3.13.33 \(\int \frac {e^{-2 \tanh ^{-1}(a x)}}{(c-a^2 c x^2)^2} \, dx\) [1233]

Optimal. Leaf size=49 \[ -\frac {1}{4 a c^2 (1+a x)^2}-\frac {1}{4 a c^2 (1+a x)}+\frac {\tanh ^{-1}(a x)}{4 a c^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.04, antiderivative size = 49, 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 = {6275, 46, 213} \begin {gather*} -\frac {1}{4 a c^2 (a x+1)}-\frac {1}{4 a c^2 (a x+1)^2}+\frac {\tanh ^{-1}(a x)}{4 a c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^2),x]

[Out]

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

Rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 6275

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[(1 - a*x)^(p - n/2)*
(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, 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} \, dx &=\frac {\int \frac {1}{(1-a x) (1+a x)^3} \, dx}{c^2}\\ &=\frac {\int \left (\frac {1}{2 (1+a x)^3}+\frac {1}{4 (1+a x)^2}-\frac {1}{4 \left (-1+a^2 x^2\right )}\right ) \, dx}{c^2}\\ &=-\frac {1}{4 a c^2 (1+a x)^2}-\frac {1}{4 a c^2 (1+a x)}-\frac {\int \frac {1}{-1+a^2 x^2} \, dx}{4 c^2}\\ &=-\frac {1}{4 a c^2 (1+a x)^2}-\frac {1}{4 a c^2 (1+a x)}+\frac {\tanh ^{-1}(a x)}{4 a c^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 33, normalized size = 0.67 \begin {gather*} \frac {-2-a x+(1+a x)^2 \tanh ^{-1}(a x)}{4 a (c+a c x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^2),x]

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.07, size = 52, normalized size = 1.06

method	result	size
risch	\(\frac {-\frac {x}{4}-\frac {1}{2 a}}{\left (a x +1\right )^{2} c^{2}}-\frac {\ln \left (a x -1\right )}{8 a \,c^{2}}+\frac {\ln \left (-a x -1\right )}{8 a \,c^{2}}\)	\(51\)
default	\(\frac {-\frac {1}{4 a \left (a x +1\right )^{2}}-\frac {1}{4 a \left (a x +1\right )}+\frac {\ln \left (a x +1\right )}{8 a}-\frac {\ln \left (a x -1\right )}{8 a}}{c^{2}}\)	\(52\)
norman	\(\frac {-\frac {a \,x^{2}}{2 c}-\frac {3 x}{4 c}+\frac {3 a^{2} x^{3}}{4 c}+\frac {a^{3} x^{4}}{2 c}}{\left (a x -1\right ) \left (a x +1\right )^{3} c}-\frac {\ln \left (a x -1\right )}{8 a \,c^{2}}+\frac {\ln \left (a x +1\right )}{8 a \,c^{2}}\)	\(86\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 63, normalized size = 1.29 \begin {gather*} -\frac {a x + 2}{4 \, {\left (a^{3} c^{2} x^{2} + 2 \, a^{2} c^{2} x + a c^{2}\right )}} + \frac {\log \left (a x + 1\right )}{8 \, a c^{2}} - \frac {\log \left (a x - 1\right )}{8 \, a c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.33, size = 76, normalized size = 1.55 \begin {gather*} -\frac {2 \, a x - {\left (a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (a x + 1\right ) + {\left (a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (a x - 1\right ) + 4}{8 \, {\left (a^{3} c^{2} x^{2} + 2 \, a^{2} c^{2} x + a c^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]
time = 0.16, size = 56, normalized size = 1.14 \begin {gather*} - \frac {a x + 2}{4 a^{3} c^{2} x^{2} + 8 a^{2} c^{2} x + 4 a c^{2}} - \frac {\frac {\log {\left (x - \frac {1}{a} \right )}}{8} - \frac {\log {\left (x + \frac {1}{a} \right )}}{8}}{a c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 0.42, size = 55, normalized size = 1.12 \begin {gather*} -\frac {\log \left ({\left | -\frac {2}{a x + 1} + 1 \right |}\right )}{8 \, a c^{2}} - \frac {\frac {a c^{2}}{a x + 1} + \frac {a c^{2}}{{\left (a x + 1\right )}^{2}}}{4 \, a^{2} c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 0.07, size = 47, normalized size = 0.96 \begin {gather*} \frac {\mathrm {atanh}\left (a\,x\right )}{4\,a\,c^2}-\frac {\frac {x}{4}+\frac {1}{2\,a}}{a^2\,c^2\,x^2+2\,a\,c^2\,x+c^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________