3.59 \(\int \frac{e^{-3 \tanh ^{-1}(a x)}}{x^5} \, dx\)

Optimal. Leaf size=135 \[ \frac{4 a^4 \sqrt{1-a^2 x^2}}{a x+1}+\frac{6 a^3 \sqrt{1-a^2 x^2}}{x}-\frac{19 a^2 \sqrt{1-a^2 x^2}}{8 x^2}+\frac{a \sqrt{1-a^2 x^2}}{x^3}-\frac{\sqrt{1-a^2 x^2}}{4 x^4}-\frac{51}{8} a^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]

[Out]

-Sqrt[1 - a^2*x^2]/(4*x^4) + (a*Sqrt[1 - a^2*x^2])/x^3 - (19*a^2*Sqrt[1 - a^2*x^2])/(8*x^2) + (6*a^3*Sqrt[1 -
a^2*x^2])/x + (4*a^4*Sqrt[1 - a^2*x^2])/(1 + a*x) - (51*a^4*ArcTanh[Sqrt[1 - a^2*x^2]])/8

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Rubi [A]  time = 0.822491, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 9, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {6124, 6742, 266, 51, 63, 208, 271, 264, 651} \[ \frac{4 a^4 \sqrt{1-a^2 x^2}}{a x+1}+\frac{6 a^3 \sqrt{1-a^2 x^2}}{x}-\frac{19 a^2 \sqrt{1-a^2 x^2}}{8 x^2}+\frac{a \sqrt{1-a^2 x^2}}{x^3}-\frac{\sqrt{1-a^2 x^2}}{4 x^4}-\frac{51}{8} a^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(E^(3*ArcTanh[a*x])*x^5),x]

[Out]

-Sqrt[1 - a^2*x^2]/(4*x^4) + (a*Sqrt[1 - a^2*x^2])/x^3 - (19*a^2*Sqrt[1 - a^2*x^2])/(8*x^2) + (6*a^3*Sqrt[1 -
a^2*x^2])/x + (4*a^4*Sqrt[1 - a^2*x^2])/(1 + a*x) - (51*a^4*ArcTanh[Sqrt[1 - a^2*x^2]])/8

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 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 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 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^5} \, dx &=\int \frac{(1-a x)^2}{x^5 (1+a x) \sqrt{1-a^2 x^2}} \, dx\\ &=\int \left (\frac{1}{x^5 \sqrt{1-a^2 x^2}}-\frac{3 a}{x^4 \sqrt{1-a^2 x^2}}+\frac{4 a^2}{x^3 \sqrt{1-a^2 x^2}}-\frac{4 a^3}{x^2 \sqrt{1-a^2 x^2}}+\frac{4 a^4}{x \sqrt{1-a^2 x^2}}-\frac{4 a^5}{(1+a x) \sqrt{1-a^2 x^2}}\right ) \, dx\\ &=-\left ((3 a) \int \frac{1}{x^4 \sqrt{1-a^2 x^2}} \, dx\right )+\left (4 a^2\right ) \int \frac{1}{x^3 \sqrt{1-a^2 x^2}} \, dx-\left (4 a^3\right ) \int \frac{1}{x^2 \sqrt{1-a^2 x^2}} \, dx+\left (4 a^4\right ) \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx-\left (4 a^5\right ) \int \frac{1}{(1+a x) \sqrt{1-a^2 x^2}} \, dx+\int \frac{1}{x^5 \sqrt{1-a^2 x^2}} \, dx\\ &=\frac{a \sqrt{1-a^2 x^2}}{x^3}+\frac{4 a^3 \sqrt{1-a^2 x^2}}{x}+\frac{4 a^4 \sqrt{1-a^2 x^2}}{1+a x}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{1-a^2 x}} \, dx,x,x^2\right )+\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1-a^2 x}} \, dx,x,x^2\right )-\left (2 a^3\right ) \int \frac{1}{x^2 \sqrt{1-a^2 x^2}} \, dx+\left (2 a^4\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}}{4 x^4}+\frac{a \sqrt{1-a^2 x^2}}{x^3}-\frac{2 a^2 \sqrt{1-a^2 x^2}}{x^2}+\frac{6 a^3 \sqrt{1-a^2 x^2}}{x}+\frac{4 a^4 \sqrt{1-a^2 x^2}}{1+a x}+\frac{1}{8} \left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1-a^2 x}} \, dx,x,x^2\right )-\left (4 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )+a^4 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{1-a^2 x^2}}{4 x^4}+\frac{a \sqrt{1-a^2 x^2}}{x^3}-\frac{19 a^2 \sqrt{1-a^2 x^2}}{8 x^2}+\frac{6 a^3 \sqrt{1-a^2 x^2}}{x}+\frac{4 a^4 \sqrt{1-a^2 x^2}}{1+a x}-4 a^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\left (2 a^2\right ) \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}{16} \left (3 a^4\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}}{4 x^4}+\frac{a \sqrt{1-a^2 x^2}}{x^3}-\frac{19 a^2 \sqrt{1-a^2 x^2}}{8 x^2}+\frac{6 a^3 \sqrt{1-a^2 x^2}}{x}+\frac{4 a^4 \sqrt{1-a^2 x^2}}{1+a x}-6 a^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\frac{1}{8} \left (3 a^2\right ) \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}}{4 x^4}+\frac{a \sqrt{1-a^2 x^2}}{x^3}-\frac{19 a^2 \sqrt{1-a^2 x^2}}{8 x^2}+\frac{6 a^3 \sqrt{1-a^2 x^2}}{x}+\frac{4 a^4 \sqrt{1-a^2 x^2}}{1+a x}-\frac{51}{8} a^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )\\ \end{align*}

