### 3.348 $$\int \frac{e^{-2 \coth ^{-1}(a x)} \sqrt{c-a c x}}{x^4} \, dx$$

Optimal. Leaf size=127 $\frac{19 a^2 \sqrt{c-a c x}}{8 x}-\frac{45}{8} a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{c}}\right )+4 \sqrt{2} a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )-\frac{13 a \sqrt{c-a c x}}{12 x^2}+\frac{\sqrt{c-a c x}}{3 x^3}$

[Out]

Sqrt[c - a*c*x]/(3*x^3) - (13*a*Sqrt[c - a*c*x])/(12*x^2) + (19*a^2*Sqrt[c - a*c*x])/(8*x) - (45*a^3*Sqrt[c]*A
rcTanh[Sqrt[c - a*c*x]/Sqrt[c]])/8 + 4*Sqrt[2]*a^3*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])]

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Rubi [A]  time = 0.277919, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.391, Rules used = {6167, 6130, 21, 98, 151, 156, 63, 208, 206} $\frac{19 a^2 \sqrt{c-a c x}}{8 x}-\frac{45}{8} a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{c}}\right )+4 \sqrt{2} a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )-\frac{13 a \sqrt{c-a c x}}{12 x^2}+\frac{\sqrt{c-a c x}}{3 x^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sqrt[c - a*c*x]/(E^(2*ArcCoth[a*x])*x^4),x]

[Out]

Sqrt[c - a*c*x]/(3*x^3) - (13*a*Sqrt[c - a*c*x])/(12*x^2) + (19*a^2*Sqrt[c - a*c*x])/(8*x) - (45*a^3*Sqrt[c]*A
rcTanh[Sqrt[c - a*c*x]/Sqrt[c]])/8 + 4*Sqrt[2]*a^3*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])]

Rule 6167

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]

Rule 6130

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 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 98

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 151

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] && LtQ[m, -1] && IntegerQ[m]

Rule 156

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 63

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 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]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{e^{-2 \coth ^{-1}(a x)} \sqrt{c-a c x}}{x^4} \, dx &=-\int \frac{e^{-2 \tanh ^{-1}(a x)} \sqrt{c-a c x}}{x^4} \, dx\\ &=-\int \frac{(1-a x) \sqrt{c-a c x}}{x^4 (1+a x)} \, dx\\ &=-\frac{\int \frac{(c-a c x)^{3/2}}{x^4 (1+a x)} \, dx}{c}\\ &=\frac{\sqrt{c-a c x}}{3 x^3}+\frac{\int \frac{\frac{13 a c^2}{2}-\frac{11}{2} a^2 c^2 x}{x^3 (1+a x) \sqrt{c-a c x}} \, dx}{3 c}\\ &=\frac{\sqrt{c-a c x}}{3 x^3}-\frac{13 a \sqrt{c-a c x}}{12 x^2}-\frac{\int \frac{\frac{57 a^2 c^3}{4}-\frac{39}{4} a^3 c^3 x}{x^2 (1+a x) \sqrt{c-a c x}} \, dx}{6 c^2}\\ &=\frac{\sqrt{c-a c x}}{3 x^3}-\frac{13 a \sqrt{c-a c x}}{12 x^2}+\frac{19 a^2 \sqrt{c-a c x}}{8 x}+\frac{\int \frac{\frac{135 a^3 c^4}{8}-\frac{57}{8} a^4 c^4 x}{x (1+a x) \sqrt{c-a c x}} \, dx}{6 c^3}\\ &=\frac{\sqrt{c-a c x}}{3 x^3}-\frac{13 a \sqrt{c-a c x}}{12 x^2}+\frac{19 a^2 \sqrt{c-a c x}}{8 x}+\frac{1}{16} \left (45 a^3 c\right ) \int \frac{1}{x \sqrt{c-a c x}} \, dx-\left (4 a^4 c\right ) \int \frac{1}{(1+a x) \sqrt{c-a c x}} \, dx\\ &=\frac{\sqrt{c-a c x}}{3 x^3}-\frac{13 a \sqrt{c-a c x}}{12 x^2}+\frac{19 a^2 \sqrt{c-a c x}}{8 x}-\frac{1}{8} \left (45 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a}-\frac{x^2}{a c}} \, dx,x,\sqrt{c-a c x}\right )+\left (8 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{2-\frac{x^2}{c}} \, dx,x,\sqrt{c-a c x}\right )\\ &=\frac{\sqrt{c-a c x}}{3 x^3}-\frac{13 a \sqrt{c-a c x}}{12 x^2}+\frac{19 a^2 \sqrt{c-a c x}}{8 x}-\frac{45}{8} a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{c}}\right )+4 \sqrt{2} a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0902037, size = 101, normalized size = 0.8 $\frac{\left (57 a^2 x^2-26 a x+8\right ) \sqrt{c-a c x}}{24 x^3}-\frac{45}{8} a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{c}}\right )+4 \sqrt{2} a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sqrt[c - a*c*x]/(E^(2*ArcCoth[a*x])*x^4),x]

[Out]

