3.11.47 \(\int \frac {e^{2 \tanh ^{-1}(a x)} (c-a^2 c x^2)^3}{x^4} \, dx\) [1047]

Optimal. Leaf size=72 \[ -\frac {c^3}{3 x^3}-\frac {a c^3}{x^2}+\frac {a^2 c^3}{x}-a^4 c^3 x+a^5 c^3 x^2+\frac {1}{3} a^6 c^3 x^3-4 a^3 c^3 \log (x) \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6285, 90} \begin {gather*} \frac {1}{3} a^6 c^3 x^3+a^5 c^3 x^2-a^4 c^3 x-4 a^3 c^3 \log (x)+\frac {a^2 c^3}{x}-\frac {a c^3}{x^2}-\frac {c^3}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^3)/x^4,x]

[Out]

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

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 6285

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[x^m*(1 -
a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p
] || GtQ[c, 0])

Rubi steps

\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^3}{x^4} \, dx &=c^3 \int \frac {(1-a x)^2 (1+a x)^4}{x^4} \, dx\\ &=c^3 \int \left (-a^4+\frac {1}{x^4}+\frac {2 a}{x^3}-\frac {a^2}{x^2}-\frac {4 a^3}{x}+2 a^5 x+a^6 x^2\right ) \, dx\\ &=-\frac {c^3}{3 x^3}-\frac {a c^3}{x^2}+\frac {a^2 c^3}{x}-a^4 c^3 x+a^5 c^3 x^2+\frac {1}{3} a^6 c^3 x^3-4 a^3 c^3 \log (x)\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 55, normalized size = 0.76 \begin {gather*} c^3 \left (-\frac {1}{3 x^3}-\frac {a}{x^2}+\frac {a^2}{x}-a^4 x+a^5 x^2+\frac {a^6 x^3}{3}-4 a^3 \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^3)/x^4,x]

[Out]

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

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Maple [A]
time = 0.07, size = 52, normalized size = 0.72

method	result	size
default	\(c^{3} \left (\frac {a^{6} x^{3}}{3}+a^{5} x^{2}-a^{4} x +\frac {a^{2}}{x}-\frac {1}{3 x^{3}}-4 a^{3} \ln \left (x \right )-\frac {a}{x^{2}}\right )\)	\(52\)
risch	\(\frac {a^{6} c^{3} x^{3}}{3}+a^{5} c^{3} x^{2}-a^{4} c^{3} x +\frac {a^{2} c^{3} x^{2}-a \,c^{3} x -\frac {1}{3} c^{3}}{x^{3}}-4 a^{3} c^{3} \ln \left (x \right )\)	\(69\)
norman	\(\frac {a^{2} c^{3} x^{2}+a^{5} c^{3} x^{5}-\frac {1}{3} c^{3}-a \,c^{3} x -a^{4} c^{3} x^{4}+\frac {1}{3} a^{6} c^{3} x^{6}}{x^{3}}-4 a^{3} c^{3} \ln \left (x \right )\)	\(71\)
meijerg	\(-\frac {a^{4} c^{3} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {5}{2}} \left (5 a^{2} x^{2}+15\right )}{15 a^{4}}+\frac {2 \left (-a^{2}\right )^{\frac {5}{2}} \arctanh \left (a x \right )}{a^{5}}\right )}{2 \sqrt {-a^{2}}}-\frac {a^{4} c^{3} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2}}+\frac {2 \left (-a^{2}\right )^{\frac {3}{2}} \arctanh \left (a x \right )}{a^{3}}\right )}{\sqrt {-a^{2}}}+\frac {a^{4} c^{3} \left (-\frac {2}{x \sqrt {-a^{2}}}+\frac {2 a \arctanh \left (a x \right )}{\sqrt {-a^{2}}}\right )}{\sqrt {-a^{2}}}-a^{3} c^{3} \left (-a^{2} x^{2}-\ln \left (-a^{2} x^{2}+1\right )\right )-3 a^{3} c^{3} \ln \left (-a^{2} x^{2}+1\right )-3 a^{3} c^{3} \left (-\ln \left (-a^{2} x^{2}+1\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right )-a^{3} c^{3} \left (\ln \left (-a^{2} x^{2}+1\right )-2 \ln \left (x \right )-\ln \left (-a^{2}\right )+\frac {1}{a^{2} x^{2}}\right )+\frac {a^{4} c^{3} \left (-\frac {2 a^{2}}{x \left (-a^{2}\right )^{\frac {3}{2}}}-\frac {2}{3 x^{3} \left (-a^{2}\right )^{\frac {3}{2}}}+\frac {2 a^{3} \arctanh \left (a x \right )}{\left (-a^{2}\right )^{\frac {3}{2}}}\right )}{2 \sqrt {-a^{2}}}\)	\(322\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^3/x^4,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.26, size = 70, normalized size = 0.97 \begin {gather*} \frac {1}{3} \, a^{6} c^{3} x^{3} + a^{5} c^{3} x^{2} - a^{4} c^{3} x - 4 \, a^{3} c^{3} \log \left (x\right ) + \frac {3 \, a^{2} c^{3} x^{2} - 3 \, a c^{3} x - c^{3}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^3/x^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.35, size = 74, normalized size = 1.03 \begin {gather*} \frac {a^{6} c^{3} x^{6} + 3 \, a^{5} c^{3} x^{5} - 3 \, a^{4} c^{3} x^{4} - 12 \, a^{3} c^{3} x^{3} \log \left (x\right ) + 3 \, a^{2} c^{3} x^{2} - 3 \, a c^{3} x - c^{3}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^3/x^4,x, algorithm="fricas")

[Out]

1/3*(a^6*c^3*x^6 + 3*a^5*c^3*x^5 - 3*a^4*c^3*x^4 - 12*a^3*c^3*x^3*log(x) + 3*a^2*c^3*x^2 - 3*a*c^3*x - c^3)/x^
3

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Sympy [A]
time = 0.13, size = 70, normalized size = 0.97 \begin {gather*} \frac {a^{6} c^{3} x^{3}}{3} + a^{5} c^{3} x^{2} - a^{4} c^{3} x - 4 a^{3} c^{3} \log {\left (x \right )} + \frac {3 a^{2} c^{3} x^{2} - 3 a c^{3} x - c^{3}}{3 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**3/x**4,x)

[Out]

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

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Giac [A]
time = 0.40, size = 71, normalized size = 0.99 \begin {gather*} \frac {1}{3} \, a^{6} c^{3} x^{3} + a^{5} c^{3} x^{2} - a^{4} c^{3} x - 4 \, a^{3} c^{3} \log \left ({\left | x \right |}\right ) + \frac {3 \, a^{2} c^{3} x^{2} - 3 \, a c^{3} x - c^{3}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^3/x^4,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.04, size = 69, normalized size = 0.96 \begin {gather*} a^5\,c^3\,x^2-a^4\,c^3\,x-\frac {-a^2\,c^3\,x^2+a\,c^3\,x+\frac {c^3}{3}}{x^3}+\frac {a^6\,c^3\,x^3}{3}-4\,a^3\,c^3\,\ln \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((c - a^2*c*x^2)^3*(a*x + 1)^2)/(x^4*(a^2*x^2 - 1)),x)

[Out]

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

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