3.11.42 \(\int e^{2 \tanh ^{-1}(a x)} x (c-a^2 c x^2)^3 \, dx\) [1042]

Optimal. Leaf size=69 \[ -\frac {4 c^3 (1+a x)^5}{5 a^2}+\frac {4 c^3 (1+a x)^6}{3 a^2}-\frac {5 c^3 (1+a x)^7}{7 a^2}+\frac {c^3 (1+a x)^8}{8 a^2} \]

[Out]

-4/5*c^3*(a*x+1)^5/a^2+4/3*c^3*(a*x+1)^6/a^2-5/7*c^3*(a*x+1)^7/a^2+1/8*c^3*(a*x+1)^8/a^2

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Rubi [A]
time = 0.05, antiderivative size = 69, 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, 78} \begin {gather*} \frac {c^3 (a x+1)^8}{8 a^2}-\frac {5 c^3 (a x+1)^7}{7 a^2}+\frac {4 c^3 (a x+1)^6}{3 a^2}-\frac {4 c^3 (a x+1)^5}{5 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*c^3*(1 + a*x)^5)/(5*a^2) + (4*c^3*(1 + a*x)^6)/(3*a^2) - (5*c^3*(1 + a*x)^7)/(7*a^2) + (c^3*(1 + a*x)^8)/(
8*a^2)

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 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 )^3 \, dx &=c^3 \int x (1-a x)^2 (1+a x)^4 \, dx\\ &=c^3 \int \left (-\frac {4 (1+a x)^4}{a}+\frac {8 (1+a x)^5}{a}-\frac {5 (1+a x)^6}{a}+\frac {(1+a x)^7}{a}\right ) \, dx\\ &=-\frac {4 c^3 (1+a x)^5}{5 a^2}+\frac {4 c^3 (1+a x)^6}{3 a^2}-\frac {5 c^3 (1+a x)^7}{7 a^2}+\frac {c^3 (1+a x)^8}{8 a^2}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 70, normalized size = 1.01 \begin {gather*} c^3 \left (\frac {x^2}{2}+\frac {2 a x^3}{3}-\frac {a^2 x^4}{4}-\frac {4 a^3 x^5}{5}-\frac {a^4 x^6}{6}+\frac {2 a^5 x^7}{7}+\frac {a^6 x^8}{8}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

c^3*(x^2/2 + (2*a*x^3)/3 - (a^2*x^4)/4 - (4*a^3*x^5)/5 - (a^4*x^6)/6 + (2*a^5*x^7)/7 + (a^6*x^8)/8)

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Maple [A]
time = 0.08, size = 57, normalized size = 0.83

method	result	size
gosper	\(\frac {c^{3} x^{2} \left (105 x^{6} a^{6}+240 x^{5} a^{5}-140 a^{4} x^{4}-672 a^{3} x^{3}-210 a^{2} x^{2}+560 a x +420\right )}{840}\)	\(55\)
default	\(c^{3} \left (\frac {1}{8} a^{6} x^{8}+\frac {2}{7} a^{5} x^{7}-\frac {1}{6} a^{4} x^{6}-\frac {4}{5} a^{3} x^{5}-\frac {1}{4} x^{4} a^{2}+\frac {2}{3} a \,x^{3}+\frac {1}{2} x^{2}\right )\)	\(57\)
norman	\(\frac {1}{2} c^{3} x^{2}+\frac {2}{3} a \,c^{3} x^{3}-\frac {1}{4} a^{2} c^{3} x^{4}-\frac {4}{5} a^{3} c^{3} x^{5}-\frac {1}{6} a^{4} c^{3} x^{6}+\frac {2}{7} a^{5} c^{3} x^{7}+\frac {1}{8} a^{6} c^{3} x^{8}\)	\(74\)
risch	\(\frac {1}{2} c^{3} x^{2}+\frac {2}{3} a \,c^{3} x^{3}-\frac {1}{4} a^{2} c^{3} x^{4}-\frac {4}{5} a^{3} c^{3} x^{5}-\frac {1}{6} a^{4} c^{3} x^{6}+\frac {2}{7} a^{5} c^{3} x^{7}+\frac {1}{8} a^{6} c^{3} x^{8}\)	\(74\)
meijerg	\(\frac {c^{3} \left (\frac {x^{2} a^{2} \left (15 x^{6} a^{6}+20 a^{4} x^{4}+30 a^{2} x^{2}+60\right )}{60}+\ln \left (-a^{2} x^{2}+1\right )\right )}{2 a^{2}}+\frac {c^{3} \left (-\frac {x^{2} a^{2} \left (4 a^{4} x^{4}+6 a^{2} x^{2}+12\right )}{12}-\ln \left (-a^{2} x^{2}+1\right )\right )}{a^{2}}-\frac {c^{3} \left (-a^{2} x^{2}-\ln \left (-a^{2} x^{2}+1\right )\right )}{a^{2}}-\frac {c^{3} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {9}{2}} \left (45 x^{6} a^{6}+63 a^{4} x^{4}+105 a^{2} x^{2}+315\right )}{315 a^{8}}+\frac {2 \left (-a^{2}\right )^{\frac {9}{2}} \arctanh \left (a x \right )}{a^{9}}\right )}{a \sqrt {-a^{2}}}-\frac {3 c^{3} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {7}{2}} \left (21 a^{4} x^{4}+35 a^{2} x^{2}+105\right )}{105 a^{6}}+\frac {2 \left (-a^{2}\right )^{\frac {7}{2}} \arctanh \left (a x \right )}{a^{7}}\right )}{a \sqrt {-a^{2}}}-\frac {3 c^{3} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {5}{2}} \left (5 a^{2} x^{2}+15\right )}{15 a^{4}}+\frac {2 \left (-a^{2}\right )^{\frac {5}{2}} \arctanh \left (a x \right )}{a^{5}}\right )}{a \sqrt {-a^{2}}}-\frac {c^{3} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2}}+\frac {2 \left (-a^{2}\right )^{\frac {3}{2}} \arctanh \left (a x \right )}{a^{3}}\right )}{a \sqrt {-a^{2}}}-\frac {c^{3} \ln \left (-a^{2} x^{2}+1\right )}{2 a^{2}}\)	\(386\)

