3.11.44 \(\int \frac {e^{2 \tanh ^{-1}(a x)} (c-a^2 c x^2)^3}{x} \, dx\) [1044]

Optimal. Leaf size=79 \[ 2 a c^3 x-\frac {1}{2} a^2 c^3 x^2-\frac {4}{3} a^3 c^3 x^3-\frac {1}{4} a^4 c^3 x^4+\frac {2}{5} a^5 c^3 x^5+\frac {1}{6} a^6 c^3 x^6+c^3 \log (x) \]

[Out]

2*a*c^3*x-1/2*a^2*c^3*x^2-4/3*a^3*c^3*x^3-1/4*a^4*c^3*x^4+2/5*a^5*c^3*x^5+1/6*a^6*c^3*x^6+c^3*ln(x)

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Rubi [A]
time = 0.06, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6285, 90} \begin {gather*} \frac {1}{6} a^6 c^3 x^6+\frac {2}{5} a^5 c^3 x^5-\frac {1}{4} a^4 c^3 x^4-\frac {4}{3} a^3 c^3 x^3-\frac {1}{2} a^2 c^3 x^2+2 a c^3 x+c^3 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*a*c^3*x - (a^2*c^3*x^2)/2 - (4*a^3*c^3*x^3)/3 - (a^4*c^3*x^4)/4 + (2*a^5*c^3*x^5)/5 + (a^6*c^3*x^6)/6 + c^3*
Log[x]

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 6285

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

Rubi steps

\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^3}{x} \, dx &=c^3 \int \frac {(1-a x)^2 (1+a x)^4}{x} \, dx\\ &=c^3 \int \left (2 a+\frac {1}{x}-a^2 x-4 a^3 x^2-a^4 x^3+2 a^5 x^4+a^6 x^5\right ) \, dx\\ &=2 a c^3 x-\frac {1}{2} a^2 c^3 x^2-\frac {4}{3} a^3 c^3 x^3-\frac {1}{4} a^4 c^3 x^4+\frac {2}{5} a^5 c^3 x^5+\frac {1}{6} a^6 c^3 x^6+c^3 \log (x)\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 51, normalized size = 0.65 \begin {gather*} c^3 \left (\frac {1}{60} a x \left (120-30 a x-80 a^2 x^2-15 a^3 x^3+24 a^4 x^4+10 a^5 x^5\right )+\log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

c^3*((a*x*(120 - 30*a*x - 80*a^2*x^2 - 15*a^3*x^3 + 24*a^4*x^4 + 10*a^5*x^5))/60 + Log[x])

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Maple [A]
time = 0.09, size = 52, normalized size = 0.66

