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

Optimal. Leaf size=163 \[ -\frac{2 c^3 (1-a x)^5}{a \sqrt{1-a^2 x^2}}-\frac{9 c^3 \sqrt{1-a^2 x^2} (1-a x)^3}{4 a}-\frac{21 c^3 \sqrt{1-a^2 x^2} (1-a x)^2}{4 a}-\frac{105 c^3 \sqrt{1-a^2 x^2} (1-a x)}{8 a}-\frac{315 c^3 \sqrt{1-a^2 x^2}}{8 a}-\frac{315 c^3 \sin ^{-1}(a x)}{8 a} \]

[Out]

(-2*c^3*(1 - a*x)^5)/(a*Sqrt[1 - a^2*x^2]) - (315*c^3*Sqrt[1 - a^2*x^2])/(8*a) - (105*c^3*(1 - a*x)*Sqrt[1 - a
^2*x^2])/(8*a) - (21*c^3*(1 - a*x)^2*Sqrt[1 - a^2*x^2])/(4*a) - (9*c^3*(1 - a*x)^3*Sqrt[1 - a^2*x^2])/(4*a) -
(315*c^3*ArcSin[a*x])/(8*a)

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Rubi [A]  time = 0.125341, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {6127, 669, 671, 641, 216} \[ -\frac{2 c^3 (1-a x)^5}{a \sqrt{1-a^2 x^2}}-\frac{9 c^3 \sqrt{1-a^2 x^2} (1-a x)^3}{4 a}-\frac{21 c^3 \sqrt{1-a^2 x^2} (1-a x)^2}{4 a}-\frac{105 c^3 \sqrt{1-a^2 x^2} (1-a x)}{8 a}-\frac{315 c^3 \sqrt{1-a^2 x^2}}{8 a}-\frac{315 c^3 \sin ^{-1}(a x)}{8 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*c^3*(1 - a*x)^5)/(a*Sqrt[1 - a^2*x^2]) - (315*c^3*Sqrt[1 - a^2*x^2])/(8*a) - (105*c^3*(1 - a*x)*Sqrt[1 - a
^2*x^2])/(8*a) - (21*c^3*(1 - a*x)^2*Sqrt[1 - a^2*x^2])/(4*a) - (9*c^3*(1 - a*x)^3*Sqrt[1 - a^2*x^2])/(4*a) -
(315*c^3*ArcSin[a*x])/(8*a)

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 669

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

Rule 671

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

Rule 641

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

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]

Rubi steps

\begin{align*} \int e^{-3 \tanh ^{-1}(a x)} (c-a c x)^3 \, dx &=\frac{\int \frac{(c-a c x)^6}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=-\frac{2 c^3 (1-a x)^5}{a \sqrt{1-a^2 x^2}}-\frac{9 \int \frac{(c-a c x)^4}{\sqrt{1-a^2 x^2}} \, dx}{c}\\ &=-\frac{2 c^3 (1-a x)^5}{a \sqrt{1-a^2 x^2}}-\frac{9 c^3 (1-a x)^3 \sqrt{1-a^2 x^2}}{4 a}-\frac{63}{4} \int \frac{(c-a c x)^3}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{2 c^3 (1-a x)^5}{a \sqrt{1-a^2 x^2}}-\frac{21 c^3 (1-a x)^2 \sqrt{1-a^2 x^2}}{4 a}-\frac{9 c^3 (1-a x)^3 \sqrt{1-a^2 x^2}}{4 a}-\frac{1}{4} (105 c) \int \frac{(c-a c x)^2}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{2 c^3 (1-a x)^5}{a \sqrt{1-a^2 x^2}}-\frac{105 c^3 (1-a x) \sqrt{1-a^2 x^2}}{8 a}-\frac{21 c^3 (1-a x)^2 \sqrt{1-a^2 x^2}}{4 a}-\frac{9 c^3 (1-a x)^3 \sqrt{1-a^2 x^2}}{4 a}-\frac{1}{8} \left (315 c^2\right ) \int \frac{c-a c x}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{2 c^3 (1-a x)^5}{a \sqrt{1-a^2 x^2}}-\frac{315 c^3 \sqrt{1-a^2 x^2}}{8 a}-\frac{105 c^3 (1-a x) \sqrt{1-a^2 x^2}}{8 a}-\frac{21 c^3 (1-a x)^2 \sqrt{1-a^2 x^2}}{4 a}-\frac{9 c^3 (1-a x)^3 \sqrt{1-a^2 x^2}}{4 a}-\frac{1}{8} \left (315 c^3\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{2 c^3 (1-a x)^5}{a \sqrt{1-a^2 x^2}}-\frac{315 c^3 \sqrt{1-a^2 x^2}}{8 a}-\frac{105 c^3 (1-a x) \sqrt{1-a^2 x^2}}{8 a}-\frac{21 c^3 (1-a x)^2 \sqrt{1-a^2 x^2}}{4 a}-\frac{9 c^3 (1-a x)^3 \sqrt{1-a^2 x^2}}{4 a}-\frac{315 c^3 \sin ^{-1}(a x)}{8 a}\\ \end{align*}

Mathematica [C]  time = 0.0196711, size = 45, normalized size = 0.28 \[ -\frac{c^3 (1-a x)^{11/2} \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{11}{2},\frac{13}{2},\frac{1}{2} (1-a x)\right )}{11 \sqrt{2} a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(c^3*(1 - a*x)^(11/2)*Hypergeometric2F1[3/2, 11/2, 13/2, (1 - a*x)/2])/(11*Sqrt[2]*a)

