3.2.19 \(\int \frac {e^{-\frac {5}{2} \tanh ^{-1}(a x)}}{x^5} \, dx\) [119]

Optimal. Leaf size=194 \[ \frac {2467 a^4 \sqrt [4]{1-a x}}{192 \sqrt [4]{1+a x}}-\frac {\sqrt [4]{1-a x}}{4 x^4 \sqrt [4]{1+a x}}+\frac {17 a \sqrt [4]{1-a x}}{24 x^3 \sqrt [4]{1+a x}}-\frac {113 a^2 \sqrt [4]{1-a x}}{96 x^2 \sqrt [4]{1+a x}}+\frac {521 a^3 \sqrt [4]{1-a x}}{192 x \sqrt [4]{1+a x}}+\frac {475}{64} a^4 \text {ArcTan}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-\frac {475}{64} a^4 \tanh ^{-1}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right ) \]

[Out]

2467/192*a^4*(-a*x+1)^(1/4)/(a*x+1)^(1/4)-1/4*(-a*x+1)^(1/4)/x^4/(a*x+1)^(1/4)+17/24*a*(-a*x+1)^(1/4)/x^3/(a*x
+1)^(1/4)-113/96*a^2*(-a*x+1)^(1/4)/x^2/(a*x+1)^(1/4)+521/192*a^3*(-a*x+1)^(1/4)/x/(a*x+1)^(1/4)+475/64*a^4*ar
ctan((a*x+1)^(1/4)/(-a*x+1)^(1/4))-475/64*a^4*arctanh((a*x+1)^(1/4)/(-a*x+1)^(1/4))

________________________________________________________________________________________

Rubi [A]
time = 0.07, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6261, 100, 156, 160, 12, 95, 304, 209, 212} \begin {gather*} \frac {475}{64} a^4 \text {ArcTan}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )+\frac {2467 a^4 \sqrt [4]{1-a x}}{192 \sqrt [4]{a x+1}}-\frac {475}{64} a^4 \tanh ^{-1}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )+\frac {521 a^3 \sqrt [4]{1-a x}}{192 x \sqrt [4]{a x+1}}-\frac {113 a^2 \sqrt [4]{1-a x}}{96 x^2 \sqrt [4]{a x+1}}-\frac {\sqrt [4]{1-a x}}{4 x^4 \sqrt [4]{a x+1}}+\frac {17 a \sqrt [4]{1-a x}}{24 x^3 \sqrt [4]{a x+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(E^((5*ArcTanh[a*x])/2)*x^5),x]

[Out]

(2467*a^4*(1 - a*x)^(1/4))/(192*(1 + a*x)^(1/4)) - (1 - a*x)^(1/4)/(4*x^4*(1 + a*x)^(1/4)) + (17*a*(1 - a*x)^(
1/4))/(24*x^3*(1 + a*x)^(1/4)) - (113*a^2*(1 - a*x)^(1/4))/(96*x^2*(1 + a*x)^(1/4)) + (521*a^3*(1 - a*x)^(1/4)
)/(192*x*(1 + a*x)^(1/4)) + (475*a^4*ArcTan[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)])/64 - (475*a^4*ArcTanh[(1 + a*x)^
(1/4)/(1 - a*x)^(1/4)])/64

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Dist[1/(b*(b*e - a*
f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 156

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]

Rule 160

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m + n + p + 2, 0] && NeQ[m, -1] && (Sum
SimplerQ[m, 1] || ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) &&  !(NeQ[p, -1] && SumSimplerQ[p, 1])))

Rule 209

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

Rule 212

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

Rule 304

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}
, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !
GtQ[a/b, 0]