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)^3,x,method=_RETURNVERBOSE)

[Out]

c^3*(1/8*a^6*x^8+2/7*a^5*x^7-1/6*a^4*x^6-4/5*a^3*x^5-1/4*x^4*a^2+2/3*a*x^3+1/2*x^2)

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Maxima [A]
time = 0.27, size = 73, normalized size = 1.06 \begin {gather*} \frac {1}{8} \, a^{6} c^{3} x^{8} + \frac {2}{7} \, a^{5} c^{3} x^{7} - \frac {1}{6} \, a^{4} c^{3} x^{6} - \frac {4}{5} \, a^{3} c^{3} x^{5} - \frac {1}{4} \, a^{2} c^{3} x^{4} + \frac {2}{3} \, a c^{3} x^{3} + \frac {1}{2} \, c^{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)*x*(-a^2*c*x^2+c)^3,x, algorithm="maxima")

[Out]

1/8*a^6*c^3*x^8 + 2/7*a^5*c^3*x^7 - 1/6*a^4*c^3*x^6 - 4/5*a^3*c^3*x^5 - 1/4*a^2*c^3*x^4 + 2/3*a*c^3*x^3 + 1/2*
c^3*x^2

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Fricas [A]
time = 0.33, size = 73, normalized size = 1.06 \begin {gather*} \frac {1}{8} \, a^{6} c^{3} x^{8} + \frac {2}{7} \, a^{5} c^{3} x^{7} - \frac {1}{6} \, a^{4} c^{3} x^{6} - \frac {4}{5} \, a^{3} c^{3} x^{5} - \frac {1}{4} \, a^{2} c^{3} x^{4} + \frac {2}{3} \, a c^{3} x^{3} + \frac {1}{2} \, c^{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)*x*(-a^2*c*x^2+c)^3,x, algorithm="fricas")

[Out]

1/8*a^6*c^3*x^8 + 2/7*a^5*c^3*x^7 - 1/6*a^4*c^3*x^6 - 4/5*a^3*c^3*x^5 - 1/4*a^2*c^3*x^4 + 2/3*a*c^3*x^3 + 1/2*
c^3*x^2

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Sympy [A]
time = 0.03, size = 82, normalized size = 1.19 \begin {gather*} \frac {a^{6} c^{3} x^{8}}{8} + \frac {2 a^{5} c^{3} x^{7}}{7} - \frac {a^{4} c^{3} x^{6}}{6} - \frac {4 a^{3} c^{3} x^{5}}{5} - \frac {a^{2} c^{3} x^{4}}{4} + \frac {2 a c^{3} x^{3}}{3} + \frac {c^{3} 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)**3,x)

[Out]

a**6*c**3*x**8/8 + 2*a**5*c**3*x**7/7 - a**4*c**3*x**6/6 - 4*a**3*c**3*x**5/5 - a**2*c**3*x**4/4 + 2*a*c**3*x*
*3/3 + c**3*x**2/2

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Giac [A]
time = 0.41, size = 73, normalized size = 1.06 \begin {gather*} \frac {1}{8} \, a^{6} c^{3} x^{8} + \frac {2}{7} \, a^{5} c^{3} x^{7} - \frac {1}{6} \, a^{4} c^{3} x^{6} - \frac {4}{5} \, a^{3} c^{3} x^{5} - \frac {1}{4} \, a^{2} c^{3} x^{4} + \frac {2}{3} \, a c^{3} x^{3} + \frac {1}{2} \, c^{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)*x*(-a^2*c*x^2+c)^3,x, algorithm="giac")

[Out]

1/8*a^6*c^3*x^8 + 2/7*a^5*c^3*x^7 - 1/6*a^4*c^3*x^6 - 4/5*a^3*c^3*x^5 - 1/4*a^2*c^3*x^4 + 2/3*a*c^3*x^3 + 1/2*
c^3*x^2

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Mupad [B]
time = 0.03, size = 73, normalized size = 1.06 \begin {gather*} \frac {a^6\,c^3\,x^8}{8}+\frac {2\,a^5\,c^3\,x^7}{7}-\frac {a^4\,c^3\,x^6}{6}-\frac {4\,a^3\,c^3\,x^5}{5}-\frac {a^2\,c^3\,x^4}{4}+\frac {2\,a\,c^3\,x^3}{3}+\frac {c^3\,x^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c^3*x^2)/2 + (2*a*c^3*x^3)/3 - (a^2*c^3*x^4)/4 - (4*a^3*c^3*x^5)/5 - (a^4*c^3*x^6)/6 + (2*a^5*c^3*x^7)/7 + (a
^6*c^3*x^8)/8

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