3.1024 \(\int \frac {e^{2 \tanh ^{-1}(a x)} (c-a^2 c x^2)}{x} \, dx\)

Optimal. Leaf size=21 \[ \frac {1}{2} a^2 c x^2+2 a c x+c \log (x) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {6150, 43} \[ \frac {1}{2} a^2 c x^2+2 a c x+c \log (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*a*c*x + (a^2*c*x^2)/2 + c*Log[x]

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 6150

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 19, normalized size = 0.90 \[ c \left (\frac {a^2 x^2}{2}+2 a x+\log (x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

c*(2*a*x + (a^2*x^2)/2 + Log[x])

________________________________________________________________________________________

fricas [A]  time = 1.02, size = 19, normalized size = 0.90 \[ \frac {1}{2} \, a^{2} c x^{2} + 2 \, a c x + c \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^2*c*x^2 + 2*a*c*x + c*log(x)

________________________________________________________________________________________

giac [A]  time = 0.29, size = 20, normalized size = 0.95 \[ \frac {1}{2} \, a^{2} c x^{2} + 2 \, a c x + c \log \left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.03, size = 20, normalized size = 0.95 \[ 2 a c x +\frac {a^{2} c \,x^{2}}{2}+c \ln \relax (x ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.32, size = 19, normalized size = 0.90 \[ \frac {1}{2} \, a^{2} c x^{2} + 2 \, a c x + c \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^2*c*x^2 + 2*a*c*x + c*log(x)

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 19, normalized size = 0.90 \[ \frac {c\,\left (2\,\ln \relax (x)+4\,a\,x+a^2\,x^2\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c*(2*log(x) + 4*a*x + a^2*x^2))/2

________________________________________________________________________________________

sympy [A]  time = 0.10, size = 20, normalized size = 0.95 \[ \frac {a^{2} c x^{2}}{2} + 2 a c x + c \log {\relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*c*x**2/2 + 2*a*c*x + c*log(x)

________________________________________________________________________________________