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

Optimal. Leaf size=89 \[ \frac{11 a^2 \sqrt{c-a c x}}{8 x}+\frac{11}{8} a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{c}}\right )+\frac{11 a \sqrt{c-a c x}}{12 x^2}+\frac{\sqrt{c-a c x}}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.219242, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {6167, 6130, 21, 78, 51, 63, 208} \[ \frac{11 a^2 \sqrt{c-a c x}}{8 x}+\frac{11}{8} a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{c}}\right )+\frac{11 a \sqrt{c-a c x}}{12 x^2}+\frac{\sqrt{c-a c x}}{3 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 78

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

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, 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]

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

Mathematica [A]  time = 0.0581621, size = 63, normalized size = 0.71 \[ \frac{\left (33 a^2 x^2+22 a x+8\right ) \sqrt{c-a c x}}{24 x^3}+\frac{11}{8} a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{c}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c - a*c*x]*(8 + 22*a*x + 33*a^2*x^2))/(24*x^3) + (11*a^3*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]])/8

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Maple [A]  time = 0.059, size = 80, normalized size = 0.9 \begin{align*} -2\,{c}^{3}{a}^{3} \left ( -{\frac{1}{{x}^{3}{a}^{3}{c}^{3}} \left ({\frac{11\, \left ( -acx+c \right ) ^{5/2}}{16\,{c}^{2}}}-{\frac{11\, \left ( -acx+c \right ) ^{3/2}}{6\,c}}+{\frac{21\,\sqrt{-acx+c}}{16}} \right ) }-{\frac{11}{16\,{c}^{5/2}}{\it Artanh} \left ({\frac{\sqrt{-acx+c}}{\sqrt{c}}} \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*c^3*a^3*(-(11/16/c^2*(-a*c*x+c)^(5/2)-11/6/c*(-a*c*x+c)^(3/2)+21/16*(-a*c*x+c)^(1/2))/x^3/a^3/c^3-11/16/c^(
5/2)*arctanh((-a*c*x+c)^(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(1/(a*x-1)*(a*x+1)*(-a*c*x+c)^(1/2)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.50698, size = 332, normalized size = 3.73 \begin{align*} \left [\frac{33 \, 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 (33 \, a^{2} x^{2} + 22 \, a x + 8\right )} \sqrt{-a c x + c}}{48 \, x^{3}}, -\frac{33 \, a^{3} \sqrt{-c} x^{3} \arctan \left (\frac{\sqrt{-a c x + c} \sqrt{-c}}{c}\right ) -{\left (33 \, a^{2} x^{2} + 22 \, 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(1/(a*x-1)*(a*x+1)*(-a*c*x+c)^(1/2)/x^4,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 18.9856, size = 439, normalized size = 4.93 \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{10 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{6 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{3 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{3 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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)*(a*x+1)*(-a*c*x+c)**(1/2)/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) + 10*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 - 6*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) - 3*a**3*c*
*3*sqrt(c**(-5))*log(-c**3*sqrt(c**(-5)) + sqrt(-a*c*x + c))/8 + 3*a**3*c**3*sqrt(c**(-5))*log(c**3*sqrt(c**(-
5)) + sqrt(-a*c*x + c))/8

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Giac [A]  time = 1.12548, size = 134, normalized size = 1.51 \begin{align*} -\frac{11 \, a^{3} c \arctan \left (\frac{\sqrt{-a c x + c}}{\sqrt{-c}}\right )}{8 \, \sqrt{-c}} + \frac{33 \,{\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} + 63 \, \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(1/(a*x-1)*(a*x+1)*(-a*c*x+c)^(1/2)/x^4,x, algorithm="giac")

[Out]

-11/8*a^3*c*arctan(sqrt(-a*c*x + c)/sqrt(-c))/sqrt(-c) + 1/24*(33*(a*c*x - c)^2*sqrt(-a*c*x + c)*a^3*c - 88*(-
a*c*x + c)^(3/2)*a^3*c^2 + 63*sqrt(-a*c*x + c)*a^3*c^3)/(a^3*c^3*x^3)