3.11.36 \(\int \frac {e^{2 \tanh ^{-1}(a x)} (c-a^2 c x^2)^2}{x^4} \, dx\) [1036]

Optimal. Leaf size=39 \[ -\frac {c^2}{3 x^3}-\frac {a c^2}{x^2}-a^4 c^2 x-2 a^3 c^2 \log (x) \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 39, 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, 76} \begin {gather*} a^4 \left (-c^2\right ) x-2 a^3 c^2 \log (x)-\frac {a c^2}{x^2}-\frac {c^2}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] &&  !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 0])

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

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Mathematica [A]
time = 0.01, size = 39, normalized size = 1.00 \begin {gather*} -\frac {c^2}{3 x^3}-\frac {a c^2}{x^2}-a^4 c^2 x-2 a^3 c^2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.06, size = 30, normalized size = 0.77

method	result	size
default	\(c^{2} \left (-a^{4} x -\frac {1}{3 x^{3}}-2 a^{3} \ln \left (x \right )-\frac {a}{x^{2}}\right )\)	\(30\)
risch	\(-a^{4} c^{2} x +\frac {-a \,c^{2} x -\frac {1}{3} c^{2}}{x^{3}}-2 a^{3} c^{2} \ln \left (x \right )\)	\(38\)
norman	\(\frac {-\frac {1}{3} c^{2}-a \,c^{2} x -a^{4} c^{2} x^{4}}{x^{3}}-2 a^{3} c^{2} \ln \left (x \right )\)	\(40\)
meijerg	\(-\frac {a^{4} c^{2} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2}}+\frac {2 \left (-a^{2}\right )^{\frac {3}{2}} \arctanh \left (a x \right )}{a^{3}}\right )}{2 \sqrt {-a^{2}}}-a^{3} c^{2} \arctanh \left (a x \right )+\frac {a^{4} c^{2} \left (-\frac {2}{x \sqrt {-a^{2}}}+\frac {2 a \arctanh \left (a x \right )}{\sqrt {-a^{2}}}\right )}{2 \sqrt {-a^{2}}}-a^{3} c^{2} \ln \left (-a^{2} x^{2}+1\right )-2 a^{3} c^{2} \left (-\ln \left (-a^{2} x^{2}+1\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right )-a^{3} c^{2} \left (\ln \left (-a^{2} x^{2}+1\right )-2 \ln \left (x \right )-\ln \left (-a^{2}\right )+\frac {1}{a^{2} x^{2}}\right )+\frac {a^{4} c^{2} \left (-\frac {2 a^{2}}{x \left (-a^{2}\right )^{\frac {3}{2}}}-\frac {2}{3 x^{3} \left (-a^{2}\right )^{\frac {3}{2}}}+\frac {2 a^{3} \arctanh \left (a x \right )}{\left (-a^{2}\right )^{\frac {3}{2}}}\right )}{2 \sqrt {-a^{2}}}\)	\(250\)

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)^2/x^4,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.25, size = 36, normalized size = 0.92 \begin {gather*} -a^{4} c^{2} x - 2 \, a^{3} c^{2} \log \left (x\right ) - \frac {3 \, a c^{2} x + c^{2}}{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)^2/x^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.32, size = 40, normalized size = 1.03 \begin {gather*} -\frac {3 \, a^{4} c^{2} x^{4} + 6 \, a^{3} c^{2} x^{3} \log \left (x\right ) + 3 \, a c^{2} x + c^{2}}{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)^2/x^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.08, size = 37, normalized size = 0.95 \begin {gather*} - a^{4} c^{2} x - 2 a^{3} c^{2} \log {\left (x \right )} - \frac {3 a c^{2} x + c^{2}}{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)**2/x**4,x)

[Out]

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

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Giac [A]
time = 0.42, size = 37, normalized size = 0.95 \begin {gather*} -a^{4} c^{2} x - 2 \, a^{3} c^{2} \log \left ({\left | x \right |}\right ) - \frac {3 \, a c^{2} x + c^{2}}{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)^2/x^4,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.04, size = 32, normalized size = 0.82 \begin {gather*} -\frac {c^2\,\left (3\,a\,x+3\,a^4\,x^4+6\,a^3\,x^3\,\ln \left (x\right )+1\right )}{3\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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