3.2.68 \(\int e^{2 \tanh ^{-1}(a x)} (c-a c x)^5 \, dx\) [168]

Optimal. Leaf size=37 \[ -\frac {2 c^5 (1-a x)^5}{5 a}+\frac {c^5 (1-a x)^6}{6 a} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 37, 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^5 (1-a x)^6}{6 a}-\frac {2 c^5 (1-a x)^5}{5 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcTanh[a*x])*(c - a*c*x)^5,x]

[Out]

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

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Mathematica [A]
time = 0.01, size = 23, normalized size = 0.62 \begin {gather*} \frac {c^5 (-1+a x)^5 (7+5 a x)}{30 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(2*ArcTanh[a*x])*(c - a*c*x)^5,x]

[Out]

(c^5*(-1 + a*x)^5*(7 + 5*a*x))/(30*a)

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Maple [A]
time = 1.24, size = 45, normalized size = 1.22

method	result	size
gosper	\(\frac {\left (5 a^{5} x^{5}-18 a^{4} x^{4}+15 a^{3} x^{3}+20 a^{2} x^{2}-45 a x +30\right ) x \,c^{5}}{30}\)	\(45\)
default	\(c^{5} \left (\frac {1}{6} x^{6} a^{5}-\frac {3}{5} a^{4} x^{5}+\frac {1}{2} a^{3} x^{4}+\frac {2}{3} a^{2} x^{3}-\frac {3}{2} a \,x^{2}+x \right )\)	\(45\)
norman	\(x \,c^{5}-\frac {3}{2} a \,c^{5} x^{2}+\frac {2}{3} a^{2} c^{5} x^{3}+\frac {1}{2} a^{3} c^{5} x^{4}-\frac {3}{5} c^{5} a^{4} x^{5}+\frac {1}{6} c^{5} a^{5} x^{6}\)	\(60\)
risch	\(x \,c^{5}-\frac {3}{2} a \,c^{5} x^{2}+\frac {2}{3} a^{2} c^{5} x^{3}+\frac {1}{2} a^{3} c^{5} x^{4}-\frac {3}{5} c^{5} a^{4} x^{5}+\frac {1}{6} c^{5} a^{5} x^{6}\)	\(60\)
meijerg	\(-\frac {c^{5} \left (-\frac {a^{2} x^{2} \left (4 a^{4} x^{4}+6 a^{2} x^{2}+12\right )}{12}-\ln \left (-a^{2} x^{2}+1\right )\right )}{2 a}-\frac {3 c^{5} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {7}{2}} \left (21 a^{4} x^{4}+35 a^{2} x^{2}+105\right )}{105 a^{6}}+\frac {2 \left (-a^{2}\right )^{\frac {7}{2}} \arctanh \left (a x \right )}{a^{7}}\right )}{2 \sqrt {-a^{2}}}+\frac {c^{5} \left (\frac {a^{2} x^{2} \left (3 a^{2} x^{2}+6\right )}{6}+\ln \left (-a^{2} x^{2}+1\right )\right )}{2 a}-\frac {5 c^{5} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {5}{2}} \left (5 a^{2} x^{2}+15\right )}{15 a^{4}}+\frac {2 \left (-a^{2}\right )^{\frac {5}{2}} \arctanh \left (a x \right )}{a^{5}}\right )}{2 \sqrt {-a^{2}}}+\frac {5 c^{5} \left (-a^{2} x^{2}-\ln \left (-a^{2} x^{2}+1\right )\right )}{2 a}-\frac {c^{5} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2}}+\frac {2 \left (-a^{2}\right )^{\frac {3}{2}} \arctanh \left (a x \right )}{a^{3}}\right )}{2 \sqrt {-a^{2}}}+\frac {3 c^{5} \ln \left (-a^{2} x^{2}+1\right )}{2 a}+\frac {c^{5} \arctanh \left (a x \right )}{a}\)	\(302\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^2/(-a^2*x^2+1)*(-a*c*x+c)^5,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.26, size = 59, normalized size = 1.59 \begin {gather*} \frac {1}{6} \, a^{5} c^{5} x^{6} - \frac {3}{5} \, a^{4} c^{5} x^{5} + \frac {1}{2} \, a^{3} c^{5} x^{4} + \frac {2}{3} \, a^{2} c^{5} x^{3} - \frac {3}{2} \, a c^{5} x^{2} + c^{5} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*(-a*c*x+c)^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.35, size = 59, normalized size = 1.59 \begin {gather*} \frac {1}{6} \, a^{5} c^{5} x^{6} - \frac {3}{5} \, a^{4} c^{5} x^{5} + \frac {1}{2} \, a^{3} c^{5} x^{4} + \frac {2}{3} \, a^{2} c^{5} x^{3} - \frac {3}{2} \, a c^{5} x^{2} + c^{5} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*(-a*c*x+c)^5,x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (27) = 54\).
time = 0.03, size = 66, normalized size = 1.78 \begin {gather*} \frac {a^{5} c^{5} x^{6}}{6} - \frac {3 a^{4} c^{5} x^{5}}{5} + \frac {a^{3} c^{5} x^{4}}{2} + \frac {2 a^{2} c^{5} x^{3}}{3} - \frac {3 a c^{5} x^{2}}{2} + c^{5} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**2/(-a**2*x**2+1)*(-a*c*x+c)**5,x)

[Out]

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

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Giac [A]
time = 0.42, size = 59, normalized size = 1.59 \begin {gather*} \frac {1}{6} \, a^{5} c^{5} x^{6} - \frac {3}{5} \, a^{4} c^{5} x^{5} + \frac {1}{2} \, a^{3} c^{5} x^{4} + \frac {2}{3} \, a^{2} c^{5} x^{3} - \frac {3}{2} \, a c^{5} x^{2} + c^{5} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*(-a*c*x+c)^5,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.04, size = 59, normalized size = 1.59 \begin {gather*} \frac {a^5\,c^5\,x^6}{6}-\frac {3\,a^4\,c^5\,x^5}{5}+\frac {a^3\,c^5\,x^4}{2}+\frac {2\,a^2\,c^5\,x^3}{3}-\frac {3\,a\,c^5\,x^2}{2}+c^5\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((c - a*c*x)^5*(a*x + 1)^2)/(a^2*x^2 - 1),x)

[Out]

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

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