3.1.57 \(\int \frac {e^{-3 \tanh ^{-1}(a x)}}{x^3} \, dx\) [57]

Optimal. Leaf size=90 \[ -\frac {\sqrt {1-a^2 x^2}}{2 x^2}+\frac {3 a \sqrt {1-a^2 x^2}}{x}+\frac {4 a^2 \sqrt {1-a^2 x^2}}{1+a x}-\frac {9}{2} a^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]

[Out]

-9/2*a^2*arctanh((-a^2*x^2+1)^(1/2))-1/2*(-a^2*x^2+1)^(1/2)/x^2+3*a*(-a^2*x^2+1)^(1/2)/x+4*a^2*(-a^2*x^2+1)^(1
/2)/(a*x+1)

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Rubi [A]
time = 0.49, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6259, 6874, 272, 44, 65, 214, 270, 665} \begin {gather*} \frac {4 a^2 \sqrt {1-a^2 x^2}}{a x+1}+\frac {3 a \sqrt {1-a^2 x^2}}{x}-\frac {\sqrt {1-a^2 x^2}}{2 x^2}-\frac {9}{2} a^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*Sqrt[1 - a^2*x^2]/x^2 + (3*a*Sqrt[1 - a^2*x^2])/x + (4*a^2*Sqrt[1 - a^2*x^2])/(1 + a*x) - (9*a^2*ArcTanh[
Sqrt[1 - a^2*x^2]])/2

Rule 44

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] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 65

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 214

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

Rule 270

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 272

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 665

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 6259

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 6874

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^3} \, dx &=\int \frac {(1-a x)^2}{x^3 (1+a x) \sqrt {1-a^2 x^2}} \, dx\\ &=\int \left (\frac {1}{x^3 \sqrt {1-a^2 x^2}}-\frac {3 a}{x^2 \sqrt {1-a^2 x^2}}+\frac {4 a^2}{x \sqrt {1-a^2 x^2}}-\frac {4 a^3}{(1+a x) \sqrt {1-a^2 x^2}}\right ) \, dx\\ &=-\left ((3 a) \int \frac {1}{x^2 \sqrt {1-a^2 x^2}} \, dx\right )+\left (4 a^2\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx-\left (4 a^3\right ) \int \frac {1}{(1+a x) \sqrt {1-a^2 x^2}} \, dx+\int \frac {1}{x^3 \sqrt {1-a^2 x^2}} \, dx\\ &=\frac {3 a \sqrt {1-a^2 x^2}}{x}+\frac {4 a^2 \sqrt {1-a^2 x^2}}{1+a x}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1-a^2 x}} \, dx,x,x^2\right )+\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {1-a^2 x^2}}{2 x^2}+\frac {3 a \sqrt {1-a^2 x^2}}{x}+\frac {4 a^2 \sqrt {1-a^2 x^2}}{1+a x}-4 \text {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} a^2 \text {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {1-a^2 x^2}}{2 x^2}+\frac {3 a \sqrt {1-a^2 x^2}}{x}+\frac {4 a^2 \sqrt {1-a^2 x^2}}{1+a x}-4 a^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\frac {1}{2} \text {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}}{2 x^2}+\frac {3 a \sqrt {1-a^2 x^2}}{x}+\frac {4 a^2 \sqrt {1-a^2 x^2}}{1+a x}-\frac {9}{2} a^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 75, normalized size = 0.83 \begin {gather*} \sqrt {1-a^2 x^2} \left (-\frac {1}{2 x^2}+\frac {3 a}{x}+\frac {4 a^2}{1+a x}\right )+\frac {9}{2} a^2 \log (x)-\frac {9}{2} a^2 \log \left (1+\sqrt {1-a^2 x^2}\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Sqrt[1 - a^2*x^2]*(-1/2*1/x^2 + (3*a)/x + (4*a^2)/(1 + a*x)) + (9*a^2*Log[x])/2 - (9*a^2*Log[1 + Sqrt[1 - a^2*
x^2]])/2

