3.3.8 \(\int e^{-2 \tanh ^{-1}(a x)} (c-a c x)^3 \, dx\) [208]

Optimal. Leaf size=73 \[ -8 c^3 x+\frac {2 c^3 (1-a x)^2}{a}+\frac {2 c^3 (1-a x)^3}{3 a}+\frac {c^3 (1-a x)^4}{4 a}+\frac {16 c^3 \log (1+a x)}{a} \]

[Out]

-8*c^3*x+2*c^3*(-a*x+1)^2/a+2/3*c^3*(-a*x+1)^3/a+1/4*c^3*(-a*x+1)^4/a+16*c^3*ln(a*x+1)/a

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Rubi [A]
time = 0.03, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6264, 45} \begin {gather*} \frac {c^3 (1-a x)^4}{4 a}+\frac {2 c^3 (1-a x)^3}{3 a}+\frac {2 c^3 (1-a x)^2}{a}+\frac {16 c^3 \log (a x+1)}{a}-8 c^3 x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-8*c^3*x + (2*c^3*(1 - a*x)^2)/a + (2*c^3*(1 - a*x)^3)/(3*a) + (c^3*(1 - a*x)^4)/(4*a) + (16*c^3*Log[1 + a*x])
/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 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])

Rubi steps

\begin {align*} \int e^{-2 \tanh ^{-1}(a x)} (c-a c x)^3 \, dx &=c^3 \int \frac {(1-a x)^4}{1+a x} \, dx\\ &=c^3 \int \left (-8-4 (1-a x)-2 (1-a x)^2-(1-a x)^3+\frac {16}{1+a x}\right ) \, dx\\ &=-8 c^3 x+\frac {2 c^3 (1-a x)^2}{a}+\frac {2 c^3 (1-a x)^3}{3 a}+\frac {c^3 (1-a x)^4}{4 a}+\frac {16 c^3 \log (1+a x)}{a}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 48, normalized size = 0.66 \begin {gather*} \frac {c^3 \left (35-180 a x+66 a^2 x^2-20 a^3 x^3+3 a^4 x^4+192 \log (1+a x)\right )}{12 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c^3*(35 - 180*a*x + 66*a^2*x^2 - 20*a^3*x^3 + 3*a^4*x^4 + 192*Log[1 + a*x]))/(12*a)

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Maple [A]
time = 0.93, size = 42, normalized size = 0.58

method	result	size
default	\(c^{3} \left (\frac {a^{3} x^{4}}{4}-\frac {5 a^{2} x^{3}}{3}+\frac {11 a \,x^{2}}{2}-15 x +\frac {16 \ln \left (a x +1\right )}{a}\right )\)	\(42\)
risch	\(\frac {a^{3} c^{3} x^{4}}{4}-\frac {5 a^{2} c^{3} x^{3}}{3}+\frac {11 a \,c^{3} x^{2}}{2}-15 x \,c^{3}+\frac {16 c^{3} \ln \left (a x +1\right )}{a}\)	\(53\)
norman	\(\frac {-15 x \,c^{3}-\frac {19}{2} a \,c^{3} x^{2}+\frac {23}{6} a^{2} c^{3} x^{3}-\frac {17}{12} a^{3} c^{3} x^{4}+\frac {1}{4} c^{3} a^{4} x^{5}}{a x +1}+\frac {16 c^{3} \ln \left (a x +1\right )}{a}\)	\(73\)
meijerg	\(\frac {c^{3} \left (-\frac {a x \left (-3 a^{4} x^{4}+5 a^{3} x^{3}-10 a^{2} x^{2}+30 a x +60\right )}{12 \left (a x +1\right )}+5 \ln \left (a x +1\right )\right )}{a}+\frac {2 c^{3} \left (-\frac {a x \left (-2 a^{2} x^{2}+6 a x +12\right )}{4 \left (a x +1\right )}+3 \ln \left (a x +1\right )\right )}{a}-\frac {3 c^{3} \left (\frac {a x \left (5 a^{3} x^{3}-10 a^{2} x^{2}+30 a x +60\right )}{15 a x +15}-4 \ln \left (a x +1\right )\right )}{a}+\frac {2 c^{3} \left (\frac {a x \left (3 a x +6\right )}{3 a x +3}-2 \ln \left (a x +1\right )\right )}{a}-\frac {3 c^{3} \left (-\frac {a x}{a x +1}+\ln \left (a x +1\right )\right )}{a}+\frac {c^{3} x}{a x +1}\)	\(223\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^3*(1/4*a^3*x^4-5/3*a^2*x^3+11/2*a*x^2-15*x+16/a*ln(a*x+1))

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Maxima [A]
time = 0.25, size = 52, normalized size = 0.71 \begin {gather*} \frac {1}{4} \, a^{3} c^{3} x^{4} - \frac {5}{3} \, a^{2} c^{3} x^{3} + \frac {11}{2} \, a c^{3} x^{2} - 15 \, c^{3} x + \frac {16 \, c^{3} \log \left (a x + 1\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a^3*c^3*x^4 - 5/3*a^2*c^3*x^3 + 11/2*a*c^3*x^2 - 15*c^3*x + 16*c^3*log(a*x + 1)/a

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Fricas [A]
time = 0.34, size = 57, normalized size = 0.78 \begin {gather*} \frac {3 \, a^{4} c^{3} x^{4} - 20 \, a^{3} c^{3} x^{3} + 66 \, a^{2} c^{3} x^{2} - 180 \, a c^{3} x + 192 \, c^{3} \log \left (a x + 1\right )}{12 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*a^4*c^3*x^4 - 20*a^3*c^3*x^3 + 66*a^2*c^3*x^2 - 180*a*c^3*x + 192*c^3*log(a*x + 1))/a

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Sympy [A]
time = 0.07, size = 56, normalized size = 0.77 \begin {gather*} \frac {a^{3} c^{3} x^{4}}{4} - \frac {5 a^{2} c^{3} x^{3}}{3} + \frac {11 a c^{3} x^{2}}{2} - 15 c^{3} x + \frac {16 c^{3} \log {\left (a x + 1 \right )}}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*c**3*x**4/4 - 5*a**2*c**3*x**3/3 + 11*a*c**3*x**2/2 - 15*c**3*x + 16*c**3*log(a*x + 1)/a

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Giac [A]
time = 0.40, size = 82, normalized size = 1.12 \begin {gather*} \frac {{\left (3 \, c^{3} - \frac {32 \, c^{3}}{a x + 1} + \frac {144 \, c^{3}}{{\left (a x + 1\right )}^{2}} - \frac {384 \, c^{3}}{{\left (a x + 1\right )}^{3}}\right )} {\left (a x + 1\right )}^{4}}{12 \, a} - \frac {16 \, c^{3} \log \left (\frac {{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2} {\left | a \right |}}\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*c^3 - 32*c^3/(a*x + 1) + 144*c^3/(a*x + 1)^2 - 384*c^3/(a*x + 1)^3)*(a*x + 1)^4/a - 16*c^3*log(abs(a*x
+ 1)/((a*x + 1)^2*abs(a)))/a

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Mupad [B]
time = 0.79, size = 52, normalized size = 0.71 \begin {gather*} \frac {11\,a\,c^3\,x^2}{2}-15\,c^3\,x-\frac {5\,a^2\,c^3\,x^3}{3}+\frac {a^3\,c^3\,x^4}{4}+\frac {16\,c^3\,\ln \left (a\,x+1\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(11*a*c^3*x^2)/2 - 15*c^3*x - (5*a^2*c^3*x^3)/3 + (a^3*c^3*x^4)/4 + (16*c^3*log(a*x + 1))/a

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