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Maple [A]  time = 0.049, size = 245, normalized size = 1.5 \begin{align*} -{\frac{{c}^{3}x}{4} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{c}^{3}x}{8}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{3\,{c}^{3}}{8}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}-28\,{\frac{{c}^{3} \left ( -{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) \right ) ^{5/2}}{{a}^{3} \left ( x+{a}^{-1} \right ) ^{2}}}-26\,{\frac{{c}^{3} \left ( -{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) \right ) ^{3/2}}{a}}-39\,{c}^{3}\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }x-39\,{\frac{{c}^{3}}{\sqrt{{a}^{2}}}\arctan \left ({\frac{\sqrt{{a}^{2}}x}{\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}} \right ) }-8\,{\frac{{c}^{3} \left ( -{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) \right ) ^{5/2}}{{a}^{4} \left ( x+{a}^{-1} \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*c^3*x*(-a^2*x^2+1)^(3/2)-3/8*c^3*x*(-a^2*x^2+1)^(1/2)-3/8*c^3/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+
1)^(1/2))-28*c^3/a^3/(x+1/a)^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(5/2)-26*c^3/a*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(3/2)-
39*c^3*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)*x-39*c^3/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*(x+1/a)^2+2*a*(x+1/a
))^(1/2))-8*c^3/a^4/(x+1/a)^3*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(5/2)

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Maxima [C]  time = 1.49614, size = 315, normalized size = 1.93 \begin{align*} -\frac{1}{4} \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} c^{3} x + 3 \, \sqrt{a^{2} x^{2} + 4 \, a x + 3} c^{3} x - \frac{3}{8} \, \sqrt{-a^{2} x^{2} + 1} c^{3} x + \frac{8 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} c^{3}}{a^{3} x^{2} + 2 \, a^{2} x + a} - \frac{6 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} c^{3}}{a^{2} x + a} + \frac{2 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} c^{3}}{a} - \frac{3 i \, c^{3} \arcsin \left (a x + 2\right )}{a} - \frac{339 \, c^{3} \arcsin \left (a x\right )}{8 \, a} - \frac{48 \, \sqrt{-a^{2} x^{2} + 1} c^{3}}{a^{2} x + a} + \frac{6 \, \sqrt{a^{2} x^{2} + 4 \, a x + 3} c^{3}}{a} - \frac{18 \, \sqrt{-a^{2} x^{2} + 1} c^{3}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(-a^2*x^2 + 1)^(3/2)*c^3*x + 3*sqrt(a^2*x^2 + 4*a*x + 3)*c^3*x - 3/8*sqrt(-a^2*x^2 + 1)*c^3*x + 8*(-a^2*x
^2 + 1)^(3/2)*c^3/(a^3*x^2 + 2*a^2*x + a) - 6*(-a^2*x^2 + 1)^(3/2)*c^3/(a^2*x + a) + 2*(-a^2*x^2 + 1)^(3/2)*c^
3/a - 3*I*c^3*arcsin(a*x + 2)/a - 339/8*c^3*arcsin(a*x)/a - 48*sqrt(-a^2*x^2 + 1)*c^3/(a^2*x + a) + 6*sqrt(a^2
*x^2 + 4*a*x + 3)*c^3/a - 18*sqrt(-a^2*x^2 + 1)*c^3/a

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Fricas [A]  time = 1.67829, size = 267, normalized size = 1.64 \begin{align*} -\frac{496 \, a c^{3} x + 496 \, c^{3} - 630 \,{\left (a c^{3} x + c^{3}\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) -{\left (2 \, a^{4} c^{3} x^{4} - 14 \, a^{3} c^{3} x^{3} + 51 \, a^{2} c^{3} x^{2} - 173 \, a c^{3} x - 496 \, c^{3}\right )} \sqrt{-a^{2} x^{2} + 1}}{8 \,{\left (a^{2} x + a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(496*a*c^3*x + 496*c^3 - 630*(a*c^3*x + c^3)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (2*a^4*c^3*x^4 - 14
*a^3*c^3*x^3 + 51*a^2*c^3*x^2 - 173*a*c^3*x - 496*c^3)*sqrt(-a^2*x^2 + 1))/(a^2*x + a)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - c^{3} \left (\int - \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx + \int \frac{3 a x \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx + \int - \frac{2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx + \int - \frac{2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx + \int \frac{3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx + \int - \frac{a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c**3*(Integral(-sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1), x) + Integral(3*a*x*sqrt(-a**2*x*
*2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1), x) + Integral(-2*a**2*x**2*sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*
a**2*x**2 + 3*a*x + 1), x) + Integral(-2*a**3*x**3*sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1),
x) + Integral(3*a**4*x**4*sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1), x) + Integral(-a**5*x**
5*sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1), x))

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Giac [A]  time = 1.28941, size = 139, normalized size = 0.85 \begin{align*} -\frac{315 \, c^{3} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{8 \,{\left | a \right |}} - \frac{1}{8} \, \sqrt{-a^{2} x^{2} + 1}{\left (\frac{240 \, c^{3}}{a} -{\left (67 \, c^{3} + 2 \,{\left (a^{2} c^{3} x - 8 \, a c^{3}\right )} x\right )} x\right )} + \frac{64 \, c^{3}}{{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} + 1\right )}{\left | a \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-315/8*c^3*arcsin(a*x)*sgn(a)/abs(a) - 1/8*sqrt(-a^2*x^2 + 1)*(240*c^3/a - (67*c^3 + 2*(a^2*c^3*x - 8*a*c^3)*x
)*x) + 64*c^3/(((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) + 1)*abs(a))