Mathematica [A]  time = 0.0787101, size = 89, normalized size = 0.66 \[ \frac{1}{8} \left (\frac{\sqrt{1-a^2 x^2} \left (80 a^4 x^4+29 a^3 x^3-11 a^2 x^2+6 a x-2\right )}{x^4 (a x+1)}-51 a^4 \log \left (\sqrt{1-a^2 x^2}+1\right )+51 a^4 \log (x)\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(E^(3*ArcTanh[a*x])*x^5),x]

[Out]

((Sqrt[1 - a^2*x^2]*(-2 + 6*a*x - 11*a^2*x^2 + 29*a^3*x^3 + 80*a^4*x^4))/(x^4*(1 + a*x)) + 51*a^4*Log[x] - 51*
a^4*Log[1 + Sqrt[1 - a^2*x^2]])/8

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Maple [B]  time = 0.066, size = 359, normalized size = 2.7 \begin{align*}{\frac{a}{ \left ( x+{a}^{-1} \right ) ^{3}} \left ( -{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{a}{{x}^{3}} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{5}{2}}}}-3\,{\frac{{a}^{2} \left ( -{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) \right ) ^{5/2}}{ \left ( x+{a}^{-1} \right ) ^{2}}}-12\,{a}^{5}\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }x-12\,{\frac{{a}^{5}}{\sqrt{{a}^{2}}}\arctan \left ({\frac{\sqrt{{a}^{2}}x}{\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}} \right ) }+8\,{\frac{{a}^{3} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{5/2}}{x}}+8\,{a}^{5}x \left ( -{a}^{2}{x}^{2}+1 \right ) ^{3/2}+12\,{a}^{5}x\sqrt{-{a}^{2}{x}^{2}+1}+12\,{\frac{{a}^{5}}{\sqrt{{a}^{2}}}\arctan \left ({\frac{\sqrt{{a}^{2}}x}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }-{\frac{23\,{a}^{2}}{8\,{x}^{2}} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{5}{2}}}}-{\frac{1}{4\,{x}^{4}} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{5}{2}}}}+{\frac{17\,{a}^{4}}{8} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{\frac{51\,{a}^{4}}{8}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{51\,{a}^{4}}{8}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }-8\,{a}^{4} \left ( -{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) \right ) ^{3/2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^5,x)

