3.10 \(\int \frac {e^{\tanh ^{-1}(a x)}}{x^5} \, dx\)

Optimal. Leaf size=114 \[ -\frac {3 a^2 \sqrt {1-a^2 x^2}}{8 x^2}-\frac {\sqrt {1-a^2 x^2}}{4 x^4}-\frac {a \sqrt {1-a^2 x^2}}{3 x^3}-\frac {3}{8} a^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\frac {2 a^3 \sqrt {1-a^2 x^2}}{3 x} \]

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 114, 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 = {6124, 835, 807, 266, 63, 208} \[ -\frac {2 a^3 \sqrt {1-a^2 x^2}}{3 x}-\frac {3 a^2 \sqrt {1-a^2 x^2}}{8 x^2}-\frac {a \sqrt {1-a^2 x^2}}{3 x^3}-\frac {\sqrt {1-a^2 x^2}}{4 x^4}-\frac {3}{8} a^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 63

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 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 266

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 807

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0]
&& EqQ[Simplify[m + 2*p + 3], 0]

Rule 835

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((e*f - d*g)
*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[2*m, 2*p])

Rule 6124

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

Rubi steps

\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{x^5} \, dx &=\int \frac {1+a x}{x^5 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1-a^2 x^2}}{4 x^4}-\frac {1}{4} \int \frac {-4 a-3 a^2 x}{x^4 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1-a^2 x^2}}{4 x^4}-\frac {a \sqrt {1-a^2 x^2}}{3 x^3}+\frac {1}{12} \int \frac {9 a^2+8 a^3 x}{x^3 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1-a^2 x^2}}{4 x^4}-\frac {a \sqrt {1-a^2 x^2}}{3 x^3}-\frac {3 a^2 \sqrt {1-a^2 x^2}}{8 x^2}-\frac {1}{24} \int \frac {-16 a^3-9 a^4 x}{x^2 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1-a^2 x^2}}{4 x^4}-\frac {a \sqrt {1-a^2 x^2}}{3 x^3}-\frac {3 a^2 \sqrt {1-a^2 x^2}}{8 x^2}-\frac {2 a^3 \sqrt {1-a^2 x^2}}{3 x}+\frac {1}{8} \left (3 a^4\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1-a^2 x^2}}{4 x^4}-\frac {a \sqrt {1-a^2 x^2}}{3 x^3}-\frac {3 a^2 \sqrt {1-a^2 x^2}}{8 x^2}-\frac {2 a^3 \sqrt {1-a^2 x^2}}{3 x}+\frac {1}{16} \left (3 a^4\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {1-a^2 x^2}}{4 x^4}-\frac {a \sqrt {1-a^2 x^2}}{3 x^3}-\frac {3 a^2 \sqrt {1-a^2 x^2}}{8 x^2}-\frac {2 a^3 \sqrt {1-a^2 x^2}}{3 x}-\frac {1}{8} \left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )\\ &=-\frac {\sqrt {1-a^2 x^2}}{4 x^4}-\frac {a \sqrt {1-a^2 x^2}}{3 x^3}-\frac {3 a^2 \sqrt {1-a^2 x^2}}{8 x^2}-\frac {2 a^3 \sqrt {1-a^2 x^2}}{3 x}-\frac {3}{8} a^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 75, normalized size = 0.66 \[ \frac {1}{24} \left (9 a^4 \log (x)-9 a^4 \log \left (\sqrt {1-a^2 x^2}+1\right )-\frac {\sqrt {1-a^2 x^2} \left (16 a^3 x^3+9 a^2 x^2+8 a x+6\right )}{x^4}\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-((Sqrt[1 - a^2*x^2]*(6 + 8*a*x + 9*a^2*x^2 + 16*a^3*x^3))/x^4) + 9*a^4*Log[x] - 9*a^4*Log[1 + Sqrt[1 - a^2*x
^2]])/24

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fricas [A]  time = 0.45, size = 69, normalized size = 0.61 \[ \frac {9 \, a^{4} x^{4} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (16 \, a^{3} x^{3} + 9 \, a^{2} x^{2} + 8 \, a x + 6\right )} \sqrt {-a^{2} x^{2} + 1}}{24 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(9*a^4*x^4*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (16*a^3*x^3 + 9*a^2*x^2 + 8*a*x + 6)*sqrt(-a^2*x^2 + 1))/x^4

