3.1.8 \(\int \frac {e^{\coth ^{-1}(a x)}}{x^4} \, dx\) [8]

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

[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 = 0.04, 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 = {6304, 811, 655, 201, 222} \begin {gather*} -\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}} \end {gather*}

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 201

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 222

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 655

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 811

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 6304

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 &=-\text {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 \text {Subst}\left (\int \frac {1+\frac {x}{a}}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\right )+a^2 \text {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 \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )+a^2 \text {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 \text {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 = 0.05, size = 51, normalized size = 0.68 \begin {gather*} \frac {1}{6} a \left (\frac {\sqrt {1-\frac {1}{a^2 x^2}} \left (2+3 a x+4 a^2 x^2\right )}{x^2}-3 a^2 \text {ArcSin}\left (\frac {1}{a x}\right )\right ) \end {gather*}

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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(283\) vs. \(2(63)=126\).
time = 0.10, size = 284, normalized size = 3.79

method	result	size
risch	\(\frac {\left (a x -1\right ) \left (4 a^{2} x^{2}+3 a x +2\right )}{6 x^{3} \sqrt {\frac {a x -1}{a x +1}}}-\frac {a^{3} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{2 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\)	\(93\)
default	\(-\frac {\left (a x -1\right ) \left (-6 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{4} x^{4}+6 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}+3 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{3} x^{3}+6 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+3 a^{3} x^{3} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )-6 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{3} x^{3}-6 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+3 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a x +2 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{6 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, x^{3} \sqrt {a^{2}}}\)	\(284\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(a*x-1)*(-6*(a^2*x^2-1)^(1/2)*(a^2)^(1/2)*a^4*x^4+6*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*a^2*x^2+3*(a^2*x^2-1)^(
1/2)*(a^2)^(1/2)*a^3*x^3+6*ln((a^2*x+(a^2*x^2-1)^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*a^4*x^3+3*a^3*x^3*(a^2)^(1/2)
*arctan(1/(a^2*x^2-1)^(1/2))-6*(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2)*a^3*x^3-6*ln((a^2*x+(a^2)^(1/2)*((a*x+1)*(a
*x-1))^(1/2))/(a^2)^(1/2))*a^4*x^3+3*(a^2)^(1/2)*(a^2*x^2-1)^(3/2)*a*x+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] Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (63) = 126\).
time = 0.48, size = 136, normalized size = 1.81 \begin {gather*} \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 \end {gather*}

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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Fricas [A]
time = 0.38, size = 68, normalized size = 0.91 \begin {gather*} \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}} \end {gather*}

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

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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (63) = 126\).
time = 0.42, size = 148, normalized size = 1.97 \begin {gather*} \frac {a^{3} \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right )}{\mathrm {sgn}\left (a x + 1\right )} - \frac {3 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{5} a^{3} - 12 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} a^{2} {\left | a \right |} - 3 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )} a^{3} - 4 \, a^{2} {\left | a \right |}}{3 \, {\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )}^{3} \mathrm {sgn}\left (a x + 1\right )} \end {gather*}

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]

a^3*arctan(-x*abs(a) + sqrt(a^2*x^2 - 1))/sgn(a*x + 1) - 1/3*(3*(x*abs(a) - sqrt(a^2*x^2 - 1))^5*a^3 - 12*(x*a
bs(a) - sqrt(a^2*x^2 - 1))^2*a^2*abs(a) - 3*(x*abs(a) - sqrt(a^2*x^2 - 1))*a^3 - 4*a^2*abs(a))/(((x*abs(a) - s
qrt(a^2*x^2 - 1))^2 + 1)^3*sgn(a*x + 1))

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Mupad [B]
time = 0.06, size = 105, normalized size = 1.40 \begin {gather*} \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} \end {gather*}

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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