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

Optimal. Leaf size=99 \[ -\frac{\left (1-a^2 x^2\right )^{5/2}}{3 a c (1-a x)^4}+\frac{8 \left (1-a^2 x^2\right )^{3/2}}{3 a c (1-a x)^2}+\frac{4 \sqrt{1-a^2 x^2}}{a c}-\frac{4 \sin ^{-1}(a x)}{a c} \]

[Out]

(4*Sqrt[1 - a^2*x^2])/(a*c) + (8*(1 - a^2*x^2)^(3/2))/(3*a*c*(1 - a*x)^2) - (1 - a^2*x^2)^(5/2)/(3*a*c*(1 - a*
x)^4) - (4*ArcSin[a*x])/(a*c)

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Rubi [A]  time = 0.120271, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {6131, 6128, 793, 663, 665, 216} \[ -\frac{\left (1-a^2 x^2\right )^{5/2}}{3 a c (1-a x)^4}+\frac{8 \left (1-a^2 x^2\right )^{3/2}}{3 a c (1-a x)^2}+\frac{4 \sqrt{1-a^2 x^2}}{a c}-\frac{4 \sin ^{-1}(a x)}{a c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[1 - a^2*x^2])/(a*c) + (8*(1 - a^2*x^2)^(3/2))/(3*a*c*(1 - a*x)^2) - (1 - a^2*x^2)^(5/2)/(3*a*c*(1 - a*
x)^4) - (4*ArcSin[a*x])/(a*c)

Rule 6131

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

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 665

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 + 2*p + 1)), x] - Dist[(2*c*d*p)/(e^2*(m + 2*p + 1)), Int[(d + e*x)^(m + 1)*(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] && (LeQ[-2, m, 0] || EqQ[m + p + 1, 0]) && NeQ[
m + 2*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^{3 \tanh ^{-1}(a x)}}{c-\frac{c}{a x}} \, dx &=-\frac{a \int \frac{e^{3 \tanh ^{-1}(a x)} x}{1-a x} \, dx}{c}\\ &=-\frac{a \int \frac{x \left (1-a^2 x^2\right )^{3/2}}{(1-a x)^4} \, dx}{c}\\ &=-\frac{\left (1-a^2 x^2\right )^{5/2}}{3 a c (1-a x)^4}+\frac{4 \int \frac{\left (1-a^2 x^2\right )^{3/2}}{(1-a x)^3} \, dx}{3 c}\\ &=\frac{8 \left (1-a^2 x^2\right )^{3/2}}{3 a c (1-a x)^2}-\frac{\left (1-a^2 x^2\right )^{5/2}}{3 a c (1-a x)^4}-\frac{4 \int \frac{\sqrt{1-a^2 x^2}}{1-a x} \, dx}{c}\\ &=\frac{4 \sqrt{1-a^2 x^2}}{a c}+\frac{8 \left (1-a^2 x^2\right )^{3/2}}{3 a c (1-a x)^2}-\frac{\left (1-a^2 x^2\right )^{5/2}}{3 a c (1-a x)^4}-\frac{4 \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{c}\\ &=\frac{4 \sqrt{1-a^2 x^2}}{a c}+\frac{8 \left (1-a^2 x^2\right )^{3/2}}{3 a c (1-a x)^2}-\frac{\left (1-a^2 x^2\right )^{5/2}}{3 a c (1-a x)^4}-\frac{4 \sin ^{-1}(a x)}{a c}\\ \end{align*}

Mathematica [C]  time = 0.0338071, size = 62, normalized size = 0.63 \[ -\frac{16 \sqrt{2} (a x-1) \text{Hypergeometric2F1}\left (-\frac{3}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2} (1-a x)\right )+(a x+1)^{5/2}}{3 a c (1-a x)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((1 + a*x)^(5/2) + 16*Sqrt[2]*(-1 + a*x)*Hypergeometric2F1[-3/2, -1/2, 1/2, (1 - a*x)/2])/(3*a*c*(1 - a*x)^(3
/2))

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Maple [A]  time = 0.046, size = 168, normalized size = 1.7 \begin{align*} -{\frac{a{x}^{2}}{c}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+9\,{\frac{1}{ac\sqrt{-{a}^{2}{x}^{2}+1}}}+12\,{\frac{x}{c\sqrt{-{a}^{2}{x}^{2}+1}}}-4\,{\frac{1}{c\sqrt{{a}^{2}}}\arctan \left ({\frac{\sqrt{{a}^{2}}x}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }+{\frac{8}{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{16\,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/x),x)

[Out]

-a/c*x^2/(-a^2*x^2+1)^(1/2)+9/a/c/(-a^2*x^2+1)^(1/2)+12/c*x/(-a^2*x^2+1)^(1/2)-4/c/(a^2)^(1/2)*arctan((a^2)^(1
/2)*x/(-a^2*x^2+1)^(1/2))+8/3/a^2/c/(x-1/a)/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)-16/3/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 x}\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/x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.17749, size = 236, normalized size = 2.38 \begin{align*} \frac{19 \, a^{2} x^{2} - 38 \, a x + 24 \,{\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} - 26 \, a x + 19\right )} \sqrt{-a^{2} x^{2} + 1} + 19}{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/x),x, algorithm="fricas")

[Out]

1/3*(19*a^2*x^2 - 38*a*x + 24*(a^2*x^2 - 2*a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (3*a^2*x^2 - 26*a
*x + 19)*sqrt(-a^2*x^2 + 1) + 19)/(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 \left (\int \frac{x}{- 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{3 a 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{3 a^{2} 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^{3} 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/x),x)

[Out]

a*(Integral(x/(-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) - s
qrt(-a**2*x**2 + 1)), x) + Integral(3*a*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(3*a**2*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**3*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 = 1.16348, size = 170, normalized size = 1.72 \begin{align*} -\frac{4 \, \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{c{\left | a \right |}} + \frac{\sqrt{-a^{2} x^{2} + 1}}{a c} - \frac{8 \,{\left (\frac{9 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}}{a^{2} x} - \frac{3 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} - 4\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/x),x, algorithm="giac")

[Out]

-4*arcsin(a*x)*sgn(a)/(c*abs(a)) + sqrt(-a^2*x^2 + 1)/(a*c) - 8/3*(9*(sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) -
3*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2/(a^4*x^2) - 4)/(c*((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 1)^3*abs(a))