \sqrt {c-a c x}}{x^3} \, dx\) [417]

Optimal. Leaf size=112 \[ -\frac {c \sqrt {1-a^2 x^2}}{2 x^2 \sqrt {c-a c x}}+\frac {7 a c \sqrt {1-a^2 x^2}}{4 x \sqrt {c-a c x}}-\frac {7}{4} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.14, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {6263, 893, 887, 889, 214} \begin {gather*} \frac {7 a c \sqrt {1-a^2 x^2}}{4 x \sqrt {c-a c x}}-\frac {c \sqrt {1-a^2 x^2}}{2 x^2 \sqrt {c-a c x}}-\frac {7}{4} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(c*Sqrt[1 - a^2*x^2])/(x^2*Sqrt[c - a*c*x]) + (7*a*c*Sqrt[1 - a^2*x^2])/(4*x*Sqrt[c - a*c*x]) - (7*a^2*Sq
rt[c]*ArcTanh[(Sqrt[c]*Sqrt[1 - a^2*x^2])/Sqrt[c - a*c*x]])/4

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 6263

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

Rubi steps

\begin {align*} \int \frac {e^{-\tanh ^{-1}(a x)} \sqrt {c-a c x}}{x^3} \, dx &=\frac {\int \frac {(c-a c x)^{3/2}}{x^3 \sqrt {1-a^2 x^2}} \, dx}{c}\\ &=-\frac {c \sqrt {1-a^2 x^2}}{2 x^2 \sqrt {c-a c x}}-\frac {1}{4} (7 a) \int \frac {\sqrt {c-a c x}}{x^2 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {c \sqrt {1-a^2 x^2}}{2 x^2 \sqrt {c-a c x}}+\frac {7 a c \sqrt {1-a^2 x^2}}{4 x \sqrt {c-a c x}}+\frac {1}{8} \left (7 a^2\right ) \int \frac {\sqrt {c-a c x}}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {c \sqrt {1-a^2 x^2}}{2 x^2 \sqrt {c-a c x}}+\frac {7 a c \sqrt {1-a^2 x^2}}{4 x \sqrt {c-a c x}}+\frac {1}{4} \left (7 a^4 c^2\right ) \text {Subst}\left (\int \frac {1}{-a^2 c+a^2 c^2 x^2} \, dx,x,\frac {\sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right )\\ &=-\frac {c \sqrt {1-a^2 x^2}}{2 x^2 \sqrt {c-a c x}}+\frac {7 a c \sqrt {1-a^2 x^2}}{4 x \sqrt {c-a c x}}-\frac {7}{4} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 64, normalized size = 0.57 \begin {gather*} -\frac {c \sqrt {1-a x} \left ((2-7 a x) \sqrt {1+a x}+7 a^2 x^2 \tanh ^{-1}\left (\sqrt {1+a x}\right )\right )}{4 x^2 \sqrt {c-a c x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(c*Sqrt[1 - a*x]*((2 - 7*a*x)*Sqrt[1 + a*x] + 7*a^2*x^2*ArcTanh[Sqrt[1 + a*x]]))/(x^2*Sqrt[c - a*c*x])

________________________________________________________________________________________

Maple [A]
time = 1.15, size = 101, normalized size = 0.90

method	result	size
default	\(\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, \left (7 c \arctanh \left (\frac {\sqrt {\left (a x +1\right ) c}}{\sqrt {c}}\right ) a^{2} x^{2}-7 \sqrt {c}\, \sqrt {\left (a x +1\right ) c}\, a x +2 \sqrt {\left (a x +1\right ) c}\, \sqrt {c}\right )}{4 \sqrt {c}\, \left (a x -1\right ) \sqrt {\left (a x +1\right ) c}\, x^{2}}\)	\(101\)
risch	\(-\frac {\left (7 a^{2} x^{2}+5 a x -2\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right ) c}{4 x^{2} \sqrt {\left (a x +1\right ) c}\, \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}+\frac {7 a^{2} \sqrt {c}\, \arctanh \left (\frac {\sqrt {c x a +c}}{\sqrt {c}}\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right )}{4 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}\)	\(150\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.36, size = 232, normalized size = 2.07 \begin {gather*} \left [\frac {7 \, {\left (a^{3} x^{3} - a^{2} x^{2}\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + a c x + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{a x^{2} - x}\right ) - 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} {\left (7 \, a x - 2\right )}}{8 \, {\left (a x^{3} - x^{2}\right )}}, -\frac {7 \, {\left (a^{3} x^{3} - a^{2} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} {\left (7 \, a x - 2\right )}}{4 \, {\left (a x^{3} - x^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- c \left (a x - 1\right )} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{x^{3} \left (a x + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 0.41, size = 117, normalized size = 1.04 \begin {gather*} \frac {1}{4} \, a^{2} c {\left (\frac {7 \, \arctan \left (\frac {\sqrt {a c x + c}}{\sqrt {-c}}\right )}{\sqrt {-c} c} + \frac {7 \, {\left (a c x + c\right )}^{\frac {3}{2}} - 9 \, \sqrt {a c x + c} c}{a^{2} c^{3} x^{2}}\right )} {\left | c \right |} - \frac {7 \, a^{2} c {\left | c \right |} \arctan \left (\frac {\sqrt {2} \sqrt {c}}{\sqrt {-c}}\right ) + 5 \, \sqrt {2} a^{2} \sqrt {-c} \sqrt {c} {\left | c \right |}}{4 \, \sqrt {-c} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a^2*c*(7*arctan(sqrt(a*c*x + c)/sqrt(-c))/(sqrt(-c)*c) + (7*(a*c*x + c)^(3/2) - 9*sqrt(a*c*x + c)*c)/(a^2*
c^3*x^2))*abs(c) - 1/4*(7*a^2*c*abs(c)*arctan(sqrt(2)*sqrt(c)/sqrt(-c)) + 5*sqrt(2)*a^2*sqrt(-c)*sqrt(c)*abs(c
))/(sqrt(-c)*c)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {1-a^2\,x^2}\,\sqrt {c-a\,c\,x}}{x^3\,\left (a\,x+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________