[Out]

a/(x+1/a)^3*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(5/2)+a/x^3*(-a^2*x^2+1)^(5/2)-3*a^2/(x+1/a)^2*(-a^2*(x+1/a)^2+2*a*(x
+1/a))^(5/2)-12*a^5*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)*x-12*a^5/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*(x+1/a)
^2+2*a*(x+1/a))^(1/2))+8*a^3/x*(-a^2*x^2+1)^(5/2)+8*a^5*x*(-a^2*x^2+1)^(3/2)+12*a^5*x*(-a^2*x^2+1)^(1/2)+12*a^
5/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))-23/8*a^2/x^2*(-a^2*x^2+1)^(5/2)-1/4/x^4*(-a^2*x^2+1)^(5
/2)+17/8*a^4*(-a^2*x^2+1)^(3/2)+51/8*a^4*(-a^2*x^2+1)^(1/2)-51/8*a^4*arctanh(1/(-a^2*x^2+1)^(1/2))-8*a^4*(-a^2
*(x+1/a)^2+2*a*(x+1/a))^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{{\left (a x + 1\right )}^{3} x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^5,x, algorithm="maxima")

[Out]

integrate((-a^2*x^2 + 1)^(3/2)/((a*x + 1)^3*x^5), x)

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Fricas [A]  time = 1.78333, size = 235, normalized size = 1.74 \begin{align*} \frac{32 \, a^{5} x^{5} + 32 \, a^{4} x^{4} + 51 \,{\left (a^{5} x^{5} + a^{4} x^{4}\right )} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) +{\left (80 \, a^{4} x^{4} + 29 \, a^{3} x^{3} - 11 \, a^{2} x^{2} + 6 \, a x - 2\right )} \sqrt{-a^{2} x^{2} + 1}}{8 \,{\left (a x^{5} + x^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^5,x, algorithm="fricas")

[Out]

1/8*(32*a^5*x^5 + 32*a^4*x^4 + 51*(a^5*x^5 + a^4*x^4)*log((sqrt(-a^2*x^2 + 1) - 1)/x) + (80*a^4*x^4 + 29*a^3*x
^3 - 11*a^2*x^2 + 6*a*x - 2)*sqrt(-a^2*x^2 + 1))/(a*x^5 + x^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/x**5,x)

[Out]

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

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Giac [B]  time = 1.20077, size = 440, normalized size = 3.26 \begin{align*} \frac{{\left (a^{5} - \frac{7 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{3}}{x} + \frac{32 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} a}{x^{2}} - \frac{160 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}}{a x^{3}} - \frac{712 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4}}{a^{3} x^{4}}\right )} a^{8} x^{4}}{64 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4}{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} + 1\right )}{\left | a \right |}} - \frac{51 \, a^{5} \log \left (\frac{{\left | -2 \, \sqrt{-a^{2} x^{2} + 1}{\left | a \right |} - 2 \, a \right |}}{2 \, a^{2}{\left | x \right |}}\right )}{8 \,{\left | a \right |}} + \frac{\frac{200 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{5}{\left | a \right |}}{x} - \frac{40 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} a^{3}{\left | a \right |}}{x^{2}} + \frac{8 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3} a{\left | a \right |}}{x^{3}} - \frac{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4}{\left | a \right |}}{a x^{4}}}{64 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^5,x, algorithm="giac")

[Out]

1/64*(a^5 - 7*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^3/x + 32*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*a/x^2 - 160*(sqrt(-
a^2*x^2 + 1)*abs(a) + a)^3/(a*x^3) - 712*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4/(a^3*x^4))*a^8*x^4/((sqrt(-a^2*x^2
+ 1)*abs(a) + a)^4*((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) + 1)*abs(a)) - 51/8*a^5*log(1/2*abs(-2*sqrt(-a^2*x
^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs(a) + 1/64*(200*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^5*abs(a)/x - 40*(sqrt
(-a^2*x^2 + 1)*abs(a) + a)^2*a^3*abs(a)/x^2 + 8*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*a*abs(a)/x^3 - (sqrt(-a^2*x^
2 + 1)*abs(a) + a)^4*abs(a)/(a*x^4))/a^4