Rule 6261

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

Rubi steps

\begin {align*} \int \frac {e^{-\frac {5}{2} \tanh ^{-1}(a x)}}{x^5} \, dx &=\int \frac {(1-a x)^{5/4}}{x^5 (1+a x)^{5/4}} \, dx\\ &=-\frac {\sqrt [4]{1-a x}}{4 x^4 \sqrt [4]{1+a x}}-\frac {1}{4} \int \frac {\frac {17 a}{2}-8 a^2 x}{x^4 (1-a x)^{3/4} (1+a x)^{5/4}} \, dx\\ &=-\frac {\sqrt [4]{1-a x}}{4 x^4 \sqrt [4]{1+a x}}+\frac {17 a \sqrt [4]{1-a x}}{24 x^3 \sqrt [4]{1+a x}}+\frac {1}{12} \int \frac {\frac {113 a^2}{4}-\frac {51 a^3 x}{2}}{x^3 (1-a x)^{3/4} (1+a x)^{5/4}} \, dx\\ &=-\frac {\sqrt [4]{1-a x}}{4 x^4 \sqrt [4]{1+a x}}+\frac {17 a \sqrt [4]{1-a x}}{24 x^3 \sqrt [4]{1+a x}}-\frac {113 a^2 \sqrt [4]{1-a x}}{96 x^2 \sqrt [4]{1+a x}}-\frac {1}{24} \int \frac {\frac {521 a^3}{8}-\frac {113 a^4 x}{2}}{x^2 (1-a x)^{3/4} (1+a x)^{5/4}} \, dx\\ &=-\frac {\sqrt [4]{1-a x}}{4 x^4 \sqrt [4]{1+a x}}+\frac {17 a \sqrt [4]{1-a x}}{24 x^3 \sqrt [4]{1+a x}}-\frac {113 a^2 \sqrt [4]{1-a x}}{96 x^2 \sqrt [4]{1+a x}}+\frac {521 a^3 \sqrt [4]{1-a x}}{192 x \sqrt [4]{1+a x}}+\frac {1}{24} \int \frac {\frac {1425 a^4}{16}-\frac {521 a^5 x}{8}}{x (1-a x)^{3/4} (1+a x)^{5/4}} \, dx\\ &=\frac {2467 a^4 \sqrt [4]{1-a x}}{192 \sqrt [4]{1+a x}}-\frac {\sqrt [4]{1-a x}}{4 x^4 \sqrt [4]{1+a x}}+\frac {17 a \sqrt [4]{1-a x}}{24 x^3 \sqrt [4]{1+a x}}-\frac {113 a^2 \sqrt [4]{1-a x}}{96 x^2 \sqrt [4]{1+a x}}+\frac {521 a^3 \sqrt [4]{1-a x}}{192 x \sqrt [4]{1+a x}}+\frac {\int \frac {1425 a^5}{32 x (1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx}{12 a}\\ &=\frac {2467 a^4 \sqrt [4]{1-a x}}{192 \sqrt [4]{1+a x}}-\frac {\sqrt [4]{1-a x}}{4 x^4 \sqrt [4]{1+a x}}+\frac {17 a \sqrt [4]{1-a x}}{24 x^3 \sqrt [4]{1+a x}}-\frac {113 a^2 \sqrt [4]{1-a x}}{96 x^2 \sqrt [4]{1+a x}}+\frac {521 a^3 \sqrt [4]{1-a x}}{192 x \sqrt [4]{1+a x}}+\frac {1}{128} \left (475 a^4\right ) \int \frac {1}{x (1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx\\ &=\frac {2467 a^4 \sqrt [4]{1-a x}}{192 \sqrt [4]{1+a x}}-\frac {\sqrt [4]{1-a x}}{4 x^4 \sqrt [4]{1+a x}}+\frac {17 a \sqrt [4]{1-a x}}{24 x^3 \sqrt [4]{1+a x}}-\frac {113 a^2 \sqrt [4]{1-a x}}{96 x^2 \sqrt [4]{1+a x}}+\frac {521 a^3 \sqrt [4]{1-a x}}{192 x \sqrt [4]{1+a x}}+\frac {1}{32} \left (475 a^4\right ) \text {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ &=\frac {2467 a^4 \sqrt [4]{1-a x}}{192 \sqrt [4]{1+a x}}-\frac {\sqrt [4]{1-a x}}{4 x^4 \sqrt [4]{1+a x}}+\frac {17 a \sqrt [4]{1-a x}}{24 x^3 \sqrt [4]{1+a x}}-\frac {113 a^2 \sqrt [4]{1-a x}}{96 x^2 \sqrt [4]{1+a x}}+\frac {521 a^3 \sqrt [4]{1-a x}}{192 x \sqrt [4]{1+a x}}-\frac {1}{64} \left (475 a^4\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )+\frac {1}{64} \left (475 a^4\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ &=\frac {2467 a^4 \sqrt [4]{1-a x}}{192 \sqrt [4]{1+a x}}-\frac {\sqrt [4]{1-a x}}{4 x^4 \sqrt [4]{1+a x}}+\frac {17 a \sqrt [4]{1-a x}}{24 x^3 \sqrt [4]{1+a x}}-\frac {113 a^2 \sqrt [4]{1-a x}}{96 x^2 \sqrt [4]{1+a x}}+\frac {521 a^3 \sqrt [4]{1-a x}}{192 x \sqrt [4]{1+a x}}+\frac {475}{64} a^4 \tan ^{-1}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-\frac {475}{64} a^4 \tanh ^{-1}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.02, size = 86, normalized size = 0.44 \begin {gather*} \frac {\sqrt [4]{1-a x} \left (-48+136 a x-226 a^2 x^2+521 a^3 x^3+2467 a^4 x^4-2850 a^4 x^4 \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {1-a x}{1+a x}\right )\right )}{192 x^4 \sqrt [4]{1+a x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(E^((5*ArcTanh[a*x])/2)*x^5),x]

[Out]

((1 - a*x)^(1/4)*(-48 + 136*a*x - 226*a^2*x^2 + 521*a^3*x^3 + 2467*a^4*x^4 - 2850*a^4*x^4*Hypergeometric2F1[1/
4, 1, 5/4, (1 - a*x)/(1 + a*x)]))/(192*x^4*(1 + a*x)^(1/4))

________________________________________________________________________________________

Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )^{\frac {5}{2}} x^{5}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.36, size = 208, normalized size = 1.07 \begin {gather*} \frac {2 \, {\left (2467 \, a^{4} x^{4} + 521 \, a^{3} x^{3} - 226 \, a^{2} x^{2} + 136 \, a x - 48\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} + 2850 \, {\left (a^{5} x^{5} + a^{4} x^{4}\right )} \arctan \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) - 1425 \, {\left (a^{5} x^{5} + a^{4} x^{4}\right )} \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) + 1425 \, {\left (a^{5} x^{5} + a^{4} x^{4}\right )} \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - 1\right )}{384 \, {\left (a x^{5} + x^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/384*(2*(2467*a^4*x^4 + 521*a^3*x^3 - 226*a^2*x^2 + 136*a*x - 48)*sqrt(-a^2*x^2 + 1)*sqrt(-sqrt(-a^2*x^2 + 1)
/(a*x - 1)) + 2850*(a^5*x^5 + a^4*x^4)*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - 1425*(a^5*x^5 + a^4*x^4)*
log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) + 1425*(a^5*x^5 + a^4*x^4)*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)
) - 1))/(a*x^5 + x^4)

________________________________________________________________________________________

Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3064 deep

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^5\,{\left (\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________