3.182 \(\int \frac {e^{3 \tanh ^{-1}(a x)}}{c-a c x} \, dx\)

Optimal. Leaf size=74 \[ \frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a c (1-a x)^3}-\frac {2 \sqrt {1-a^2 x^2}}{a c (1-a x)}+\frac {\sin ^{-1}(a x)}{a c} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6127, 663, 216} \[ \frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a c (1-a x)^3}-\frac {2 \sqrt {1-a^2 x^2}}{a c (1-a x)}+\frac {\sin ^{-1}(a x)}{a c} \]

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcTanh[a*x])/(c - a*c*x),x]

[Out]

(-2*Sqrt[1 - a^2*x^2])/(a*c*(1 - a*x)) + (2*(1 - a^2*x^2)^(3/2))/(3*a*c*(1 - a*x)^3) + ArcSin[a*x]/(a*c)

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 663

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

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]

Rubi steps

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

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Mathematica [C]  time = 0.01, size = 45, normalized size = 0.61 \[ \frac {4 \sqrt {2} \, _2F_1\left (-\frac {3}{2},-\frac {3}{2};-\frac {1}{2};\frac {1}{2} (1-a x)\right )}{3 a c (1-a x)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(4*Sqrt[2]*Hypergeometric2F1[-3/2, -3/2, -1/2, (1 - a*x)/2])/(3*a*c*(1 - a*x)^(3/2))

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fricas [A]  time = 0.44, size = 94, normalized size = 1.27 \[ -\frac {2 \, {\left (2 \, a^{2} x^{2} - 4 \, a x + 3 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - 2 \, \sqrt {-a^{2} x^{2} + 1} {\left (2 \, a x - 1\right )} + 2\right )}}{3 \, {\left (a^{3} c x^{2} - 2 \, a^{2} c x + a c\right )}} \]

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),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.25, size = 79, normalized size = 1.07 \[ \frac {\arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{c {\left | a \right |}} + \frac {8 \, {\left (\frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{a^{2} x} - 1\right )}}{3 \, c {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{3} {\left | a \right |}} \]

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),x, algorithm="giac")

[Out]

arcsin(a*x)*sgn(a)/(c*abs(a)) + 8/3*(3*(sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 1)/(c*((sqrt(-a^2*x^2 + 1)*ab
s(a) + a)/(a^2*x) - 1)^3*abs(a))

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maple [B]  time = 0.04, size = 146, normalized size = 1.97 \[ -\frac {8 x}{c \sqrt {-a^{2} x^{2}+1}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{c \sqrt {a^{2}}}-\frac {4}{c a \sqrt {-a^{2} x^{2}+1}}-\frac {8}{3 c \,a^{2} \left (x -\frac {1}{a}\right ) \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {16 x}{3 c \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}} \]

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),x)

[Out]

-8/c*x/(-a^2*x^2+1)^(1/2)+1/c/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))-4/c/a/(-a^2*x^2+1)^(1/2)-8/
3/c/a^2/(x-1/a)/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)+16/3/c/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)*x

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maxima [A]  time = 0.43, size = 89, normalized size = 1.20 \[ -\frac {8 \, x}{3 \, \sqrt {-a^{2} x^{2} + 1} c} - \frac {8}{3 \, {\left (\sqrt {-a^{2} x^{2} + 1} a^{2} c x - \sqrt {-a^{2} x^{2} + 1} a c\right )}} + \frac {\arcsin \left (a x\right )}{a c} - \frac {4}{\sqrt {-a^{2} x^{2} + 1} a c} \]

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),x, algorithm="maxima")

[Out]

-8/3*x/(sqrt(-a^2*x^2 + 1)*c) - 8/3/(sqrt(-a^2*x^2 + 1)*a^2*c*x - sqrt(-a^2*x^2 + 1)*a*c) + arcsin(a*x)/(a*c)
- 4/(sqrt(-a^2*x^2 + 1)*a*c)

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mupad [B]  time = 0.07, size = 114, normalized size = 1.54 \[ -\frac {4\,a\,\sqrt {1-a^2\,x^2}+3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}-8\,a^2\,x\,\sqrt {1-a^2\,x^2}+3\,a^2\,x^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}-6\,a\,x\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}{3\,a^2\,c\,{\left (a\,x-1\right )}^2} \]

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)),x)

[Out]

-(4*a*(1 - a^2*x^2)^(1/2) + 3*asinh(x*(-a^2)^(1/2))*(-a^2)^(1/2) - 8*a^2*x*(1 - a^2*x^2)^(1/2) + 3*a^2*x^2*asi
nh(x*(-a^2)^(1/2))*(-a^2)^(1/2) - 6*a*x*asinh(x*(-a^2)^(1/2))*(-a^2)^(1/2))/(3*a^2*c*(a*x - 1)^2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \frac {3 a x}{- a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {3 a^{2} x^{2}}{- a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a^{3} x^{3}}{- a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{- a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx}{c} \]

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),x)

[Out]

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

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