### 3.198 $$\int e^{-\coth ^{-1}(a x)} (c-a c x)^3 \, dx$$

Optimal. Leaf size=127 $-\frac{1}{4} a^3 c^3 x^4 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{4}{3} a^2 c^3 x^3 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{27}{8} a c^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{20}{3} c^3 x \sqrt{1-\frac{1}{a^2 x^2}}-\frac{35 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{8 a}$

[Out]

(20*c^3*Sqrt[1 - 1/(a^2*x^2)]*x)/3 - (27*a*c^3*Sqrt[1 - 1/(a^2*x^2)]*x^2)/8 + (4*a^2*c^3*Sqrt[1 - 1/(a^2*x^2)]
*x^3)/3 - (a^3*c^3*Sqrt[1 - 1/(a^2*x^2)]*x^4)/4 - (35*c^3*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(8*a)

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Rubi [A]  time = 0.351761, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.389, Rules used = {6175, 6178, 1807, 807, 266, 63, 208} $-\frac{1}{4} a^3 c^3 x^4 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{4}{3} a^2 c^3 x^3 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{27}{8} a c^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{20}{3} c^3 x \sqrt{1-\frac{1}{a^2 x^2}}-\frac{35 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{8 a}$

Antiderivative was successfully veriﬁed.

[In]

Int[(c - a*c*x)^3/E^ArcCoth[a*x],x]

[Out]

(20*c^3*Sqrt[1 - 1/(a^2*x^2)]*x)/3 - (27*a*c^3*Sqrt[1 - 1/(a^2*x^2)]*x^2)/8 + (4*a^2*c^3*Sqrt[1 - 1/(a^2*x^2)]
*x^3)/3 - (a^3*c^3*Sqrt[1 - 1/(a^2*x^2)]*x^4)/4 - (35*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 1807

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, c*x, x],
R = PolynomialRemainder[Pq, c*x, x]}, Simp[(R*(c*x)^(m + 1)*(a + b*x^2)^(p + 1))/(a*c*(m + 1)), x] + Dist[1/(
a*c*(m + 1)), Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*(m + 2*p + 3)*x, x], x], x]] /;
FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && LtQ[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])

Rule 807

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0]
&& EqQ[Simplify[m + 2*p + 3], 0]

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 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^{-\coth ^{-1}(a x)} (c-a c x)^3 \, dx &=-\left (\left (a^3 c^3\right ) \int e^{-\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}{a}\right )^4}{x^5 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{4} a^3 c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^4-\frac{1}{4} \left (a^3 c^3\right ) \operatorname{Subst}\left (\int \frac{\frac{16}{a}-\frac{27 x}{a^2}+\frac{16 x^2}{a^3}-\frac{4 x^3}{a^4}}{x^4 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{4}{3} a^2 c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^3-\frac{1}{4} a^3 c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^4+\frac{1}{12} \left (a^3 c^3\right ) \operatorname{Subst}\left (\int \frac{\frac{81}{a^2}-\frac{80 x}{a^3}+\frac{12 x^2}{a^4}}{x^3 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{27}{8} a c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{4}{3} a^2 c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^3-\frac{1}{4} a^3 c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^4-\frac{1}{24} \left (a^3 c^3\right ) \operatorname{Subst}\left (\int \frac{\frac{160}{a^3}-\frac{105 x}{a^4}}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{20}{3} c^3 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{27}{8} a c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{4}{3} a^2 c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^3-\frac{1}{4} a^3 c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^4+\frac{\left (35 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{8 a}\\ &=\frac{20}{3} c^3 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{27}{8} a c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{4}{3} a^2 c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^3-\frac{1}{4} a^3 c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^4+\frac{\left (35 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{20}{3} c^3 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{27}{8} a c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{4}{3} a^2 c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^3-\frac{1}{4} a^3 c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^4-\frac{1}{8} \left (35 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{20}{3} c^3 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{27}{8} a c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{4}{3} a^2 c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^3-\frac{1}{4} a^3 c^3 \sqrt{1-\frac{1}{a^2 x^2}} x^4-\frac{35 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{8 a}\\ \end{align*}

