3.5.59 \(\int e^{2 \tanh ^{-1}(a x)} (c-\frac {c}{a x})^3 \, dx\) [459]

Optimal. Leaf size=40 \[ \frac {c^3}{2 a^3 x^2}-\frac {c^3}{a^2 x}-c^3 x+\frac {c^3 \log (x)}{a} \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 40, 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, 76} \begin {gather*} \frac {c^3}{2 a^3 x^2}-\frac {c^3}{a^2 x}+\frac {c^3 \log (x)}{a}+c^3 (-x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

c^3/(2*a^3*x^2) - c^3/(a^2*x) - c^3*x + (c^3*Log[x])/a

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] &&  !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 0])

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

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Mathematica [A]
time = 0.08, size = 42, normalized size = 1.05 \begin {gather*} \frac {c^3}{2 a^3 x^2}-\frac {c^3}{a^2 x}-c^3 x+\frac {c^3 \log (a x)}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

c^3/(2*a^3*x^2) - c^3/(a^2*x) - c^3*x + (c^3*Log[a*x])/a

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Maple [A]
time = 1.08, size = 32, normalized size = 0.80

method	result	size
default	\(\frac {c^{3} \left (-a^{3} x +\frac {1}{2 x^{2}}-\frac {a}{x}+a^{2} \ln \left (x \right )\right )}{a^{3}}\)	\(32\)
risch	\(-x \,c^{3}+\frac {-a \,c^{3} x +\frac {1}{2} c^{3}}{a^{3} x^{2}}+\frac {c^{3} \ln \left (x \right )}{a}\)	\(37\)
norman	\(\frac {\frac {c^{3}}{2 a}-x \,c^{3}-a^{2} c^{3} x^{3}}{a^{2} x^{2}}+\frac {c^{3} \ln \left (x \right )}{a}\)	\(44\)
meijerg	\(-\frac {c^{3} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2}}+\frac {2 \left (-a^{2}\right )^{\frac {3}{2}} \arctanh \left (a x \right )}{a^{3}}\right )}{2 \sqrt {-a^{2}}}+\frac {c^{3} \ln \left (-a^{2} x^{2}+1\right )}{2 a}-\frac {2 c^{3} \arctanh \left (a x \right )}{a}+\frac {c^{3} \left (-\ln \left (-a^{2} x^{2}+1\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right )}{a}-\frac {c^{3} \left (-\frac {2}{x \sqrt {-a^{2}}}+\frac {2 a \arctanh \left (a x \right )}{\sqrt {-a^{2}}}\right )}{2 \sqrt {-a^{2}}}+\frac {c^{3} \left (\ln \left (-a^{2} x^{2}+1\right )-2 \ln \left (x \right )-\ln \left (-a^{2}\right )+\frac {1}{a^{2} x^{2}}\right )}{2 a}\)	\(184\)

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]

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

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Maxima [A]
time = 0.25, size = 37, normalized size = 0.92 \begin {gather*} -c^{3} x + \frac {c^{3} \log \left (x\right )}{a} - \frac {2 \, a c^{3} x - c^{3}}{2 \, a^{3} x^{2}} \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]

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

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Fricas [A]
time = 0.38, size = 45, normalized size = 1.12 \begin {gather*} -\frac {2 \, a^{3} c^{3} x^{3} - 2 \, a^{2} c^{3} x^{2} \log \left (x\right ) + 2 \, a c^{3} x - c^{3}}{2 \, a^{3} x^{2}} \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]

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

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Sympy [A]
time = 0.08, size = 37, normalized size = 0.92 \begin {gather*} \frac {- a^{3} c^{3} x + a^{2} c^{3} \log {\left (x \right )} - \frac {2 a c^{3} x - c^{3}}{2 x^{2}}}{a^{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]

(-a**3*c**3*x + a**2*c**3*log(x) - (2*a*c**3*x - c**3)/(2*x**2))/a**3

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Giac [A]
time = 0.40, size = 38, normalized size = 0.95 \begin {gather*} -c^{3} x + \frac {c^{3} \log \left ({\left | x \right |}\right )}{a} - \frac {2 \, a c^{3} x - c^{3}}{2 \, a^{3} x^{2}} \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]

-c^3*x + c^3*log(abs(x))/a - 1/2*(2*a*c^3*x - c^3)/(a^3*x^2)

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Mupad [B]
time = 0.05, size = 35, normalized size = 0.88 \begin {gather*} -\frac {c^3\,\left (2\,a\,x+2\,a^3\,x^3-2\,a^2\,x^2\,\ln \left (x\right )-1\right )}{2\,a^3\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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