3.436 \(\int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-\frac {c}{a x})^4} \, dx\)

Optimal. Leaf size=111 \[ -\frac {x \left (4 a+\frac {3}{x}\right )}{3 a c^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {8 x \sqrt {1-\frac {1}{a^2 x^2}}}{3 c^4}-\frac {a x}{3 c^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right )}+\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.15, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6177, 857, 823, 807, 266, 63, 208} \[ -\frac {x \left (4 a+\frac {3}{x}\right )}{3 a c^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {8 x \sqrt {1-\frac {1}{a^2 x^2}}}{3 c^4}-\frac {a x}{3 c^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right )}+\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^4} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^(3*ArcCoth[a*x])*(c - c/(a*x))^4),x]

[Out]

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

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 823

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(
m + 1)*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), x] + Di
st[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^2*
(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 857

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

Rule 6177

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.), x_Symbol] :> -Dist[c^n, Subst[Int[((c + d*x)^(p -
n)*(1 - x^2/a^2)^(n/2))/x^2, x], x, 1/x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] && IntegerQ[(n - 1)
/2] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p, n/2 + 1]) && IntegerQ[2*p]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.07, size = 94, normalized size = 0.85 \[ \frac {3 a^3 x^3-7 a^2 x^2+3 a x \sqrt {1-\frac {1}{a^2 x^2}} (a x-1) \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )-5 a x+8}{3 a^2 c^4 x \sqrt {1-\frac {1}{a^2 x^2}} (a x-1)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(E^(3*ArcCoth[a*x])*(c - c/(a*x))^4),x]

[Out]

(8 - 5*a*x - 7*a^2*x^2 + 3*a^3*x^3 + 3*a*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(3
*a^2*c^4*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x))

________________________________________________________________________________________

fricas [A]  time = 0.51, size = 134, normalized size = 1.21 \[ \frac {3 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 3 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (3 \, a^{3} x^{3} - 7 \, a^{2} x^{2} - 5 \, a x + 8\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{3 \, {\left (a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3*(a^2*x^2 - 2*a*x + 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 3*(a^2*x^2 - 2*a*x + 1)*log(sqrt((a*x - 1)/(
a*x + 1)) - 1) + (3*a^3*x^3 - 7*a^2*x^2 - 5*a*x + 8)*sqrt((a*x - 1)/(a*x + 1)))/(a^3*c^4*x^2 - 2*a^2*c^4*x + a
*c^4)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [B]  time = 0.06, size = 523, normalized size = 4.71 \[ -\frac {\left (-45 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{5} a^{5}-24 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{5} a^{6}+21 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x^{3} a^{3}+45 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{4} a^{4}+24 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{4} a^{5}+11 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x^{2} a^{2}+90 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{3} a^{3}+48 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{3} a^{4}-5 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x a -90 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x^{2} a^{2}-48 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{2} a^{3}-19 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-45 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x a -24 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x \,a^{2}+45 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}+24 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{24 a \sqrt {a^{2}}\, c^{4} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x -1\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(-45*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^5*a^5-24*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)
^(1/2))*x^5*a^6+21*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x^3*a^3+45*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^4*a^4+
24*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^4*a^5+11*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*
x^2*a^2+90*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^3*a^3+48*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)
^(1/2))*x^3*a^4-5*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x*a-90*((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2)*x^2*a^2-48*ln
((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^2*a^3-19*((a*x-1)*(a*x+1))^(3/2)*(a^2)^(1/2)-45*((
a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2)*x*a-24*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x*a^2+45*
((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2)+24*a*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2)))/a*((a*x-
1)/(a*x+1))^(3/2)/(a^2)^(1/2)/c^4/((a*x-1)*(a*x+1))^(1/2)/(a*x-1)^4

________________________________________________________________________________________

maxima [A]  time = 0.31, size = 160, normalized size = 1.44 \[ \frac {1}{12} \, a {\left (\frac {\frac {17 \, {\left (a x - 1\right )}}{a x + 1} - \frac {42 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1}{a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} + \frac {12 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{4}} - \frac {12 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{4}} + \frac {3 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*a*((17*(a*x - 1)/(a*x + 1) - 42*(a*x - 1)^2/(a*x + 1)^2 + 1)/(a^2*c^4*((a*x - 1)/(a*x + 1))^(5/2) - a^2*c
^4*((a*x - 1)/(a*x + 1))^(3/2)) + 12*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^4) - 12*log(sqrt((a*x - 1)/(a*x
+ 1)) - 1)/(a^2*c^4) + 3*sqrt((a*x - 1)/(a*x + 1))/(a^2*c^4))

________________________________________________________________________________________

mupad [B]  time = 1.24, size = 128, normalized size = 1.15 \[ \frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{4\,a\,c^4}-\frac {\frac {17\,\left (a\,x-1\right )}{3\,\left (a\,x+1\right )}-\frac {14\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {1}{3}}{4\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}-4\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}+\frac {2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((a*x - 1)/(a*x + 1))^(1/2)/(4*a*c^4) - ((17*(a*x - 1))/(3*(a*x + 1)) - (14*(a*x - 1)^2)/(a*x + 1)^2 + 1/3)/(4
*a*c^4*((a*x - 1)/(a*x + 1))^(3/2) - 4*a*c^4*((a*x - 1)/(a*x + 1))^(5/2)) + (2*atanh(((a*x - 1)/(a*x + 1))^(1/
2)))/(a*c^4)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {a^{4} \left (\int \left (- \frac {x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{5} x^{5} - 3 a^{4} x^{4} + 2 a^{3} x^{3} + 2 a^{2} x^{2} - 3 a x + 1}\right )\, dx + \int \frac {a x^{5} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{5} x^{5} - 3 a^{4} x^{4} + 2 a^{3} x^{3} + 2 a^{2} x^{2} - 3 a x + 1}\, dx\right )}{c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**4*(Integral(-x**4*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a**5*x**5 - 3*a**4*x**4 + 2*a**3*x**3 + 2*a**2*x**2 -
3*a*x + 1), x) + Integral(a*x**5*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a**5*x**5 - 3*a**4*x**4 + 2*a**3*x**3 + 2*
a**2*x**2 - 3*a*x + 1), x))/c**4

________________________________________________________________________________________