3.1.15 \(\int \frac {\tanh ^{-1}(\frac {\sqrt {e} x}{\sqrt {d+e x^2}})}{x^6} \, dx\) [15]

Optimal. Leaf size=111 \[ -\frac {\sqrt {e} \sqrt {d+e x^2}}{20 d x^4}+\frac {3 e^{3/2} \sqrt {d+e x^2}}{40 d^2 x^2}-\frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{5 x^5}-\frac {3 e^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{40 d^{5/2}} \]

[Out]

-1/5*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/x^5-3/40*e^(5/2)*arctanh((e*x^2+d)^(1/2)/d^(1/2))/d^(5/2)+3/40*e^(3/2)
*(e*x^2+d)^(1/2)/d^2/x^2-1/20*e^(1/2)*(e*x^2+d)^(1/2)/d/x^4

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Rubi [A]
time = 0.04, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {6356, 272, 44, 65, 214} \begin {gather*} -\frac {3 e^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{40 d^{5/2}}+\frac {3 e^{3/2} \sqrt {d+e x^2}}{40 d^2 x^2}-\frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{5 x^5}-\frac {\sqrt {e} \sqrt {d+e x^2}}{20 d x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]]/x^6,x]

[Out]

-1/20*(Sqrt[e]*Sqrt[d + e*x^2])/(d*x^4) + (3*e^(3/2)*Sqrt[d + e*x^2])/(40*d^2*x^2) - ArcTanh[(Sqrt[e]*x)/Sqrt[
d + e*x^2]]/(5*x^5) - (3*e^(5/2)*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(40*d^(5/2))

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 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 6356

Int[ArcTanh[((c_.)*(x_))/Sqrt[(a_.) + (b_.)*(x_)^2]]*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*(ArcT
anh[(c*x)/Sqrt[a + b*x^2]]/(d*(m + 1))), x] - Dist[c/(d*(m + 1)), Int[(d*x)^(m + 1)/Sqrt[a + b*x^2], x], x] /;
FreeQ[{a, b, c, d, m}, x] && EqQ[b, c^2] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{x^6} \, dx &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{5 x^5}+\frac {1}{5} \sqrt {e} \int \frac {1}{x^5 \sqrt {d+e x^2}} \, dx\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{5 x^5}+\frac {1}{10} \sqrt {e} \text {Subst}\left (\int \frac {1}{x^3 \sqrt {d+e x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {e} \sqrt {d+e x^2}}{20 d x^4}-\frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{5 x^5}-\frac {\left (3 e^{3/2}\right ) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {d+e x}} \, dx,x,x^2\right )}{40 d}\\ &=-\frac {\sqrt {e} \sqrt {d+e x^2}}{20 d x^4}+\frac {3 e^{3/2} \sqrt {d+e x^2}}{40 d^2 x^2}-\frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{5 x^5}+\frac {\left (3 e^{5/2}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^2\right )}{80 d^2}\\ &=-\frac {\sqrt {e} \sqrt {d+e x^2}}{20 d x^4}+\frac {3 e^{3/2} \sqrt {d+e x^2}}{40 d^2 x^2}-\frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{5 x^5}+\frac {\left (3 e^{3/2}\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )}{40 d^2}\\ &=-\frac {\sqrt {e} \sqrt {d+e x^2}}{20 d x^4}+\frac {3 e^{3/2} \sqrt {d+e x^2}}{40 d^2 x^2}-\frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{5 x^5}-\frac {3 e^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{40 d^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 107, normalized size = 0.96 \begin {gather*} \frac {-8 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )+\frac {\sqrt {e} x \left (\sqrt {d} \sqrt {d+e x^2} \left (-2 d+3 e x^2\right )+3 e^2 x^4 \log (x)-3 e^2 x^4 \log \left (d+\sqrt {d} \sqrt {d+e x^2}\right )\right )}{d^{5/2}}}{40 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]]/x^6,x]

[Out]

