### 3.179 $$\int e^{3 \coth ^{-1}(a x)} (c-a c x)^3 \, dx$$

Optimal. Leaf size=78 $-\frac{1}{4} a^3 c^3 x^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}+\frac{3}{8} a c^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{3 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{8 a}$

[Out]

(3*a*c^3*Sqrt[1 - 1/(a^2*x^2)]*x^2)/8 - (a^3*c^3*(1 - 1/(a^2*x^2))^(3/2)*x^4)/4 - (3*c^3*ArcTanh[Sqrt[1 - 1/(a
^2*x^2)]])/(8*a)

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Rubi [A]  time = 0.141355, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.333, Rules used = {6175, 6178, 266, 47, 63, 208} $-\frac{1}{4} a^3 c^3 x^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}+\frac{3}{8} a c^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{3 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{8 a}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(3*a*c^3*Sqrt[1 - 1/(a^2*x^2)]*x^2)/8 - (a^3*c^3*(1 - 1/(a^2*x^2))^(3/2)*x^4)/4 - (3*c^3*ArcTanh[Sqrt[1 - 1/(a
^2*x^2)]])/(8*a)

Rule 6175

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> Dist[d^p, Int[u*x^p*(1 + c/(d*
x))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c^2 - d^2, 0] &&  !IntegerQ[n/2] && Inte
gerQ[p]

Rule 6178

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_.), x_Symbol] :> -Dist[c^n, Subst[Int[((c
+ d*x)^(p - n)*(1 - x^2/a^2)^(n/2))/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] &&
IntegerQ[(n - 1)/2] && IntegerQ[m] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p, n/2 + 1] || LtQ[-5, m, -1]) && In
tegerQ[2*p]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 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 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 e^{3 \coth ^{-1}(a x)} (c-a c x)^3 \, dx &=-\left (\left (a^3 c^3\right ) \int e^{3 \coth ^{-1}(a x)} \left (1-\frac{1}{a x}\right )^3 x^3 \, dx\right )\\ &=\left (a^3 c^3\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x^2}{a^2}\right )^{3/2}}{x^5} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{2} \left (a^3 c^3\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a^2}\right )^{3/2}}{x^3} \, dx,x,\frac{1}{x^2}\right )\\ &=-\frac{1}{4} a^3 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^4-\frac{1}{8} \left (3 a c^3\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x}{a^2}}}{x^2} \, dx,x,\frac{1}{x^2}\right )\\ &=\frac{3}{8} a c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^2-\frac{1}{4} a^3 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^4+\frac{\left (3 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{16 a}\\ &=\frac{3}{8} a c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^2-\frac{1}{4} a^3 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^4-\frac{1}{8} \left (3 a c^3\right ) \operatorname{Subst}\left (\int \frac{1}{a^2-a^2 x^2} \, dx,x,\sqrt{1-\frac{1}{a^2 x^2}}\right )\\ &=\frac{3}{8} a c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^2-\frac{1}{4} a^3 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^4-\frac{3 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{8 a}\\ \end{align*}

Mathematica [A]  time = 0.136916, size = 64, normalized size = 0.82 $\frac{c^3 \left (a^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}} \left (5-2 a^2 x^2\right )-3 \log \left (a x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )\right )}{8 a}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^3*(a^2*Sqrt[1 - 1/(a^2*x^2)]*x^2*(5 - 2*a^2*x^2) - 3*Log[a*(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(8*a)

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Maple [A]  time = 0.165, size = 124, normalized size = 1.6 \begin{align*} -{\frac{ \left ( ax-1 \right ) ^{2}{c}^{3}}{8\,ax+8} \left ( 2\,x \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}-3\,x\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}+3\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ) \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}}{\frac{1}{\sqrt{{a}^{2}}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(a*x-1)^2*c^3*(2*x*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)-3*x*(a^2*x^2-1)^(1/2)*(a^2)^(1/2)+3*ln((a^2*x+(a^2*x^2-1
)^(1/2)*(a^2)^(1/2))/(a^2)^(1/2)))/((a*x-1)/(a*x+1))^(3/2)/(a*x+1)/((a*x-1)*(a*x+1))^(1/2)/(a^2)^(1/2)

