3.2.59 \(\int e^{\coth ^{-1}(a x)} (c-a c x)^3 \, dx\) [159]

Optimal. Leaf size=105 \[ -\frac {5}{8} a c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {2}{3} a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3-\frac {1}{4} a^3 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^4+\frac {5 c^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{8 a} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.15, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6310, 6313, 1821, 821, 272, 43, 65, 214} \begin {gather*} -\frac {5}{8} a c^3 x^2 \sqrt {1-\frac {1}{a^2 x^2}}+\frac {5 c^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{8 a}+\frac {2}{3} a^2 c^3 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}-\frac {1}{4} a^3 c^3 x^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^ArcCoth[a*x]*(c - a*c*x)^3,x]

[Out]

(-5*a*c^3*Sqrt[1 - 1/(a^2*x^2)]*x^2)/8 + (2*a^2*c^3*(1 - 1/(a^2*x^2))^(3/2)*x^3)/3 - (a^3*c^3*(1 - 1/(a^2*x^2)
)^(3/2)*x^4)/4 + (5*c^3*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(8*a)

Rule 43

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, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 65

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 214

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

Rule 272

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 821

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 1821

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 6310

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 6313

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]

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 ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^2 \sqrt {1-\frac {x^2}{a^2}}}{x^5} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {1}{4} a^3 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^4-\frac {1}{4} \left (a^3 c^3\right ) \text {Subst}\left (\int \frac {\left (\frac {8}{a}-\frac {5 x}{a^2}\right ) \sqrt {1-\frac {x^2}{a^2}}}{x^4} \, dx,x,\frac {1}{x}\right )\\ &=\frac {2}{3} a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3-\frac {1}{4} a^3 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^4+\frac {1}{4} \left (5 a c^3\right ) \text {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a^2}}}{x^3} \, dx,x,\frac {1}{x}\right )\\ &=\frac {2}{3} a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3-\frac {1}{4} a^3 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^4+\frac {1}{8} \left (5 a c^3\right ) \text {Subst}\left (\int \frac {\sqrt {1-\frac {x}{a^2}}}{x^2} \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {5}{8} a c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {2}{3} a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3-\frac {1}{4} a^3 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^4-\frac {\left (5 c^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{16 a}\\ &=-\frac {5}{8} a c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {2}{3} a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3-\frac {1}{4} a^3 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^4+\frac {1}{8} \left (5 a c^3\right ) \text {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right )\\ &=-\frac {5}{8} a c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {2}{3} a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3-\frac {1}{4} a^3 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^4+\frac {5 c^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{8 a}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.09, size = 73, normalized size = 0.70 \begin {gather*} \frac {c^3 \left (-a \sqrt {1-\frac {1}{a^2 x^2}} x \left (16+9 a x-16 a^2 x^2+6 a^3 x^3\right )+15 \log \left (a \left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{24 a} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^3*(-(a*Sqrt[1 - 1/(a^2*x^2)]*x*(16 + 9*a*x - 16*a^2*x^2 + 6*a^3*x^3)) + 15*Log[a*(1 + Sqrt[1 - 1/(a^2*x^2)]
)*x]))/(24*a)

________________________________________________________________________________________

Maple [A]
time = 0.08, size = 141, normalized size = 1.34

method	result	size
risch	\(-\frac {\left (6 a^{3} x^{3}-16 a^{2} x^{2}+9 a x +16\right ) \left (a x -1\right ) c^{3}}{24 a \sqrt {\frac {a x -1}{a x +1}}}+\frac {5 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right ) c^{3} \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{8 \sqrt {a^{2}}\, \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\)	\(120\)
default	\(-\frac {\left (a x -1\right ) c^{3} \left (6 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a x +15 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a x -16 \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-15 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a \right )}{24 a \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}}\)	\(141\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(a*x-1)*c^3/a*(6*(a^2)^(1/2)*(a^2*x^2-1)^(3/2)*a*x+15*(a^2)^(1/2)*(a^2*x^2-1)^(1/2)*a*x-16*((a*x+1)*(a*x
-1))^(3/2)*(a^2)^(1/2)-15*ln((a^2*x+(a^2*x^2-1)^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*a)/((a*x-1)/(a*x+1))^(1/2)/((a
*x+1)*(a*x-1))^(1/2)/(a^2)^(1/2)

