3.5.79 \(\int e^{4 \tanh ^{-1}(a x)} (c-\frac {c}{a x}) \, dx\) [479]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.04, 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 = {6266, 6264, 84} \begin {gather*} -\frac {c \log (x)}{a}+\frac {4 c \log (1-a x)}{a}+c x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(4*ArcTanh[a*x])*(c - c/(a*x)),x]

[Out]

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

Rule 84

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 6264

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 6266

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^{4 \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx &=-\frac {c \int \frac {e^{4 \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*}

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 25, normalized size = 1.00 \begin {gather*} c x-\frac {c \log (x)}{a}+\frac {4 c \log (1-a x)}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(4*ArcTanh[a*x])*(c - c/(a*x)),x]

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.85, size = 22, normalized size = 0.88

method	result	size
default	\(\frac {c \left (a x -\ln \left (x \right )+4 \ln \left (a x -1\right )\right )}{a}\)	\(22\)
risch	\(c x -\frac {c \ln \left (x \right )}{a}+\frac {4 c \ln \left (-a x +1\right )}{a}\)	\(26\)
norman	\(\frac {c \,x^{3} a^{2}-c x}{a^{2} x^{2}-1}-\frac {c \ln \left (x \right )}{a}+\frac {4 c \ln \left (a x -1\right )}{a}\)	\(47\)
meijerg	\(\frac {c \left (\frac {x \left (-a^{2}\right )^{\frac {5}{2}} \left (-10 a^{2} x^{2}+15\right )}{5 a^{4} \left (-a^{2} x^{2}+1\right )}-\frac {3 \left (-a^{2}\right )^{\frac {5}{2}} \arctanh \left (a x \right )}{a^{5}}\right )}{2 \sqrt {-a^{2}}}+\frac {3 c \left (\frac {a^{2} x^{2}}{-a^{2} x^{2}+1}+\ln \left (-a^{2} x^{2}+1\right )\right )}{2 a}-\frac {c \left (\frac {x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2} \left (-a^{2} x^{2}+1\right )}-\frac {\left (-a^{2}\right )^{\frac {3}{2}} \arctanh \left (a x \right )}{a^{3}}\right )}{\sqrt {-a^{2}}}-\frac {a c \,x^{2}}{-a^{2} x^{2}+1}-\frac {3 c \left (\frac {2 x \sqrt {-a^{2}}}{-2 a^{2} x^{2}+2}+\frac {\sqrt {-a^{2}}\, \arctanh \left (a x \right )}{a}\right )}{2 \sqrt {-a^{2}}}-\frac {c \left (\frac {2 a^{2} x^{2}}{-2 a^{2} x^{2}+2}-\ln \left (-a^{2} x^{2}+1\right )+1+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right )}{2 a}\)	\(270\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^4/(-a^2*x^2+1)^2*(c-c/a/x),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.26, size = 24, normalized size = 0.96 \begin {gather*} c x + \frac {4 \, c \log \left (a x - 1\right )}{a} - \frac {c \log \left (x\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^4/(-a^2*x^2+1)^2*(c-c/a/x),x, algorithm="maxima")

[Out]

c*x + 4*c*log(a*x - 1)/a - c*log(x)/a

________________________________________________________________________________________

Fricas [A]
time = 0.34, size = 23, normalized size = 0.92 \begin {gather*} \frac {a c x + 4 \, c \log \left (a x - 1\right ) - c \log \left (x\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^4/(-a^2*x^2+1)^2*(c-c/a/x),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]
time = 0.13, size = 17, normalized size = 0.68 \begin {gather*} c x + \frac {c \left (- \log {\left (x \right )} + 4 \log {\left (x - \frac {1}{a} \right )}\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*x + c*(-log(x) + 4*log(x - 1/a))/a

________________________________________________________________________________________

Giac [A]
time = 0.40, size = 26, normalized size = 1.04 \begin {gather*} c x + \frac {4 \, c \log \left ({\left | a x - 1 \right |}\right )}{a} - \frac {c \log \left ({\left | x \right |}\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^4/(-a^2*x^2+1)^2*(c-c/a/x),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 0.08, size = 24, normalized size = 0.96 \begin {gather*} c\,x-\frac {c\,\ln \left (x\right )}{a}+\frac {4\,c\,\ln \left (a\,x-1\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________