Mathematica [A]  time = 0.188479, size = 72, normalized size = 0.57 $\frac{c^3 \left (a x \sqrt{1-\frac{1}{a^2 x^2}} \left (-6 a^3 x^3+32 a^2 x^2-81 a x+160\right )-105 \log \left (a x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )\right )}{24 a}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^3*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(160 - 81*a*x + 32*a^2*x^2 - 6*a^3*x^3) - 105*Log[a*(1 + Sqrt[1 - 1/(a^2*x^2)]
)*x]))/(24*a)

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Maple [A]  time = 0.132, size = 196, normalized size = 1.5 \begin{align*} -{\frac{ \left ( ax+1 \right ){c}^{3}}{24\,a}\sqrt{{\frac{ax-1}{ax+1}}} \left ( 6\,\sqrt{{a}^{2}} \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}xa+87\,\sqrt{{a}^{2}}\sqrt{{a}^{2}{x}^{2}-1}xa-32\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}-87\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ) a-192\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }+192\,a\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) \right ){\frac{1}{\sqrt{{a}^{2}}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*((a*x-1)/(a*x+1))^(1/2)*(a*x+1)*c^3/a*(6*(a^2)^(1/2)*(a^2*x^2-1)^(3/2)*x*a+87*(a^2)^(1/2)*(a^2*x^2-1)^(1
/2)*x*a-32*((a*x-1)*(a*x+1))^(3/2)*(a^2)^(1/2)-87*ln((a^2*x+(a^2*x^2-1)^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*a-192*
(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)+192*a*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2)))/((a*x-1
)*(a*x+1))^(1/2)/(a^2)^(1/2)

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Maxima [B]  time = 1.05021, size = 298, normalized size = 2.35 \begin{align*} -\frac{1}{24} \,{\left (\frac{105 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{105 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac{2 \,{\left (279 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} - 511 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + 385 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - 105 \, 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((-a*c*x+c)^3*((a*x-1)/(a*x+1))^(1/2),x, algorithm="maxima")

[Out]

-1/24*(105*c^3*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 - 105*c^3*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/a^2 - 2*(27
9*c^3*((a*x - 1)/(a*x + 1))^(7/2) - 511*c^3*((a*x - 1)/(a*x + 1))^(5/2) + 385*c^3*((a*x - 1)/(a*x + 1))^(3/2)
- 105*c^3*sqrt((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 = 1.59482, size = 270, normalized size = 2.13 \begin{align*} -\frac{105 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 105 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) +{\left (6 \, a^{4} c^{3} x^{4} - 26 \, a^{3} c^{3} x^{3} + 49 \, a^{2} c^{3} x^{2} - 79 \, a c^{3} x - 160 \, c^{3}\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{24 \, a} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(105*c^3*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 105*c^3*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (6*a^4*c^3*x^
4 - 26*a^3*c^3*x^3 + 49*a^2*c^3*x^2 - 79*a*c^3*x - 160*c^3)*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 3 a x \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}\, dx + \int - 3 a^{2} x^{2} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}\, dx + \int a^{3} x^{3} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}\, dx + \int - \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}\, dx\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.14968, size = 147, normalized size = 1.16 \begin{align*} \frac{35 \, c^{3} \log \left ({\left | -x{\left | a \right |} + \sqrt{a^{2} x^{2} - 1} \right |}\right ) \mathrm{sgn}\left (a x + 1\right )}{8 \,{\left | a \right |}} + \frac{1}{24} \, \sqrt{a^{2} x^{2} - 1}{\left (\frac{160 \, c^{3} \mathrm{sgn}\left (a x + 1\right )}{a} -{\left (81 \, c^{3} \mathrm{sgn}\left (a x + 1\right ) + 2 \,{\left (3 \, a^{2} c^{3} x \mathrm{sgn}\left (a x + 1\right ) - 16 \, a c^{3} \mathrm{sgn}\left (a x + 1\right )\right )} x\right )} x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

35/8*c^3*log(abs(-x*abs(a) + sqrt(a^2*x^2 - 1)))*sgn(a*x + 1)/abs(a) + 1/24*sqrt(a^2*x^2 - 1)*(160*c^3*sgn(a*x
+ 1)/a - (81*c^3*sgn(a*x + 1) + 2*(3*a^2*c^3*x*sgn(a*x + 1) - 16*a*c^3*sgn(a*x + 1))*x)*x)