(-8*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]] + (Sqrt[e]*x*(Sqrt[d]*Sqrt[d + e*x^2]*(-2*d + 3*e*x^2) + 3*e^2*x^4*Lo
g[x] - 3*e^2*x^4*Log[d + Sqrt[d]*Sqrt[d + e*x^2]]))/d^(5/2))/(40*x^5)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(170\) vs. \(2(83)=166\).
time = 0.01, size = 171, normalized size = 1.54

method	result	size
default	\(-\frac {\arctanh \left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right )}{5 x^{5}}-\frac {e^{\frac {3}{2}} \left (-\frac {\sqrt {e \,x^{2}+d}}{2 x^{2} d}+\frac {e \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {e \,x^{2}+d}}{x}\right )}{2 d^{\frac {3}{2}}}\right )}{5 d}+\frac {\sqrt {e}\, \left (-\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{4 d \,x^{4}}-\frac {e \left (-\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{2 d \,x^{2}}+\frac {e \left (\sqrt {e \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {e \,x^{2}+d}}{x}\right )\right )}{2 d}\right )}{4 d}\right )}{5 d}\)	\(171\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/x^6,x,method=_RETURNVERBOSE)

[Out]

-1/5*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/x^5-1/5*e^(3/2)/d*(-1/2*(e*x^2+d)^(1/2)/x^2/d+1/2*e/d^(3/2)*ln((2*d+2*
d^(1/2)*(e*x^2+d)^(1/2))/x))+1/5*e^(1/2)/d*(-1/4/d/x^4*(e*x^2+d)^(3/2)-1/4*e/d*(-1/2/d/x^2*(e*x^2+d)^(3/2)+1/2
*e/d*((e*x^2+d)^(1/2)-d^(1/2)*ln((2*d+2*d^(1/2)*(e*x^2+d)^(1/2))/x))))

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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(arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/x^6,x, algorithm="maxima")

[Out]

d*integrate(-1/5*e^(1/2*log(x^2*e + d) + 1/2)/(x^9*e^2 + d*x^7*e - (x^7*e + d*x^5)*(x^2*e + d)), x) - 1/10*(lo
g(x*e^(1/2) + sqrt(x^2*e + d)) - log(-x*e^(1/2) + sqrt(x^2*e + d)))/x^5

