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

Optimal. Leaf size=105 \[ -\frac {2 \left (a+\frac {1}{x}\right )}{3 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {6 a+\frac {7}{x}}{3 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c^3}+\frac {2 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^3} \]

[Out]

-2/3*(a+1/x)/a^2/c^3/(1-1/a^2/x^2)^(3/2)+2*arctanh((1-1/a^2/x^2)^(1/2))/a/c^3+1/3*(-6*a-7/x)/a^2/c^3/(1-1/a^2/
x^2)^(1/2)+x*(1-1/a^2/x^2)^(1/2)/c^3

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Rubi [A]  time = 0.29, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6177, 852, 1805, 807, 266, 63, 208} \[ -\frac {2 \left (a+\frac {1}{x}\right )}{3 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {6 a+\frac {7}{x}}{3 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c^3}+\frac {2 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(a + x^(-1)))/(3*a^2*c^3*(1 - 1/(a^2*x^2))^(3/2)) - (6*a + 7/x)/(3*a^2*c^3*Sqrt[1 - 1/(a^2*x^2)]) + (Sqrt[
1 - 1/(a^2*x^2)]*x)/c^3 + (2*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(a*c^3)

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 852

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

Rule 1805

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(c*x)^m*Pq,
a + b*x^2, x], f = Coeff[PolynomialRemainder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[
(c*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[((a*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*
(p + 1)), Int[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x],
x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 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^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1}{x^2 \left (c-\frac {c x}{a}\right )^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\left (c+\frac {c x}{a}\right )^2}{x^2 \left (1-\frac {x^2}{a^2}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{c^5}\\ &=-\frac {2 \left (a+\frac {1}{x}\right )}{3 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}+\frac {\operatorname {Subst}\left (\int \frac {-3 c^2-\frac {6 c^2 x}{a}-\frac {4 c^2 x^2}{a^2}}{x^2 \left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{3 c^5}\\ &=-\frac {2 \left (a+\frac {1}{x}\right )}{3 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {6 a+\frac {7}{x}}{3 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {\operatorname {Subst}\left (\int \frac {3 c^2+\frac {6 c^2 x}{a}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{3 c^5}\\ &=-\frac {2 \left (a+\frac {1}{x}\right )}{3 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {6 a+\frac {7}{x}}{3 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^3}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a c^3}\\ &=-\frac {2 \left (a+\frac {1}{x}\right )}{3 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {6 a+\frac {7}{x}}{3 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^3}-\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{a c^3}\\ &=-\frac {2 \left (a+\frac {1}{x}\right )}{3 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {6 a+\frac {7}{x}}{3 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^3}+\frac {(2 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^3}\\ &=-\frac {2 \left (a+\frac {1}{x}\right )}{3 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {6 a+\frac {7}{x}}{3 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^3}+\frac {2 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^3}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 94, normalized size = 0.90 \[ \frac {3 a^3 x^3-11 a^2 x^2+6 a x \sqrt {1-\frac {1}{a^2 x^2}} (a x-1) \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )-4 a x+10}{3 a^2 c^3 x \sqrt {1-\frac {1}{a^2 x^2}} (a x-1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10 - 4*a*x - 11*a^2*x^2 + 3*a^3*x^3 + 6*a*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/
(3*a^2*c^3*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x))

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fricas [A]  time = 0.56, size = 134, normalized size = 1.28 \[ \frac {6 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 6 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (3 \, a^{3} x^{3} - 11 \, a^{2} x^{2} - 4 \, a x + 10\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{3 \, {\left (a^{3} c^{3} x^{2} - 2 \, a^{2} c^{3} x + a c^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(6*(a^2*x^2 - 2*a*x + 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 6*(a^2*x^2 - 2*a*x + 1)*log(sqrt((a*x - 1)/(
a*x + 1)) - 1) + (3*a^3*x^3 - 11*a^2*x^2 - 4*a*x + 10)*sqrt((a*x - 1)/(a*x + 1)))/(a^3*c^3*x^2 - 2*a^2*c^3*x +
a*c^3)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(a*x+1)]sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [B]  time = 0.06, size = 344, normalized size = 3.28 \[ \frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \left (24 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{3} a^{4}+27 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{3} a^{3}-72 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{2} a^{3}-15 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x a -81 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x^{2} a^{2}+72 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x \,a^{2}+13 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+81 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x a -24 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )-27 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\right )}{12 a \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, c^{3} \sqrt {a^{2}}\, \left (a x -1\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*((a*x-1)/(a*x+1))^(1/2)*(a*x+1)/a*(24*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^3*a^4
+27*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^3*a^3-72*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))
*x^2*a^3-15*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x*a-81*((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2)*x^2*a^2+72*ln((a^2*
x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x*a^2+13*((a*x-1)*(a*x+1))^(3/2)*(a^2)^(1/2)+81*((a*x-1)*(
a*x+1))^(1/2)*(a^2)^(1/2)*x*a-24*a*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))-27*((a*x-1)*(a*
x+1))^(1/2)*(a^2)^(1/2))/((a*x-1)*(a*x+1))^(1/2)/c^3/(a^2)^(1/2)/(a*x-1)^3

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maxima [A]  time = 0.31, size = 137, normalized size = 1.30 \[ \frac {1}{6} \, a {\left (\frac {\frac {14 \, {\left (a x - 1\right )}}{a x + 1} - \frac {27 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1}{a^{2} c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - a^{2} c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} + \frac {12 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{3}} - \frac {12 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*a*((14*(a*x - 1)/(a*x + 1) - 27*(a*x - 1)^2/(a*x + 1)^2 + 1)/(a^2*c^3*((a*x - 1)/(a*x + 1))^(5/2) - a^2*c^
3*((a*x - 1)/(a*x + 1))^(3/2)) + 12*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^3) - 12*log(sqrt((a*x - 1)/(a*x
+ 1)) - 1)/(a^2*c^3))

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mupad [B]  time = 1.25, size = 105, normalized size = 1.00 \[ \frac {4\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c^3}-\frac {\frac {14\,\left (a\,x-1\right )}{3\,\left (a\,x+1\right )}-\frac {9\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {1}{3}}{2\,a\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}-2\,a\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/(a*c^3) - ((14*(a*x - 1))/(3*(a*x + 1)) - (9*(a*x - 1)^2)/(a*x + 1)^2 +
1/3)/(2*a*c^3*((a*x - 1)/(a*x + 1))^(3/2) - 2*a*c^3*((a*x - 1)/(a*x + 1))^(5/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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