3.13.29 \(\int e^{-2 \tanh ^{-1}(a x)} (c-a^2 c x^2)^3 \, dx\) [1229]

Optimal. Leaf size=55 \[ -\frac {4 c^3 (1-a x)^5}{5 a}+\frac {2 c^3 (1-a x)^6}{3 a}-\frac {c^3 (1-a x)^7}{7 a} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6275, 45} \begin {gather*} -\frac {c^3 (1-a x)^7}{7 a}+\frac {2 c^3 (1-a x)^6}{3 a}-\frac {4 c^3 (1-a x)^5}{5 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 6275

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[(1 - a*x)^(p - n/2)*
(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, 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)} \left (c-a^2 c x^2\right )^3 \, dx &=c^3 \int (1-a x)^4 (1+a x)^2 \, dx\\ &=c^3 \int \left (4 (1-a x)^4-4 (1-a x)^5+(1-a x)^6\right ) \, dx\\ &=-\frac {4 c^3 (1-a x)^5}{5 a}+\frac {2 c^3 (1-a x)^6}{3 a}-\frac {c^3 (1-a x)^7}{7 a}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 31, normalized size = 0.56 \begin {gather*} \frac {c^3 (-1+a x)^5 \left (29+40 a x+15 a^2 x^2\right )}{105 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c^3*(-1 + a*x)^5*(29 + 40*a*x + 15*a^2*x^2))/(105*a)

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Maple [A]
time = 0.07, size = 52, normalized size = 0.95

method	result	size
default	\(c^{3} \left (\frac {1}{7} a^{6} x^{7}-\frac {1}{3} a^{5} x^{6}-\frac {1}{5} a^{4} x^{5}+a^{3} x^{4}-\frac {1}{3} a^{2} x^{3}-x^{2} a +x \right )\)	\(52\)
gosper	\(\frac {c^{3} x \left (15 x^{6} a^{6}-35 x^{5} a^{5}-21 a^{4} x^{4}+105 a^{3} x^{3}-35 a^{2} x^{2}-105 a x +105\right )}{105}\)	\(53\)
risch	\(\frac {1}{7} a^{6} c^{3} x^{7}-\frac {1}{3} a^{5} c^{3} x^{6}-\frac {1}{5} a^{4} c^{3} x^{5}+a^{3} c^{3} x^{4}-\frac {1}{3} a^{2} c^{3} x^{3}-a \,c^{3} x^{2}+c^{3} x\)	\(70\)
norman	\(\frac {-\frac {c^{3}}{a}-\frac {4 a^{2} c^{3} x^{3}}{3}+\frac {2 a^{3} c^{3} x^{4}}{3}+\frac {4 a^{4} c^{3} x^{5}}{5}-\frac {8 a^{5} c^{3} x^{6}}{15}-\frac {4 a^{6} c^{3} x^{7}}{21}+\frac {a^{7} c^{3} x^{8}}{7}}{a x +1}\)	\(84\)
meijerg	\(\frac {c^{3} \left (\frac {a x \left (45 a^{7} x^{7}-60 x^{6} a^{6}+84 x^{5} a^{5}-126 a^{4} x^{4}+210 a^{3} x^{3}-420 a^{2} x^{2}+1260 a x +2520\right )}{315 a x +315}-8 \ln \left (a x +1\right )\right )}{a}-\frac {4 c^{3} \left (\frac {a x \left (14 x^{5} a^{5}-21 a^{4} x^{4}+35 a^{3} x^{3}-70 a^{2} x^{2}+210 a x +420\right )}{70 a x +70}-6 \ln \left (a x +1\right )\right )}{a}+\frac {6 c^{3} \left (\frac {x a \left (5 a^{3} x^{3}-10 a^{2} x^{2}+30 a x +60\right )}{15 a x +15}-4 \ln \left (a x +1\right )\right )}{a}-\frac {4 c^{3} \left (\frac {a x \left (3 a x +6\right )}{3 a x +3}-2 \ln \left (a x +1\right )\right )}{a}+\frac {c^{3} x}{a x +1}\)	\(245\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.25, size = 69, normalized size = 1.25 \begin {gather*} \frac {1}{7} \, a^{6} c^{3} x^{7} - \frac {1}{3} \, a^{5} c^{3} x^{6} - \frac {1}{5} \, a^{4} c^{3} x^{5} + a^{3} c^{3} x^{4} - \frac {1}{3} \, a^{2} c^{3} x^{3} - a c^{3} x^{2} + c^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.33, size = 69, normalized size = 1.25 \begin {gather*} \frac {1}{7} \, a^{6} c^{3} x^{7} - \frac {1}{3} \, a^{5} c^{3} x^{6} - \frac {1}{5} \, a^{4} c^{3} x^{5} + a^{3} c^{3} x^{4} - \frac {1}{3} \, a^{2} c^{3} x^{3} - a c^{3} x^{2} + c^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.02, size = 70, normalized size = 1.27 \begin {gather*} \frac {a^{6} c^{3} x^{7}}{7} - \frac {a^{5} c^{3} x^{6}}{3} - \frac {a^{4} c^{3} x^{5}}{5} + a^{3} c^{3} x^{4} - \frac {a^{2} c^{3} x^{3}}{3} - a c^{3} x^{2} + c^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.41, size = 66, normalized size = 1.20 \begin {gather*} \frac {{\left (15 \, c^{3} - \frac {140 \, c^{3}}{a x + 1} + \frac {504 \, c^{3}}{{\left (a x + 1\right )}^{2}} - \frac {840 \, c^{3}}{{\left (a x + 1\right )}^{3}} + \frac {560 \, c^{3}}{{\left (a x + 1\right )}^{4}}\right )} {\left (a x + 1\right )}^{7}}{105 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(15*c^3 - 140*c^3/(a*x + 1) + 504*c^3/(a*x + 1)^2 - 840*c^3/(a*x + 1)^3 + 560*c^3/(a*x + 1)^4)*(a*x + 1)
^7/a

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Mupad [B]
time = 0.04, size = 69, normalized size = 1.25 \begin {gather*} \frac {a^6\,c^3\,x^7}{7}-\frac {a^5\,c^3\,x^6}{3}-\frac {a^4\,c^3\,x^5}{5}+a^3\,c^3\,x^4-\frac {a^2\,c^3\,x^3}{3}-a\,c^3\,x^2+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^2*x^2 - 1))/(a*x + 1)^2,x)

[Out]

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

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