3.467 \(\int e^{3 \tanh ^{-1}(a x)} (c-\frac{c}{a x})^3 \, dx\)

Optimal. Leaf size=77 \[ \frac{c^3 \left (1-a^2 x^2\right )^{3/2}}{2 a^3 x^2}+\frac{3 c^3 \sqrt{1-a^2 x^2}}{2 a}-\frac{3 c^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{2 a} \]

[Out]

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

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Rubi [A]  time = 0.121125, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {6131, 6128, 266, 47, 50, 63, 208} \[ \frac{c^3 \left (1-a^2 x^2\right )^{3/2}}{2 a^3 x^2}+\frac{3 c^3 \sqrt{1-a^2 x^2}}{2 a}-\frac{3 c^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{2 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6131

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

Rule 6128

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[c^n,
Int[(e + f*x)^m*(c + d*x)^(p - n)*(1 - a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, e, f, m, p}, x] && EqQ[a*c +
d, 0] && IntegerQ[(n - 1)/2] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p - n/2 - 1, 0]) && IntegerQ[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 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 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 \tanh ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^3 \, dx &=-\frac{c^3 \int \frac{e^{3 \tanh ^{-1}(a x)} (1-a x)^3}{x^3} \, dx}{a^3}\\ &=-\frac{c^3 \int \frac{\left (1-a^2 x^2\right )^{3/2}}{x^3} \, dx}{a^3}\\ &=-\frac{c^3 \operatorname{Subst}\left (\int \frac{\left (1-a^2 x\right )^{3/2}}{x^2} \, dx,x,x^2\right )}{2 a^3}\\ &=\frac{c^3 \left (1-a^2 x^2\right )^{3/2}}{2 a^3 x^2}+\frac{\left (3 c^3\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-a^2 x}}{x} \, dx,x,x^2\right )}{4 a}\\ &=\frac{3 c^3 \sqrt{1-a^2 x^2}}{2 a}+\frac{c^3 \left (1-a^2 x^2\right )^{3/2}}{2 a^3 x^2}+\frac{\left (3 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )}{4 a}\\ &=\frac{3 c^3 \sqrt{1-a^2 x^2}}{2 a}+\frac{c^3 \left (1-a^2 x^2\right )^{3/2}}{2 a^3 x^2}-\frac{\left (3 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )}{2 a^3}\\ &=\frac{3 c^3 \sqrt{1-a^2 x^2}}{2 a}+\frac{c^3 \left (1-a^2 x^2\right )^{3/2}}{2 a^3 x^2}-\frac{3 c^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{2 a}\\ \end{align*}

Mathematica [A]  time = 0.129802, size = 77, normalized size = 1. \[ \frac{\sqrt{1-a^2 x^2} \left (\frac{c^3}{2 a^2 x^2}+c^3\right )}{a}-\frac{3 c^3 \log \left (\sqrt{1-a^2 x^2}+1\right )}{2 a}+\frac{3 c^3 \log (a x)}{2 a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.043, size = 118, normalized size = 1.5 \begin{align*}{\frac{{c}^{3}}{{a}^{3}} \left ({a}^{6} \left ( -{\frac{{x}^{2}}{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+2\,{\frac{1}{{a}^{4}\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) -3\,{\frac{{a}^{2}}{\sqrt{-{a}^{2}{x}^{2}+1}}}+{\frac{3\,{a}^{2}}{2} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}-{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) \right ) }+{\frac{1}{2\,{x}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^3/a^3*(a^6*(-x^2/a^2/(-a^2*x^2+1)^(1/2)+2/a^4/(-a^2*x^2+1)^(1/2))-3*a^2/(-a^2*x^2+1)^(1/2)+3/2*a^2*(1/(-a^2*
x^2+1)^(1/2)-arctanh(1/(-a^2*x^2+1)^(1/2)))+1/2/x^2/(-a^2*x^2+1)^(1/2))

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Maxima [B]  time = 1.44512, size = 254, normalized size = 3.3 \begin{align*} -a^{3} c^{3}{\left (\frac{x^{2}}{\sqrt{-a^{2} x^{2} + 1} a^{2}} - \frac{2}{\sqrt{-a^{2} x^{2} + 1} a^{4}}\right )} + \frac{3 \, c^{3}{\left (\frac{1}{\sqrt{-a^{2} x^{2} + 1}} - \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right )\right )}}{a} - \frac{3 \, c^{3}}{\sqrt{-a^{2} x^{2} + 1} a} + \frac{{\left (3 \, a^{2} \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) - \frac{3 \, a^{2}}{\sqrt{-a^{2} x^{2} + 1}} + \frac{1}{\sqrt{-a^{2} x^{2} + 1} x^{2}}\right )} c^{3}}{2 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.08806, size = 165, normalized size = 2.14 \begin{align*} \frac{3 \, a^{2} c^{3} x^{2} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) + 2 \, a^{2} c^{3} x^{2} +{\left (2 \, a^{2} c^{3} x^{2} + c^{3}\right )} \sqrt{-a^{2} x^{2} + 1}}{2 \, a^{3} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 21.5522, size = 104, normalized size = 1.35 \begin{align*} \frac{2 c^{3} \sqrt{- a^{2} x^{2} + 1} + \frac{3 c^{3} \log{\left (-1 + \frac{1}{\sqrt{- a^{2} x^{2} + 1}} \right )}}{2} - \frac{3 c^{3} \log{\left (1 + \frac{1}{\sqrt{- a^{2} x^{2} + 1}} \right )}}{2} + \frac{c^{3}}{2 \left (1 + \frac{1}{\sqrt{- a^{2} x^{2} + 1}}\right )} + \frac{c^{3}}{2 \left (-1 + \frac{1}{\sqrt{- a^{2} x^{2} + 1}}\right )}}{2 a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.16145, size = 107, normalized size = 1.39 \begin{align*} \frac{c^{3}{\left (4 \, \sqrt{-a^{2} x^{2} + 1} + \frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{a^{2} x^{2}} - 3 \, \log \left (\sqrt{-a^{2} x^{2} + 1} + 1\right ) + 3 \, \log \left (-\sqrt{-a^{2} x^{2} + 1} + 1\right )\right )}}{4 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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