3.418 \(\int \frac{e^{-\tanh ^{-1}(a x)} \sqrt{c-a c x}}{x^4} \, dx\)

Optimal. Leaf size=148 \[ -\frac{11 a^2 c \sqrt{1-a^2 x^2}}{8 x \sqrt{c-a c x}}+\frac{11 a c \sqrt{1-a^2 x^2}}{12 x^2 \sqrt{c-a c x}}-\frac{c \sqrt{1-a^2 x^2}}{3 x^3 \sqrt{c-a c x}}+\frac{11}{8} a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.236645, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {6128, 879, 873, 875, 208} \[ -\frac{11 a^2 c \sqrt{1-a^2 x^2}}{8 x \sqrt{c-a c x}}+\frac{11 a c \sqrt{1-a^2 x^2}}{12 x^2 \sqrt{c-a c x}}-\frac{c \sqrt{1-a^2 x^2}}{3 x^3 \sqrt{c-a c x}}+\frac{11}{8} a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c - a*c*x]/(E^ArcTanh[a*x]*x^4),x]

[Out]

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

Rule 6128

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]

Rule 879

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 873

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] && IntegerQ[
2*p]

Rule 875

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 208

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

Rubi steps

\begin{align*} \int \frac{e^{-\tanh ^{-1}(a x)} \sqrt{c-a c x}}{x^4} \, dx &=\frac{\int \frac{(c-a c x)^{3/2}}{x^4 \sqrt{1-a^2 x^2}} \, dx}{c}\\ &=-\frac{c \sqrt{1-a^2 x^2}}{3 x^3 \sqrt{c-a c x}}-\frac{1}{6} (11 a) \int \frac{\sqrt{c-a c x}}{x^3 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{c \sqrt{1-a^2 x^2}}{3 x^3 \sqrt{c-a c x}}+\frac{11 a c \sqrt{1-a^2 x^2}}{12 x^2 \sqrt{c-a c x}}+\frac{1}{8} \left (11 a^2\right ) \int \frac{\sqrt{c-a c x}}{x^2 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{c \sqrt{1-a^2 x^2}}{3 x^3 \sqrt{c-a c x}}+\frac{11 a c \sqrt{1-a^2 x^2}}{12 x^2 \sqrt{c-a c x}}-\frac{11 a^2 c \sqrt{1-a^2 x^2}}{8 x \sqrt{c-a c x}}-\frac{1}{16} \left (11 a^3\right ) \int \frac{\sqrt{c-a c x}}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{c \sqrt{1-a^2 x^2}}{3 x^3 \sqrt{c-a c x}}+\frac{11 a c \sqrt{1-a^2 x^2}}{12 x^2 \sqrt{c-a c x}}-\frac{11 a^2 c \sqrt{1-a^2 x^2}}{8 x \sqrt{c-a c x}}-\frac{1}{8} \left (11 a^5 c^2\right ) \operatorname{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}}{3 x^3 \sqrt{c-a c x}}+\frac{11 a c \sqrt{1-a^2 x^2}}{12 x^2 \sqrt{c-a c x}}-\frac{11 a^2 c \sqrt{1-a^2 x^2}}{8 x \sqrt{c-a c x}}+\frac{11}{8} a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}\right )\\ \end{align*}

Mathematica [C]  time = 0.0223248, size = 56, normalized size = 0.38 \[ \frac{c \sqrt{1-a^2 x^2} \left (11 a^3 x^3 \text{Hypergeometric2F1}\left (\frac{1}{2},3,\frac{3}{2},a x+1\right )-1\right )}{3 x^3 \sqrt{c-a c x}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sqrt[c - a*c*x]/(E^ArcTanh[a*x]*x^4),x]

[Out]

(c*Sqrt[1 - a^2*x^2]*(-1 + 11*a^3*x^3*Hypergeometric2F1[1/2, 3, 3/2, 1 + a*x]))/(3*x^3*Sqrt[c - a*c*x])

