3.8 \(\int \frac {e^{\coth ^{-1}(a x)}}{x^4} \, dx\)

Optimal. Leaf size=75 \[ -\frac {1}{2} a^3 \csc ^{-1}(a x)+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}-\frac {1}{3} a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}+a^3 \sqrt {1-\frac {1}{a^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6169, 797, 641, 195, 216} \[ -\frac {1}{3} a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}+a^3 \sqrt {1-\frac {1}{a^2 x^2}}+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}-\frac {1}{2} a^3 \csc ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^3*Sqrt[1 - 1/(a^2*x^2)] - (a^3*(1 - 1/(a^2*x^2))^(3/2))/3 + (a^2*Sqrt[1 - 1/(a^2*x^2)])/(2*x) - (a^3*ArcCsc[
a*x])/2

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rule 797

Int[(x_)^2*((f_) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/c, Int[(f + g*x)*(a + c*x^2)^(p
+ 1), x], x] - Dist[a/c, Int[(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, f, g, p}, x] && EqQ[a*g^2 + f^2*
c, 0]

Rule 6169

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(x_)^(m_.), x_Symbol] :> -Subst[Int[(1 + x/a)^((n + 1)/2)/(x^(m + 2)*(1 - x/
a)^((n - 1)/2)*Sqrt[1 - x^2/a^2]), x], x, 1/x] /; FreeQ[a, x] && IntegerQ[(n - 1)/2] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {e^{\coth ^{-1}(a x)}}{x^4} \, dx &=-\operatorname {Subst}\left (\int \frac {x^2 \left (1+\frac {x}{a}\right )}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-\left (a^2 \operatorname {Subst}\left (\int \frac {1+\frac {x}{a}}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\right )+a^2 \operatorname {Subst}\left (\int \left (1+\frac {x}{a}\right ) \sqrt {1-\frac {x^2}{a^2}} \, dx,x,\frac {1}{x}\right )\\ &=a^3 \sqrt {1-\frac {1}{a^2 x^2}}-\frac {1}{3} a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}-a^2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )+a^2 \operatorname {Subst}\left (\int \sqrt {1-\frac {x^2}{a^2}} \, dx,x,\frac {1}{x}\right )\\ &=a^3 \sqrt {1-\frac {1}{a^2 x^2}}-\frac {1}{3} a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}-a^3 \csc ^{-1}(a x)+\frac {1}{2} a^2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=a^3 \sqrt {1-\frac {1}{a^2 x^2}}-\frac {1}{3} a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}-\frac {1}{2} a^3 \csc ^{-1}(a x)\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 51, normalized size = 0.68 \[ \frac {1}{6} a \left (\frac {\sqrt {1-\frac {1}{a^2 x^2}} \left (4 a^2 x^2+3 a x+2\right )}{x^2}-3 a^2 \sin ^{-1}\left (\frac {1}{a x}\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a*((Sqrt[1 - 1/(a^2*x^2)]*(2 + 3*a*x + 4*a^2*x^2))/x^2 - 3*a^2*ArcSin[1/(a*x)]))/6

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fricas [A]  time = 0.51, size = 68, normalized size = 0.91 \[ \frac {6 \, a^{3} x^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + {\left (4 \, a^{3} x^{3} + 7 \, a^{2} x^{2} + 5 \, a x + 2\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(6*a^3*x^3*arctan(sqrt((a*x - 1)/(a*x + 1))) + (4*a^3*x^3 + 7*a^2*x^2 + 5*a*x + 2)*sqrt((a*x - 1)/(a*x + 1
)))/x^3

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giac [B]  time = 0.15, size = 130, normalized size = 1.73 \[ \frac {1}{3} \, {\left (3 \, a^{2} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + \frac {\frac {4 \, {\left (a x - 1\right )} a^{2} \sqrt {\frac {a x - 1}{a x + 1}}}{a x + 1} + \frac {3 \, {\left (a x - 1\right )}^{2} a^{2} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} + 9 \, a^{2} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (\frac {a x - 1}{a x + 1} + 1\right )}^{3}}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.06, size = 284, normalized size = 3.79 \[ -\frac {\left (a x -1\right ) \left (-6 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{4} a^{4}+6 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{2} a^{2}+3 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{3} a^{3}+3 a^{3} x^{3} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+6 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{3} a^{4}-6 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{3} a^{3}-6 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{3} a^{4}+3 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x a +2 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{6 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{3} \sqrt {a^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(a*x-1)*(-6*(a^2*x^2-1)^(1/2)*(a^2)^(1/2)*x^4*a^4+6*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*x^2*a^2+3*(a^2*x^2-1)^(
1/2)*(a^2)^(1/2)*x^3*a^3+3*a^3*x^3*(a^2)^(1/2)*arctan(1/(a^2*x^2-1)^(1/2))+6*ln((a^2*x+(a^2*x^2-1)^(1/2)*(a^2)
^(1/2))/(a^2)^(1/2))*x^3*a^4-6*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^3*a^3-6*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)
*(a^2)^(1/2))/(a^2)^(1/2))*x^3*a^4+3*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*x*a+2*(a^2*x^2-1)^(3/2)*(a^2)^(1/2))/((a*x-
1)/(a*x+1))^(1/2)/((a*x-1)*(a*x+1))^(1/2)/x^3/(a^2)^(1/2)

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maxima [B]  time = 0.42, size = 136, normalized size = 1.81 \[ \frac {1}{3} \, {\left (3 \, a^{2} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + \frac {3 \, a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 4 \, a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 9 \, a^{2} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {3 \, {\left (a x - 1\right )}}{a x + 1} + \frac {3 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 1}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3*a^2*arctan(sqrt((a*x - 1)/(a*x + 1))) + (3*a^2*((a*x - 1)/(a*x + 1))^(5/2) + 4*a^2*((a*x - 1)/(a*x + 1)
)^(3/2) + 9*a^2*sqrt((a*x - 1)/(a*x + 1)))/(3*(a*x - 1)/(a*x + 1) + 3*(a*x - 1)^2/(a*x + 1)^2 + (a*x - 1)^3/(a
*x + 1)^3 + 1))*a

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mupad [B]  time = 0.06, size = 105, normalized size = 1.40 \[ \frac {2\,a^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{3}+\frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{3\,x^3}+a^3\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )+\frac {7\,a^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{6\,x}+\frac {5\,a\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{6\,x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*a^3*((a*x - 1)/(a*x + 1))^(1/2))/3 + ((a*x - 1)/(a*x + 1))^(1/2)/(3*x^3) + a^3*atan(((a*x - 1)/(a*x + 1))^(
1/2)) + (7*a^2*((a*x - 1)/(a*x + 1))^(1/2))/(6*x) + (5*a*((a*x - 1)/(a*x + 1))^(1/2))/(6*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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