3.1150 \(\int \frac{e^{3 \tanh ^{-1}(a x)} x^2}{c-a^2 c x^2} \, dx\)

Optimal. Leaf size=95 \[ \frac{(a x+1)^3}{3 a^3 c \left (1-a^2 x^2\right )^{3/2}}-\frac{2 (a x+1)^2}{a^3 c \sqrt{1-a^2 x^2}}-\frac{3 \sqrt{1-a^2 x^2}}{a^3 c}+\frac{3 \sin ^{-1}(a x)}{a^3 c} \]

[Out]

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

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Rubi [A]  time = 0.186649, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {6148, 1635, 21, 669, 641, 216} \[ \frac{(a x+1)^3}{3 a^3 c \left (1-a^2 x^2\right )^{3/2}}-\frac{2 (a x+1)^2}{a^3 c \sqrt{1-a^2 x^2}}-\frac{3 \sqrt{1-a^2 x^2}}{a^3 c}+\frac{3 \sin ^{-1}(a x)}{a^3 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6148

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

Rule 1635

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

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 669

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

Rule 641

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

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^{3 \tanh ^{-1}(a x)} x^2}{c-a^2 c x^2} \, dx &=\frac{\int \frac{x^2 (1+a x)^3}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{c}\\ &=\frac{(1+a x)^3}{3 a^3 c \left (1-a^2 x^2\right )^{3/2}}-\frac{\int \frac{\left (\frac{3}{a^2}+\frac{3 x}{a}\right ) (1+a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{3 c}\\ &=\frac{(1+a x)^3}{3 a^3 c \left (1-a^2 x^2\right )^{3/2}}-\frac{\int \frac{(1+a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{a^2 c}\\ &=\frac{(1+a x)^3}{3 a^3 c \left (1-a^2 x^2\right )^{3/2}}-\frac{2 (1+a x)^2}{a^3 c \sqrt{1-a^2 x^2}}+\frac{3 \int \frac{1+a x}{\sqrt{1-a^2 x^2}} \, dx}{a^2 c}\\ &=\frac{(1+a x)^3}{3 a^3 c \left (1-a^2 x^2\right )^{3/2}}-\frac{2 (1+a x)^2}{a^3 c \sqrt{1-a^2 x^2}}-\frac{3 \sqrt{1-a^2 x^2}}{a^3 c}+\frac{3 \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{a^2 c}\\ &=\frac{(1+a x)^3}{3 a^3 c \left (1-a^2 x^2\right )^{3/2}}-\frac{2 (1+a x)^2}{a^3 c \sqrt{1-a^2 x^2}}-\frac{3 \sqrt{1-a^2 x^2}}{a^3 c}+\frac{3 \sin ^{-1}(a x)}{a^3 c}\\ \end{align*}

Mathematica [A]  time = 0.0850362, size = 78, normalized size = 0.82 \[ \frac{3 a^3 x^3-16 a^2 x^2+9 (a x-1) \sqrt{1-a^2 x^2} \sin ^{-1}(a x)-5 a x+14}{3 a^3 c (a x-1) \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(14 - 5*a*x - 16*a^2*x^2 + 3*a^3*x^3 + 9*(-1 + a*x)*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(3*a^3*c*(-1 + a*x)*Sqrt[1
- a^2*x^2])

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Maple [B]  time = 0.04, size = 178, normalized size = 1.9 \begin{align*}{\frac{{x}^{2}}{ac}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-6\,{\frac{1}{{a}^{3}c\sqrt{-{a}^{2}{x}^{2}+1}}}-7\,{\frac{x}{{a}^{2}c\sqrt{-{a}^{2}{x}^{2}+1}}}+3\,{\frac{1}{{a}^{2}c\sqrt{{a}^{2}}}\arctan \left ({\frac{\sqrt{{a}^{2}}x}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }-{\frac{4}{3\,c{a}^{4}} \left ( x-{a}^{-1} \right ) ^{-1}{\frac{1}{\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) }}}}+{\frac{8\,x}{3\,{a}^{2}c}{\frac{1}{\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-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)*x^2/(-a^2*c*x^2+c),x)

[Out]

1/c*x^2/a/(-a^2*x^2+1)^(1/2)-6/c/a^3/(-a^2*x^2+1)^(1/2)-7/c*x/a^2/(-a^2*x^2+1)^(1/2)+3/c/a^2/(a^2)^(1/2)*arcta
n((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))-4/3/c/a^4/(x-1/a)/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)+8/3/c/a^2/(-a^2*(x-1/
a)^2-2*a*(x-1/a))^(1/2)*x

