3.3.7 \(\int e^{-2 \tanh ^{-1}(a x)} (c-a c x)^4 \, dx\) [207]

Optimal. Leaf size=91 \[ -16 c^4 x+\frac {4 c^4 (1-a x)^2}{a}+\frac {4 c^4 (1-a x)^3}{3 a}+\frac {c^4 (1-a x)^4}{2 a}+\frac {c^4 (1-a x)^5}{5 a}+\frac {32 c^4 \log (1+a x)}{a} \]

[Out]

-16*c^4*x+4*c^4*(-a*x+1)^2/a+4/3*c^4*(-a*x+1)^3/a+1/2*c^4*(-a*x+1)^4/a+1/5*c^4*(-a*x+1)^5/a+32*c^4*ln(a*x+1)/a

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Rubi [A]
time = 0.03, antiderivative size = 91, 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^4 (1-a x)^5}{5 a}+\frac {c^4 (1-a x)^4}{2 a}+\frac {4 c^4 (1-a x)^3}{3 a}+\frac {4 c^4 (1-a x)^2}{a}+\frac {32 c^4 \log (a x+1)}{a}-16 c^4 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c - a*c*x)^4/E^(2*ArcTanh[a*x]),x]

[Out]

-16*c^4*x + (4*c^4*(1 - a*x)^2)/a + (4*c^4*(1 - a*x)^3)/(3*a) + (c^4*(1 - a*x)^4)/(2*a) + (c^4*(1 - a*x)^5)/(5
*a) + (32*c^4*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)^4 \, dx &=c^4 \int \frac {(1-a x)^5}{1+a x} \, dx\\ &=c^4 \int \left (-16-8 (1-a x)-4 (1-a x)^2-2 (1-a x)^3-(1-a x)^4+\frac {32}{1+a x}\right ) \, dx\\ &=-16 c^4 x+\frac {4 c^4 (1-a x)^2}{a}+\frac {4 c^4 (1-a x)^3}{3 a}+\frac {c^4 (1-a x)^4}{2 a}+\frac {c^4 (1-a x)^5}{5 a}+\frac {32 c^4 \log (1+a x)}{a}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 56, normalized size = 0.62 \begin {gather*} -\frac {c^4 \left (-181+930 a x-390 a^2 x^2+160 a^3 x^3-45 a^4 x^4+6 a^5 x^5-960 \log (1+a x)\right )}{30 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c - a*c*x)^4/E^(2*ArcTanh[a*x]),x]

[Out]

-1/30*(c^4*(-181 + 930*a*x - 390*a^2*x^2 + 160*a^3*x^3 - 45*a^4*x^4 + 6*a^5*x^5 - 960*Log[1 + a*x]))/a

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Maple [A]
time = 0.77, size = 50, normalized size = 0.55

method	result	size
default	\(c^{4} \left (-\frac {a^{4} x^{5}}{5}+\frac {3 a^{3} x^{4}}{2}-\frac {16 a^{2} x^{3}}{3}+13 a \,x^{2}-31 x +\frac {32 \ln \left (a x +1\right )}{a}\right )\)	\(50\)
risch	\(-\frac {a^{4} c^{4} x^{5}}{5}+\frac {3 c^{4} x^{4} a^{3}}{2}-\frac {16 a^{2} c^{4} x^{3}}{3}+13 c^{4} a \,x^{2}-31 c^{4} x +\frac {32 c^{4} \ln \left (a x +1\right )}{a}\)	\(64\)
norman	\(\frac {-31 c^{4} x +\frac {23}{3} a^{2} c^{4} x^{3}+\frac {13}{10} a^{4} c^{4} x^{5}-18 c^{4} a \,x^{2}-\frac {1}{5} c^{4} a^{5} x^{6}-\frac {23}{6} c^{4} x^{4} a^{3}}{a x +1}+\frac {32 c^{4} \ln \left (a x +1\right )}{a}\)	\(84\)
meijerg	\(-\frac {c^{4} \left (\frac {a x \left (14 a^{5} x^{5}-21 a^{4} x^{4}+35 a^{3} x^{3}-70 a^{2} x^{2}+210 a x +420\right )}{70 a x +70}-6 \ln \left (a x +1\right )\right )}{a}-\frac {5 c^{4} \left (\frac {a x \left (5 a^{3} x^{3}-10 a^{2} x^{2}+30 a x +60\right )}{15 a x +15}-4 \ln \left (a x +1\right )\right )}{a}+\frac {4 c^{4} \left (-\frac {a x \left (-3 a^{4} x^{4}+5 a^{3} x^{3}-10 a^{2} x^{2}+30 a x +60\right )}{12 \left (a x +1\right )}+5 \ln \left (a x +1\right )\right )}{a}+\frac {5 c^{4} \left (\frac {a x \left (3 a x +6\right )}{3 a x +3}-2 \ln \left (a x +1\right )\right )}{a}-\frac {4 c^{4} \left (-\frac {a x}{a x +1}+\ln \left (a x +1\right )\right )}{a}+\frac {c^{4} x}{a x +1}\)	\(248\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a*c*x+c)^4/(a*x+1)^2*(-a^2*x^2+1),x,method=_RETURNVERBOSE)

