3.2.83 \(\int \frac {e^{3 \tanh ^{-1}(a x)}}{(c-a c x)^2} \, dx\) [183]

Optimal. Leaf size=32 \[ \frac {\left (1-a^2 x^2\right )^{5/2}}{5 a c^2 (1-a x)^5} \]

[Out]

1/5*(-a^2*x^2+1)^(5/2)/a/c^2/(-a*x+1)^5

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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 = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6262, 665} \begin {gather*} \frac {\left (1-a^2 x^2\right )^{5/2}}{5 a c^2 (1-a x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcTanh[a*x])/(c - a*c*x)^2,x]

[Out]

(1 - a^2*x^2)^(5/2)/(5*a*c^2*(1 - a*x)^5)

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^{3 \tanh ^{-1}(a x)}}{(c-a c x)^2} \, dx &=c^3 \int \frac {\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^5} \, dx\\ &=\frac {\left (1-a^2 x^2\right )^{5/2}}{5 a c^2 (1-a x)^5}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 29, normalized size = 0.91 \begin {gather*} \frac {(1+a x)^{5/2}}{5 a c^2 (1-a x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(3*ArcTanh[a*x])/(c - a*c*x)^2,x]

[Out]

(1 + a*x)^(5/2)/(5*a*c^2*(1 - a*x)^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(271\) vs. \(2(28)=56\).
time = 1.10, size = 272, normalized size = 8.50

method	result	size
gosper	\(-\frac {\left (a x +1\right )^{4}}{5 \left (a x -1\right ) c^{2} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} a}\)	\(35\)
trager	\(-\frac {\left (a^{2} x^{2}+2 a x +1\right ) \sqrt {-a^{2} x^{2}+1}}{5 c^{2} \left (a x -1\right )^{3} a}\)	\(41\)
default	\(\frac {\frac {1}{a \sqrt {-a^{2} x^{2}+1}}+\frac {5 x}{\sqrt {-a^{2} x^{2}+1}}+\frac {\frac {8}{5 a \left (x -\frac {1}{a}\right )^{2} \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {24 a \left (\frac {1}{3 a \left (x -\frac {1}{a}\right ) \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {-2 a^{2} \left (x -\frac {1}{a}\right )-2 a}{3 a \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{5}}{a^{2}}+\frac {\frac {4}{a \left (x -\frac {1}{a}\right ) \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {4 \left (-2 a^{2} \left (x -\frac {1}{a}\right )-2 a \right )}{a \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}}{a}}{c^{2}}\)	\(272\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^2,x,method=_RETURNVERBOSE)

[Out]

1/c^2*(1/a/(-a^2*x^2+1)^(1/2)+5*x/(-a^2*x^2+1)^(1/2)+8/a^2*(1/5/a/(x-1/a)^2/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)
-3/5*a*(1/3/a/(x-1/a)/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)+1/3/a*(-2*a^2*(x-1/a)-2*a)/(-a^2*(x-1/a)^2-2*a*(x-1/a
))^(1/2)))+12/a*(1/3/a/(x-1/a)/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)+1/3/a*(-2*a^2*(x-1/a)-2*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. 146 vs. \(2 (27) = 54\).
time = 0.26, size = 146, normalized size = 4.56 \begin {gather*} \frac {8}{5 \, {\left (\sqrt {-a^{2} x^{2} + 1} a^{3} c^{2} x^{2} - 2 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{2} x + \sqrt {-a^{2} x^{2} + 1} a c^{2}\right )}} + \frac {12}{5 \, {\left (\sqrt {-a^{2} x^{2} + 1} a^{2} c^{2} x - \sqrt {-a^{2} x^{2} + 1} a c^{2}\right )}} + \frac {x}{5 \, \sqrt {-a^{2} x^{2} + 1} c^{2}} + \frac {1}{\sqrt {-a^{2} x^{2} + 1} a c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^2,x, algorithm="maxima")

[Out]

8/5/(sqrt(-a^2*x^2 + 1)*a^3*c^2*x^2 - 2*sqrt(-a^2*x^2 + 1)*a^2*c^2*x + sqrt(-a^2*x^2 + 1)*a*c^2) + 12/5/(sqrt(
-a^2*x^2 + 1)*a^2*c^2*x - sqrt(-a^2*x^2 + 1)*a*c^2) + 1/5*x/(sqrt(-a^2*x^2 + 1)*c^2) + 1/(sqrt(-a^2*x^2 + 1)*a
*c^2)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (27) = 54\).
time = 0.41, size = 89, normalized size = 2.78 \begin {gather*} \frac {a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - {\left (a^{2} x^{2} + 2 \, a x + 1\right )} \sqrt {-a^{2} x^{2} + 1} - 1}{5 \, {\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, 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)^3/(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^2,x, algorithm="fricas")

[Out]

1/5*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - (a^2*x^2 + 2*a*x + 1)*sqrt(-a^2*x^2 + 1) - 1)/(a^4*c^2*x^3 - 3*a^3*c^2*x^2
+ 3*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 {3 a x}{- a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 2 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {3 a^{2} x^{2}}{- a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 2 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a^{3} x^{3}}{- a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 2 a^{3} x^{3} \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^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 2 a^{3} x^{3} \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)**3/(-a**2*x**2+1)**(3/2)/(-a*c*x+c)**2,x)

[Out]

(Integral(3*a*x/(-a**4*x**4*sqrt(-a**2*x**2 + 1) + 2*a**3*x**3*sqrt(-a**2*x**2 + 1) - 2*a*x*sqrt(-a**2*x**2 +
1) + sqrt(-a**2*x**2 + 1)), x) + Integral(3*a**2*x**2/(-a**4*x**4*sqrt(-a**2*x**2 + 1) + 2*a**3*x**3*sqrt(-a**
2*x**2 + 1) - 2*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a**3*x**3/(-a**4*x**4*sqrt(-a*
*2*x**2 + 1) + 2*a**3*x**3*sqrt(-a**2*x**2 + 1) - 2*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Int
egral(1/(-a**4*x**4*sqrt(-a**2*x**2 + 1) + 2*a**3*x**3*sqrt(-a**2*x**2 + 1) - 2*a*x*sqrt(-a**2*x**2 + 1) + sqr
t(-a**2*x**2 + 1)), x))/c**2

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Giac [C] Result contains complex when optimal does not.
time = 0.45, size = 83, normalized size = 2.59 \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 )}^{2} \sqrt {-\frac {2 \, c}{a c x - c} - 1}}{\mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (c\right )}}{5 \, c^{2} {\left | a \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^2,x, algorithm="giac")

[Out]

-1/5*(-I*sgn(1/(a*c*x - c))*sgn(a)*sgn(c) + (2*c/(a*c*x - c) + 1)^2*sqrt(-2*c/(a*c*x - c) - 1)/(sgn(1/(a*c*x -
c))*sgn(a)*sgn(c)))/(c^2*abs(a))

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Mupad [B]
time = 0.08, size = 34, normalized size = 1.06 \begin {gather*} -\frac {\sqrt {1-a^2\,x^2}\,{\left (a\,x+1\right )}^2}{5\,a\,c^2\,{\left (a\,x-1\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((1 - a^2*x^2)^(1/2)*(a*x + 1)^2)/(5*a*c^2*(a*x - 1)^3)

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