3.341 \(\int e^{-2 \coth ^{-1}(a x)} x^3 \sqrt{c-a c x} \, dx\)

Optimal. Leaf size=139 \[ \frac{2 (c-a c x)^{9/2}}{9 a^4 c^4}-\frac{2 (c-a c x)^{7/2}}{7 a^4 c^3}+\frac{2 (c-a c x)^{5/2}}{5 a^4 c^2}+\frac{2 (c-a c x)^{3/2}}{3 a^4 c}+\frac{4 \sqrt{c-a c x}}{a^4}-\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )}{a^4} \]

[Out]

(4*Sqrt[c - a*c*x])/a^4 + (2*(c - a*c*x)^(3/2))/(3*a^4*c) + (2*(c - a*c*x)^(5/2))/(5*a^4*c^2) - (2*(c - a*c*x)
^(7/2))/(7*a^4*c^3) + (2*(c - a*c*x)^(9/2))/(9*a^4*c^4) - (4*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*
Sqrt[c])])/a^4

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Rubi [A]  time = 0.264182, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {6167, 6130, 21, 88, 50, 63, 206} \[ \frac{2 (c-a c x)^{9/2}}{9 a^4 c^4}-\frac{2 (c-a c x)^{7/2}}{7 a^4 c^3}+\frac{2 (c-a c x)^{5/2}}{5 a^4 c^2}+\frac{2 (c-a c x)^{3/2}}{3 a^4 c}+\frac{4 \sqrt{c-a c x}}{a^4}-\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )}{a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[c - a*c*x])/a^4 + (2*(c - a*c*x)^(3/2))/(3*a^4*c) + (2*(c - a*c*x)^(5/2))/(5*a^4*c^2) - (2*(c - a*c*x)
^(7/2))/(7*a^4*c^3) + (2*(c - a*c*x)^(9/2))/(9*a^4*c^4) - (4*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*
Sqrt[c])])/a^4

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 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && 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 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 e^{-2 \coth ^{-1}(a x)} x^3 \sqrt{c-a c x} \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} x^3 \sqrt{c-a c x} \, dx\\ &=-\int \frac{x^3 (1-a x) \sqrt{c-a c x}}{1+a x} \, dx\\ &=-\frac{\int \frac{x^3 (c-a c x)^{3/2}}{1+a x} \, dx}{c}\\ &=-\frac{\int \left (\frac{(c-a c x)^{3/2}}{a^3}-\frac{(c-a c x)^{3/2}}{a^3 (1+a x)}-\frac{(c-a c x)^{5/2}}{a^3 c}+\frac{(c-a c x)^{7/2}}{a^3 c^2}\right ) \, dx}{c}\\ &=\frac{2 (c-a c x)^{5/2}}{5 a^4 c^2}-\frac{2 (c-a c x)^{7/2}}{7 a^4 c^3}+\frac{2 (c-a c x)^{9/2}}{9 a^4 c^4}+\frac{\int \frac{(c-a c x)^{3/2}}{1+a x} \, dx}{a^3 c}\\ &=\frac{2 (c-a c x)^{3/2}}{3 a^4 c}+\frac{2 (c-a c x)^{5/2}}{5 a^4 c^2}-\frac{2 (c-a c x)^{7/2}}{7 a^4 c^3}+\frac{2 (c-a c x)^{9/2}}{9 a^4 c^4}+\frac{2 \int \frac{\sqrt{c-a c x}}{1+a x} \, dx}{a^3}\\ &=\frac{4 \sqrt{c-a c x}}{a^4}+\frac{2 (c-a c x)^{3/2}}{3 a^4 c}+\frac{2 (c-a c x)^{5/2}}{5 a^4 c^2}-\frac{2 (c-a c x)^{7/2}}{7 a^4 c^3}+\frac{2 (c-a c x)^{9/2}}{9 a^4 c^4}+\frac{(4 c) \int \frac{1}{(1+a x) \sqrt{c-a c x}} \, dx}{a^3}\\ &=\frac{4 \sqrt{c-a c x}}{a^4}+\frac{2 (c-a c x)^{3/2}}{3 a^4 c}+\frac{2 (c-a c x)^{5/2}}{5 a^4 c^2}-\frac{2 (c-a c x)^{7/2}}{7 a^4 c^3}+\frac{2 (c-a c x)^{9/2}}{9 a^4 c^4}-\frac{8 \operatorname{Subst}\left (\int \frac{1}{2-\frac{x^2}{c}} \, dx,x,\sqrt{c-a c x}\right )}{a^4}\\ &=\frac{4 \sqrt{c-a c x}}{a^4}+\frac{2 (c-a c x)^{3/2}}{3 a^4 c}+\frac{2 (c-a c x)^{5/2}}{5 a^4 c^2}-\frac{2 (c-a c x)^{7/2}}{7 a^4 c^3}+\frac{2 (c-a c x)^{9/2}}{9 a^4 c^4}-\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )}{a^4}\\ \end{align*}

Mathematica [A]  time = 0.126347, size = 85, normalized size = 0.61 \[ \frac{2 \left (\left (35 a^4 x^4-95 a^3 x^3+138 a^2 x^2-236 a x+788\right ) \sqrt{c-a c x}-630 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )\right )}{315 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(Sqrt[c - a*c*x]*(788 - 236*a*x + 138*a^2*x^2 - 95*a^3*x^3 + 35*a^4*x^4) - 630*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt
[c - a*c*x]/(Sqrt[2]*Sqrt[c])]))/(315*a^4)