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Maxima [B]  time = 1.92079, size = 782, normalized size = 8.23 \begin{align*} \frac{a^{2} c{\left (\frac{\left (a^{2} c^{2}\right )^{\frac{3}{2}}}{\sqrt{a^{2} c^{2}} \sqrt{-a^{2} x^{2} + 1} a^{7} c^{3} x + \sqrt{-a^{2} x^{2} + 1} a^{7} c^{4}} - \frac{\left (a^{2} c^{2}\right )^{\frac{3}{2}}}{\sqrt{a^{2} c^{2}} \sqrt{-a^{2} x^{2} + 1} a^{7} c^{3} x - \sqrt{-a^{2} x^{2} + 1} a^{7} c^{4}} - \frac{4 \, c}{\sqrt{a^{2} c^{2}} \sqrt{-a^{2} x^{2} + 1} a^{4} c x + \sqrt{-a^{2} x^{2} + 1} a^{4} c^{2}} - \frac{4 \, c}{\sqrt{a^{2} c^{2}} \sqrt{-a^{2} x^{2} + 1} a^{4} c x - \sqrt{-a^{2} x^{2} + 1} a^{4} c^{2}} + \frac{3 \, \sqrt{a^{2} c^{2}}}{\sqrt{a^{2} c^{2}} \sqrt{-a^{2} x^{2} + 1} a^{5} c x + \sqrt{-a^{2} x^{2} + 1} a^{5} c^{2}} - \frac{3 \, \sqrt{a^{2} c^{2}}}{\sqrt{a^{2} c^{2}} \sqrt{-a^{2} x^{2} + 1} a^{5} c x - \sqrt{-a^{2} x^{2} + 1} a^{5} c^{2}} + \frac{16 \, x}{\sqrt{a^{2} c^{2}} \sqrt{-a^{2} x^{2} + 1} a^{2}} + \frac{6 \, \sqrt{a^{2} c^{2}} x^{2}}{\sqrt{-a^{2} x^{2} + 1} a^{3} c^{2}} - \frac{16}{\sqrt{a^{2} c^{2}} \sqrt{-a^{2} x^{2} + 1} a^{3}} - \frac{42 \, \sqrt{a^{2} c^{2}} x}{\sqrt{-a^{2} x^{2} + 1} a^{4} c^{2}} - \frac{21 \, \sqrt{a^{2} c^{2}}}{\sqrt{-a^{2} x^{2} + 1} a^{5} c^{2}} + \frac{18 \, \sqrt{a^{2} c^{2}} \arcsin \left (\frac{x}{c \sqrt{\frac{1}{a^{2} c^{2}}}}\right )}{a^{6} c^{3} \sqrt{\frac{1}{a^{2} c^{2}}}} + \frac{\left (a^{2} c^{2}\right )^{\frac{3}{2}}}{\sqrt{-a^{2} x^{2} + 1} a^{7} c^{4}}\right )}}{6 \, \sqrt{a^{2} c^{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)*x^2/(-a^2*c*x^2+c),x, algorithm="maxima")

[Out]

1/6*a^2*c*((a^2*c^2)^(3/2)/(sqrt(a^2*c^2)*sqrt(-a^2*x^2 + 1)*a^7*c^3*x + sqrt(-a^2*x^2 + 1)*a^7*c^4) - (a^2*c^
2)^(3/2)/(sqrt(a^2*c^2)*sqrt(-a^2*x^2 + 1)*a^7*c^3*x - sqrt(-a^2*x^2 + 1)*a^7*c^4) - 4*c/(sqrt(a^2*c^2)*sqrt(-
a^2*x^2 + 1)*a^4*c*x + sqrt(-a^2*x^2 + 1)*a^4*c^2) - 4*c/(sqrt(a^2*c^2)*sqrt(-a^2*x^2 + 1)*a^4*c*x - sqrt(-a^2
*x^2 + 1)*a^4*c^2) + 3*sqrt(a^2*c^2)/(sqrt(a^2*c^2)*sqrt(-a^2*x^2 + 1)*a^5*c*x + sqrt(-a^2*x^2 + 1)*a^5*c^2) -
3*sqrt(a^2*c^2)/(sqrt(a^2*c^2)*sqrt(-a^2*x^2 + 1)*a^5*c*x - sqrt(-a^2*x^2 + 1)*a^5*c^2) + 16*x/(sqrt(a^2*c^2)
*sqrt(-a^2*x^2 + 1)*a^2) + 6*sqrt(a^2*c^2)*x^2/(sqrt(-a^2*x^2 + 1)*a^3*c^2) - 16/(sqrt(a^2*c^2)*sqrt(-a^2*x^2
+ 1)*a^3) - 42*sqrt(a^2*c^2)*x/(sqrt(-a^2*x^2 + 1)*a^4*c^2) - 21*sqrt(a^2*c^2)/(sqrt(-a^2*x^2 + 1)*a^5*c^2) +
18*sqrt(a^2*c^2)*arcsin(x/(c*sqrt(1/(a^2*c^2))))/(a^6*c^3*sqrt(1/(a^2*c^2))) + (a^2*c^2)^(3/2)/(sqrt(-a^2*x^2
+ 1)*a^7*c^4))/sqrt(a^2*c^2)

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Fricas [A]  time = 2.58919, size = 240, normalized size = 2.53 \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 x^{2} - 2 \, a^{4} c x + a^{3} c\right )}} \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)*x^2/(-a^2*c*x^2+c),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*x^2 - 2*a^4*c*x + a^3*c)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{x^{2}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{3 a x^{3}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{3 a^{2} x^{4}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a^{3} x^{5}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c} \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)*x**2/(-a**2*c*x**2+c),x)

[Out]

(Integral(x**2/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x)
+ Integral(3*a*x**3/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1))
, x) + Integral(3*a**2*x**4/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x*
*2 + 1)), x) + Integral(a**3*x**5/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a
**2*x**2 + 1)), x))/c

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Giac [A]  time = 1.17991, size = 180, normalized size = 1.89 \begin{align*} \frac{3 \, \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{a^{2} c{\left | a \right |}} - \frac{\sqrt{-a^{2} x^{2} + 1}}{a^{3} c} + \frac{2 \,{\left (\frac{24 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}}{a^{2} x} - \frac{9 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} - 11\right )}}{3 \, a^{2} c{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} - 1\right )}^{3}{\left | a \right |}} \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)*x^2/(-a^2*c*x^2+c),x, algorithm="giac")

[Out]

3*arcsin(a*x)*sgn(a)/(a^2*c*abs(a)) - sqrt(-a^2*x^2 + 1)/(a^3*c) + 2/3*(24*(sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^
2*x) - 9*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2/(a^4*x^2) - 11)/(a^2*c*((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 1
)^3*abs(a))