3.5.77 \(\int e^{4 \tanh ^{-1}(a x)} (c-\frac {c}{a x})^3 \, dx\) [477]

Optimal. Leaf size=38 \[ \frac {c^3}{2 a^3 x^2}+\frac {c^3}{a^2 x}+c^3 x+\frac {c^3 \log (x)}{a} \]

[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 = 38, 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^(4*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^{4 \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx &=-\frac {c^3 \int \frac {e^{4 \tanh ^{-1}(a x)} (1-a x)^3}{x^3} \, dx}{a^3}\\ &=-\frac {c^3 \int \frac {(1-a x) (1+a x)^2}{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 = 0.04, size = 40, 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^(4*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 = 1.54, size = 30, normalized size = 0.79

method	result	size
default	\(\frac {c^{3} \left (a^{3} x +\frac {1}{2 x^{2}}+\frac {a}{x}+a^{2} \ln \left (x \right )\right )}{a^{3}}\)	\(30\)
risch	\(x \,c^{3}+\frac {a \,c^{3} x +\frac {1}{2} c^{3}}{a^{3} x^{2}}+\frac {c^{3} \ln \left (x \right )}{a}\)	\(35\)
norman	\(\frac {\frac {a^{3} c^{3} x^{4}}{2}+c^{3} a^{4} x^{5}-\frac {c^{3}}{2 a}-x \,c^{3}}{x^{2} \left (a^{2} x^{2}-1\right ) a^{2}}+\frac {c^{3} \ln \left (x \right )}{a}\)	\(65\)
meijerg	\(\frac {c^{3} \left (\frac {x \left (-a^{2}\right )^{\frac {5}{2}} \left (-10 a^{2} x^{2}+15\right )}{5 a^{4} \left (-a^{2} x^{2}+1\right )}-\frac {3 \left (-a^{2}\right )^{\frac {5}{2}} \arctanh \left (a x \right )}{a^{5}}\right )}{2 \sqrt {-a^{2}}}+\frac {c^{3} \left (\frac {a^{2} x^{2}}{-a^{2} x^{2}+1}+\ln \left (-a^{2} x^{2}+1\right )\right )}{2 a}+\frac {3 c^{3} \left (\frac {x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2} \left (-a^{2} x^{2}+1\right )}-\frac {\left (-a^{2}\right )^{\frac {3}{2}} \arctanh \left (a x \right )}{a^{3}}\right )}{2 \sqrt {-a^{2}}}-\frac {3 a \,c^{3} x^{2}}{2 \left (-a^{2} x^{2}+1\right )}+\frac {3 c^{3} \left (\frac {2 x \sqrt {-a^{2}}}{-2 a^{2} x^{2}+2}+\frac {\sqrt {-a^{2}}\, \arctanh \left (a x \right )}{a}\right )}{2 \sqrt {-a^{2}}}+\frac {3 c^{3} \left (\frac {2 a^{2} x^{2}}{-2 a^{2} x^{2}+2}-\ln \left (-a^{2} x^{2}+1\right )+1+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right )}{2 a}+\frac {c^{3} \left (-\frac {2 \left (-3 a^{2} x^{2}+2\right )}{x \sqrt {-a^{2}}\, \left (-2 a^{2} x^{2}+2\right )}+\frac {3 a \arctanh \left (a x \right )}{\sqrt {-a^{2}}}\right )}{2 \sqrt {-a^{2}}}+\frac {c^{3} \left (-\frac {3 a^{2} x^{2}}{-3 a^{2} x^{2}+3}+2 \ln \left (-a^{2} x^{2}+1\right )-1-4 \ln \left (x \right )-2 \ln \left (-a^{2}\right )+\frac {1}{a^{2} x^{2}}\right )}{2 a}\)	\(405\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^4/(-a^2*x^2+1)^2*(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 = 0.26, size = 34, normalized size = 0.89 \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)^4/(-a^2*x^2+1)^2*(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 = 0.35, size = 43, normalized size = 1.13 \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)^4/(-a^2*x^2+1)^2*(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 = 0.97 \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)**4/(-a**2*x**2+1)**2*(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 = 0.43, size = 35, normalized size = 0.92 \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)^4/(-a^2*x^2+1)^2*(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 = 0.81, size = 31, normalized size = 0.82 \begin {gather*} \frac {c^3\,\left (a\,x+a^3\,x^3+a^2\,x^2\,\ln \left (x\right )+\frac {1}{2}\right )}{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)^4)/(a^2*x^2 - 1)^2,x)

[Out]

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

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