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

Optimal. Leaf size=25 \[ -\frac {c \log (x)}{a}+\frac {4 c \log (a x+1)}{a}-c x \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6131, 6129, 72} \[ -\frac {c \log (x)}{a}+\frac {4 c \log (a x+1)}{a}-c x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rule 6129

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> Dist[c^p, Int[(u*(1 + (d*x)/c)
^p*(1 + a*x)^(n/2))/(1 - a*x)^(n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && (IntegerQ
[p] || GtQ[c, 0])

Rule 6131

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_.), x_Symbol] :> Dist[d^p, Int[(u*(1 + (c*x)/d)
^p*E^(n*ArcTanh[a*x]))/x^p, x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[c^2 - a^2*d^2, 0] && IntegerQ[p]

Rubi steps

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

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Mathematica [A]  time = 0.04, size = 25, normalized size = 1.00 \[ -\frac {c \log (x)}{a}+\frac {4 c \log (a x+1)}{a}-c x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.51, size = 23, normalized size = 0.92 \[ -\frac {a c x - 4 \, c \log \left (a x + 1\right ) + c \log \relax (x)}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.16, size = 56, normalized size = 2.24 \[ -\frac {{\left (a x + 1\right )} c}{a} - \frac {3 \, c \log \left (\frac {{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2} {\left | a \right |}}\right )}{a} - \frac {c \log \left ({\left | -\frac {1}{a x + 1} + 1 \right |}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.03, size = 26, normalized size = 1.04 \[ -c x -\frac {c \ln \relax (x )}{a}+\frac {4 c \ln \left (a x +1\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.31, size = 25, normalized size = 1.00 \[ -c x + \frac {4 \, c \log \left (a x + 1\right )}{a} - \frac {c \log \relax (x)}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.07, size = 25, normalized size = 1.00 \[ \frac {4\,c\,\ln \left (a\,x+1\right )}{a}-\frac {c\,\ln \relax (x)}{a}-c\,x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.22, size = 19, normalized size = 0.76 \[ - c x - \frac {c \left (\log {\relax (x )} - 4 \log {\left (x + \frac {1}{a} \right )}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c*x - c*(log(x) - 4*log(x + 1/a))/a

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