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

Optimal. Leaf size=155 \[ -\frac{(a x+1)^6}{7 a c^3 \left (1-a^2 x^2\right )^{7/2}}+\frac{4 (a x+1)^5}{7 a c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{(a x+1)^4}{a c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{4 (a x+1)^2}{a c^3 \sqrt{1-a^2 x^2}}+\frac{6 \sqrt{1-a^2 x^2}}{a c^3}-\frac{6 \sin ^{-1}(a x)}{a c^3} \]

[Out]

-(1 + a*x)^6/(7*a*c^3*(1 - a^2*x^2)^(7/2)) + (4*(1 + a*x)^5)/(7*a*c^3*(1 - a^2*x^2)^(5/2)) - (1 + a*x)^4/(a*c^
3*(1 - a^2*x^2)^(3/2)) + (4*(1 + a*x)^2)/(a*c^3*Sqrt[1 - a^2*x^2]) + (6*Sqrt[1 - a^2*x^2])/(a*c^3) - (6*ArcSin
[a*x])/(a*c^3)

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Rubi [A]  time = 0.334214, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {6131, 6128, 852, 1635, 789, 669, 641, 216} \[ -\frac{(a x+1)^6}{7 a c^3 \left (1-a^2 x^2\right )^{7/2}}+\frac{4 (a x+1)^5}{7 a c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{(a x+1)^4}{a c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{4 (a x+1)^2}{a c^3 \sqrt{1-a^2 x^2}}+\frac{6 \sqrt{1-a^2 x^2}}{a c^3}-\frac{6 \sin ^{-1}(a x)}{a c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(1 + a*x)^6/(7*a*c^3*(1 - a^2*x^2)^(7/2)) + (4*(1 + a*x)^5)/(7*a*c^3*(1 - a^2*x^2)^(5/2)) - (1 + a*x)^4/(a*c^
3*(1 - a^2*x^2)^(3/2)) + (4*(1 + a*x)^2)/(a*c^3*Sqrt[1 - a^2*x^2]) + (6*Sqrt[1 - a^2*x^2])/(a*c^3) - (6*ArcSin
[a*x])/(a*c^3)

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 852

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

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 789

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*(p + 1)), x] - Dist[(e*(m*(d*g + e*f) + 2*e*f*(p + 1)))/(2*c*d*(p + 1)
), Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && EqQ[c*d^2 + a*e^2, 0]
&& LtQ[p, -1] && GtQ[m, 0]

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)}}{\left (c-\frac{c}{a x}\right )^3} \, dx &=-\frac{a^3 \int \frac{e^{3 \tanh ^{-1}(a x)} x^3}{(1-a x)^3} \, dx}{c^3}\\ &=-\frac{a^3 \int \frac{x^3 \left (1-a^2 x^2\right )^{3/2}}{(1-a x)^6} \, dx}{c^3}\\ &=-\frac{a^3 \int \frac{x^3 (1+a x)^6}{\left (1-a^2 x^2\right )^{9/2}} \, dx}{c^3}\\ &=-\frac{(1+a x)^6}{7 a c^3 \left (1-a^2 x^2\right )^{7/2}}+\frac{a^3 \int \frac{(1+a x)^5 \left (\frac{6}{a^3}+\frac{7 x}{a^2}+\frac{7 x^2}{a}\right )}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{7 c^3}\\ &=-\frac{(1+a x)^6}{7 a c^3 \left (1-a^2 x^2\right )^{7/2}}+\frac{4 (1+a x)^5}{7 a c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{a^3 \int \frac{\left (\frac{70}{a^3}+\frac{35 x}{a^2}\right ) (1+a x)^4}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{35 c^3}\\ &=-\frac{(1+a x)^6}{7 a c^3 \left (1-a^2 x^2\right )^{7/2}}+\frac{4 (1+a x)^5}{7 a c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{(1+a x)^4}{a c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{2 \int \frac{(1+a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=-\frac{(1+a x)^6}{7 a c^3 \left (1-a^2 x^2\right )^{7/2}}+\frac{4 (1+a x)^5}{7 a c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{(1+a x)^4}{a c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{4 (1+a x)^2}{a c^3 \sqrt{1-a^2 x^2}}-\frac{6 \int \frac{1+a x}{\sqrt{1-a^2 x^2}} \, dx}{c^3}\\ &=-\frac{(1+a x)^6}{7 a c^3 \left (1-a^2 x^2\right )^{7/2}}+\frac{4 (1+a x)^5}{7 a c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{(1+a x)^4}{a c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{4 (1+a x)^2}{a c^3 \sqrt{1-a^2 x^2}}+\frac{6 \sqrt{1-a^2 x^2}}{a c^3}-\frac{6 \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{c^3}\\ &=-\frac{(1+a x)^6}{7 a c^3 \left (1-a^2 x^2\right )^{7/2}}+\frac{4 (1+a x)^5}{7 a c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{(1+a x)^4}{a c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{4 (1+a x)^2}{a c^3 \sqrt{1-a^2 x^2}}+\frac{6 \sqrt{1-a^2 x^2}}{a c^3}-\frac{6 \sin ^{-1}(a x)}{a c^3}\\ \end{align*}

