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

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

[Out]

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

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Rubi [A]  time = 0.226167, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {6157, 6148, 1635, 21, 669, 641, 216} \[ -\frac{(a x+1)^3}{3 a c \left (1-a^2 x^2\right )^{3/2}}+\frac{2 (a x+1)^2}{a c \sqrt{1-a^2 x^2}}+\frac{3 \sqrt{1-a^2 x^2}}{a c}-\frac{3 \sin ^{-1}(a x)}{a c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6157

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

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

Mathematica [A]  time = 0.0756841, size = 78, normalized size = 0.82 \[ \frac{-3 a^3 x^3+16 a^2 x^2-9 (a x-1) \sqrt{1-a^2 x^2} \sin ^{-1}(a x)+5 a x-14}{3 a c (a x-1) \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

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Maple [A]  time = 0.043, size = 168, normalized size = 1.8 \begin{align*} -{\frac{a{x}^{2}}{c}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+6\,{\frac{1}{ac\sqrt{-{a}^{2}{x}^{2}+1}}}+7\,{\frac{x}{c\sqrt{-{a}^{2}{x}^{2}+1}}}-3\,{\frac{1}{c\sqrt{{a}^{2}}}\arctan \left ({\frac{\sqrt{{a}^{2}}x}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }+{\frac{4}{3\,{a}^{2}c} \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\,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)/(c-c/a^2/x^2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\left (c - \frac{c}{a^{2} x^{2}}\right )}}\,{d x} \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)/(c-c/a^2/x^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.12014, size = 236, normalized size = 2.48 \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^{3} c x^{2} - 2 \, a^{2} c x + a 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)/(c-c/a^2/x^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^3*c*x^2 - 2*a^2*c*x + a*c)

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

[Out]

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

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Giac [A]  time = 1.23167, size = 170, normalized size = 1.79 \begin{align*} -\frac{3 \, \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{c{\left | a \right |}} + \frac{\sqrt{-a^{2} x^{2} + 1}}{a 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 \, 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)/(c-c/a^2/x^2),x, algorithm="giac")

[Out]

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