3.1064 \(\int \frac {e^{2 \tanh ^{-1}(a x)}}{x^2 (c-a^2 c x^2)^2} \, dx\)

Optimal. Leaf size=78 \[ \frac {5 a}{4 c^2 (1-a x)}+\frac {a}{4 c^2 (1-a x)^2}+\frac {2 a \log (x)}{c^2}-\frac {17 a \log (1-a x)}{8 c^2}+\frac {a \log (a x+1)}{8 c^2}-\frac {1}{c^2 x} \]

[Out]

-1/c^2/x+1/4*a/c^2/(-a*x+1)^2+5/4*a/c^2/(-a*x+1)+2*a*ln(x)/c^2-17/8*a*ln(-a*x+1)/c^2+1/8*a*ln(a*x+1)/c^2

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Rubi [A]  time = 0.11, antiderivative size = 78, 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 = {6150, 88} \[ \frac {5 a}{4 c^2 (1-a x)}+\frac {a}{4 c^2 (1-a x)^2}+\frac {2 a \log (x)}{c^2}-\frac {17 a \log (1-a x)}{8 c^2}+\frac {a \log (a x+1)}{8 c^2}-\frac {1}{c^2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(1/(c^2*x)) + a/(4*c^2*(1 - a*x)^2) + (5*a)/(4*c^2*(1 - a*x)) + (2*a*Log[x])/c^2 - (17*a*Log[1 - a*x])/(8*c^2
) + (a*Log[1 + a*x])/(8*c^2)

Rule 88

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 6150

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)}}{x^2 \left (c-a^2 c x^2\right )^2} \, dx &=\frac {\int \frac {1}{x^2 (1-a x)^3 (1+a x)} \, dx}{c^2}\\ &=\frac {\int \left (\frac {1}{x^2}+\frac {2 a}{x}-\frac {a^2}{2 (-1+a x)^3}+\frac {5 a^2}{4 (-1+a x)^2}-\frac {17 a^2}{8 (-1+a x)}+\frac {a^2}{8 (1+a x)}\right ) \, dx}{c^2}\\ &=-\frac {1}{c^2 x}+\frac {a}{4 c^2 (1-a x)^2}+\frac {5 a}{4 c^2 (1-a x)}+\frac {2 a \log (x)}{c^2}-\frac {17 a \log (1-a x)}{8 c^2}+\frac {a \log (1+a x)}{8 c^2}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 57, normalized size = 0.73 \[ \frac {\frac {10 a}{1-a x}+\frac {2 a}{(a x-1)^2}+16 a \log (x)-17 a \log (1-a x)+a \log (a x+1)-\frac {8}{x}}{8 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8/x + (10*a)/(1 - a*x) + (2*a)/(-1 + a*x)^2 + 16*a*Log[x] - 17*a*Log[1 - a*x] + a*Log[1 + a*x])/(8*c^2)

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fricas [A]  time = 0.65, size = 120, normalized size = 1.54 \[ -\frac {18 \, a^{2} x^{2} - 28 \, a x - {\left (a^{3} x^{3} - 2 \, a^{2} x^{2} + a x\right )} \log \left (a x + 1\right ) + 17 \, {\left (a^{3} x^{3} - 2 \, a^{2} x^{2} + a x\right )} \log \left (a x - 1\right ) - 16 \, {\left (a^{3} x^{3} - 2 \, a^{2} x^{2} + a x\right )} \log \relax (x) + 8}{8 \, {\left (a^{2} c^{2} x^{3} - 2 \, a c^{2} x^{2} + c^{2} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(18*a^2*x^2 - 28*a*x - (a^3*x^3 - 2*a^2*x^2 + a*x)*log(a*x + 1) + 17*(a^3*x^3 - 2*a^2*x^2 + a*x)*log(a*x
- 1) - 16*(a^3*x^3 - 2*a^2*x^2 + a*x)*log(x) + 8)/(a^2*c^2*x^3 - 2*a*c^2*x^2 + c^2*x)