Mathematica [A]  time = 0.148317, size = 69, normalized size = 0.45 \[ \frac{\frac{\sqrt{1-a^2 x^2} \left (7 a^4 x^4-116 a^3 x^3+261 a^2 x^2-222 a x+66\right )}{(a x-1)^4}-42 \sin ^{-1}(a x)}{7 a c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((Sqrt[1 - a^2*x^2]*(66 - 222*a*x + 261*a^2*x^2 - 116*a^3*x^3 + 7*a^4*x^4))/(-1 + a*x)^4 - 42*ArcSin[a*x])/(7*
a*c^3)

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Maple [A]  time = 0.052, size = 256, normalized size = 1.7 \begin{align*} -{\frac{a{x}^{2}}{{c}^{3}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+20\,{\frac{1}{a{c}^{3}\sqrt{-{a}^{2}{x}^{2}+1}}}+44\,{\frac{x}{{c}^{3}\sqrt{-{a}^{2}{x}^{2}+1}}}-6\,{\frac{1}{{c}^{3}\sqrt{{a}^{2}}}\arctan \left ({\frac{\sqrt{{a}^{2}}x}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }+{\frac{8}{7\,{a}^{4}{c}^{3}} \left ( x-{a}^{-1} \right ) ^{-3}{\frac{1}{\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) }}}}+{\frac{44}{7\,{a}^{3}{c}^{3}} \left ( x-{a}^{-1} \right ) ^{-2}{\frac{1}{\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) }}}}+{\frac{110}{7\,{a}^{2}{c}^{3}} \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{220\,x}{7\,{c}^{3}}{\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)^3,x)

[Out]

-a/c^3*x^2/(-a^2*x^2+1)^(1/2)+20/a/c^3/(-a^2*x^2+1)^(1/2)+44*x/c^3/(-a^2*x^2+1)^(1/2)-6/c^3/(a^2)^(1/2)*arctan
((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+8/7/a^4/c^3/(x-1/a)^3/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)+44/7/a^3/c^3/(x-1/
a)^2/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)+110/7/a^2/c^3/(x-1/a)/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)-220/7/c^3/(-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 )}^{3}}\,{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)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.20865, size = 398, normalized size = 2.57 \begin{align*} \frac{66 \, a^{4} x^{4} - 264 \, a^{3} x^{3} + 396 \, a^{2} x^{2} - 264 \, a x + 84 \,{\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) +{\left (7 \, a^{4} x^{4} - 116 \, a^{3} x^{3} + 261 \, a^{2} x^{2} - 222 \, a x + 66\right )} \sqrt{-a^{2} x^{2} + 1} + 66}{7 \,{\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\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)^3,x, algorithm="fricas")

[Out]

1/7*(66*a^4*x^4 - 264*a^3*x^3 + 396*a^2*x^2 - 264*a*x + 84*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*arcta
n((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (7*a^4*x^4 - 116*a^3*x^3 + 261*a^2*x^2 - 222*a*x + 66)*sqrt(-a^2*x^2 + 1)
+ 66)/(a^5*c^3*x^4 - 4*a^4*c^3*x^3 + 6*a^3*c^3*x^2 - 4*a^2*c^3*x + a*c^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{a^{3} \left (\int \frac{x^{3}}{- a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 3 a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{3 a x^{4}}{- a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 3 a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{3 a^{2} x^{5}}{- a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 3 a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a^{3} x^{6}}{- a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 3 a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx\right )}{c^{3}} \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)**3,x)

[Out]

a**3*(Integral(x**3/(-a**5*x**5*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**3*x**3*sqrt(-a*
*2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Int
egral(3*a*x**4/(-a**5*x**5*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**3*x**3*sqrt(-a**2*x*
*2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral
(3*a**2*x**5/(-a**5*x**5*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**3*x**3*sqrt(-a**2*x**2
+ 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(a
**3*x**6/(-a**5*x**5*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**3*x**3*sqrt(-a**2*x**2 + 1
) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x))/c**3

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Giac [A]  time = 1.22897, size = 316, normalized size = 2.04 \begin{align*} -\frac{6 \, \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{c^{3}{\left | a \right |}} + \frac{\sqrt{-a^{2} x^{2} + 1}}{a c^{3}} - \frac{2 \,{\left (\frac{371 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}}{a^{2} x} - \frac{952 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + \frac{1246 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}}{a^{6} x^{3}} - \frac{819 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4}}{a^{8} x^{4}} + \frac{287 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{5}}{a^{10} x^{5}} - \frac{42 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{6}}{a^{12} x^{6}} - 59\right )}}{7 \, c^{3}{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} - 1\right )}^{7}{\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)^3,x, algorithm="giac")

[Out]

-6*arcsin(a*x)*sgn(a)/(c^3*abs(a)) + sqrt(-a^2*x^2 + 1)/(a*c^3) - 2/7*(371*(sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^
2*x) - 952*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2/(a^4*x^2) + 1246*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3/(a^6*x^3) - 81
9*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4/(a^8*x^4) + 287*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^5/(a^10*x^5) - 42*(sqrt(-a
^2*x^2 + 1)*abs(a) + a)^6/(a^12*x^6) - 59)/(c^3*((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 1)^7*abs(a))