method	result	size
default	\(c^{3} \left (\frac {x^{6} a^{6}}{6}+\frac {2 x^{5} a^{5}}{5}-\frac {a^{4} x^{4}}{4}-\frac {4 a^{3} x^{3}}{3}-\frac {a^{2} x^{2}}{2}+2 a x +\ln \left (x \right )\right )\)	\(52\)
norman	\(2 a \,c^{3} x -\frac {a^{2} c^{3} x^{2}}{2}-\frac {4 a^{3} c^{3} x^{3}}{3}-\frac {a^{4} c^{3} x^{4}}{4}+\frac {2 a^{5} c^{3} x^{5}}{5}+\frac {a^{6} c^{3} x^{6}}{6}+c^{3} \ln \left (x \right )\)	\(70\)
risch	\(2 a \,c^{3} x -\frac {a^{2} c^{3} x^{2}}{2}-\frac {4 a^{3} c^{3} x^{3}}{3}-\frac {a^{4} c^{3} x^{4}}{4}+\frac {2 a^{5} c^{3} x^{5}}{5}+\frac {a^{6} c^{3} x^{6}}{6}+c^{3} \ln \left (x \right )\)	\(70\)
meijerg	\(-\frac {c^{3} \left (-\frac {x^{2} a^{2} \left (4 a^{4} x^{4}+6 a^{2} x^{2}+12\right )}{12}-\ln \left (-a^{2} x^{2}+1\right )\right )}{2}-c^{3} \left (\frac {x^{2} a^{2} \left (3 a^{2} x^{2}+6\right )}{6}+\ln \left (-a^{2} x^{2}+1\right )\right )+c^{3} \ln \left (-a^{2} x^{2}+1\right )+\frac {a \,c^{3} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {7}{2}} \left (21 a^{4} x^{4}+35 a^{2} x^{2}+105\right )}{105 a^{6}}+\frac {2 \left (-a^{2}\right )^{\frac {7}{2}} \arctanh \left (a x \right )}{a^{7}}\right )}{\sqrt {-a^{2}}}+\frac {3 a \,c^{3} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {5}{2}} \left (5 a^{2} x^{2}+15\right )}{15 a^{4}}+\frac {2 \left (-a^{2}\right )^{\frac {5}{2}} \arctanh \left (a x \right )}{a^{5}}\right )}{\sqrt {-a^{2}}}+\frac {3 a \,c^{3} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2}}+\frac {2 \left (-a^{2}\right )^{\frac {3}{2}} \arctanh \left (a x \right )}{a^{3}}\right )}{\sqrt {-a^{2}}}+2 c^{3} \arctanh \left (a x \right )+\frac {c^{3} \left (-\ln \left (-a^{2} x^{2}+1\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right )}{2}\)	\(291\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^3*(1/6*x^6*a^6+2/5*x^5*a^5-1/4*a^4*x^4-4/3*a^3*x^3-1/2*a^2*x^2+2*a*x+ln(x))

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Maxima [A]
time = 0.26, size = 69, normalized size = 0.87 \begin {gather*} \frac {1}{6} \, a^{6} c^{3} x^{6} + \frac {2}{5} \, a^{5} c^{3} x^{5} - \frac {1}{4} \, a^{4} c^{3} x^{4} - \frac {4}{3} \, a^{3} c^{3} x^{3} - \frac {1}{2} \, a^{2} c^{3} x^{2} + 2 \, a c^{3} x + c^{3} \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.32, size = 69, normalized size = 0.87 \begin {gather*} \frac {1}{6} \, a^{6} c^{3} x^{6} + \frac {2}{5} \, a^{5} c^{3} x^{5} - \frac {1}{4} \, a^{4} c^{3} x^{4} - \frac {4}{3} \, a^{3} c^{3} x^{3} - \frac {1}{2} \, a^{2} c^{3} x^{2} + 2 \, a c^{3} x + c^{3} \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.06, size = 76, normalized size = 0.96 \begin {gather*} \frac {a^{6} c^{3} x^{6}}{6} + \frac {2 a^{5} c^{3} x^{5}}{5} - \frac {a^{4} c^{3} x^{4}}{4} - \frac {4 a^{3} c^{3} x^{3}}{3} - \frac {a^{2} c^{3} x^{2}}{2} + 2 a c^{3} x + c^{3} \log {\left (x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**6*c**3*x**6/6 + 2*a**5*c**3*x**5/5 - a**4*c**3*x**4/4 - 4*a**3*c**3*x**3/3 - a**2*c**3*x**2/2 + 2*a*c**3*x
+ c**3*log(x)

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Giac [A]
time = 0.41, size = 70, normalized size = 0.89 \begin {gather*} \frac {1}{6} \, a^{6} c^{3} x^{6} + \frac {2}{5} \, a^{5} c^{3} x^{5} - \frac {1}{4} \, a^{4} c^{3} x^{4} - \frac {4}{3} \, a^{3} c^{3} x^{3} - \frac {1}{2} \, a^{2} c^{3} x^{2} + 2 \, a c^{3} x + c^{3} \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.04, size = 69, normalized size = 0.87 \begin {gather*} c^3\,\ln \left (x\right )-\frac {a^2\,c^3\,x^2}{2}-\frac {4\,a^3\,c^3\,x^3}{3}-\frac {a^4\,c^3\,x^4}{4}+\frac {2\,a^5\,c^3\,x^5}{5}+\frac {a^6\,c^3\,x^6}{6}+2\,a\,c^3\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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