3.13.28 \(\int e^{-2 \tanh ^{-1}(a x)} (c-a^2 c x^2)^4 \, dx\) [1228]

Optimal. Leaf size=73 \[ -\frac {4 c^4 (1-a x)^6}{3 a}+\frac {12 c^4 (1-a x)^7}{7 a}-\frac {3 c^4 (1-a x)^8}{4 a}+\frac {c^4 (1-a x)^9}{9 a} \]

[Out]

-4/3*c^4*(-a*x+1)^6/a+12/7*c^4*(-a*x+1)^7/a-3/4*c^4*(-a*x+1)^8/a+1/9*c^4*(-a*x+1)^9/a

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Rubi [A]
time = 0.04, antiderivative size = 73, 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^4 (1-a x)^9}{9 a}-\frac {3 c^4 (1-a x)^8}{4 a}+\frac {12 c^4 (1-a x)^7}{7 a}-\frac {4 c^4 (1-a x)^6}{3 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*c^4*(1 - a*x)^6)/(3*a) + (12*c^4*(1 - a*x)^7)/(7*a) - (3*c^4*(1 - a*x)^8)/(4*a) + (c^4*(1 - a*x)^9)/(9*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 )^4 \, dx &=c^4 \int (1-a x)^5 (1+a x)^3 \, dx\\ &=c^4 \int \left (8 (1-a x)^5-12 (1-a x)^6+6 (1-a x)^7-(1-a x)^8\right ) \, dx\\ &=-\frac {4 c^4 (1-a x)^6}{3 a}+\frac {12 c^4 (1-a x)^7}{7 a}-\frac {3 c^4 (1-a x)^8}{4 a}+\frac {c^4 (1-a x)^9}{9 a}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 39, normalized size = 0.53 \begin {gather*} -\frac {c^4 (-1+a x)^6 \left (65+138 a x+105 a^2 x^2+28 a^3 x^3\right )}{252 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/252*(c^4*(-1 + a*x)^6*(65 + 138*a*x + 105*a^2*x^2 + 28*a^3*x^3))/a

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Maple [A]
time = 0.09, size = 61, normalized size = 0.84

method	result	size
gosper	\(-\frac {c^{4} x \left (28 a^{8} x^{8}-63 a^{7} x^{7}-72 x^{6} a^{6}+252 x^{5} a^{5}-378 a^{3} x^{3}+168 a^{2} x^{2}+252 a x -252\right )}{252}\)	\(61\)
default	\(c^{4} \left (-\frac {1}{9} a^{8} x^{9}+\frac {1}{4} a^{7} x^{8}+\frac {2}{7} a^{6} x^{7}-a^{5} x^{6}+\frac {3}{2} a^{3} x^{4}-\frac {2}{3} a^{2} x^{3}-x^{2} a +x \right )\)	\(61\)
risch	\(-\frac {1}{9} a^{8} c^{4} x^{9}+\frac {1}{4} a^{7} c^{4} x^{8}+\frac {2}{7} c^{4} a^{6} x^{7}-a^{5} c^{4} x^{6}+\frac {3}{2} a^{3} c^{4} x^{4}-\frac {2}{3} a^{2} c^{4} x^{3}-a \,c^{4} x^{2}+c^{4} x\)	\(82\)
norman	\(\frac {c^{4} x -\frac {5}{3} a^{2} c^{4} x^{3}+\frac {5}{6} a^{3} c^{4} x^{4}+\frac {3}{2} a^{4} c^{4} x^{5}-a^{5} c^{4} x^{6}+\frac {15}{28} a^{7} c^{4} x^{8}+\frac {5}{36} a^{8} c^{4} x^{9}-\frac {1}{9} a^{9} c^{4} x^{10}-\frac {5}{7} c^{4} a^{6} x^{7}}{a x +1}\)	\(103\)
meijerg	\(-\frac {c^{4} \left (\frac {x a \left (308 a^{9} x^{9}-385 a^{8} x^{8}+495 a^{7} x^{7}-660 x^{6} a^{6}+924 x^{5} a^{5}-1386 a^{4} x^{4}+2310 a^{3} x^{3}-4620 a^{2} x^{2}+13860 a x +27720\right )}{2772 a x +2772}-10 \ln \left (a x +1\right )\right )}{a}+\frac {5 c^{4} \left (\frac {a x \left (45 a^{7} x^{7}-60 x^{6} a^{6}+84 x^{5} a^{5}-126 a^{4} x^{4}+210 a^{3} x^{3}-420 a^{2} x^{2}+1260 a x +2520\right )}{315 a x +315}-8 \ln \left (a x +1\right )\right )}{a}-\frac {10 c^{4} \left (\frac {a x \left (14 x^{5} a^{5}-21 a^{4} x^{4}+35 a^{3} x^{3}-70 a^{2} x^{2}+210 a x +420\right )}{70 a x +70}-6 \ln \left (a x +1\right )\right )}{a}+\frac {10 c^{4} \left (\frac {x a \left (5 a^{3} x^{3}-10 a^{2} x^{2}+30 a x +60\right )}{15 a x +15}-4 \ln \left (a x +1\right )\right )}{a}-\frac {5 c^{4} \left (\frac {a x \left (3 a x +6\right )}{3 a x +3}-2 \ln \left (a x +1\right )\right )}{a}+\frac {c^{4} x}{a x +1}\)	\(344\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^4*(-1/9*a^8*x^9+1/4*a^7*x^8+2/7*a^6*x^7-a^5*x^6+3/2*a^3*x^4-2/3*a^2*x^3-x^2*a+x)