(Sqrt[c - a*c*x]*(8 - 26*a*x + 57*a^2*x^2))/(24*x^3) - (45*a^3*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]])/8 + 4
*Sqrt[2]*a^3*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])]

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Maple [A]  time = 0.056, size = 110, normalized size = 0.9 \begin{align*} -2\,{c}^{3}{a}^{3} \left ( -{\frac{1}{{c}^{2}} \left ( -{\frac{1}{{x}^{3}{a}^{3}{c}^{3}} \left ( -{\frac{19\, \left ( -acx+c \right ) ^{5/2}}{16}}+{\frac{11\,c \left ( -acx+c \right ) ^{3/2}}{6}}-{\frac{13\,\sqrt{-acx+c}{c}^{2}}{16}} \right ) }-{\frac{45}{16\,\sqrt{c}}{\it Artanh} \left ({\frac{\sqrt{-acx+c}}{\sqrt{c}}} \right ) } \right ) }-2\,{\frac{\sqrt{2}}{{c}^{5/2}}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{-acx+c}\sqrt{2}}{\sqrt{c}}} \right ) } \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*c^3*a^3*(-1/c^2*(-(-19/16*(-a*c*x+c)^(5/2)+11/6*c*(-a*c*x+c)^(3/2)-13/16*(-a*c*x+c)^(1/2)*c^2)/x^3/a^3/c^3-
45/16/c^(1/2)*arctanh((-a*c*x+c)^(1/2)/c^(1/2)))-2/c^(5/2)*2^(1/2)*arctanh(1/2*(-a*c*x+c)^(1/2)*2^(1/2)/c^(1/2
)))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^(1/2)*(a*x-1)/(a*x+1)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.37009, size = 568, normalized size = 4.47 \begin{align*} \left [\frac{96 \, \sqrt{2} a^{3} \sqrt{c} x^{3} \log \left (\frac{a c x - 2 \, \sqrt{2} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a x + 1}\right ) + 135 \, a^{3} \sqrt{c} x^{3} \log \left (\frac{a c x + 2 \, \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{x}\right ) + 2 \,{\left (57 \, a^{2} x^{2} - 26 \, a x + 8\right )} \sqrt{-a c x + c}}{48 \, x^{3}}, -\frac{96 \, \sqrt{2} a^{3} \sqrt{-c} x^{3} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{-c}}{2 \, c}\right ) - 135 \, a^{3} \sqrt{-c} x^{3} \arctan \left (\frac{\sqrt{-a c x + c} \sqrt{-c}}{c}\right ) -{\left (57 \, a^{2} x^{2} - 26 \, a x + 8\right )} \sqrt{-a c x + c}}{24 \, x^{3}}\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*x+1)/x^4,x, algorithm="fricas")

[Out]

[1/48*(96*sqrt(2)*a^3*sqrt(c)*x^3*log((a*c*x - 2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a*x + 1)) + 135*a^3*
sqrt(c)*x^3*log((a*c*x + 2*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/x) + 2*(57*a^2*x^2 - 26*a*x + 8)*sqrt(-a*c*x + c))/
x^3, -1/24*(96*sqrt(2)*a^3*sqrt(-c)*x^3*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(-c)/c) - 135*a^3*sqrt(-c)*x^3
*arctan(sqrt(-a*c*x + c)*sqrt(-c)/c) - (57*a^2*x^2 - 26*a*x + 8)*sqrt(-a*c*x + c))/x^3]

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Sympy [B]  time = 16.9645, size = 614, normalized size = 4.83 \begin{align*} - \frac{66 a^{3} c^{6} \sqrt{- a c x + c}}{- 144 a c^{6} x + 96 c^{6} - 144 c^{4} \left (- a c x + c\right )^{2} + 48 c^{3} \left (- a c x + c\right )^{3}} + \frac{80 a^{3} c^{5} \left (- a c x + c\right )^{\frac{3}{2}}}{- 144 a c^{6} x + 96 c^{6} - 144 c^{4} \left (- a c x + c\right )^{2} + 48 c^{3} \left (- a c x + c\right )^{3}} - \frac{30 a^{3} c^{4} \left (- a c x + c\right )^{\frac{5}{2}}}{- 144 a c^{6} x + 96 c^{6} - 144 c^{4} \left (- a c x + c\right )^{2} + 48 c^{3} \left (- a c x + c\right )^{3}} - \frac{30 a^{3} c^{4} \sqrt{- a c x + c}}{16 a c^{4} x - 8 c^{4} + 8 c^{2} \left (- a c x + c\right )^{2}} - \frac{5 a^{3} c^{4} \sqrt{\frac{1}{c^{7}}} \log{\left (- c^{4} \sqrt{\frac{1}{c^{7}}} + \sqrt{- a c x + c} \right )}}{16} + \frac{5 a^{3} c^{4} \sqrt{\frac{1}{c^{7}}} \log{\left (c^{4} \sqrt{\frac{1}{c^{7}}} + \sqrt{- a c x + c} \right )}}{16} + \frac{18 a^{3} c^{3} \left (- a c x + c\right )^{\frac{3}{2}}}{16 a c^{4} x - 8 c^{4} + 8 c^{2} \left (- a c x + c\right )^{2}} + \frac{9 a^{3} c^{3} \sqrt{\frac{1}{c^{5}}} \log{\left (- c^{3} \sqrt{\frac{1}{c^{5}}} + \sqrt{- a c x + c} \right )}}{8} - \frac{9 a^{3} c^{3} \sqrt{\frac{1}{c^{5}}} \log{\left (c^{3} \sqrt{\frac{1}{c^{5}}} + \sqrt{- a c x + c} \right )}}{8} - 2 a^{3} c^{2} \sqrt{\frac{1}{c^{3}}} \log{\left (- c^{2} \sqrt{\frac{1}{c^{3}}} + \sqrt{- a c x + c} \right )} + 2 a^{3} c^{2} \sqrt{\frac{1}{c^{3}}} \log{\left (c^{2} \sqrt{\frac{1}{c^{3}}} + \sqrt{- a c x + c} \right )} + \frac{8 a^{3} c \operatorname{atan}{\left (\frac{\sqrt{- a c x + c}}{\sqrt{- c}} \right )}}{\sqrt{- c}} - \frac{4 \sqrt{2} a^{3} c \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{- a c x + c}}{2 \sqrt{- c}} \right )}}{\sqrt{- c}} + \frac{4 a^{2} \sqrt{- a c x + c}}{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*x+1)/x**4,x)