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Maxima [B]  time = 1.06614, size = 298, normalized size = 3.82 \begin{align*} -\frac{1}{8} \,{\left (\frac{3 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{3 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}} + \frac{2 \,{\left (3 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} - 11 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} - 11 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + 3 \, c^{3} \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{\frac{4 \,{\left (a x - 1\right )} a^{2}}{a x + 1} - \frac{6 \,{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac{4 \,{\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac{{\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} - a^{2}}\right )} a \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(3*c^3*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 - 3*c^3*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/a^2 + 2*(3*c^3*(
(a*x - 1)/(a*x + 1))^(7/2) - 11*c^3*((a*x - 1)/(a*x + 1))^(5/2) - 11*c^3*((a*x - 1)/(a*x + 1))^(3/2) + 3*c^3*s
qrt((a*x - 1)/(a*x + 1)))/(4*(a*x - 1)*a^2/(a*x + 1) - 6*(a*x - 1)^2*a^2/(a*x + 1)^2 + 4*(a*x - 1)^3*a^2/(a*x
+ 1)^3 - (a*x - 1)^4*a^2/(a*x + 1)^4 - a^2))*a

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Fricas [A]  time = 1.52967, size = 246, normalized size = 3.15 \begin{align*} -\frac{3 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 3 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) +{\left (2 \, a^{4} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{3} - 5 \, a^{2} c^{3} x^{2} - 5 \, a c^{3} x\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{8 \, a} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(3*c^3*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 3*c^3*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (2*a^4*c^3*x^4 + 2
*a^3*c^3*x^3 - 5*a^2*c^3*x^2 - 5*a*c^3*x)*sqrt((a*x - 1)/(a*x + 1)))/a

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - c^{3} \left (\int \frac{3 a x}{\frac{a x \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} - \frac{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1}}\, dx + \int - \frac{3 a^{2} x^{2}}{\frac{a x \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} - \frac{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1}}\, dx + \int \frac{a^{3} x^{3}}{\frac{a x \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} - \frac{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1}}\, dx + \int - \frac{1}{\frac{a x \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} - \frac{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1}}\, dx\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.23131, size = 275, normalized size = 3.53 \begin{align*} -\frac{1}{16} \,{\left (\frac{3 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + \frac{1}{\sqrt{\frac{a x - 1}{a x + 1}}} + 2\right )}{a^{2}} - \frac{3 \, c^{3} \log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} + \frac{1}{\sqrt{\frac{a x - 1}{a x + 1}}} - 2 \right |}\right )}{a^{2}} - \frac{4 \,{\left (3 \, c^{3}{\left (\sqrt{\frac{a x - 1}{a x + 1}} + \frac{1}{\sqrt{\frac{a x - 1}{a x + 1}}}\right )}^{3} - 20 \, c^{3}{\left (\sqrt{\frac{a x - 1}{a x + 1}} + \frac{1}{\sqrt{\frac{a x - 1}{a x + 1}}}\right )}\right )}}{{\left ({\left (\sqrt{\frac{a x - 1}{a x + 1}} + \frac{1}{\sqrt{\frac{a x - 1}{a x + 1}}}\right )}^{2} - 4\right )}^{2} a^{2}}\right )} a \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(3*c^3*log(sqrt((a*x - 1)/(a*x + 1)) + 1/sqrt((a*x - 1)/(a*x + 1)) + 2)/a^2 - 3*c^3*log(abs(sqrt((a*x -
1)/(a*x + 1)) + 1/sqrt((a*x - 1)/(a*x + 1)) - 2))/a^2 - 4*(3*c^3*(sqrt((a*x - 1)/(a*x + 1)) + 1/sqrt((a*x - 1)
/(a*x + 1)))^3 - 20*c^3*(sqrt((a*x - 1)/(a*x + 1)) + 1/sqrt((a*x - 1)/(a*x + 1))))/(((sqrt((a*x - 1)/(a*x + 1)
) + 1/sqrt((a*x - 1)/(a*x + 1)))^2 - 4)^2*a^2))*a