3.2.63 \(\int \frac {e^{\tanh ^{-1}(a x)}}{(c-a c x)^2} \, dx\) [163]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, 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 = {6262, 665} \begin {gather*} \frac {\left (1-a^2 x^2\right )^{3/2}}{3 a c^2 (1-a x)^3} \end {gather*}

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 665

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]

Rule 6262

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]

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*}

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Mathematica [A]
time = 0.01, size = 29, normalized size = 0.91 \begin {gather*} \frac {(1+a x)^{3/2}}{3 a c^2 (1-a x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[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 [B] Leaf count of result is larger than twice the leaf count of optimal. \(130\) vs. \(2(28)=56\).
time = 0.76, size = 131, normalized size = 4.09

method	result	size
trager	\(\frac {\left (a x +1\right ) \sqrt {-a^{2} x^{2}+1}}{3 c^{2} \left (a x -1\right )^{2} a}\)	\(33\)
gosper	\(-\frac {\left (a x +1\right )^{2}}{3 \left (a x -1\right ) c^{2} a \sqrt {-a^{2} x^{2}+1}}\)	\(35\)
default	\(\frac {\frac {\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}}{a^{2}}+\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{a^{2} \left (x -\frac {1}{a}\right )}}{c^{2}}\)	\(131\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (27) = 54\).
time = 0.47, size = 73, normalized size = 2.28 \begin {gather*} \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{3 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}} + \frac {\sqrt {-a^{2} x^{2} + 1}}{3 \, {\left (a^{2} c^{2} x - a c^{2}\right )}} \end {gather*}

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]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (27) = 54\).
time = 0.34, size = 60, normalized size = 1.88 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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 [C] Result contains complex when optimal does not.
time = 0.44, size = 66, normalized size = 2.06 \begin {gather*} -\frac {i \, \mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (c\right ) + \frac {{\left (-\frac {2 \, c}{a c x - c} - 1\right )}^{\frac {3}{2}}}{\mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (c\right )}}{3 \, c^{2} {\left | a \right |}} \end {gather*}

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]

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

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Mupad [B]
time = 0.82, size = 32, normalized size = 1.00 \begin {gather*} \frac {\sqrt {1-a^2\,x^2}\,\left (a\,x+1\right )}{3\,a\,c^2\,{\left (a\,x-1\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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