3.40 \(\int \frac {e^{\text {sech}^{-1}(a x)}}{x^4} \, dx\)

Optimal. Leaf size=132 \[ \frac {1}{8} a^3 \sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )+\frac {\sqrt {1-a x}}{12 a x^4 \sqrt {\frac {1}{a x+1}}}+\frac {1}{12 a x^4}-\frac {e^{\text {sech}^{-1}(a x)}}{3 x^3}+\frac {a \sqrt {1-a x}}{8 x^2 \sqrt {\frac {1}{a x+1}}} \]

[Out]

1/12/a/x^4-1/3*(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))/x^3+1/12*(-a*x+1)^(1/2)/a/x^4/(1/(a*x+1))^(1/2)+1/8*a*(
-a*x+1)^(1/2)/x^2/(1/(a*x+1))^(1/2)+1/8*a^3*arctanh((-a*x+1)^(1/2)*(a*x+1)^(1/2))*(1/(a*x+1))^(1/2)*(a*x+1)^(1
/2)

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Rubi [A]  time = 0.06, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6335, 30, 103, 12, 92, 208} \[ \frac {1}{8} a^3 \sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )+\frac {a \sqrt {1-a x}}{8 x^2 \sqrt {\frac {1}{a x+1}}}+\frac {\sqrt {1-a x}}{12 a x^4 \sqrt {\frac {1}{a x+1}}}+\frac {1}{12 a x^4}-\frac {e^{\text {sech}^{-1}(a x)}}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcSech[a*x]/x^4,x]

[Out]

1/(12*a*x^4) - E^ArcSech[a*x]/(3*x^3) + Sqrt[1 - a*x]/(12*a*x^4*Sqrt[(1 + a*x)^(-1)]) + (a*Sqrt[1 - a*x])/(8*x
^2*Sqrt[(1 + a*x)^(-1)]) + (a^3*Sqrt[(1 + a*x)^(-1)]*Sqrt[1 + a*x]*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/8

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 92

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 103

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

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 6335

Int[E^ArcSech[(a_.)*(x_)^(p_.)]*(x_)^(m_.), x_Symbol] :> Simp[(x^(m + 1)*E^ArcSech[a*x^p])/(m + 1), x] + (Dist
[p/(a*(m + 1)), Int[x^(m - p), x], x] + Dist[(p*Sqrt[1 + a*x^p]*Sqrt[1/(1 + a*x^p)])/(a*(m + 1)), Int[x^(m - p
)/(Sqrt[1 + a*x^p]*Sqrt[1 - a*x^p]), x], x]) /; FreeQ[{a, m, p}, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {e^{\text {sech}^{-1}(a x)}}{x^4} \, dx &=-\frac {e^{\text {sech}^{-1}(a x)}}{3 x^3}-\frac {\int \frac {1}{x^5} \, dx}{3 a}-\frac {\left (\sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {1}{x^5 \sqrt {1-a x} \sqrt {1+a x}} \, dx}{3 a}\\ &=\frac {1}{12 a x^4}-\frac {e^{\text {sech}^{-1}(a x)}}{3 x^3}+\frac {\sqrt {1-a x}}{12 a x^4 \sqrt {\frac {1}{1+a x}}}+\frac {\left (\sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int -\frac {3 a^2}{x^3 \sqrt {1-a x} \sqrt {1+a x}} \, dx}{12 a}\\ &=\frac {1}{12 a x^4}-\frac {e^{\text {sech}^{-1}(a x)}}{3 x^3}+\frac {\sqrt {1-a x}}{12 a x^4 \sqrt {\frac {1}{1+a x}}}-\frac {1}{4} \left (a \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {1}{x^3 \sqrt {1-a x} \sqrt {1+a x}} \, dx\\ &=\frac {1}{12 a x^4}-\frac {e^{\text {sech}^{-1}(a x)}}{3 x^3}+\frac {\sqrt {1-a x}}{12 a x^4 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{8 x^2 \sqrt {\frac {1}{1+a x}}}-\frac {1}{8} \left (a \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {a^2}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx\\ &=\frac {1}{12 a x^4}-\frac {e^{\text {sech}^{-1}(a x)}}{3 x^3}+\frac {\sqrt {1-a x}}{12 a x^4 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{8 x^2 \sqrt {\frac {1}{1+a x}}}-\frac {1}{8} \left (a^3 \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {1}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx\\ &=\frac {1}{12 a x^4}-\frac {e^{\text {sech}^{-1}(a x)}}{3 x^3}+\frac {\sqrt {1-a x}}{12 a x^4 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{8 x^2 \sqrt {\frac {1}{1+a x}}}+\frac {1}{8} \left (a^4 \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int \frac {1}{a-a x^2} \, dx,x,\sqrt {1-a x} \sqrt {1+a x}\right )\\ &=\frac {1}{12 a x^4}-\frac {e^{\text {sech}^{-1}(a x)}}{3 x^3}+\frac {\sqrt {1-a x}}{12 a x^4 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{8 x^2 \sqrt {\frac {1}{1+a x}}}+\frac {1}{8} a^3 \sqrt {\frac {1}{1+a x}} \sqrt {1+a x} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {1+a x}\right )\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 110, normalized size = 0.83 \[ \frac {-a^4 x^4 \log (x)+a^4 x^4 \log \left (a x \sqrt {\frac {1-a x}{a x+1}}+\sqrt {\frac {1-a x}{a x+1}}+1\right )+\sqrt {\frac {1-a x}{a x+1}} \left (a^3 x^3+a^2 x^2-2 a x-2\right )-2}{8 a x^4} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcSech[a*x]/x^4,x]