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giac [A]  time = 0.20, size = 65, normalized size = 0.83 \[ \frac {a \log \left ({\left | a x + 1 \right |}\right )}{8 \, c^{2}} - \frac {17 \, a \log \left ({\left | a x - 1 \right |}\right )}{8 \, c^{2}} + \frac {2 \, a \log \left ({\left | x \right |}\right )}{c^{2}} - \frac {9 \, a^{2} x^{2} - 14 \, a x + 4}{4 \, {\left (a x - 1\right )}^{2} c^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*a*log(abs(a*x + 1))/c^2 - 17/8*a*log(abs(a*x - 1))/c^2 + 2*a*log(abs(x))/c^2 - 1/4*(9*a^2*x^2 - 14*a*x + 4
)/((a*x - 1)^2*c^2*x)

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maple [A]  time = 0.04, size = 68, normalized size = 0.87 \[ -\frac {1}{c^{2} x}+\frac {2 a \ln \relax (x )}{c^{2}}+\frac {a}{4 c^{2} \left (a x -1\right )^{2}}-\frac {5 a}{4 c^{2} \left (a x -1\right )}-\frac {17 a \ln \left (a x -1\right )}{8 c^{2}}+\frac {a \ln \left (a x +1\right )}{8 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/c^2/x+2*a*ln(x)/c^2+1/4/c^2*a/(a*x-1)^2-5/4/c^2*a/(a*x-1)-17/8/c^2*a*ln(a*x-1)+1/8*a*ln(a*x+1)/c^2

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maxima [A]  time = 0.31, size = 76, normalized size = 0.97 \[ -\frac {9 \, a^{2} x^{2} - 14 \, a x + 4}{4 \, {\left (a^{2} c^{2} x^{3} - 2 \, a c^{2} x^{2} + c^{2} x\right )}} + \frac {a \log \left (a x + 1\right )}{8 \, c^{2}} - \frac {17 \, a \log \left (a x - 1\right )}{8 \, c^{2}} + \frac {2 \, a \log \relax (x)}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(9*a^2*x^2 - 14*a*x + 4)/(a^2*c^2*x^3 - 2*a*c^2*x^2 + c^2*x) + 1/8*a*log(a*x + 1)/c^2 - 17/8*a*log(a*x -
1)/c^2 + 2*a*log(x)/c^2

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mupad [B]  time = 0.11, size = 76, normalized size = 0.97 \[ \frac {2\,a\,\ln \relax (x)}{c^2}-\frac {\frac {9\,a^2\,x^2}{4}-\frac {7\,a\,x}{2}+1}{a^2\,c^2\,x^3-2\,a\,c^2\,x^2+c^2\,x}-\frac {17\,a\,\ln \left (a\,x-1\right )}{8\,c^2}+\frac {a\,\ln \left (a\,x+1\right )}{8\,c^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*a*log(x))/c^2 - ((9*a^2*x^2)/4 - (7*a*x)/2 + 1)/(c^2*x - 2*a*c^2*x^2 + a^2*c^2*x^3) - (17*a*log(a*x - 1))/(
8*c^2) + (a*log(a*x + 1))/(8*c^2)

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sympy [A]  time = 0.58, size = 76, normalized size = 0.97 \[ - \frac {9 a^{2} x^{2} - 14 a x + 4}{4 a^{2} c^{2} x^{3} - 8 a c^{2} x^{2} + 4 c^{2} x} - \frac {- 2 a \log {\relax (x )} + \frac {17 a \log {\left (x - \frac {1}{a} \right )}}{8} - \frac {a \log {\left (x + \frac {1}{a} \right )}}{8}}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(9*a**2*x**2 - 14*a*x + 4)/(4*a**2*c**2*x**3 - 8*a*c**2*x**2 + 4*c**2*x) - (-2*a*log(x) + 17*a*log(x - 1/a)/8
- a*log(x + 1/a)/8)/c**2

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