3.25 \(\int \frac{e^{3 \tanh ^{-1}(a x)}}{x^4} \, dx\)

Optimal. Leaf size=117 \[ \frac{4 a^3 \sqrt{1-a^2 x^2}}{1-a x}-\frac{14 a^2 \sqrt{1-a^2 x^2}}{3 x}-\frac{3 a \sqrt{1-a^2 x^2}}{2 x^2}-\frac{\sqrt{1-a^2 x^2}}{3 x^3}-\frac{11}{2} a^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]

[Out]

-Sqrt[1 - a^2*x^2]/(3*x^3) - (3*a*Sqrt[1 - a^2*x^2])/(2*x^2) - (14*a^2*Sqrt[1 - a^2*x^2])/(3*x) + (4*a^3*Sqrt[
1 - a^2*x^2])/(1 - a*x) - (11*a^3*ArcTanh[Sqrt[1 - a^2*x^2]])/2

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Rubi [A]  time = 0.737876, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {6124, 6742, 271, 264, 266, 51, 63, 208, 651} \[ \frac{4 a^3 \sqrt{1-a^2 x^2}}{1-a x}-\frac{14 a^2 \sqrt{1-a^2 x^2}}{3 x}-\frac{3 a \sqrt{1-a^2 x^2}}{2 x^2}-\frac{\sqrt{1-a^2 x^2}}{3 x^3}-\frac{11}{2} a^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcTanh[a*x])/x^4,x]

[Out]

-Sqrt[1 - a^2*x^2]/(3*x^3) - (3*a*Sqrt[1 - a^2*x^2])/(2*x^2) - (14*a^2*Sqrt[1 - a^2*x^2])/(3*x) + (4*a^3*Sqrt[
1 - a^2*x^2])/(1 - a*x) - (11*a^3*ArcTanh[Sqrt[1 - a^2*x^2]])/2

Rule 6124

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.), x_Symbol] :> Int[x^m*((1 + a*x)^((n + 1)/2)/((1 - a*x)^((n - 1)/
2)*Sqrt[1 - a^2*x^2])), x] /; FreeQ[{a, m}, x] && IntegerQ[(n - 1)/2]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 651

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^m*(a + c*x^2)^(p + 1))
/(2*c*d*(p + 1)), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p
+ 2, 0]

Rubi steps

\begin{align*} \int \frac{e^{3 \tanh ^{-1}(a x)}}{x^4} \, dx &=\int \frac{(1+a x)^2}{x^4 (1-a x) \sqrt{1-a^2 x^2}} \, dx\\ &=\int \left (\frac{1}{x^4 \sqrt{1-a^2 x^2}}+\frac{3 a}{x^3 \sqrt{1-a^2 x^2}}+\frac{4 a^2}{x^2 \sqrt{1-a^2 x^2}}+\frac{4 a^3}{x \sqrt{1-a^2 x^2}}-\frac{4 a^4}{(-1+a x) \sqrt{1-a^2 x^2}}\right ) \, dx\\ &=(3 a) \int \frac{1}{x^3 \sqrt{1-a^2 x^2}} \, dx+\left (4 a^2\right ) \int \frac{1}{x^2 \sqrt{1-a^2 x^2}} \, dx+\left (4 a^3\right ) \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx-\left (4 a^4\right ) \int \frac{1}{(-1+a x) \sqrt{1-a^2 x^2}} \, dx+\int \frac{1}{x^4 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{\sqrt{1-a^2 x^2}}{3 x^3}-\frac{4 a^2 \sqrt{1-a^2 x^2}}{x}+\frac{4 a^3 \sqrt{1-a^2 x^2}}{1-a x}+\frac{1}{2} (3 a) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1-a^2 x}} \, dx,x,x^2\right )+\frac{1}{3} \left (2 a^2\right ) \int \frac{1}{x^2 \sqrt{1-a^2 x^2}} \, dx+\left (2 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{1-a^2 x^2}}{3 x^3}-\frac{3 a \sqrt{1-a^2 x^2}}{2 x^2}-\frac{14 a^2 \sqrt{1-a^2 x^2}}{3 x}+\frac{4 a^3 \sqrt{1-a^2 x^2}}{1-a x}-(4 a) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )+\frac{1}{4} \left (3 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{1-a^2 x^2}}{3 x^3}-\frac{3 a \sqrt{1-a^2 x^2}}{2 x^2}-\frac{14 a^2 \sqrt{1-a^2 x^2}}{3 x}+\frac{4 a^3 \sqrt{1-a^2 x^2}}{1-a x}-4 a^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\frac{1}{2} (3 a) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )\\ &=-\frac{\sqrt{1-a^2 x^2}}{3 x^3}-\frac{3 a \sqrt{1-a^2 x^2}}{2 x^2}-\frac{14 a^2 \sqrt{1-a^2 x^2}}{3 x}+\frac{4 a^3 \sqrt{1-a^2 x^2}}{1-a x}-\frac{11}{2} a^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )\\ \end{align*}