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 369 vs. \(2 (83) = 166\).
time = 0.39, size = 727, normalized size = 6.55 \begin {gather*} \left [\frac {8 \, d^{3} x^{5} \log \left (-x \cosh \left (\frac {1}{2}\right ) - x \sinh \left (\frac {1}{2}\right ) + \sqrt {\frac {{\left (x^{2} + d\right )} \cosh \left (\frac {1}{2}\right ) + {\left (x^{2} - d\right )} \sinh \left (\frac {1}{2}\right )}{\cosh \left (\frac {1}{2}\right ) - \sinh \left (\frac {1}{2}\right )}}\right ) + 3 \, {\left (x^{5} \cosh \left (\frac {1}{2}\right )^{5} + 5 \, x^{5} \cosh \left (\frac {1}{2}\right )^{4} \sinh \left (\frac {1}{2}\right ) + 10 \, x^{5} \cosh \left (\frac {1}{2}\right )^{3} \sinh \left (\frac {1}{2}\right )^{2} + 10 \, x^{5} \cosh \left (\frac {1}{2}\right )^{2} \sinh \left (\frac {1}{2}\right )^{3} + 5 \, x^{5} \cosh \left (\frac {1}{2}\right ) \sinh \left (\frac {1}{2}\right )^{4} + x^{5} \sinh \left (\frac {1}{2}\right )^{5}\right )} \sqrt {d} \log \left (-\frac {\sqrt {d} - \sqrt {\frac {{\left (x^{2} + d\right )} \cosh \left (\frac {1}{2}\right ) + {\left (x^{2} - d\right )} \sinh \left (\frac {1}{2}\right )}{\cosh \left (\frac {1}{2}\right ) - \sinh \left (\frac {1}{2}\right )}}}{x}\right ) + 4 \, {\left (d^{3} x^{5} - d^{3}\right )} \log \left (\frac {2 \, x^{2} \cosh \left (\frac {1}{2}\right )^{2} + 4 \, x^{2} \cosh \left (\frac {1}{2}\right ) \sinh \left (\frac {1}{2}\right ) + 2 \, x^{2} \sinh \left (\frac {1}{2}\right )^{2} + 2 \, {\left (x \cosh \left (\frac {1}{2}\right ) + x \sinh \left (\frac {1}{2}\right )\right )} \sqrt {\frac {{\left (x^{2} + d\right )} \cosh \left (\frac {1}{2}\right ) + {\left (x^{2} - d\right )} \sinh \left (\frac {1}{2}\right )}{\cosh \left (\frac {1}{2}\right ) - \sinh \left (\frac {1}{2}\right )}} + d}{d}\right ) + {\left (3 \, d x^{3} \cosh \left (\frac {1}{2}\right )^{3} + 9 \, d x^{3} \cosh \left (\frac {1}{2}\right ) \sinh \left (\frac {1}{2}\right )^{2} + 3 \, d x^{3} \sinh \left (\frac {1}{2}\right )^{3} - 2 \, d^{2} x \cosh \left (\frac {1}{2}\right ) + {\left (9 \, d x^{3} \cosh \left (\frac {1}{2}\right )^{2} - 2 \, d^{2} x\right )} \sinh \left (\frac {1}{2}\right )\right )} \sqrt {\frac {{\left (x^{2} + d\right )} \cosh \left (\frac {1}{2}\right ) + {\left (x^{2} - d\right )} \sinh \left (\frac {1}{2}\right )}{\cosh \left (\frac {1}{2}\right ) - \sinh \left (\frac {1}{2}\right )}}}{40 \, d^{3} x^{5}}, \frac {8 \, d^{3} x^{5} \log \left (-x \cosh \left (\frac {1}{2}\right ) - x \sinh \left (\frac {1}{2}\right ) + \sqrt {\frac {{\left (x^{2} + d\right )} \cosh \left (\frac {1}{2}\right ) + {\left (x^{2} - d\right )} \sinh \left (\frac {1}{2}\right )}{\cosh \left (\frac {1}{2}\right ) - \sinh \left (\frac {1}{2}\right )}}\right ) + 6 \, {\left (x^{5} \cosh \left (\frac {1}{2}\right )^{5} + 5 \, x^{5} \cosh \left (\frac {1}{2}\right )^{4} \sinh \left (\frac {1}{2}\right ) + 10 \, x^{5} \cosh \left (\frac {1}{2}\right )^{3} \sinh \left (\frac {1}{2}\right )^{2} + 10 \, x^{5} \cosh \left (\frac {1}{2}\right )^{2} \sinh \left (\frac {1}{2}\right )^{3} + 5 \, x^{5} \cosh \left (\frac {1}{2}\right ) \sinh \left (\frac {1}{2}\right )^{4} + x^{5} \sinh \left (\frac {1}{2}\right )^{5}\right )} \sqrt {-d} \arctan \left (-\frac {{\left (x \cosh \left (\frac {1}{2}\right ) + x \sinh \left (\frac {1}{2}\right ) - \sqrt {\frac {{\left (x^{2} + d\right )} \cosh \left (\frac {1}{2}\right ) + {\left (x^{2} - d\right )} \sinh \left (\frac {1}{2}\right )}{\cosh \left (\frac {1}{2}\right ) - \sinh \left (\frac {1}{2}\right )}}\right )} \sqrt {-d}}{d}\right ) + 4 \, {\left (d^{3} x^{5} - d^{3}\right )} \log \left (\frac {2 \, x^{2} \cosh \left (\frac {1}{2}\right )^{2} + 4 \, x^{2} \cosh \left (\frac {1}{2}\right ) \sinh \left (\frac {1}{2}\right ) + 2 \, x^{2} \sinh \left (\frac {1}{2}\right )^{2} + 2 \, {\left (x \cosh \left (\frac {1}{2}\right ) + x \sinh \left (\frac {1}{2}\right )\right )} \sqrt {\frac {{\left (x^{2} + d\right )} \cosh \left (\frac {1}{2}\right ) + {\left (x^{2} - d\right )} \sinh \left (\frac {1}{2}\right )}{\cosh \left (\frac {1}{2}\right ) - \sinh \left (\frac {1}{2}\right )}} + d}{d}\right ) + {\left (3 \, d x^{3} \cosh \left (\frac {1}{2}\right )^{3} + 9 \, d x^{3} \cosh \left (\frac {1}{2}\right ) \sinh \left (\frac {1}{2}\right )^{2} + 3 \, d x^{3} \sinh \left (\frac {1}{2}\right )^{3} - 2 \, d^{2} x \cosh \left (\frac {1}{2}\right ) + {\left (9 \, d x^{3} \cosh \left (\frac {1}{2}\right )^{2} - 2 \, d^{2} x\right )} \sinh \left (\frac {1}{2}\right )\right )} \sqrt {\frac {{\left (x^{2} + d\right )} \cosh \left (\frac {1}{2}\right ) + {\left (x^{2} - d\right )} \sinh \left (\frac {1}{2}\right )}{\cosh \left (\frac {1}{2}\right ) - \sinh \left (\frac {1}{2}\right )}}}{40 \, d^{3} x^{5}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/x^6,x, algorithm="fricas")

