3.11.31 \(\int e^{2 \tanh ^{-1}(a x)} x (c-a^2 c x^2)^2 \, dx\) [1031]

Optimal. Leaf size=48 \[ \frac {c^2 x^2}{2}+\frac {2}{3} a c^2 x^3-\frac {2}{5} a^3 c^2 x^5-\frac {1}{6} a^4 c^2 x^6 \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {6285, 76} \begin {gather*} -\frac {1}{6} a^4 c^2 x^6-\frac {2}{5} a^3 c^2 x^5+\frac {2}{3} a c^2 x^3+\frac {c^2 x^2}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]
time = 0.01, size = 32, normalized size = 0.67 \begin {gather*} -\frac {1}{30} c^2 x^2 \left (-15-20 a x+12 a^3 x^3+5 a^4 x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/30*(c^2*x^2*(-15 - 20*a*x + 12*a^3*x^3 + 5*a^4*x^4))

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Maple [A]
time = 0.07, size = 33, normalized size = 0.69

method	result	size
gosper	\(-\frac {c^{2} x^{2} \left (5 a^{4} x^{4}+12 a^{3} x^{3}-20 a x -15\right )}{30}\)	\(31\)
default	\(c^{2} \left (-\frac {1}{6} a^{4} x^{6}-\frac {2}{5} a^{3} x^{5}+\frac {2}{3} a \,x^{3}+\frac {1}{2} x^{2}\right )\)	\(33\)
norman	\(\frac {1}{2} c^{2} x^{2}+\frac {2}{3} a \,c^{2} x^{3}-\frac {2}{5} a^{3} c^{2} x^{5}-\frac {1}{6} a^{4} c^{2} x^{6}\)	\(41\)
risch	\(\frac {1}{2} c^{2} x^{2}+\frac {2}{3} a \,c^{2} x^{3}-\frac {2}{5} a^{3} c^{2} x^{5}-\frac {1}{6} a^{4} c^{2} x^{6}\)	\(41\)
meijerg	\(\frac {c^{2} \left (-\frac {x^{2} a^{2} \left (4 a^{4} x^{4}+6 a^{2} x^{2}+12\right )}{12}-\ln \left (-a^{2} x^{2}+1\right )\right )}{2 a^{2}}+\frac {c^{2} \left (\frac {x^{2} a^{2} \left (3 a^{2} x^{2}+6\right )}{6}+\ln \left (-a^{2} x^{2}+1\right )\right )}{2 a^{2}}-\frac {c^{2} \left (-a^{2} x^{2}-\ln \left (-a^{2} x^{2}+1\right )\right )}{2 a^{2}}-\frac {c^{2} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {7}{2}} \left (21 a^{4} x^{4}+35 a^{2} x^{2}+105\right )}{105 a^{6}}+\frac {2 \left (-a^{2}\right )^{\frac {7}{2}} \arctanh \left (a x \right )}{a^{7}}\right )}{a \sqrt {-a^{2}}}-\frac {2 c^{2} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {5}{2}} \left (5 a^{2} x^{2}+15\right )}{15 a^{4}}+\frac {2 \left (-a^{2}\right )^{\frac {5}{2}} \arctanh \left (a x \right )}{a^{5}}\right )}{a \sqrt {-a^{2}}}-\frac {c^{2} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2}}+\frac {2 \left (-a^{2}\right )^{\frac {3}{2}} \arctanh \left (a x \right )}{a^{3}}\right )}{a \sqrt {-a^{2}}}-\frac {c^{2} \ln \left (-a^{2} x^{2}+1\right )}{2 a^{2}}\)	\(300\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 40, normalized size = 0.83 \begin {gather*} -\frac {1}{6} \, a^{4} c^{2} x^{6} - \frac {2}{5} \, a^{3} c^{2} x^{5} + \frac {2}{3} \, a c^{2} x^{3} + \frac {1}{2} \, c^{2} 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)*x*(-a^2*c*x^2+c)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.33, size = 40, normalized size = 0.83 \begin {gather*} -\frac {1}{6} \, a^{4} c^{2} x^{6} - \frac {2}{5} \, a^{3} c^{2} x^{5} + \frac {2}{3} \, a c^{2} x^{3} + \frac {1}{2} \, c^{2} 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)*x*(-a^2*c*x^2+c)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.02, size = 44, normalized size = 0.92 \begin {gather*} - \frac {a^{4} c^{2} x^{6}}{6} - \frac {2 a^{3} c^{2} x^{5}}{5} + \frac {2 a c^{2} x^{3}}{3} + \frac {c^{2} x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.42, size = 40, normalized size = 0.83 \begin {gather*} -\frac {1}{6} \, a^{4} c^{2} x^{6} - \frac {2}{5} \, a^{3} c^{2} x^{5} + \frac {2}{3} \, a c^{2} x^{3} + \frac {1}{2} \, c^{2} 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)*x*(-a^2*c*x^2+c)^2,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.05, size = 40, normalized size = 0.83 \begin {gather*} -\frac {a^4\,c^2\,x^6}{6}-\frac {2\,a^3\,c^2\,x^5}{5}+\frac {2\,a\,c^2\,x^3}{3}+\frac {c^2\,x^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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