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

Optimal. Leaf size=74 \[ \frac{\left (1-a^2 x^2\right )^{3/2}}{3 a^2 c^2 (1-a x)^3}-\frac{2 \sqrt{1-a^2 x^2}}{a^2 c^2 (1-a x)}+\frac{\sin ^{-1}(a x)}{a^2 c^2} \]

[Out]

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

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Rubi [A]  time = 0.0803656, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {6128, 793, 663, 216} \[ \frac{\left (1-a^2 x^2\right )^{3/2}}{3 a^2 c^2 (1-a x)^3}-\frac{2 \sqrt{1-a^2 x^2}}{a^2 c^2 (1-a x)}+\frac{\sin ^{-1}(a x)}{a^2 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - a^2*x^2])/(a^2*c^2*(1 - a*x)) + (1 - a^2*x^2)^(3/2)/(3*a^2*c^2*(1 - a*x)^3) + ArcSin[a*x]/(a^2*c^
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 793

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

Rule 663

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

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]

Rubi steps

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

Mathematica [C]  time = 0.0657945, size = 57, normalized size = 0.77 \[ -\frac{(a x+1)^{3/2}-4 \sqrt{2} \text{Hypergeometric2F1}\left (-\frac{3}{2},-\frac{3}{2},-\frac{1}{2},\frac{1}{2} (1-a x)\right )}{3 a^2 c^2 (1-a x)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((1 + a*x)^(3/2) - 4*Sqrt[2]*Hypergeometric2F1[-3/2, -3/2, -1/2, (1 - a*x)/2])/(3*a^2*c^2*(1 - a*x)^(3/2))

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Maple [A]  time = 0.042, size = 122, normalized size = 1.7 \begin{align*}{\frac{1}{a{c}^{2}}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{\frac{2}{3\,{a}^{4}{c}^{2}}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-2}}+{\frac{7}{3\,{c}^{2}{a}^{3}}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-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)*x/(-a*c*x+c)^2,x)

[Out]

1/c^2/a/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+2/3/c^2/a^4/(x-1/a)^2*(-a^2*(x-1/a)^2-2*a*(x-1/a)
)^(1/2)+7/3/c^2/a^3/(x-1/a)*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.57973, size = 225, normalized size = 3.04 \begin{align*} -\frac{5 \, a^{2} x^{2} - 10 \, a x + 6 \,{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) - \sqrt{-a^{2} x^{2} + 1}{\left (7 \, a x - 5\right )} + 5}{3 \,{\left (a^{4} c^{2} x^{2} - 2 \, a^{3} c^{2} x + a^{2} c^{2}\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)*x/(-a*c*x+c)^2,x, algorithm="fricas")

[Out]

-1/3*(5*a^2*x^2 - 10*a*x + 6*(a^2*x^2 - 2*a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - sqrt(-a^2*x^2 + 1)
*(7*a*x - 5) + 5)/(a^4*c^2*x^2 - 2*a^3*c^2*x + a^2*c^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{x}{a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 2 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{2}}{a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 2 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{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)*x/(-a*c*x+c)**2,x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x + 1\right )} x}{\sqrt{-a^{2} x^{2} + 1}{\left (a c x - c\right )}^{2}}\,{d 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)*x/(-a*c*x+c)^2,x, algorithm="giac")

[Out]

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