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(447\) vs. \(2(78)=156\).
time = 0.82, size = 448, normalized size = 4.98

method	result	size
risch	\(-\frac {6 a^{3} x^{3}-a^{2} x^{2}-6 a x +1}{2 x^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {a^{2} \left (-9 \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {8 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{a \left (x +\frac {1}{a}\right )}\right )}{2}\)	\(97\)
default	\(-6 a^{2} \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )\right )-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{2 x^{2}}+\frac {9 a^{2} \left (\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3}+\sqrt {-a^{2} x^{2}+1}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}+\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{3}}-a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{2}}+3 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )\right )\right )-3 a \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{x}-4 a^{2} \left (\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} x}{4}+\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 \sqrt {a^{2}}}\right )\right )\)	\(448\)

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^3,x,method=_RETURNVERBOSE)

[Out]

-6*a^2*(1/3*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(3/2)+a*(-1/4*(-2*a^2*(x+1/a)+2*a)/a^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(
1/2)+1/2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2))))-1/2/x^2*(-a^2*x^2+1)^(5/2)+9/2
*a^2*(1/3*(-a^2*x^2+1)^(3/2)+(-a^2*x^2+1)^(1/2)-arctanh(1/(-a^2*x^2+1)^(1/2)))+1/a/(x+1/a)^3*(-a^2*(x+1/a)^2+2
*a*(x+1/a))^(5/2)-a*(1/a/(x+1/a)^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(5/2)+3*a*(1/3*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(3
/2)+a*(-1/4*(-2*a^2*(x+1/a)+2*a)/a^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)+1/2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(
-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)))))-3*a*(-1/x*(-a^2*x^2+1)^(5/2)-4*a^2*(1/4*(-a^2*x^2+1)^(3/2)*x+3/8*x*(-a^2
*x^2+1)^(1/2)+3/8/(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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.46, size = 93, normalized size = 1.03 \begin {gather*} \frac {8 \, a^{3} x^{3} + 8 \, a^{2} x^{2} + 9 \, {\left (a^{3} x^{3} + a^{2} x^{2}\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + {\left (14 \, a^{2} x^{2} + 5 \, a x - 1\right )} \sqrt {-a^{2} x^{2} + 1}}{2 \, {\left (a x^{3} + x^{2}\right )}} \end {gather*}

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^3,x, algorithm="fricas")

[Out]

1/2*(8*a^3*x^3 + 8*a^2*x^2 + 9*(a^3*x^3 + a^2*x^2)*log((sqrt(-a^2*x^2 + 1) - 1)/x) + (14*a^2*x^2 + 5*a*x - 1)*
sqrt(-a^2*x^2 + 1))/(a*x^3 + x^2)

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

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**3,x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 214 vs. \(2 (78) = 156\).
time = 0.43, size = 214, normalized size = 2.38 \begin {gather*} \frac {{\left (a^{3} - \frac {11 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a}{x} - \frac {76 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a x^{2}}\right )} a^{4} x^{2}}{8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )} {\left | a \right |}} - \frac {9 \, a^{3} \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 {12 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a {\left | a \right |}}{x} - \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} {\left | a \right |}}{a x^{2}}}{8 \, a^{2}} \end {gather*}

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^3,x, algorithm="giac")

[Out]

1/8*(a^3 - 11*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a/x - 76*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2/(a*x^2))*a^4*x^2/((sq
rt(-a^2*x^2 + 1)*abs(a) + a)^2*((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) + 1)*abs(a)) - 9/2*a^3*log(1/2*abs(-2*
sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs(a) + 1/8*(12*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a*abs(a)/x - (
sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*abs(a)/(a*x^2))/a^2

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Mupad [B]
time = 0.81, size = 105, normalized size = 1.17 \begin {gather*} \frac {3\,a\,\sqrt {1-a^2\,x^2}}{x}-\frac {\sqrt {1-a^2\,x^2}}{2\,x^2}-\frac {4\,a^3\,\sqrt {1-a^2\,x^2}}{\left (x\,\sqrt {-a^2}+\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}+\frac {a^2\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,9{}\mathrm {i}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^2*atan((1 - a^2*x^2)^(1/2)*1i)*9i)/2 - (1 - a^2*x^2)^(1/2)/(2*x^2) + (3*a*(1 - a^2*x^2)^(1/2))/x - (4*a^3*(
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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