[Out]

(-2 + Sqrt[(1 - a*x)/(1 + a*x)]*(-2 - 2*a*x + a^2*x^2 + a^3*x^3) - a^4*x^4*Log[x] + a^4*x^4*Log[1 + Sqrt[(1 -
a*x)/(1 + a*x)] + a*x*Sqrt[(1 - a*x)/(1 + a*x)]])/(8*a*x^4)

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fricas [A]  time = 0.65, size = 138, normalized size = 1.05 \[ \frac {a^{4} x^{4} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 1\right ) - a^{4} x^{4} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 1\right ) + 2 \, {\left (a^{3} x^{3} - 2 \, a x\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 4}{16 \, a x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))/x^4,x, algorithm="fricas")

[Out]

1/16*(a^4*x^4*log(a*x*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) + 1) - a^4*x^4*log(a*x*sqrt((a*x + 1)/(a*x)
)*sqrt(-(a*x - 1)/(a*x)) - 1) + 2*(a^3*x^3 - 2*a*x)*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) - 4)/(a*x^4)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}}{x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))/x^4,x, algorithm="giac")

[Out]

integrate((sqrt(1/(a*x) + 1)*sqrt(1/(a*x) - 1) + 1/(a*x))/x^4, x)

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maple [A]  time = 0.06, size = 110, normalized size = 0.83 \[ \frac {\sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, \left (\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right ) x^{4} a^{4}+a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}-2 \sqrt {-a^{2} x^{2}+1}\right )}{8 x^{3} \sqrt {-a^{2} x^{2}+1}}-\frac {1}{4 x^{4} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))/x^4,x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\frac {1}{8} \, a^{4} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {1}{8} \, \sqrt {-a^{2} x^{2} + 1} a^{4} - \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2}}{8 \, x^{2}} - \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{4 \, x^{4}}}{a} - \frac {1}{4 \, a x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))/x^4,x, algorithm="maxima")

[Out]

integrate(sqrt(a*x + 1)*sqrt(-a*x + 1)/x^5, x)/a - 1/4/(a*x^4)