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (89) = 178\).
time = 0.27, size = 221, normalized size = 2.10 \begin {gather*} \frac {1}{24} \, {\left (\frac {15 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {15 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} + \frac {2 \, {\left (15 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 73 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - 55 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 15 \, 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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(15*c^3*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 - 15*c^3*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/a^2 + 2*(15*c^
3*((a*x - 1)/(a*x + 1))^(7/2) + 73*c^3*((a*x - 1)/(a*x + 1))^(5/2) - 55*c^3*((a*x - 1)/(a*x + 1))^(3/2) + 15*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

________________________________________________________________________________________

Fricas [A]
time = 0.43, size = 115, normalized size = 1.10 \begin {gather*} \frac {15 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 15 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (6 \, a^{4} c^{3} x^{4} - 10 \, a^{3} c^{3} x^{3} - 7 \, a^{2} c^{3} x^{2} + 25 \, a c^{3} x + 16 \, c^{3}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{24 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(15*c^3*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 15*c^3*log(sqrt((a*x - 1)/(a*x + 1)) - 1) - (6*a^4*c^3*x^4 -
10*a^3*c^3*x^3 - 7*a^2*c^3*x^2 + 25*a*c^3*x + 16*c^3)*sqrt((a*x - 1)/(a*x + 1)))/a

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - c^{3} \left (\int \frac {3 a x}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \left (- \frac {3 a^{2} x^{2}}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\right )\, dx + \int \frac {a^{3} x^{3}}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \left (- \frac {1}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\right )\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

Giac [A]
time = 0.41, size = 118, normalized size = 1.12 \begin {gather*} -\frac {5 \, c^{3} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{8 \, {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} - \frac {1}{24} \, \sqrt {a^{2} x^{2} - 1} {\left ({\left (2 \, {\left (\frac {3 \, a^{2} c^{3} x}{\mathrm {sgn}\left (a x + 1\right )} - \frac {8 \, a c^{3}}{\mathrm {sgn}\left (a x + 1\right )}\right )} x + \frac {9 \, c^{3}}{\mathrm {sgn}\left (a x + 1\right )}\right )} x + \frac {16 \, c^{3}}{a \mathrm {sgn}\left (a x + 1\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/8*c^3*log(abs(-x*abs(a) + sqrt(a^2*x^2 - 1)))/(abs(a)*sgn(a*x + 1)) - 1/24*sqrt(a^2*x^2 - 1)*((2*(3*a^2*c^3
*x/sgn(a*x + 1) - 8*a*c^3/sgn(a*x + 1))*x + 9*c^3/sgn(a*x + 1))*x + 16*c^3/(a*sgn(a*x + 1)))

________________________________________________________________________________________

Mupad [B]
time = 0.08, size = 177, normalized size = 1.69 \begin {gather*} \frac {5\,c^3\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{4\,a}-\frac {\frac {5\,c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{4}-\frac {55\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{12}+\frac {73\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{12}+\frac {5\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{4}}{a-\frac {4\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {6\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {4\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(5*c^3*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/(4*a) - ((5*c^3*((a*x - 1)/(a*x + 1))^(1/2))/4 - (55*c^3*((a*x - 1)
/(a*x + 1))^(3/2))/12 + (73*c^3*((a*x - 1)/(a*x + 1))^(5/2))/12 + (5*c^3*((a*x - 1)/(a*x + 1))^(7/2))/4)/(a -
(4*a*(a*x - 1))/(a*x + 1) + (6*a*(a*x - 1)^2)/(a*x + 1)^2 - (4*a*(a*x - 1)^3)/(a*x + 1)^3 + (a*(a*x - 1)^4)/(a
*x + 1)^4)

________________________________________________________________________________________