3.1.9 \(\int \frac {e^{\coth ^{-1}(a x)}}{x^5} \, dx\) [9]

Optimal. Leaf size=88 \[ \frac {1}{24} a^3 \sqrt {1-\frac {1}{a^2 x^2}} \left (16 a+\frac {9}{x}\right )+\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}-\frac {3}{8} a^4 \csc ^{-1}(a x) \]

[Out]

-3/8*a^4*arccsc(a*x)+1/24*a^3*(16*a+9/x)*(1-1/a^2/x^2)^(1/2)+1/4*a*(1-1/a^2/x^2)^(1/2)/x^3+1/3*a^2*(1-1/a^2/x^
2)^(1/2)/x^2

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Rubi [A]
time = 0.06, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6304, 847, 794, 222} \begin {gather*} -\frac {3}{8} a^4 \csc ^{-1}(a x)+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}+\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}+\frac {1}{24} a^3 \sqrt {1-\frac {1}{a^2 x^2}} \left (16 a+\frac {9}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^ArcCoth[a*x]/x^5,x]

[Out]

(a^3*Sqrt[1 - 1/(a^2*x^2)]*(16*a + 9/x))/24 + (a*Sqrt[1 - 1/(a^2*x^2)])/(4*x^3) + (a^2*Sqrt[1 - 1/(a^2*x^2)])/
(3*x^2) - (3*a^4*ArcCsc[a*x])/8

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 794

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*((a + c*x^2)^(p + 1)/(2*c*(p + 1)*(2*p + 3))), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rule 847

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g*(d + e*x)^
m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*
Simp[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p
}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p]) &&  !(IGtQ[m, 0] && EqQ[f, 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^5} \, dx &=-\text {Subst}\left (\int \frac {x^3 \left (1+\frac {x}{a}\right )}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}+\frac {1}{4} a^2 \text {Subst}\left (\int \frac {x^2 \left (-\frac {3}{a}-\frac {4 x}{a^2}\right )}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}-\frac {1}{12} a^4 \text {Subst}\left (\int \frac {x \left (\frac {8}{a^2}+\frac {9 x}{a^3}\right )}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{24} a^3 \sqrt {1-\frac {1}{a^2 x^2}} \left (16 a+\frac {9}{x}\right )+\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}-\frac {1}{8} \left (3 a^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{24} a^3 \sqrt {1-\frac {1}{a^2 x^2}} \left (16 a+\frac {9}{x}\right )+\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}-\frac {3}{8} a^4 \csc ^{-1}(a x)\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 59, normalized size = 0.67 \begin {gather*} \frac {1}{24} a \left (\frac {\sqrt {1-\frac {1}{a^2 x^2}} \left (6+8 a x+9 a^2 x^2+16 a^3 x^3\right )}{x^3}-9 a^3 \text {ArcSin}\left (\frac {1}{a x}\right )\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcCoth[a*x]/x^5,x]

[Out]

(a*((Sqrt[1 - 1/(a^2*x^2)]*(6 + 8*a*x + 9*a^2*x^2 + 16*a^3*x^3))/x^3 - 9*a^3*ArcSin[1/(a*x)]))/24

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(307\) vs. \(2(74)=148\).
time = 0.10, size = 308, normalized size = 3.50

method	result	size
risch	\(\frac {\left (a x -1\right ) \left (16 a^{3} x^{3}+9 a^{2} x^{2}+8 a x +6\right )}{24 x^{4} \sqrt {\frac {a x -1}{a x +1}}}-\frac {3 a^{4} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{8 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\)	\(101\)
default	\(-\frac {\left (a x -1\right ) \left (-24 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{5} x^{5}+24 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}+9 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{4} x^{4}+9 a^{4} x^{4} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+24 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}-24 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{4} x^{4}-24 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}+15 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}+8 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a x +6 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{24 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, x^{4} \sqrt {a^{2}}}\)	\(308\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a*x-1)/(a*x+1))^(1/2)/x^5,x,method=_RETURNVERBOSE)

[Out]