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mupad [B]  time = 13.42, size = 602, normalized size = 4.56 \[ \frac {a^3\,\mathrm {atanh}\left (\frac {\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}}{\sqrt {\frac {1}{a\,x}+1}-1}\right )}{2}-\frac {\frac {35\,a^3\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^3}{2\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^3}+\frac {273\,a^3\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^5}{2\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^5}+\frac {715\,a^3\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^7}{2\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^7}+\frac {715\,a^3\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^9}{2\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^9}+\frac {273\,a^3\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{11}}{2\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{11}}+\frac {35\,a^3\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{13}}{2\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{13}}+\frac {a^3\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{15}}{2\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{15}}+\frac {a^3\,\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}{2\,\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}}{1+\frac {28\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}-\frac {56\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^6}+\frac {70\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^8}-\frac {56\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{10}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{10}}+\frac {28\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{12}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{12}}-\frac {8\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{14}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{14}}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{16}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{16}}-\frac {8\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}}-\frac {1}{4\,a\,x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1/(a*x) - 1)^(1/2)*(1/(a*x) + 1)^(1/2) + 1/(a*x))/x^4,x)

[Out]

(a^3*atanh(((1/(a*x) - 1)^(1/2) - 1i)/((1/(a*x) + 1)^(1/2) - 1)))/2 - ((35*a^3*((1/(a*x) - 1)^(1/2) - 1i)^3)/(
2*((1/(a*x) + 1)^(1/2) - 1)^3) + (273*a^3*((1/(a*x) - 1)^(1/2) - 1i)^5)/(2*((1/(a*x) + 1)^(1/2) - 1)^5) + (715
*a^3*((1/(a*x) - 1)^(1/2) - 1i)^7)/(2*((1/(a*x) + 1)^(1/2) - 1)^7) + (715*a^3*((1/(a*x) - 1)^(1/2) - 1i)^9)/(2
*((1/(a*x) + 1)^(1/2) - 1)^9) + (273*a^3*((1/(a*x) - 1)^(1/2) - 1i)^11)/(2*((1/(a*x) + 1)^(1/2) - 1)^11) + (35
*a^3*((1/(a*x) - 1)^(1/2) - 1i)^13)/(2*((1/(a*x) + 1)^(1/2) - 1)^13) + (a^3*((1/(a*x) - 1)^(1/2) - 1i)^15)/(2*
((1/(a*x) + 1)^(1/2) - 1)^15) + (a^3*((1/(a*x) - 1)^(1/2) - 1i))/(2*((1/(a*x) + 1)^(1/2) - 1)))/((28*((1/(a*x)
- 1)^(1/2) - 1i)^4)/((1/(a*x) + 1)^(1/2) - 1)^4 - (8*((1/(a*x) - 1)^(1/2) - 1i)^2)/((1/(a*x) + 1)^(1/2) - 1)^
2 - (56*((1/(a*x) - 1)^(1/2) - 1i)^6)/((1/(a*x) + 1)^(1/2) - 1)^6 + (70*((1/(a*x) - 1)^(1/2) - 1i)^8)/((1/(a*x
) + 1)^(1/2) - 1)^8 - (56*((1/(a*x) - 1)^(1/2) - 1i)^10)/((1/(a*x) + 1)^(1/2) - 1)^10 + (28*((1/(a*x) - 1)^(1/
2) - 1i)^12)/((1/(a*x) + 1)^(1/2) - 1)^12 - (8*((1/(a*x) - 1)^(1/2) - 1i)^14)/((1/(a*x) + 1)^(1/2) - 1)^14 + (
(1/(a*x) - 1)^(1/2) - 1i)^16/((1/(a*x) + 1)^(1/2) - 1)^16 + 1) - 1/(4*a*x^4)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{x^{5}}\, dx + \int \frac {a \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}}}{x^{4}}\, dx}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(x**(-5), x) + Integral(a*sqrt(-1 + 1/(a*x))*sqrt(1 + 1/(a*x))/x**4, x))/a

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