3.5.64 \(\int \frac {e^{2 \tanh ^{-1}(a x)}}{(c-\frac {c}{a x})^3} \, dx\) [464]

Optimal. Leaf size=74 \[ -\frac {x}{c^3}-\frac {2}{3 a c^3 (1-a x)^3}+\frac {7}{2 a c^3 (1-a x)^2}-\frac {9}{a c^3 (1-a x)}-\frac {5 \log (1-a x)}{a c^3} \]

[Out]

-x/c^3-2/3/a/c^3/(-a*x+1)^3+7/2/a/c^3/(-a*x+1)^2-9/a/c^3/(-a*x+1)-5*ln(-a*x+1)/a/c^3

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Rubi [A]
time = 0.09, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6266, 6264, 78} \begin {gather*} -\frac {9}{a c^3 (1-a x)}+\frac {7}{2 a c^3 (1-a x)^2}-\frac {2}{3 a c^3 (1-a x)^3}-\frac {5 \log (1-a x)}{a c^3}-\frac {x}{c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcTanh[a*x])/(c - c/(a*x))^3,x]

[Out]

-(x/c^3) - 2/(3*a*c^3*(1 - a*x)^3) + 7/(2*a*c^3*(1 - a*x)^2) - 9/(a*c^3*(1 - a*x)) - (5*Log[1 - a*x])/(a*c^3)

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 6264

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

Rule 6266

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

Rubi steps

\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx &=-\frac {a^3 \int \frac {e^{2 \tanh ^{-1}(a x)} x^3}{(1-a x)^3} \, dx}{c^3}\\ &=-\frac {a^3 \int \frac {x^3 (1+a x)}{(1-a x)^4} \, dx}{c^3}\\ &=-\frac {a^3 \int \left (\frac {1}{a^3}+\frac {2}{a^3 (-1+a x)^4}+\frac {7}{a^3 (-1+a x)^3}+\frac {9}{a^3 (-1+a x)^2}+\frac {5}{a^3 (-1+a x)}\right ) \, dx}{c^3}\\ &=-\frac {x}{c^3}-\frac {2}{3 a c^3 (1-a x)^3}+\frac {7}{2 a c^3 (1-a x)^2}-\frac {9}{a c^3 (1-a x)}-\frac {5 \log (1-a x)}{a c^3}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 63, normalized size = 0.85 \begin {gather*} \frac {37-81 a x+36 a^2 x^2+18 a^3 x^3-6 a^4 x^4-30 (-1+a x)^3 \log (1-a x)}{6 a c^3 (-1+a x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(2*ArcTanh[a*x])/(c - c/(a*x))^3,x]

[Out]

(37 - 81*a*x + 36*a^2*x^2 + 18*a^3*x^3 - 6*a^4*x^4 - 30*(-1 + a*x)^3*Log[1 - a*x])/(6*a*c^3*(-1 + a*x)^3)

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Maple [A]
time = 0.76, size = 62, normalized size = 0.84

method	result	size
risch	\(-\frac {x}{c^{3}}+\frac {9 a \,c^{3} x^{2}-\frac {29 x \,c^{3}}{2}+\frac {37 c^{3}}{6 a}}{c^{6} \left (a x -1\right )^{3}}-\frac {5 \ln \left (a x -1\right )}{a \,c^{3}}\)	\(57\)
default	\(\frac {a^{3} \left (-\frac {x}{a^{3}}+\frac {9}{a^{4} \left (a x -1\right )}+\frac {7}{2 \left (a x -1\right )^{2} a^{4}}+\frac {2}{3 \left (a x -1\right )^{3} a^{4}}-\frac {5 \ln \left (a x -1\right )}{a^{4}}\right )}{c^{3}}\)	\(62\)
norman	\(\frac {-\frac {a^{3} x^{4}}{c}+\frac {5 x}{c}-\frac {25 a \,x^{2}}{2 c}+\frac {55 a^{2} x^{3}}{6 c}}{\left (a x -1\right )^{3} c^{2}}-\frac {5 \ln \left (a x -1\right )}{a \,c^{3}}\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^3/c^3*(-x/a^3+9/a^4/(a*x-1)+7/2/(a*x-1)^2/a^4+2/3/(a*x-1)^3/a^4-5/a^4*ln(a*x-1))

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Maxima [A]
time = 0.25, size = 76, normalized size = 1.03 \begin {gather*} \frac {54 \, a^{2} x^{2} - 87 \, a x + 37}{6 \, {\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}} - \frac {x}{c^{3}} - \frac {5 \, \log \left (a x - 1\right )}{a c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(54*a^2*x^2 - 87*a*x + 37)/(a^4*c^3*x^3 - 3*a^3*c^3*x^2 + 3*a^2*c^3*x - a*c^3) - x/c^3 - 5*log(a*x - 1)/(a
*c^3)

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Fricas [A]
time = 0.40, size = 100, normalized size = 1.35 \begin {gather*} -\frac {6 \, a^{4} x^{4} - 18 \, a^{3} x^{3} - 36 \, a^{2} x^{2} + 81 \, a x + 30 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (a x - 1\right ) - 37}{6 \, {\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(6*a^4*x^4 - 18*a^3*x^3 - 36*a^2*x^2 + 81*a*x + 30*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*log(a*x - 1) - 37)/(
a^4*c^3*x^3 - 3*a^3*c^3*x^2 + 3*a^2*c^3*x - a*c^3)

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Sympy [A]
time = 0.20, size = 75, normalized size = 1.01 \begin {gather*} - \frac {- 54 a^{2} x^{2} + 87 a x - 37}{6 a^{4} c^{3} x^{3} - 18 a^{3} c^{3} x^{2} + 18 a^{2} c^{3} x - 6 a c^{3}} - \frac {x}{c^{3}} - \frac {5 \log {\left (a x - 1 \right )}}{a c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-54*a**2*x**2 + 87*a*x - 37)/(6*a**4*c**3*x**3 - 18*a**3*c**3*x**2 + 18*a**2*c**3*x - 6*a*c**3) - x/c**3 - 5
*log(a*x - 1)/(a*c**3)

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Giac [A]
time = 0.40, size = 51, normalized size = 0.69 \begin {gather*} -\frac {x}{c^{3}} - \frac {5 \, \log \left ({\left | a x - 1 \right |}\right )}{a c^{3}} + \frac {54 \, a^{2} x^{2} - 87 \, a x + 37}{6 \, {\left (a x - 1\right )}^{3} a c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/c^3 - 5*log(abs(a*x - 1))/(a*c^3) + 1/6*(54*a^2*x^2 - 87*a*x + 37)/((a*x - 1)^3*a*c^3)

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Mupad [B]
time = 0.84, size = 73, normalized size = 0.99 \begin {gather*} -\frac {9\,a\,x^2-\frac {29\,x}{2}+\frac {37}{6\,a}}{-a^3\,c^3\,x^3+3\,a^2\,c^3\,x^2-3\,a\,c^3\,x+c^3}-\frac {x}{c^3}-\frac {5\,\ln \left (a\,x-1\right )}{a\,c^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- (9*a*x^2 - (29*x)/2 + 37/(6*a))/(c^3 + 3*a^2*c^3*x^2 - a^3*c^3*x^3 - 3*a*c^3*x) - x/c^3 - (5*log(a*x - 1))/(
a*c^3)

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