[Out]

[1/40*(8*d^3*x^5*log(-x*cosh(1/2) - x*sinh(1/2) + sqrt(((x^2 + d)*cosh(1/2) + (x^2 - d)*sinh(1/2))/(cosh(1/2)
- sinh(1/2)))) + 3*(x^5*cosh(1/2)^5 + 5*x^5*cosh(1/2)^4*sinh(1/2) + 10*x^5*cosh(1/2)^3*sinh(1/2)^2 + 10*x^5*co
sh(1/2)^2*sinh(1/2)^3 + 5*x^5*cosh(1/2)*sinh(1/2)^4 + x^5*sinh(1/2)^5)*sqrt(d)*log(-(sqrt(d) - sqrt(((x^2 + d)
*cosh(1/2) + (x^2 - d)*sinh(1/2))/(cosh(1/2) - sinh(1/2))))/x) + 4*(d^3*x^5 - d^3)*log((2*x^2*cosh(1/2)^2 + 4*
x^2*cosh(1/2)*sinh(1/2) + 2*x^2*sinh(1/2)^2 + 2*(x*cosh(1/2) + x*sinh(1/2))*sqrt(((x^2 + d)*cosh(1/2) + (x^2 -
d)*sinh(1/2))/(cosh(1/2) - sinh(1/2))) + d)/d) + (3*d*x^3*cosh(1/2)^3 + 9*d*x^3*cosh(1/2)*sinh(1/2)^2 + 3*d*x
^3*sinh(1/2)^3 - 2*d^2*x*cosh(1/2) + (9*d*x^3*cosh(1/2)^2 - 2*d^2*x)*sinh(1/2))*sqrt(((x^2 + d)*cosh(1/2) + (x
^2 - d)*sinh(1/2))/(cosh(1/2) - sinh(1/2))))/(d^3*x^5), 1/40*(8*d^3*x^5*log(-x*cosh(1/2) - x*sinh(1/2) + sqrt(
((x^2 + d)*cosh(1/2) + (x^2 - d)*sinh(1/2))/(cosh(1/2) - sinh(1/2)))) + 6*(x^5*cosh(1/2)^5 + 5*x^5*cosh(1/2)^4
*sinh(1/2) + 10*x^5*cosh(1/2)^3*sinh(1/2)^2 + 10*x^5*cosh(1/2)^2*sinh(1/2)^3 + 5*x^5*cosh(1/2)*sinh(1/2)^4 + x
^5*sinh(1/2)^5)*sqrt(-d)*arctan(-(x*cosh(1/2) + x*sinh(1/2) - sqrt(((x^2 + d)*cosh(1/2) + (x^2 - d)*sinh(1/2))
/(cosh(1/2) - sinh(1/2))))*sqrt(-d)/d) + 4*(d^3*x^5 - d^3)*log((2*x^2*cosh(1/2)^2 + 4*x^2*cosh(1/2)*sinh(1/2)
+ 2*x^2*sinh(1/2)^2 + 2*(x*cosh(1/2) + x*sinh(1/2))*sqrt(((x^2 + d)*cosh(1/2) + (x^2 - d)*sinh(1/2))/(cosh(1/2
) - sinh(1/2))) + d)/d) + (3*d*x^3*cosh(1/2)^3 + 9*d*x^3*cosh(1/2)*sinh(1/2)^2 + 3*d*x^3*sinh(1/2)^3 - 2*d^2*x
*cosh(1/2) + (9*d*x^3*cosh(1/2)^2 - 2*d^2*x)*sinh(1/2))*sqrt(((x^2 + d)*cosh(1/2) + (x^2 - d)*sinh(1/2))/(cosh
(1/2) - sinh(1/2))))/(d^3*x^5)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {atanh}{\left (\frac {\sqrt {e} x}{\sqrt {d + e x^{2}}} \right )}}{x^{6}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(x*e**(1/2)/(e*x**2+d)**(1/2))/x**6,x)

[Out]

Integral(atanh(sqrt(e)*x/sqrt(d + e*x**2))/x**6, x)

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/x^6,x, algorithm="giac")

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {atanh}\left (\frac {\sqrt {e}\,x}{\sqrt {e\,x^2+d}}\right )}{x^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atanh((e^(1/2)*x)/(d + e*x^2)^(1/2))/x^6,x)

[Out]

int(atanh((e^(1/2)*x)/(d + e*x^2)^(1/2))/x^6, x)

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