3.301 \(\int \frac{e^{\tanh ^{-1}(a x)} (c-a c x)^2}{x^3} \, dx\)

Optimal. Leaf size=67 \[ -\frac{c^2 (1-2 a x) \sqrt{1-a^2 x^2}}{2 x^2}+\frac{1}{2} a^2 c^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+a^2 c^2 \sin ^{-1}(a x) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.104605, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {6128, 811, 844, 216, 266, 63, 208} \[ -\frac{c^2 (1-2 a x) \sqrt{1-a^2 x^2}}{2 x^2}+\frac{1}{2} a^2 c^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+a^2 c^2 \sin ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[(E^ArcTanh[a*x]*(c - a*c*x)^2)/x^3,x]

[Out]

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

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 811

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

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 \frac{e^{\tanh ^{-1}(a x)} (c-a c x)^2}{x^3} \, dx &=c \int \frac{(c-a c x) \sqrt{1-a^2 x^2}}{x^3} \, dx\\ &=-\frac{c^2 (1-2 a x) \sqrt{1-a^2 x^2}}{2 x^2}-\frac{1}{4} c \int \frac{2 a^2 c-4 a^3 c x}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{c^2 (1-2 a x) \sqrt{1-a^2 x^2}}{2 x^2}-\frac{1}{2} \left (a^2 c^2\right ) \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx+\left (a^3 c^2\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{c^2 (1-2 a x) \sqrt{1-a^2 x^2}}{2 x^2}+a^2 c^2 \sin ^{-1}(a x)-\frac{1}{4} \left (a^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac{c^2 (1-2 a x) \sqrt{1-a^2 x^2}}{2 x^2}+a^2 c^2 \sin ^{-1}(a x)+\frac{1}{2} 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 )\\ &=-\frac{c^2 (1-2 a x) \sqrt{1-a^2 x^2}}{2 x^2}+a^2 c^2 \sin ^{-1}(a x)+\frac{1}{2} a^2 c^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )\\ \end{align*}

Mathematica [B]  time = 0.100961, size = 147, normalized size = 2.19 \[ -\frac{c^2 \left (4 a^3 x^3-2 a^2 x^2+a^2 x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)+10 a^2 x^2 \sqrt{1-a^2 x^2} \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )-2 a^2 x^2 \sqrt{1-a^2 x^2} \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-4 a x+2\right )}{4 x^2 \sqrt{1-a^2 x^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(E^ArcTanh[a*x]*(c - a*c*x)^2)/x^3,x]

[Out]

-(c^2*(2 - 4*a*x - 2*a^2*x^2 + 4*a^3*x^3 + a^2*x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x] + 10*a^2*x^2*Sqrt[1 - a^2*x^2
]*ArcSin[Sqrt[1 - a*x]/Sqrt[2]] - 2*a^2*x^2*Sqrt[1 - a^2*x^2]*ArcTanh[Sqrt[1 - a^2*x^2]]))/(4*x^2*Sqrt[1 - a^2
*x^2])

________________________________________________________________________________________

Maple [A]  time = 0.048, size = 95, normalized size = 1.4 \begin{align*}{{c}^{2}{a}^{3}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{\frac{a{c}^{2}}{x}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{{a}^{2}{c}^{2}}{2}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }-{\frac{{c}^{2}}{2\,{x}^{2}}\sqrt{-{a}^{2}{x}^{2}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.42484, size = 132, normalized size = 1.97 \begin{align*} \frac{a^{3} c^{2} \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{\sqrt{a^{2}}} + \frac{1}{2} \, a^{2} c^{2} \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) + \frac{\sqrt{-a^{2} x^{2} + 1} a c^{2}}{x} - \frac{\sqrt{-a^{2} x^{2} + 1} c^{2}}{2 \, x^{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)*(-a*c*x+c)^2/x^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.64576, size = 203, normalized size = 3.03 \begin{align*} -\frac{4 \, a^{2} c^{2} x^{2} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) + a^{2} c^{2} x^{2} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) -{\left (2 \, a c^{2} x - c^{2}\right )} \sqrt{-a^{2} x^{2} + 1}}{2 \, x^{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)*(-a*c*x+c)^2/x^3,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [C]  time = 7.36336, size = 226, normalized size = 3.37 \begin{align*} a^{3} 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 ) - a^{2} 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 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 ) + c^{2} \left (\begin{cases} - \frac{a^{2} \operatorname{acosh}{\left (\frac{1}{a x} \right )}}{2} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{2 x} & \text{for}\: \frac{1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac{i a^{2} \operatorname{asin}{\left (\frac{1}{a x} \right )}}{2} - \frac{i a}{2 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{2 a x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\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)*(-a*c*x+c)**2/x**3,x)

[Out]

a**3*c**2*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 <
0)) - a**2*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)) + c**2*Piecewise((-a**2*acos
h(1/(a*x))/2 - a*sqrt(-1 + 1/(a**2*x**2))/(2*x), 1/Abs(a**2*x**2) > 1), (I*a**2*asin(1/(a*x))/2 - I*a/(2*x*sqr
t(1 - 1/(a**2*x**2))) + I/(2*a*x**3*sqrt(1 - 1/(a**2*x**2))), True))

________________________________________________________________________________________

Giac [B]  time = 1.43579, size = 259, normalized size = 3.87 \begin{align*} \frac{a^{3} c^{2} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{{\left | a \right |}} + \frac{a^{3} 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 )}{2 \,{\left | a \right |}} + \frac{{\left (a^{3} c^{2} - \frac{4 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a c^{2}}{x}\right )} a^{4} x^{2}}{8 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}{\left | a \right |}} + \frac{\frac{4 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a c^{2}{\left | a \right |}}{x} - \frac{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} c^{2}{\left | a \right |}}{a x^{2}}}{8 \, 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)*(-a*c*x+c)^2/x^3,x, algorithm="giac")

[Out]

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