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giac [B]  time = 0.18, size = 273, normalized size = 2.39 \[ \frac {{\left (3 \, a^{5} + \frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{3}}{x} + \frac {24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a}{x^{2}} + \frac {72 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a x^{3}}\right )} a^{8} x^{4}}{192 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} {\left | a \right |}} - \frac {3 \, a^{5} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{8 \, {\left | a \right |}} - \frac {\frac {72 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{5} {\left | a \right |}}{x} + \frac {24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{3} {\left | a \right |}}{x^{2}} + \frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} a {\left | a \right |}}{x^{3}} + \frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} {\left | a \right |}}{a x^{4}}}{192 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*(3*a^5 + 8*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^3/x + 24*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*a/x^2 + 72*(sqrt
(-a^2*x^2 + 1)*abs(a) + a)^3/(a*x^3))*a^8*x^4/((sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*abs(a)) - 3/8*a^5*log(1/2*abs
(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs(a) - 1/192*(72*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^5*abs(
a)/x + 24*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*a^3*abs(a)/x^2 + 8*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*a*abs(a)/x^3
+ 3*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*abs(a)/(a*x^4))/a^4

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maple [A]  time = 0.04, size = 100, normalized size = 0.88 \[ a \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{3 x^{3}}-\frac {2 a^{2} \sqrt {-a^{2} x^{2}+1}}{3 x}\right )-\frac {\sqrt {-a^{2} x^{2}+1}}{4 x^{4}}+\frac {3 a^{2} \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}-\frac {a^{2} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.41, size = 107, normalized size = 0.94 \[ -\frac {3}{8} \, a^{4} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {2 \, \sqrt {-a^{2} x^{2} + 1} a^{3}}{3 \, x} - \frac {3 \, \sqrt {-a^{2} x^{2} + 1} a^{2}}{8 \, x^{2}} - \frac {\sqrt {-a^{2} x^{2} + 1} a}{3 \, x^{3}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/8*a^4*log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2/abs(x)) - 2/3*sqrt(-a^2*x^2 + 1)*a^3/x - 3/8*sqrt(-a^2*x^2 + 1)*a
^2/x^2 - 1/3*sqrt(-a^2*x^2 + 1)*a/x^3 - 1/4*sqrt(-a^2*x^2 + 1)/x^4

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mupad [B]  time = 0.03, size = 98, normalized size = 0.86 \[ -\frac {\sqrt {1-a^2\,x^2}}{4\,x^4}-\frac {a\,\sqrt {1-a^2\,x^2}}{3\,x^3}-\frac {3\,a^2\,\sqrt {1-a^2\,x^2}}{8\,x^2}-\frac {2\,a^3\,\sqrt {1-a^2\,x^2}}{3\,x}+\frac {a^4\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^4*atan((1 - a^2*x^2)^(1/2)*1i)*3i)/8 - (1 - a^2*x^2)^(1/2)/(4*x^4) - (a*(1 - a^2*x^2)^(1/2))/(3*x^3) - (3*a
^2*(1 - a^2*x^2)^(1/2))/(8*x^2) - (2*a^3*(1 - a^2*x^2)^(1/2))/(3*x)

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sympy [C]  time = 4.44, size = 258, normalized size = 2.26 \[ a \left (\begin {cases} - \frac {2 i a^{2} \sqrt {a^{2} x^{2} - 1}}{3 x} - \frac {i \sqrt {a^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {2 a^{2} \sqrt {- a^{2} x^{2} + 1}}{3 x} - \frac {\sqrt {- a^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\right ) + \begin {cases} - \frac {3 a^{4} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{8} + \frac {3 a^{3}}{8 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {a}{8 x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{4 a x^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {3 i a^{4} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{8} - \frac {3 i a^{3}}{8 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i a}{8 x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i}{4 a x^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*Piecewise((-2*I*a**2*sqrt(a**2*x**2 - 1)/(3*x) - I*sqrt(a**2*x**2 - 1)/(3*x**3), Abs(a**2*x**2) > 1), (-2*a*
*2*sqrt(-a**2*x**2 + 1)/(3*x) - sqrt(-a**2*x**2 + 1)/(3*x**3), True)) + Piecewise((-3*a**4*acosh(1/(a*x))/8 +
3*a**3/(8*x*sqrt(-1 + 1/(a**2*x**2))) - a/(8*x**3*sqrt(-1 + 1/(a**2*x**2))) - 1/(4*a*x**5*sqrt(-1 + 1/(a**2*x*
*2))), 1/Abs(a**2*x**2) > 1), (3*I*a**4*asin(1/(a*x))/8 - 3*I*a**3/(8*x*sqrt(1 - 1/(a**2*x**2))) + I*a/(8*x**3
*sqrt(1 - 1/(a**2*x**2))) + I/(4*a*x**5*sqrt(1 - 1/(a**2*x**2))), True))

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