[Out]

c^4*(-1/5*a^4*x^5+3/2*a^3*x^4-16/3*a^2*x^3+13*a*x^2-31*x+32/a*ln(a*x+1))

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Maxima [A]
time = 0.25, size = 63, normalized size = 0.69 \begin {gather*} -\frac {1}{5} \, a^{4} c^{4} x^{5} + \frac {3}{2} \, a^{3} c^{4} x^{4} - \frac {16}{3} \, a^{2} c^{4} x^{3} + 13 \, a c^{4} x^{2} - 31 \, c^{4} x + \frac {32 \, c^{4} \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)^4/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="maxima")

[Out]

-1/5*a^4*c^4*x^5 + 3/2*a^3*c^4*x^4 - 16/3*a^2*c^4*x^3 + 13*a*c^4*x^2 - 31*c^4*x + 32*c^4*log(a*x + 1)/a

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Fricas [A]
time = 0.35, size = 68, normalized size = 0.75 \begin {gather*} -\frac {6 \, a^{5} c^{4} x^{5} - 45 \, a^{4} c^{4} x^{4} + 160 \, a^{3} c^{4} x^{3} - 390 \, a^{2} c^{4} x^{2} + 930 \, a c^{4} x - 960 \, c^{4} \log \left (a x + 1\right )}{30 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^4/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="fricas")

[Out]

-1/30*(6*a^5*c^4*x^5 - 45*a^4*c^4*x^4 + 160*a^3*c^4*x^3 - 390*a^2*c^4*x^2 + 930*a*c^4*x - 960*c^4*log(a*x + 1)
)/a

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Sympy [A]
time = 0.08, size = 68, normalized size = 0.75 \begin {gather*} - \frac {a^{4} c^{4} x^{5}}{5} + \frac {3 a^{3} c^{4} x^{4}}{2} - \frac {16 a^{2} c^{4} x^{3}}{3} + 13 a c^{4} x^{2} - 31 c^{4} x + \frac {32 c^{4} \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)**4/(a*x+1)**2*(-a**2*x**2+1),x)

[Out]

-a**4*c**4*x**5/5 + 3*a**3*c**4*x**4/2 - 16*a**2*c**4*x**3/3 + 13*a*c**4*x**2 - 31*c**4*x + 32*c**4*log(a*x +
1)/a

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Giac [A]
time = 0.42, size = 94, normalized size = 1.03 \begin {gather*} -\frac {{\left (6 \, c^{4} - \frac {75 \, c^{4}}{a x + 1} + \frac {400 \, c^{4}}{{\left (a x + 1\right )}^{2}} - \frac {1200 \, c^{4}}{{\left (a x + 1\right )}^{3}} + \frac {2400 \, c^{4}}{{\left (a x + 1\right )}^{4}}\right )} {\left (a x + 1\right )}^{5}}{30 \, a} - \frac {32 \, c^{4} \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)^4/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="giac")

[Out]

-1/30*(6*c^4 - 75*c^4/(a*x + 1) + 400*c^4/(a*x + 1)^2 - 1200*c^4/(a*x + 1)^3 + 2400*c^4/(a*x + 1)^4)*(a*x + 1)
^5/a - 32*c^4*log(abs(a*x + 1)/((a*x + 1)^2*abs(a)))/a

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Mupad [B]
time = 0.05, size = 63, normalized size = 0.69 \begin {gather*} 13\,a\,c^4\,x^2-31\,c^4\,x-\frac {16\,a^2\,c^4\,x^3}{3}+\frac {3\,a^3\,c^4\,x^4}{2}-\frac {a^4\,c^4\,x^5}{5}+\frac {32\,c^4\,\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)^4)/(a*x + 1)^2,x)

[Out]

13*a*c^4*x^2 - 31*c^4*x - (16*a^2*c^4*x^3)/3 + (3*a^3*c^4*x^4)/2 - (a^4*c^4*x^5)/5 + (32*c^4*log(a*x + 1))/a

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