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

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

[Out]

-51/8*a^4*arctanh((-a^2*x^2+1)^(1/2))-1/4*(-a^2*x^2+1)^(1/2)/x^4+a*(-a^2*x^2+1)^(1/2)/x^3-19/8*a^2*(-a^2*x^2+1
)^(1/2)/x^2+6*a^3*(-a^2*x^2+1)^(1/2)/x+4*a^4*(-a^2*x^2+1)^(1/2)/(a*x+1)

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Rubi [A]  time = 0.82, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, 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 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 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 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 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]

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]]

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*}

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Mathematica [A]  time = 0.08, size = 89, normalized size = 0.66 \[ \frac {1}{8} \left (51 a^4 \log (x)-51 a^4 \log \left (\sqrt {1-a^2 x^2}+1\right )+\frac {\sqrt {1-a^2 x^2} \left (80 a^4 x^4+29 a^3 x^3-11 a^2 x^2+6 a x-2\right )}{x^4 (a x+1)}\right ) \]

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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fricas [A]  time = 0.51, size = 109, normalized size = 0.81 \[ \frac {32 \, a^{5} x^{5} + 32 \, a^{4} x^{4} + 51 \, {\left (a^{5} x^{5} + a^{4} x^{4}\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + {\left (80 \, a^{4} x^{4} + 29 \, a^{3} x^{3} - 11 \, a^{2} x^{2} + 6 \, a x - 2\right )} \sqrt {-a^{2} x^{2} + 1}}{8 \, {\left (a x^{5} + x^{4}\right )}} \]

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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giac [B]  time = 0.19, size = 326, normalized size = 2.41 \[ \frac {{\left (a^{5} - \frac {7 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{3}}{x} + \frac {32 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a}{x^{2}} - \frac {160 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a x^{3}} - \frac {712 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4}}{a^{3} x^{4}}\right )} a^{8} x^{4}}{64 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )} {\left | a \right |}} - \frac {51 \, a^{5} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{8 \, {\left | a \right |}} + \frac {\frac {200 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{5} {\left | a \right |}}{x} - \frac {40 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{3} {\left | a \right |}}{x^{2}} + \frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} a {\left | a \right |}}{x^{3}} - \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} {\left | a \right |}}{a x^{4}}}{64 \, a^{4}} \]

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

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maple [B]  time = 0.07, size = 359, normalized size = 2.66 \[ -\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{4 x^{4}}+\frac {17 a^{4} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{8}-8 a^{4} \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}+\frac {51 a^{4} \sqrt {-a^{2} x^{2}+1}}{8}-\frac {51 a^{4} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{8}+\frac {a \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{\left (x +\frac {1}{a}\right )^{3}}-\frac {3 a^{2} \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{\left (x +\frac {1}{a}\right )^{2}}-12 a^{5} \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}\, x +\frac {8 a^{3} \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{x}+8 a^{5} x \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}+\frac {a \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{x^{3}}-\frac {23 a^{2} \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{8 x^{2}}-\frac {12 a^{5} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}}+12 a^{5} x \sqrt {-a^{2} x^{2}+1}+\frac {12 a^{5} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}} \]

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]

-1/4/x^4*(-a^2*x^2+1)^(5/2)+17/8*a^4*(-a^2*x^2+1)^(3/2)-8*a^4*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(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))+a/(x+1/a)^3*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(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+8*a^3/x*(-a^2*x^2+1)^(5/
2)+8*a^5*x*(-a^2*x^2+1)^(3/2)+a/x^3*(-a^2*x^2+1)^(5/2)-23/8*a^2/x^2*(-a^2*x^2+1)^(5/2)-12*a^5/(a^2)^(1/2)*arct
an((a^2)^(1/2)*x/(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2))+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))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3} x^{5}}\,{d x} \]

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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mupad [B]  time = 0.05, size = 144, normalized size = 1.07 \[ \frac {a\,\sqrt {1-a^2\,x^2}}{x^3}-\frac {\sqrt {1-a^2\,x^2}}{4\,x^4}-\frac {19\,a^2\,\sqrt {1-a^2\,x^2}}{8\,x^2}+\frac {6\,a^3\,\sqrt {1-a^2\,x^2}}{x}-\frac {4\,a^5\,\sqrt {1-a^2\,x^2}}{\left (x\,\sqrt {-a^2}+\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}+\frac {a^4\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,51{}\mathrm {i}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^4*atan((1 - a^2*x^2)^(1/2)*1i)*51i)/8 - (1 - a^2*x^2)^(1/2)/(4*x^4) + (a*(1 - a^2*x^2)^(1/2))/x^3 - (19*a^2
*(1 - a^2*x^2)^(1/2))/(8*x^2) + (6*a^3*(1 - a^2*x^2)^(1/2))/x - (4*a^5*(1 - a^2*x^2)^(1/2))/((x*(-a^2)^(1/2) +
(-a^2)^(1/2)/a)*(-a^2)^(1/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{x^{5} \left (a x + 1\right )^{3}}\, dx \]

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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