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

Optimal. Leaf size=104 \[ \frac{\left (1-a^2 x^2\right )^{3/2}}{a^3 c^2 (1-a x)^2}+\frac{\left (1-a^2 x^2\right )^{3/2}}{3 a^3 c^2 (1-a x)^3}-\frac{6 \sqrt{1-a^2 x^2}}{a^3 c^2 (1-a x)}+\frac{3 \sin ^{-1}(a x)}{a^3 c^2} \]

[Out]

(-6*Sqrt[1 - a^2*x^2])/(a^3*c^2*(1 - a*x)) + (1 - a^2*x^2)^(3/2)/(3*a^3*c^2*(1 - a*x)^3) + (1 - a^2*x^2)^(3/2)
/(a^3*c^2*(1 - a*x)^2) + (3*ArcSin[a*x])/(a^3*c^2)

________________________________________________________________________________________

Rubi [A]  time = 0.18841, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {6128, 1639, 793, 663, 216} \[ \frac{\left (1-a^2 x^2\right )^{3/2}}{a^3 c^2 (1-a x)^2}+\frac{\left (1-a^2 x^2\right )^{3/2}}{3 a^3 c^2 (1-a x)^3}-\frac{6 \sqrt{1-a^2 x^2}}{a^3 c^2 (1-a x)}+\frac{3 \sin ^{-1}(a x)}{a^3 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*Sqrt[1 - a^2*x^2])/(a^3*c^2*(1 - a*x)) + (1 - a^2*x^2)^(3/2)/(3*a^3*c^2*(1 - a*x)^3) + (1 - a^2*x^2)^(3/2)
/(a^3*c^2*(1 - a*x)^2) + (3*ArcSin[a*x])/(a^3*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 1639

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff
[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + c*x^2)^(p + 1))/(c*e^(q - 1)*(m + q + 2*p + 1)), x]
+ Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*
f*(m + q + 2*p + 1)*(d + e*x)^q - 2*e*f*(m + p + q)*(d + e*x)^(q - 2)*(a*e - c*d*x), x], x], x] /; NeQ[m + q +
2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

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^2}{(c-a c x)^2} \, dx &=c \int \frac{x^2 \sqrt{1-a^2 x^2}}{(c-a c x)^3} \, dx\\ &=\frac{\left (1-a^2 x^2\right )^{3/2}}{a^3 c^2 (1-a x)^2}-\frac{\int \frac{\left (2 a^2 c^2-3 a^3 c^2 x\right ) \sqrt{1-a^2 x^2}}{(c-a c x)^3} \, dx}{a^4 c}\\ &=\frac{\left (1-a^2 x^2\right )^{3/2}}{3 a^3 c^2 (1-a x)^3}+\frac{\left (1-a^2 x^2\right )^{3/2}}{a^3 c^2 (1-a x)^2}-\frac{3 \int \frac{\sqrt{1-a^2 x^2}}{(c-a c x)^2} \, dx}{a^2}\\ &=-\frac{6 \sqrt{1-a^2 x^2}}{a^3 c^2 (1-a x)}+\frac{\left (1-a^2 x^2\right )^{3/2}}{3 a^3 c^2 (1-a x)^3}+\frac{\left (1-a^2 x^2\right )^{3/2}}{a^3 c^2 (1-a x)^2}+\frac{3 \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{a^2 c^2}\\ &=-\frac{6 \sqrt{1-a^2 x^2}}{a^3 c^2 (1-a x)}+\frac{\left (1-a^2 x^2\right )^{3/2}}{3 a^3 c^2 (1-a x)^3}+\frac{\left (1-a^2 x^2\right )^{3/2}}{a^3 c^2 (1-a x)^2}+\frac{3 \sin ^{-1}(a x)}{a^3 c^2}\\ \end{align*}

Mathematica [A]  time = 0.0709114, size = 64, normalized size = 0.62 \[ \frac{\frac{\sqrt{a x+1} \left (-3 a^2 x^2+19 a x-14\right )}{(1-a x)^{3/2}}-18 \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )}{3 a^3 c^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((Sqrt[1 + a*x]*(-14 + 19*a*x - 3*a^2*x^2))/(1 - a*x)^(3/2) - 18*ArcSin[Sqrt[1 - a*x]/Sqrt[2]])/(3*a^3*c^2)

________________________________________________________________________________________

Maple [A]  time = 0.046, size = 143, normalized size = 1.4 \begin{align*} -{\frac{1}{{c}^{2}{a}^{3}}\sqrt{-{a}^{2}{x}^{2}+1}}+3\,{\frac{1}{{a}^{2}{c}^{2}\sqrt{{a}^{2}}}\arctan \left ({\frac{\sqrt{{a}^{2}}x}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }+{\frac{2}{3\,{c}^{2}{a}^{5}}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-2}}+{\frac{13}{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 ) ^{-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^2/(-a*c*x+c)^2,x)

[Out]

-1/c^2/a^3*(-a^2*x^2+1)^(1/2)+3/c^2/a^2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+2/3/c^2/a^5/(x-1/
a)^2*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)+13/3/c^2/a^4/(x-1/a)*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.61741, size = 248, normalized size = 2.38 \begin{align*} -\frac{14 \, a^{2} x^{2} - 28 \, a x + 18 \,{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) +{\left (3 \, a^{2} x^{2} - 19 \, a x + 14\right )} \sqrt{-a^{2} x^{2} + 1} + 14}{3 \,{\left (a^{5} c^{2} x^{2} - 2 \, a^{4} c^{2} x + a^{3} 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^2/(-a*c*x+c)^2,x, algorithm="fricas")

[Out]

-1/3*(14*a^2*x^2 - 28*a*x + 18*(a^2*x^2 - 2*a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (3*a^2*x^2 - 19*
a*x + 14)*sqrt(-a^2*x^2 + 1) + 14)/(a^5*c^2*x^2 - 2*a^4*c^2*x + a^3*c^2)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{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 + \int \frac{a x^{3}}{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**2/(-a*c*x+c)**2,x)

[Out]

(Integral(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) + Inte
gral(a*x**3/(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

________________________________________________________________________________________

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

[Out]

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