∫ e−2 tanh−1(ax)(c − c _

3.7.72 \(\int e^{-2 \tanh ^{-1}(a x)} (c-\frac {c}{a^2 x^2})^2 \, dx\) [672]

Optimal. Leaf size=40 \[ -\frac {c^2}{3 a^4 x^3}+\frac {c^2}{a^3 x^2}-c^2 x+\frac {2 c^2 \log (x)}{a} \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6292, 6285, 76} \begin {gather*} -\frac {c^2}{3 a^4 x^3}+\frac {c^2}{a^3 x^2}+\frac {2 c^2 \log (x)}{a}+c^2 (-x) \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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])

Rule 6292

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_.), x_Symbol] :> Dist[d^p, Int[(u/x^(2*p))*(1

- a^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[c + a^2*d, 0] && IntegerQ[p]

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 40, normalized size = 1.00 \begin {gather*} -\frac {c^2}{3 a^4 x^3}+\frac {c^2}{a^3 x^2}-c^2 x+\frac {2 c^2 \log (x)}{a} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]
time = 0.76, size = 32, normalized size = 0.80


method	result	size

default	\(\frac {c^{2} \left (-a^{4} x -\frac {1}{3 x^{3}}+\frac {a}{x^{2}}+2 a^{3} \ln \left (x \right )\right )}{a^{4}}\)	\(32\)
risch	\(-x \,c^{2}+\frac {a \,c^{2} x -\frac {1}{3} c^{2}}{a^{4} x^{3}}+\frac {2 c^{2} \ln \left (x \right )}{a}\)	\(37\)
norman	\(\frac {-a^{3} c^{2} x^{4}+a \,c^{2} x^{2}-\frac {c^{2}}{3 a}+\frac {2 x \,c^{2}}{3}-a^{4} c^{2} x^{5}}{a^{3} x^{3} \left (a x +1\right )}+\frac {2 c^{2} \ln \left (x \right )}{a}\)	\(71\)
meijerg	\(-\frac {c^{2} \left (\frac {a x \left (3 a x +6\right )}{3 a x +3}-2 \ln \left (a x +1\right )\right )}{a}+\frac {3 c^{2} x}{a x +1}-\frac {3 c^{2} \left (\frac {3 a x}{3 a x +3}+2 \ln \left (a x +1\right )-1-2 \ln \left (x \right )-2 \ln \left (a \right )-\frac {1}{a x}\right )}{a}+\frac {c^{2} \left (\frac {5 a x}{5 a x +5}+4 \ln \left (a x +1\right )-1-4 \ln \left (x \right )-4 \ln \left (a \right )-\frac {1}{3 a^{3} x^{3}}+\frac {1}{a^{2} x^{2}}-\frac {3}{a x}\right )}{a}\)	\(155\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.25, size = 38, normalized size = 0.95 \begin {gather*} -c^{2} x + \frac {2 \, c^{2} \log \left (x\right )}{a} + \frac {3 \, a c^{2} x - c^{2}}{3 \, a^{4} x^{3}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.36, size = 43, normalized size = 1.08 \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 \, a^{4} x^{3}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [A]
time = 0.09, size = 39, normalized size = 0.98 \begin {gather*} \frac {- 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}}}{a^{4}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (38) = 76\).
time = 0.40, size = 112, normalized size = 2.80 \begin {gather*} -\frac {2 \, c^{2} \log \left (\frac {{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2} {\left | a \right |}}\right )}{a} + \frac {2 \, c^{2} \log \left ({\left | -\frac {1}{a x + 1} + 1 \right |}\right )}{a} + \frac {{\left (3 \, c^{2} - \frac {5 \, c^{2}}{a x + 1} - \frac {3 \, c^{2}}{{\left (a x + 1\right )}^{2}} + \frac {6 \, c^{2}}{{\left (a x + 1\right )}^{3}}\right )} {\left (a x + 1\right )}}{3 \, a {\left (\frac {1}{a x + 1} - 1\right )}^{3}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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Mupad [B]
time = 0.89, size = 35, normalized size = 0.88 \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\,a^4\,x^3} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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