3.11.64 \(\int \frac {e^{2 \tanh ^{-1}(a x)}}{x^2 (c-a^2 c x^2)^2} \, dx\) [1064]

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

[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 = 0.08, 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 = {6285, 90} \begin {gather*} \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} \end {gather*}

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 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)}}{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 = 0.06, size = 57, normalized size = 0.73 \begin {gather*} \frac {-\frac {8}{x}+\frac {10 a}{1-a x}+\frac {2 a}{(-1+a x)^2}+16 a \log (x)-17 a \log (1-a x)+a \log (1+a x)}{8 c^2} \end {gather*}

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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Maple [A]
time = 0.06, size = 54, normalized size = 0.69

method	result	size
default	\(\frac {\frac {a \ln \left (a x +1\right )}{8}-\frac {1}{x}+2 a \ln \left (x \right )+\frac {a}{4 \left (a x -1\right )^{2}}-\frac {5 a}{4 \left (a x -1\right )}-\frac {17 a \ln \left (a x -1\right )}{8}}{c^{2}}\)	\(54\)
risch	\(\frac {-\frac {9}{4} a^{2} x^{2}+\frac {7}{2} a x -1}{c^{2} x \left (a x -1\right )^{2}}+\frac {a \ln \left (a x +1\right )}{8 c^{2}}+\frac {2 a \ln \left (-x \right )}{c^{2}}-\frac {17 a \ln \left (-a x +1\right )}{8 c^{2}}\)	\(65\)
norman	\(\frac {-\frac {1}{c}+\frac {15 a^{2} x^{2}}{4 c}-\frac {9 a^{4} x^{4}}{4 c}+\frac {2 a^{3} x^{3}}{c}-\frac {3 a^{5} x^{5}}{2 c}}{\left (a^{2} x^{2}-1\right )^{2} x c}+\frac {2 a \ln \left (x \right )}{c^{2}}-\frac {17 a \ln \left (a x -1\right )}{8 c^{2}}+\frac {a \ln \left (a x +1\right )}{8 c^{2}}\)	\(102\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.26, size = 76, normalized size = 0.97 \begin {gather*} -\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 \left (x\right )}{c^{2}} \end {gather*}

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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Fricas [A]
time = 0.33, size = 120, normalized size = 1.54 \begin {gather*} -\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 \left (x\right ) + 8}{8 \, {\left (a^{2} c^{2} x^{3} - 2 \, a c^{2} x^{2} + c^{2} x\right )}} \end {gather*}

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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Sympy [A]
time = 0.32, size = 76, normalized size = 0.97 \begin {gather*} - \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 {\left (x \right )} + \frac {17 a \log {\left (x - \frac {1}{a} \right )}}{8} - \frac {a \log {\left (x + \frac {1}{a} \right )}}{8}}{c^{2}} \end {gather*}

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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Giac [A]
time = 0.41, size = 65, normalized size = 0.83 \begin {gather*} \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} \end {gather*}

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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Mupad [B]
time = 0.11, size = 76, normalized size = 0.97 \begin {gather*} \frac {2\,a\,\ln \left (x\right )}{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} \end {gather*}

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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