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

Optimal. Leaf size=32 \[ \frac{\left (1-a^2 x^2\right )^{3/2}}{3 a c^2 (1-a x)^3} \]

[Out]

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

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Rubi [A]  time = 0.033848, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {6127, 651} \[ \frac{\left (1-a^2 x^2\right )^{3/2}}{3 a c^2 (1-a x)^3} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]/(c - a*c*x)^2,x]

[Out]

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

Rule 6127

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

Rule 651

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^m*(a + c*x^2)^(p + 1))
/(2*c*d*(p + 1)), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p
+ 2, 0]

Rubi steps

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

Mathematica [A]  time = 0.0091477, size = 29, normalized size = 0.91 \[ \frac{(a x+1)^{3/2}}{3 a c^2 (1-a x)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.03, size = 35, normalized size = 1.1 \begin{align*} -{\frac{ \left ( ax+1 \right ) ^{2}}{ \left ( 3\,ax-3 \right ){c}^{2}a}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.60685, size = 127, normalized size = 3.97 \begin{align*} \frac{a^{2} x^{2} - 2 \, a x + \sqrt{-a^{2} x^{2} + 1}{\left (a x + 1\right )} + 1}{3 \,{\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{a x}{a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 2 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 2 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(a*x/(a**2*x**2*sqrt(-a**2*x**2 + 1) - 2*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integ
ral(1/(a**2*x**2*sqrt(-a**2*x**2 + 1) - 2*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**2

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}{\left (a c x - c\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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