3.192 \(\int e^{4 \coth ^{-1}(a x)} (c-a c x) \, dx\)

Optimal. Leaf size=27 \[ -\frac {1}{2} a c x^2-\frac {4 c \log (1-a x)}{a}-3 c x \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6167, 6129, 43} \[ -\frac {1}{2} a c x^2-\frac {4 c \log (1-a x)}{a}-3 c x \]

Antiderivative was successfully verified.

[In]

Int[E^(4*ArcCoth[a*x])*(c - a*c*x),x]

[Out]

-3*c*x - (a*c*x^2)/2 - (4*c*Log[1 - a*x])/a

Rule 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 6167

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

Rubi steps

\begin {align*} \int e^{4 \coth ^{-1}(a x)} (c-a c x) \, dx &=\int e^{4 \tanh ^{-1}(a x)} (c-a c x) \, dx\\ &=c \int \frac {(1+a x)^2}{1-a x} \, dx\\ &=c \int \left (-3-a x+\frac {4}{1-a x}\right ) \, dx\\ &=-3 c x-\frac {1}{2} a c x^2-\frac {4 c \log (1-a x)}{a}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 26, normalized size = 0.96 \[ c \left (-\frac {a x^2}{2}-\frac {4 \log (1-a x)}{a}-3 x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

c*(-3*x - (a*x^2)/2 - (4*Log[1 - a*x])/a)

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fricas [A]  time = 0.57, size = 28, normalized size = 1.04 \[ -\frac {a^{2} c x^{2} + 6 \, a c x + 8 \, c \log \left (a x - 1\right )}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(a^2*c*x^2 + 6*a*c*x + 8*c*log(a*x - 1))/a

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giac [A]  time = 0.14, size = 50, normalized size = 1.85 \[ -\frac {{\left (a x - 1\right )}^{2} {\left (c + \frac {8 \, c}{a x - 1}\right )}}{2 \, a} + \frac {4 \, c \log \left (\frac {{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2} {\left | a \right |}}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.03, size = 25, normalized size = 0.93 \[ -\frac {a c \,x^{2}}{2}-3 c x -\frac {4 c \ln \left (a x -1\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a*c*x^2-3*c*x-4*c/a*ln(a*x-1)

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maxima [A]  time = 0.31, size = 24, normalized size = 0.89 \[ -\frac {1}{2} \, a c x^{2} - 3 \, c x - \frac {4 \, c \log \left (a x - 1\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.18, size = 26, normalized size = 0.96 \[ -\frac {c\,\left (8\,\ln \left (a\,x-1\right )+6\,a\,x+a^2\,x^2\right )}{2\,a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(c*(8*log(a*x - 1) + 6*a*x + a^2*x^2))/(2*a)

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sympy [A]  time = 0.11, size = 26, normalized size = 0.96 \[ - \frac {a c x^{2}}{2} - 3 c x - \frac {4 c \log {\left (a x - 1 \right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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