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Maple [A]  time = 0.046, size = 101, normalized size = 0.7 \begin{align*} 2\,{\frac{1}{{c}^{4}{a}^{4}} \left ( 1/9\, \left ( -acx+c \right ) ^{9/2}-1/7\, \left ( -acx+c \right ) ^{7/2}c+1/5\, \left ( -acx+c \right ) ^{5/2}{c}^{2}+1/3\,{c}^{3} \left ( -acx+c \right ) ^{3/2}+2\,\sqrt{-acx+c}{c}^{4}-2\,{c}^{9/2}\sqrt{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(x^3*(-a*c*x+c)^(1/2)/(a*x+1)*(a*x-1),x)

[Out]

2/c^4/a^4*(1/9*(-a*c*x+c)^(9/2)-1/7*(-a*c*x+c)^(7/2)*c+1/5*(-a*c*x+c)^(5/2)*c^2+1/3*c^3*(-a*c*x+c)^(3/2)+2*(-a
*c*x+c)^(1/2)*c^4-2*c^(9/2)*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(x^3*(-a*c*x+c)^(1/2)*(a*x-1)/(a*x+1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.59791, size = 452, normalized size = 3.25 \begin{align*} \left [\frac{2 \,{\left (315 \, \sqrt{2} \sqrt{c} \log \left (\frac{a c x + 2 \, \sqrt{2} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a x + 1}\right ) +{\left (35 \, a^{4} x^{4} - 95 \, a^{3} x^{3} + 138 \, a^{2} x^{2} - 236 \, a x + 788\right )} \sqrt{-a c x + c}\right )}}{315 \, a^{4}}, \frac{2 \,{\left (630 \, \sqrt{2} \sqrt{-c} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{-c}}{2 \, c}\right ) +{\left (35 \, a^{4} x^{4} - 95 \, a^{3} x^{3} + 138 \, a^{2} x^{2} - 236 \, a x + 788\right )} \sqrt{-a c x + c}\right )}}{315 \, a^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[2/315*(315*sqrt(2)*sqrt(c)*log((a*c*x + 2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a*x + 1)) + (35*a^4*x^4 -
95*a^3*x^3 + 138*a^2*x^2 - 236*a*x + 788)*sqrt(-a*c*x + c))/a^4, 2/315*(630*sqrt(2)*sqrt(-c)*arctan(1/2*sqrt(2
)*sqrt(-a*c*x + c)*sqrt(-c)/c) + (35*a^4*x^4 - 95*a^3*x^3 + 138*a^2*x^2 - 236*a*x + 788)*sqrt(-a*c*x + c))/a^4
]

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Sympy [A]  time = 8.78105, size = 126, normalized size = 0.91 \begin{align*} \frac{2 \left (\frac{2 \sqrt{2} c^{5} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{- a c x + c}}{2 \sqrt{- c}} \right )}}{\sqrt{- c}} + 2 c^{4} \sqrt{- a c x + c} + \frac{c^{3} \left (- a c x + c\right )^{\frac{3}{2}}}{3} + \frac{c^{2} \left (- a c x + c\right )^{\frac{5}{2}}}{5} - \frac{c \left (- a c x + c\right )^{\frac{7}{2}}}{7} + \frac{\left (- a c x + c\right )^{\frac{9}{2}}}{9}\right )}{a^{4} c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(2*sqrt(2)*c**5*atan(sqrt(2)*sqrt(-a*c*x + c)/(2*sqrt(-c)))/sqrt(-c) + 2*c**4*sqrt(-a*c*x + c) + c**3*(-a*c*
x + c)**(3/2)/3 + c**2*(-a*c*x + c)**(5/2)/5 - c*(-a*c*x + c)**(7/2)/7 + (-a*c*x + c)**(9/2)/9)/(a**4*c**4)

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Giac [A]  time = 1.1452, size = 215, normalized size = 1.55 \begin{align*} \frac{4 \, \sqrt{2} c \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c}}{2 \, \sqrt{-c}}\right )}{a^{4} \sqrt{-c}} + \frac{2 \,{\left (35 \,{\left (a c x - c\right )}^{4} \sqrt{-a c x + c} a^{32} c^{32} + 45 \,{\left (a c x - c\right )}^{3} \sqrt{-a c x + c} a^{32} c^{33} + 63 \,{\left (a c x - c\right )}^{2} \sqrt{-a c x + c} a^{32} c^{34} + 105 \,{\left (-a c x + c\right )}^{\frac{3}{2}} a^{32} c^{35} + 630 \, \sqrt{-a c x + c} a^{32} c^{36}\right )}}{315 \, a^{36} c^{36}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*sqrt(2)*c*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)/sqrt(-c))/(a^4*sqrt(-c)) + 2/315*(35*(a*c*x - c)^4*sqrt(-a*c*x
+ c)*a^32*c^32 + 45*(a*c*x - c)^3*sqrt(-a*c*x + c)*a^32*c^33 + 63*(a*c*x - c)^2*sqrt(-a*c*x + c)*a^32*c^34 +
105*(-a*c*x + c)^(3/2)*a^32*c^35 + 630*sqrt(-a*c*x + c)*a^32*c^36)/(a^36*c^36)