3.25 \(\int \frac {e^{3 \tanh ^{-1}(a x)}}{x^4} \, dx\)

Optimal. Leaf size=117 \[ -\frac {14 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {3 a \sqrt {1-a^2 x^2}}{2 x^2}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}+\frac {4 a^3 \sqrt {1-a^2 x^2}}{1-a x}-\frac {11}{2} a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]

[Out]

-11/2*a^3*arctanh((-a^2*x^2+1)^(1/2))-1/3*(-a^2*x^2+1)^(1/2)/x^3-3/2*a*(-a^2*x^2+1)^(1/2)/x^2-14/3*a^2*(-a^2*x
^2+1)^(1/2)/x+4*a^3*(-a^2*x^2+1)^(1/2)/(-a*x+1)

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Rubi [A]  time = 0.74, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6124, 6742, 271, 264, 266, 51, 63, 208, 651} \[ \frac {4 a^3 \sqrt {1-a^2 x^2}}{1-a x}-\frac {14 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {3 a \sqrt {1-a^2 x^2}}{2 x^2}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}-\frac {11}{2} a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcTanh[a*x])/x^4,x]

[Out]

-Sqrt[1 - a^2*x^2]/(3*x^3) - (3*a*Sqrt[1 - a^2*x^2])/(2*x^2) - (14*a^2*Sqrt[1 - a^2*x^2])/(3*x) + (4*a^3*Sqrt[
1 - a^2*x^2])/(1 - a*x) - (11*a^3*ArcTanh[Sqrt[1 - a^2*x^2]])/2

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

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 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

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 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 651

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^m*(a + c*x^2)^(p + 1))
/(2*c*d*(p + 1)), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p
+ 2, 0]

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]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {e^{3 \tanh ^{-1}(a x)}}{x^4} \, dx &=\int \frac {(1+a x)^2}{x^4 (1-a x) \sqrt {1-a^2 x^2}} \, dx\\ &=\int \left (\frac {1}{x^4 \sqrt {1-a^2 x^2}}+\frac {3 a}{x^3 \sqrt {1-a^2 x^2}}+\frac {4 a^2}{x^2 \sqrt {1-a^2 x^2}}+\frac {4 a^3}{x \sqrt {1-a^2 x^2}}-\frac {4 a^4}{(-1+a x) \sqrt {1-a^2 x^2}}\right ) \, dx\\ &=(3 a) \int \frac {1}{x^3 \sqrt {1-a^2 x^2}} \, dx+\left (4 a^2\right ) \int \frac {1}{x^2 \sqrt {1-a^2 x^2}} \, dx+\left (4 a^3\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx-\left (4 a^4\right ) \int \frac {1}{(-1+a x) \sqrt {1-a^2 x^2}} \, dx+\int \frac {1}{x^4 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1-a^2 x^2}}{3 x^3}-\frac {4 a^2 \sqrt {1-a^2 x^2}}{x}+\frac {4 a^3 \sqrt {1-a^2 x^2}}{1-a x}+\frac {1}{2} (3 a) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1-a^2 x}} \, dx,x,x^2\right )+\frac {1}{3} \left (2 a^2\right ) \int \frac {1}{x^2 \sqrt {1-a^2 x^2}} \, dx+\left (2 a^3\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}}{3 x^3}-\frac {3 a \sqrt {1-a^2 x^2}}{2 x^2}-\frac {14 a^2 \sqrt {1-a^2 x^2}}{3 x}+\frac {4 a^3 \sqrt {1-a^2 x^2}}{1-a x}-(4 a) \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 {1}{4} \left (3 a^3\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}}{3 x^3}-\frac {3 a \sqrt {1-a^2 x^2}}{2 x^2}-\frac {14 a^2 \sqrt {1-a^2 x^2}}{3 x}+\frac {4 a^3 \sqrt {1-a^2 x^2}}{1-a x}-4 a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\frac {1}{2} (3 a) \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}}{3 x^3}-\frac {3 a \sqrt {1-a^2 x^2}}{2 x^2}-\frac {14 a^2 \sqrt {1-a^2 x^2}}{3 x}+\frac {4 a^3 \sqrt {1-a^2 x^2}}{1-a x}-\frac {11}{2} a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 81, normalized size = 0.69 \[ \frac {1}{6} \left (33 a^3 \log (x)-33 a^3 \log \left (\sqrt {1-a^2 x^2}+1\right )+\frac {\sqrt {1-a^2 x^2} \left (-52 a^3 x^3+19 a^2 x^2+7 a x+2\right )}{x^3 (a x-1)}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(3*ArcTanh[a*x])/x^4,x]

[Out]

((Sqrt[1 - a^2*x^2]*(2 + 7*a*x + 19*a^2*x^2 - 52*a^3*x^3))/(x^3*(-1 + a*x)) + 33*a^3*Log[x] - 33*a^3*Log[1 + S
qrt[1 - a^2*x^2]])/6

