3.450 \(\int e^{\tanh ^{-1}(a x)} (c-\frac{c}{a x})^2 \, dx\)

Optimal. Leaf size=65 \[ -\frac{c^2 (a x+1) \sqrt{1-a^2 x^2}}{a^2 x}+\frac{c^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{a}-\frac{c^2 \sin ^{-1}(a x)}{a} \]

[Out]

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

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Rubi [A]  time = 0.122547, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {6131, 6128, 813, 844, 216, 266, 63, 208} \[ -\frac{c^2 (a x+1) \sqrt{1-a^2 x^2}}{a^2 x}+\frac{c^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{a}-\frac{c^2 \sin ^{-1}(a x)}{a} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]*(c - c/(a*x))^2,x]

[Out]

-((c^2*(1 + a*x)*Sqrt[1 - a^2*x^2])/(a^2*x)) - (c^2*ArcSin[a*x])/a + (c^2*ArcTanh[Sqrt[1 - a^2*x^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 813

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*(a + c*x^2)^p)/(e^2*(m + 1)*(m + 2*p + 2)), x] + Di
st[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Simp[g*(2*a*e + 2*a*e*m) + (g*(2*c
*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2,
0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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] &&  !IGtQ[m, 0]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

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^{\tanh ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^2 \, dx &=\frac{c^2 \int \frac{e^{\tanh ^{-1}(a x)} (1-a x)^2}{x^2} \, dx}{a^2}\\ &=\frac{c^2 \int \frac{(1-a x) \sqrt{1-a^2 x^2}}{x^2} \, dx}{a^2}\\ &=-\frac{c^2 (1+a x) \sqrt{1-a^2 x^2}}{a^2 x}-\frac{c^2 \int \frac{2 a+2 a^2 x}{x \sqrt{1-a^2 x^2}} \, dx}{2 a^2}\\ &=-\frac{c^2 (1+a x) \sqrt{1-a^2 x^2}}{a^2 x}-c^2 \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx-\frac{c^2 \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx}{a}\\ &=-\frac{c^2 (1+a x) \sqrt{1-a^2 x^2}}{a^2 x}-\frac{c^2 \sin ^{-1}(a x)}{a}-\frac{c^2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )}{2 a}\\ &=-\frac{c^2 (1+a x) \sqrt{1-a^2 x^2}}{a^2 x}-\frac{c^2 \sin ^{-1}(a x)}{a}+\frac{c^2 \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )}{a^3}\\ &=-\frac{c^2 (1+a x) \sqrt{1-a^2 x^2}}{a^2 x}-\frac{c^2 \sin ^{-1}(a x)}{a}+\frac{c^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{a}\\ \end{align*}

Mathematica [A]  time = 0.100301, size = 78, normalized size = 1.2 \[ -\frac{c^2 \left (a x \sqrt{1-a^2 x^2}+\sqrt{1-a^2 x^2}-a x \log \left (\sqrt{1-a^2 x^2}+1\right )+a x \log (a x)+a x \sin ^{-1}(a x)\right )}{a^2 x} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcTanh[a*x]*(c - c/(a*x))^2,x]

[Out]

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

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Maple [A]  time = 0.039, size = 95, normalized size = 1.5 \begin{align*} -{\frac{{c}^{2}}{a}\sqrt{-{a}^{2}{x}^{2}+1}}-{{c}^{2}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}-{\frac{{c}^{2}}{{a}^{2}x}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{{c}^{2}}{a}{\it Artanh} \left ({\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)/(-a^2*x^2+1)^(1/2)*(c-c/a/x)^2,x)

[Out]

-c^2*(-a^2*x^2+1)^(1/2)/a-c^2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))-c^2*(-a^2*x^2+1)^(1/2)/a^2/
x+c^2/a*arctanh(1/(-a^2*x^2+1)^(1/2))

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Maxima [A]  time = 1.44266, size = 132, normalized size = 2.03 \begin{align*} -\frac{c^{2} \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{\sqrt{a^{2}}} + \frac{c^{2} \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right )}{a} - \frac{\sqrt{-a^{2} x^{2} + 1} c^{2}}{a} - \frac{\sqrt{-a^{2} x^{2} + 1} c^{2}}{a^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.2136, size = 201, normalized size = 3.09 \begin{align*} \frac{2 \, a c^{2} x \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) - a c^{2} x \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) - a c^{2} x -{\left (a c^{2} x + c^{2}\right )} \sqrt{-a^{2} x^{2} + 1}}{a^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 6.37038, size = 151, normalized size = 2.32 \begin{align*} a c^{2} \left (\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}\right ) - c^{2} \left (\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left (x \sqrt{a^{2}} \right )} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left (x \sqrt{- a^{2}} \right )} & \text{for}\: a^{2} < 0 \end{cases}\right ) - \frac{c^{2} \left (\begin{cases} - \operatorname{acosh}{\left (\frac{1}{a x} \right )} & \text{for}\: \frac{1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname{asin}{\left (\frac{1}{a x} \right )} & \text{otherwise} \end{cases}\right )}{a} + \frac{c^{2} \left (\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right )}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*c**2*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) - c**2*Piecewise((sqrt(a**(-2))*as
in(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) - c**2*Piecewise((-acosh(1/(a*x))
, 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True))/a + c**2*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2)
> 1), (-sqrt(-a**2*x**2 + 1)/x, True))/a**2

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Giac [B]  time = 1.19354, size = 188, normalized size = 2.89 \begin{align*} \frac{a^{2} c^{2} x}{2 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}{\left | a \right |}} - \frac{c^{2} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{{\left | a \right |}} + \frac{c^{2} \log \left (\frac{{\left | -2 \, \sqrt{-a^{2} x^{2} + 1}{\left | a \right |} - 2 \, a \right |}}{2 \, a^{2}{\left | x \right |}}\right )}{{\left | a \right |}} - \frac{\sqrt{-a^{2} x^{2} + 1} c^{2}}{a} - \frac{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} c^{2}}{2 \, a^{2} x{\left | a \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^2*c^2*x/((sqrt(-a^2*x^2 + 1)*abs(a) + a)*abs(a)) - c^2*arcsin(a*x)*sgn(a)/abs(a) + c^2*log(1/2*abs(-2*sq
rt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs(a) - sqrt(-a^2*x^2 + 1)*c^2/a - 1/2*(sqrt(-a^2*x^2 + 1)*abs(a
) + a)*c^2/(a^2*x*abs(a))