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Maxima [A]
time = 0.26, size = 81, normalized size = 1.11 \begin {gather*} -\frac {1}{9} \, a^{8} c^{4} x^{9} + \frac {1}{4} \, a^{7} c^{4} x^{8} + \frac {2}{7} \, a^{6} c^{4} x^{7} - a^{5} c^{4} x^{6} + \frac {3}{2} \, a^{3} c^{4} x^{4} - \frac {2}{3} \, a^{2} c^{4} x^{3} - a c^{4} x^{2} + c^{4} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*a^8*c^4*x^9 + 1/4*a^7*c^4*x^8 + 2/7*a^6*c^4*x^7 - a^5*c^4*x^6 + 3/2*a^3*c^4*x^4 - 2/3*a^2*c^4*x^3 - a*c^4
*x^2 + c^4*x

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Fricas [A]
time = 0.36, size = 81, normalized size = 1.11 \begin {gather*} -\frac {1}{9} \, a^{8} c^{4} x^{9} + \frac {1}{4} \, a^{7} c^{4} x^{8} + \frac {2}{7} \, a^{6} c^{4} x^{7} - a^{5} c^{4} x^{6} + \frac {3}{2} \, a^{3} c^{4} x^{4} - \frac {2}{3} \, a^{2} c^{4} x^{3} - a c^{4} x^{2} + c^{4} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*a^8*c^4*x^9 + 1/4*a^7*c^4*x^8 + 2/7*a^6*c^4*x^7 - a^5*c^4*x^6 + 3/2*a^3*c^4*x^4 - 2/3*a^2*c^4*x^3 - a*c^4
*x^2 + c^4*x

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Sympy [A]
time = 0.03, size = 87, normalized size = 1.19 \begin {gather*} - \frac {a^{8} c^{4} x^{9}}{9} + \frac {a^{7} c^{4} x^{8}}{4} + \frac {2 a^{6} c^{4} x^{7}}{7} - a^{5} c^{4} x^{6} + \frac {3 a^{3} c^{4} x^{4}}{2} - \frac {2 a^{2} c^{4} x^{3}}{3} - a c^{4} x^{2} + c^{4} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a**2*c*x**2+c)**4/(a*x+1)**2*(-a**2*x**2+1),x)

[Out]

-a**8*c**4*x**9/9 + a**7*c**4*x**8/4 + 2*a**6*c**4*x**7/7 - a**5*c**4*x**6 + 3*a**3*c**4*x**4/2 - 2*a**2*c**4*
x**3/3 - a*c**4*x**2 + c**4*x

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Giac [A]
time = 0.41, size = 78, normalized size = 1.07 \begin {gather*} -\frac {{\left (28 \, c^{4} - \frac {315 \, c^{4}}{a x + 1} + \frac {1440 \, c^{4}}{{\left (a x + 1\right )}^{2}} - \frac {3360 \, c^{4}}{{\left (a x + 1\right )}^{3}} + \frac {4032 \, c^{4}}{{\left (a x + 1\right )}^{4}} - \frac {2016 \, c^{4}}{{\left (a x + 1\right )}^{5}}\right )} {\left (a x + 1\right )}^{9}}{252 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/252*(28*c^4 - 315*c^4/(a*x + 1) + 1440*c^4/(a*x + 1)^2 - 3360*c^4/(a*x + 1)^3 + 4032*c^4/(a*x + 1)^4 - 2016
*c^4/(a*x + 1)^5)*(a*x + 1)^9/a

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Mupad [B]
time = 0.05, size = 81, normalized size = 1.11 \begin {gather*} -\frac {a^8\,c^4\,x^9}{9}+\frac {a^7\,c^4\,x^8}{4}+\frac {2\,a^6\,c^4\,x^7}{7}-a^5\,c^4\,x^6+\frac {3\,a^3\,c^4\,x^4}{2}-\frac {2\,a^2\,c^4\,x^3}{3}-a\,c^4\,x^2+c^4\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^4*x - a*c^4*x^2 - (2*a^2*c^4*x^3)/3 + (3*a^3*c^4*x^4)/2 - a^5*c^4*x^6 + (2*a^6*c^4*x^7)/7 + (a^7*c^4*x^8)/4
- (a^8*c^4*x^9)/9

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