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fricas [A]  time = 0.56, size = 105, normalized size = 0.90 \[ \frac {24 \, a^{4} x^{4} - 24 \, a^{3} x^{3} + 33 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (52 \, a^{3} x^{3} - 19 \, a^{2} x^{2} - 7 \, a x - 2\right )} \sqrt {-a^{2} x^{2} + 1}}{6 \, {\left (a x^{4} - x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/x^4,x, algorithm="fricas")

[Out]

1/6*(24*a^4*x^4 - 24*a^3*x^3 + 33*(a^4*x^4 - a^3*x^3)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (52*a^3*x^3 - 19*a^2*x
^2 - 7*a*x - 2)*sqrt(-a^2*x^2 + 1))/(a*x^4 - x^3)

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giac [B]  time = 0.40, size = 265, normalized size = 2.26 \[ -\frac {{\left (a^{4} + \frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{2}}{x} + \frac {48 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{x^{2}} - \frac {249 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{2} x^{3}}\right )} a^{6} x^{3}}{24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )} {\left | a \right |}} - \frac {11 \, a^{4} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{2 \, {\left | a \right |}} - \frac {\frac {57 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{4}}{x} + \frac {9 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{2}}{x^{2}} + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{x^{3}}}{24 \, a^{2} {\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/x^4,x, algorithm="giac")

[Out]

-1/24*(a^4 + 8*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^2/x + 48*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2/x^2 - 249*(sqrt(-a
^2*x^2 + 1)*abs(a) + a)^3/(a^2*x^3))*a^6*x^3/((sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*((sqrt(-a^2*x^2 + 1)*abs(a) +
a)/(a^2*x) - 1)*abs(a)) - 11/2*a^4*log(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs(a) - 1/24
*(57*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^4/x + 9*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*a^2/x^2 + (sqrt(-a^2*x^2 + 1)
*abs(a) + a)^3/x^3)/(a^2*abs(a))

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maple [A]  time = 0.04, size = 146, normalized size = 1.25 \[ a^{3} \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )+\frac {13 a^{2} \left (-\frac {1}{x \sqrt {-a^{2} x^{2}+1}}+\frac {2 a^{2} x}{\sqrt {-a^{2} x^{2}+1}}\right )}{3}-\frac {1}{3 x^{3} \sqrt {-a^{2} x^{2}+1}}+3 a \left (-\frac {1}{2 x^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {3 a^{2} \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^3*(1/(-a^2*x^2+1)^(1/2)-arctanh(1/(-a^2*x^2+1)^(1/2)))+13/3*a^2*(-1/x/(-a^2*x^2+1)^(1/2)+2*a^2*x/(-a^2*x^2+1
)^(1/2))-1/3/x^3/(-a^2*x^2+1)^(1/2)+3*a*(-1/2/x^2/(-a^2*x^2+1)^(1/2)+3/2*a^2*(1/(-a^2*x^2+1)^(1/2)-arctanh(1/(
-a^2*x^2+1)^(1/2))))

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maxima [A]  time = 0.32, size = 122, normalized size = 1.04 \[ \frac {26 \, a^{4} x}{3 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {11}{2} \, a^{3} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {11 \, a^{3}}{2 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {13 \, a^{2}}{3 \, \sqrt {-a^{2} x^{2} + 1} x} - \frac {3 \, a}{2 \, \sqrt {-a^{2} x^{2} + 1} x^{2}} - \frac {1}{3 \, \sqrt {-a^{2} x^{2} + 1} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/x^4,x, algorithm="maxima")

[Out]

26/3*a^4*x/sqrt(-a^2*x^2 + 1) - 11/2*a^3*log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2/abs(x)) + 11/2*a^3/sqrt(-a^2*x^2
+ 1) - 13/3*a^2/(sqrt(-a^2*x^2 + 1)*x) - 3/2*a/(sqrt(-a^2*x^2 + 1)*x^2) - 1/3/(sqrt(-a^2*x^2 + 1)*x^3)

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mupad [B]  time = 0.81, size = 126, normalized size = 1.08 \[ \frac {4\,a^4\,\sqrt {1-a^2\,x^2}}{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}-\frac {3\,a\,\sqrt {1-a^2\,x^2}}{2\,x^2}-\frac {14\,a^2\,\sqrt {1-a^2\,x^2}}{3\,x}-\frac {\sqrt {1-a^2\,x^2}}{3\,x^3}+\frac {a^3\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,11{}\mathrm {i}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^3*atan((1 - a^2*x^2)^(1/2)*1i)*11i)/2 - (1 - a^2*x^2)^(1/2)/(3*x^3) - (3*a*(1 - a^2*x^2)^(1/2))/(2*x^2) - (
14*a^2*(1 - a^2*x^2)^(1/2))/(3*x) + (4*a^4*(1 - a^2*x^2)^(1/2))/((x*(-a^2)^(1/2) - (-a^2)^(1/2)/a)*(-a^2)^(1/2
))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a x + 1\right )^{3}}{x^{4} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/x**4,x)

[Out]

Integral((a*x + 1)**3/(x**4*(-(a*x - 1)*(a*x + 1))**(3/2)), x)

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