Mathematica [A]  time = 0.0803808, size = 81, normalized size = 0.69 \[ \frac{1}{6} \left (\frac{\sqrt{1-a^2 x^2} \left (-52 a^3 x^3+19 a^2 x^2+7 a x+2\right )}{x^3 (a x-1)}-33 a^3 \log \left (\sqrt{1-a^2 x^2}+1\right )+33 a^3 \log (x)\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(3*ArcTanh[a*x])/x^4,x]

[Out]

((Sqrt[1 - a^2*x^2]*(2 + 7*a*x + 19*a^2*x^2 - 52*a^3*x^3))/(x^3*(-1 + a*x)) + 33*a^3*Log[x] - 33*a^3*Log[1 + S
qrt[1 - a^2*x^2]])/6

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Maple [A]  time = 0.045, size = 146, normalized size = 1.3 \begin{align*}{\frac{13\,{a}^{2}}{3} \left ( -{\frac{1}{x}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+2\,{\frac{{a}^{2}x}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }+{a}^{3} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}-{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) \right ) +3\,a \left ( -1/2\,{\frac{1}{{x}^{2}\sqrt{-{a}^{2}{x}^{2}+1}}}+3/2\,{a}^{2} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}-{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) \right ) \right ) -{\frac{1}{3\,{x}^{3}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \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^4,x)

[Out]

13/3*a^2*(-1/x/(-a^2*x^2+1)^(1/2)+2*a^2*x/(-a^2*x^2+1)^(1/2))+a^3*(1/(-a^2*x^2+1)^(1/2)-arctanh(1/(-a^2*x^2+1)
^(1/2)))+3*a*(-1/2/x^2/(-a^2*x^2+1)^(1/2)+3/2*a^2*(1/(-a^2*x^2+1)^(1/2)-arctanh(1/(-a^2*x^2+1)^(1/2))))-1/3/x^
3/(-a^2*x^2+1)^(1/2)

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Maxima [A]  time = 0.955518, size = 165, normalized size = 1.41 \begin{align*} \frac{26 \, a^{4} x}{3 \, \sqrt{-a^{2} x^{2} + 1}} - \frac{11}{2} \, a^{3} \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) + \frac{11 \, a^{3}}{2 \, \sqrt{-a^{2} x^{2} + 1}} - \frac{13 \, a^{2}}{3 \, \sqrt{-a^{2} x^{2} + 1} x} - \frac{3 \, a}{2 \, \sqrt{-a^{2} x^{2} + 1} x^{2}} - \frac{1}{3 \, \sqrt{-a^{2} x^{2} + 1} x^{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)/x^4,x, algorithm="maxima")

[Out]

26/3*a^4*x/sqrt(-a^2*x^2 + 1) - 11/2*a^3*log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2/abs(x)) + 11/2*a^3/sqrt(-a^2*x^2
+ 1) - 13/3*a^2/(sqrt(-a^2*x^2 + 1)*x) - 3/2*a/(sqrt(-a^2*x^2 + 1)*x^2) - 1/3/(sqrt(-a^2*x^2 + 1)*x^3)

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Fricas [A]  time = 1.88863, size = 217, normalized size = 1.85 \begin{align*} \frac{24 \, a^{4} x^{4} - 24 \, a^{3} x^{3} + 33 \,{\left (a^{4} x^{4} - a^{3} x^{3}\right )} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) -{\left (52 \, a^{3} x^{3} - 19 \, a^{2} x^{2} - 7 \, a x - 2\right )} \sqrt{-a^{2} x^{2} + 1}}{6 \,{\left (a x^{4} - x^{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)/x^4,x, algorithm="fricas")

[Out]

1/6*(24*a^4*x^4 - 24*a^3*x^3 + 33*(a^4*x^4 - a^3*x^3)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (52*a^3*x^3 - 19*a^2*x
^2 - 7*a*x - 2)*sqrt(-a^2*x^2 + 1))/(a*x^4 - x^3)

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

[Out]

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

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Giac [B]  time = 1.23796, size = 358, normalized size = 3.06 \begin{align*} -\frac{{\left (a^{4} + \frac{8 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{2}}{x} + \frac{48 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{x^{2}} - \frac{249 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}}{a^{2} x^{3}}\right )} a^{6} x^{3}}{24 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} - 1\right )}{\left | a \right |}} - \frac{11 \, a^{4} \log \left (\frac{{\left | -2 \, \sqrt{-a^{2} x^{2} + 1}{\left | a \right |} - 2 \, a \right |}}{2 \, a^{2}{\left | x \right |}}\right )}{2 \,{\left | a \right |}} - \frac{\frac{57 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{4}}{x} + \frac{9 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} a^{2}}{x^{2}} + \frac{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}}{x^{3}}}{24 \, a^{2}{\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^4,x, algorithm="giac")

[Out]

-1/24*(a^4 + 8*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^2/x + 48*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2/x^2 - 249*(sqrt(-a
^2*x^2 + 1)*abs(a) + a)^3/(a^2*x^3))*a^6*x^3/((sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*((sqrt(-a^2*x^2 + 1)*abs(a) +
a)/(a^2*x) - 1)*abs(a)) - 11/2*a^4*log(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs(a) - 1/24
*(57*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^4/x + 9*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*a^2/x^2 + (sqrt(-a^2*x^2 + 1)
*abs(a) + a)^3/x^3)/(a^2*abs(a))