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Maple [A]  time = 0.105, size = 121, normalized size = 0.8 \begin{align*} -{\frac{1}{ \left ( 24\,ax-24 \right ){x}^{3}}\sqrt{-c \left ( ax-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ( 33\,c{\it Artanh} \left ({\frac{\sqrt{c \left ( ax+1 \right ) }}{\sqrt{c}}} \right ){x}^{3}{a}^{3}-33\,{x}^{2}{a}^{2}\sqrt{c \left ( ax+1 \right ) }\sqrt{c}+22\,xa\sqrt{c \left ( ax+1 \right ) }\sqrt{c}-8\,\sqrt{c \left ( ax+1 \right ) }\sqrt{c} \right ){\frac{1}{\sqrt{c \left ( ax+1 \right ) }}}{\frac{1}{\sqrt{c}}}} \end{align*}

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^4,x)

[Out]

-1/24*(-c*(a*x-1))^(1/2)*(-a^2*x^2+1)^(1/2)*(33*c*arctanh((c*(a*x+1))^(1/2)/c^(1/2))*x^3*a^3-33*x^2*a^2*(c*(a*
x+1))^(1/2)*c^(1/2)+22*x*a*(c*(a*x+1))^(1/2)*c^(1/2)-8*(c*(a*x+1))^(1/2)*c^(1/2))/c^(1/2)/(a*x-1)/(c*(a*x+1))^
(1/2)/x^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{{\left (a x + 1\right )} x^{4}}\,{d x} \end{align*}

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^4,x, algorithm="maxima")

[Out]

integrate(sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/((a*x + 1)*x^4), x)

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Fricas [A]  time = 1.96078, size = 543, normalized size = 3.67 \begin{align*} \left [\frac{33 \,{\left (a^{4} x^{4} - a^{3} x^{3}\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 \,{\left (33 \, a^{2} x^{2} - 22 \, a x + 8\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{48 \,{\left (a x^{4} - x^{3}\right )}}, \frac{33 \,{\left (a^{4} x^{4} - a^{3} x^{3}\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 ) +{\left (33 \, a^{2} x^{2} - 22 \, a x + 8\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{24 \,{\left (a x^{4} - x^{3}\right )}}\right ] \end{align*}

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^4,x, algorithm="fricas")

[Out]

[1/48*(33*(a^4*x^4 - a^3*x^3)*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*(33*a^2*x^2 - 22*a*x + 8)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a*x^4 - x^3), 1/24*(33
*(a^4*x^4 - a^3*x^3)*sqrt(-c)*arctan(sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) + (33*a^2*x
^2 - 22*a*x + 8)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a*x^4 - x^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- c \left (a x - 1\right )} \sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}{x^{4} \left (a x + 1\right )}\, dx \end{align*}

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**4,x)

[Out]

Integral(sqrt(-c*(a*x - 1))*sqrt(-(a*x - 1)*(a*x + 1))/(x**4*(a*x + 1)), x)

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Giac [A]  time = 1.31783, size = 177, normalized size = 1.2 \begin{align*} -\frac{{\left (a^{3} c^{4}{\left (\frac{33 \, \arctan \left (\frac{\sqrt{a c x + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{2}} + \frac{33 \,{\left (a c x + c\right )}^{\frac{5}{2}} - 88 \,{\left (a c x + c\right )}^{\frac{3}{2}} c + 63 \, \sqrt{a c x + c} c^{2}}{a^{3} c^{5} x^{3}}\right )} - \frac{33 \, a^{3} c^{2} \arctan \left (\frac{\sqrt{2} \sqrt{c}}{\sqrt{-c}}\right ) + 19 \, \sqrt{2} a^{3} \sqrt{-c} c^{\frac{3}{2}}}{\sqrt{-c}}\right )}{\left | c \right |}}{24 \, c^{2}} \end{align*}

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^4,x, algorithm="giac")

[Out]

-1/24*(a^3*c^4*(33*arctan(sqrt(a*c*x + c)/sqrt(-c))/(sqrt(-c)*c^2) + (33*(a*c*x + c)^(5/2) - 88*(a*c*x + c)^(3
/2)*c + 63*sqrt(a*c*x + c)*c^2)/(a^3*c^5*x^3)) - (33*a^3*c^2*arctan(sqrt(2)*sqrt(c)/sqrt(-c)) + 19*sqrt(2)*a^3
*sqrt(-c)*c^(3/2))/sqrt(-c))*abs(c)/c^2