-1/24*(a*x-1)*(-24*(a^2*x^2-1)^(1/2)*(a^2)^(1/2)*a^5*x^5+24*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*a^3*x^3+9*(a^2*x^2-1
)^(1/2)*(a^2)^(1/2)*a^4*x^4+9*a^4*x^4*(a^2)^(1/2)*arctan(1/(a^2*x^2-1)^(1/2))+24*ln((a^2*x+(a^2*x^2-1)^(1/2)*(
a^2)^(1/2))/(a^2)^(1/2))*a^5*x^4-24*(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2)*a^4*x^4-24*ln((a^2*x+(a^2)^(1/2)*((a*x
+1)*(a*x-1))^(1/2))/(a^2)^(1/2))*a^5*x^4+15*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*a^2*x^2+8*(a^2)^(1/2)*(a^2*x^2-1)^(3
/2)*a*x+6*(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^4/(a^2)^(1/2)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (74) = 148\).
time = 0.47, size = 172, normalized size = 1.95 \begin {gather*} \frac {1}{12} \, {\left (9 \, a^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + \frac {9 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 49 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 31 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 39 \, a^{3} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {4 \, {\left (a x - 1\right )}}{a x + 1} + \frac {6 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {4 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + \frac {{\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} + 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^5,x, algorithm="maxima")

[Out]

1/12*(9*a^3*arctan(sqrt((a*x - 1)/(a*x + 1))) + (9*a^3*((a*x - 1)/(a*x + 1))^(7/2) + 49*a^3*((a*x - 1)/(a*x +
1))^(5/2) + 31*a^3*((a*x - 1)/(a*x + 1))^(3/2) + 39*a^3*sqrt((a*x - 1)/(a*x + 1)))/(4*(a*x - 1)/(a*x + 1) + 6*
(a*x - 1)^2/(a*x + 1)^2 + 4*(a*x - 1)^3/(a*x + 1)^3 + (a*x - 1)^4/(a*x + 1)^4 + 1))*a

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Fricas [A]
time = 0.35, size = 76, normalized size = 0.86 \begin {gather*} \frac {18 \, a^{4} x^{4} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + {\left (16 \, a^{4} x^{4} + 25 \, a^{3} x^{3} + 17 \, a^{2} x^{2} + 14 \, a x + 6\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{24 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/2)/x^5,x, algorithm="fricas")

[Out]

1/24*(18*a^4*x^4*arctan(sqrt((a*x - 1)/(a*x + 1))) + (16*a^4*x^4 + 25*a^3*x^3 + 17*a^2*x^2 + 14*a*x + 6)*sqrt(
(a*x - 1)/(a*x + 1)))/x^4

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{5} \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**5,x)

[Out]

Integral(1/(x**5*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. 226 vs. \(2 (74) = 148\).
time = 0.41, size = 226, normalized size = 2.57 \begin {gather*} \frac {3 \, a^{4} \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right )}{4 \, \mathrm {sgn}\left (a x + 1\right )} - \frac {9 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{7} a^{4} + 33 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{5} a^{4} - 48 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{4} a^{3} {\left | a \right |} - 33 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{3} a^{4} - 64 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} a^{3} {\left | a \right |} - 9 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )} a^{4} - 16 \, a^{3} {\left | a \right |}}{12 \, {\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )}^{4} \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^5,x, algorithm="giac")

[Out]

3/4*a^4*arctan(-x*abs(a) + sqrt(a^2*x^2 - 1))/sgn(a*x + 1) - 1/12*(9*(x*abs(a) - sqrt(a^2*x^2 - 1))^7*a^4 + 33
*(x*abs(a) - sqrt(a^2*x^2 - 1))^5*a^4 - 48*(x*abs(a) - sqrt(a^2*x^2 - 1))^4*a^3*abs(a) - 33*(x*abs(a) - sqrt(a
^2*x^2 - 1))^3*a^4 - 64*(x*abs(a) - sqrt(a^2*x^2 - 1))^2*a^3*abs(a) - 9*(x*abs(a) - sqrt(a^2*x^2 - 1))*a^4 - 1
6*a^3*abs(a))/(((x*abs(a) - sqrt(a^2*x^2 - 1))^2 + 1)^4*sgn(a*x + 1))

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Mupad [B]
time = 0.08, size = 129, normalized size = 1.47 \begin {gather*} \frac {2\,a^4\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{3}+\frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{4\,x^4}+\frac {3\,a^4\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{4}+\frac {17\,a^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{24\,x^2}+\frac {25\,a^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{24\,x}+\frac {7\,a\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{12\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^5*((a*x - 1)/(a*x + 1))^(1/2)),x)

[Out]

(2*a^4*((a*x - 1)/(a*x + 1))^(1/2))/3 + ((a*x - 1)/(a*x + 1))^(1/2)/(4*x^4) + (3*a^4*atan(((a*x - 1)/(a*x + 1)
)^(1/2)))/4 + (17*a^2*((a*x - 1)/(a*x + 1))^(1/2))/(24*x^2) + (25*a^3*((a*x - 1)/(a*x + 1))^(1/2))/(24*x) + (7
*a*((a*x - 1)/(a*x + 1))^(1/2))/(12*x^3)

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