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

Optimal. Leaf size=133 \[ \frac{c^3 (1-a x)^3 \sqrt{1-a^2 x^2}}{4 a}+\frac{7 c^3 (1-a x)^2 \sqrt{1-a^2 x^2}}{12 a}+\frac{35 c^3 (1-a x) \sqrt{1-a^2 x^2}}{24 a}+\frac{35 c^3 \sqrt{1-a^2 x^2}}{8 a}+\frac{35 c^3 \sin ^{-1}(a x)}{8 a} \]

[Out]

(35*c^3*Sqrt[1 - a^2*x^2])/(8*a) + (35*c^3*(1 - a*x)*Sqrt[1 - a^2*x^2])/(24*a) + (7*c^3*(1 - a*x)^2*Sqrt[1 - a
^2*x^2])/(12*a) + (c^3*(1 - a*x)^3*Sqrt[1 - a^2*x^2])/(4*a) + (35*c^3*ArcSin[a*x])/(8*a)

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Rubi [A]  time = 0.0955488, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6127, 671, 641, 216} \[ \frac{c^3 (1-a x)^3 \sqrt{1-a^2 x^2}}{4 a}+\frac{7 c^3 (1-a x)^2 \sqrt{1-a^2 x^2}}{12 a}+\frac{35 c^3 (1-a x) \sqrt{1-a^2 x^2}}{24 a}+\frac{35 c^3 \sqrt{1-a^2 x^2}}{8 a}+\frac{35 c^3 \sin ^{-1}(a x)}{8 a} \]

Antiderivative was successfully verified.

[In]

Int[(c - a*c*x)^3/E^ArcTanh[a*x],x]

[Out]

(35*c^3*Sqrt[1 - a^2*x^2])/(8*a) + (35*c^3*(1 - a*x)*Sqrt[1 - a^2*x^2])/(24*a) + (7*c^3*(1 - a*x)^2*Sqrt[1 - a
^2*x^2])/(12*a) + (c^3*(1 - a*x)^3*Sqrt[1 - a^2*x^2])/(4*a) + (35*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 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^{-\tanh ^{-1}(a x)} (c-a c x)^3 \, dx &=\frac{\int \frac{(c-a c x)^4}{\sqrt{1-a^2 x^2}} \, dx}{c}\\ &=\frac{c^3 (1-a x)^3 \sqrt{1-a^2 x^2}}{4 a}+\frac{7}{4} \int \frac{(c-a c x)^3}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{7 c^3 (1-a x)^2 \sqrt{1-a^2 x^2}}{12 a}+\frac{c^3 (1-a x)^3 \sqrt{1-a^2 x^2}}{4 a}+\frac{1}{12} (35 c) \int \frac{(c-a c x)^2}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{35 c^3 (1-a x) \sqrt{1-a^2 x^2}}{24 a}+\frac{7 c^3 (1-a x)^2 \sqrt{1-a^2 x^2}}{12 a}+\frac{c^3 (1-a x)^3 \sqrt{1-a^2 x^2}}{4 a}+\frac{1}{8} \left (35 c^2\right ) \int \frac{c-a c x}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{35 c^3 \sqrt{1-a^2 x^2}}{8 a}+\frac{35 c^3 (1-a x) \sqrt{1-a^2 x^2}}{24 a}+\frac{7 c^3 (1-a x)^2 \sqrt{1-a^2 x^2}}{12 a}+\frac{c^3 (1-a x)^3 \sqrt{1-a^2 x^2}}{4 a}+\frac{1}{8} \left (35 c^3\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{35 c^3 \sqrt{1-a^2 x^2}}{8 a}+\frac{35 c^3 (1-a x) \sqrt{1-a^2 x^2}}{24 a}+\frac{7 c^3 (1-a x)^2 \sqrt{1-a^2 x^2}}{12 a}+\frac{c^3 (1-a x)^3 \sqrt{1-a^2 x^2}}{4 a}+\frac{35 c^3 \sin ^{-1}(a x)}{8 a}\\ \end{align*}

Mathematica [A]  time = 0.0693521, size = 80, normalized size = 0.6 \[ \frac{c^3 \left (\frac{\sqrt{a x+1} \left (6 a^4 x^4-38 a^3 x^3+113 a^2 x^2-241 a x+160\right )}{\sqrt{1-a x}}-210 \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{24 a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^3*((Sqrt[1 + a*x]*(160 - 241*a*x + 113*a^2*x^2 - 38*a^3*x^3 + 6*a^4*x^4))/Sqrt[1 - a*x] - 210*ArcSin[Sqrt[1
- a*x]/Sqrt[2]]))/(24*a)

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Maple [A]  time = 0.039, size = 160, normalized size = 1.2 \begin{align*}{\frac{{c}^{3}x}{4} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-{\frac{29\,{c}^{3}x}{8}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{29\,{c}^{3}}{8}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}-{\frac{4\,{c}^{3}}{3\,a} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+8\,{\frac{{c}^{3}\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}{a}}+8\,{\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 ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*c^3*x*(-a^2*x^2+1)^(3/2)-29/8*c^3*x*(-a^2*x^2+1)^(1/2)-29/8*c^3/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2
+1)^(1/2))-4/3*c^3*(-a^2*x^2+1)^(3/2)/a+8*c^3/a*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)+8*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))

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Maxima [A]  time = 1.4716, size = 120, normalized size = 0.9 \begin{align*} \frac{1}{4} \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} c^{3} x - \frac{29}{8} \, \sqrt{-a^{2} x^{2} + 1} c^{3} x - \frac{4 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} c^{3}}{3 \, a} + \frac{35 \, c^{3} \arcsin \left (a x\right )}{8 \, a} + \frac{8 \, \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)*(-a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

1/4*(-a^2*x^2 + 1)^(3/2)*c^3*x - 29/8*sqrt(-a^2*x^2 + 1)*c^3*x - 4/3*(-a^2*x^2 + 1)^(3/2)*c^3/a + 35/8*c^3*arc
sin(a*x)/a + 8*sqrt(-a^2*x^2 + 1)*c^3/a

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Fricas [A]  time = 1.62902, size = 182, normalized size = 1.37 \begin{align*} -\frac{210 \, c^{3} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) +{\left (6 \, a^{3} c^{3} x^{3} - 32 \, a^{2} c^{3} x^{2} + 81 \, a c^{3} x - 160 \, c^{3}\right )} \sqrt{-a^{2} x^{2} + 1}}{24 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(210*c^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (6*a^3*c^3*x^3 - 32*a^2*c^3*x^2 + 81*a*c^3*x - 160*c^3
)*sqrt(-a^2*x^2 + 1))/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 x + 1}\, dx + \int \frac{3 a x \sqrt{- a^{2} x^{2} + 1}}{a x + 1}\, dx + \int - \frac{3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1}}{a x + 1}\, dx + \int \frac{a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1}}{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)*(-a**2*x**2+1)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.189, size = 90, normalized size = 0.68 \begin{align*} \frac{35 \, c^{3} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{8 \,{\left | a \right |}} + \frac{1}{24} \, \sqrt{-a^{2} x^{2} + 1}{\left (\frac{160 \, c^{3}}{a} -{\left (81 \, c^{3} + 2 \,{\left (3 \, a^{2} c^{3} x - 16 \, a c^{3}\right )} x\right )} x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

35/8*c^3*arcsin(a*x)*sgn(a)/abs(a) + 1/24*sqrt(-a^2*x^2 + 1)*(160*c^3/a - (81*c^3 + 2*(3*a^2*c^3*x - 16*a*c^3)
*x)*x)