[Out]

-66*a**3*c**6*sqrt(-a*c*x + c)/(-144*a*c**6*x + 96*c**6 - 144*c**4*(-a*c*x + c)**2 + 48*c**3*(-a*c*x + c)**3)
+ 80*a**3*c**5*(-a*c*x + c)**(3/2)/(-144*a*c**6*x + 96*c**6 - 144*c**4*(-a*c*x + c)**2 + 48*c**3*(-a*c*x + c)*
*3) - 30*a**3*c**4*(-a*c*x + c)**(5/2)/(-144*a*c**6*x + 96*c**6 - 144*c**4*(-a*c*x + c)**2 + 48*c**3*(-a*c*x +
c)**3) - 30*a**3*c**4*sqrt(-a*c*x + c)/(16*a*c**4*x - 8*c**4 + 8*c**2*(-a*c*x + c)**2) - 5*a**3*c**4*sqrt(c**
(-7))*log(-c**4*sqrt(c**(-7)) + sqrt(-a*c*x + c))/16 + 5*a**3*c**4*sqrt(c**(-7))*log(c**4*sqrt(c**(-7)) + sqrt
(-a*c*x + c))/16 + 18*a**3*c**3*(-a*c*x + c)**(3/2)/(16*a*c**4*x - 8*c**4 + 8*c**2*(-a*c*x + c)**2) + 9*a**3*c
**3*sqrt(c**(-5))*log(-c**3*sqrt(c**(-5)) + sqrt(-a*c*x + c))/8 - 9*a**3*c**3*sqrt(c**(-5))*log(c**3*sqrt(c**(
-5)) + sqrt(-a*c*x + c))/8 - 2*a**3*c**2*sqrt(c**(-3))*log(-c**2*sqrt(c**(-3)) + sqrt(-a*c*x + c)) + 2*a**3*c*
*2*sqrt(c**(-3))*log(c**2*sqrt(c**(-3)) + sqrt(-a*c*x + c)) + 8*a**3*c*atan(sqrt(-a*c*x + c)/sqrt(-c))/sqrt(-c
) - 4*sqrt(2)*a**3*c*atan(sqrt(2)*sqrt(-a*c*x + c)/(2*sqrt(-c)))/sqrt(-c) + 4*a**2*sqrt(-a*c*x + c)/x

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Giac [A]  time = 1.13983, size = 180, normalized size = 1.42 \begin{align*} -\frac{4 \, \sqrt{2} a^{3} c \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c}}{2 \, \sqrt{-c}}\right )}{\sqrt{-c}} + \frac{45 \, a^{3} c \arctan \left (\frac{\sqrt{-a c x + c}}{\sqrt{-c}}\right )}{8 \, \sqrt{-c}} + \frac{57 \,{\left (a c x - c\right )}^{2} \sqrt{-a c x + c} a^{3} c - 88 \,{\left (-a c x + c\right )}^{\frac{3}{2}} a^{3} c^{2} + 39 \, \sqrt{-a c x + c} a^{3} c^{3}}{24 \, a^{3} c^{3} x^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^(1/2)*(a*x-1)/(a*x+1)/x^4,x, algorithm="giac")

[Out]

-4*sqrt(2)*a^3*c*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)/sqrt(-c))/sqrt(-c) + 45/8*a^3*c*arctan(sqrt(-a*c*x + c)/s
qrt(-c))/sqrt(-c) + 1/24*(57*(a*c*x - c)^2*sqrt(-a*c*x + c)*a^3*c - 88*(-a*c*x + c)^(3/2)*a^3*c^2 + 39*sqrt(-a
*c*x + c)*a^3*c^3)/(a^3*c^3*x^3)