3.1255 \(\int e^{-3 \tanh ^{-1}(a x)} (c-a^2 c x^2)^3 \, dx\)

Optimal. Leaf size=145 \[ \frac{c^3 (1-a x)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}+\frac{3 c^3 (1-a x) \left (1-a^2 x^2\right )^{5/2}}{14 a}+\frac{3 c^3 \left (1-a^2 x^2\right )^{5/2}}{10 a}+\frac{3}{8} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac{9}{16} c^3 x \sqrt{1-a^2 x^2}+\frac{9 c^3 \sin ^{-1}(a x)}{16 a} \]

[Out]

(9*c^3*x*Sqrt[1 - a^2*x^2])/16 + (3*c^3*x*(1 - a^2*x^2)^(3/2))/8 + (3*c^3*(1 - a^2*x^2)^(5/2))/(10*a) + (3*c^3
*(1 - a*x)*(1 - a^2*x^2)^(5/2))/(14*a) + (c^3*(1 - a*x)^2*(1 - a^2*x^2)^(5/2))/(7*a) + (9*c^3*ArcSin[a*x])/(16
*a)

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Rubi [A]  time = 0.0866331, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {6139, 671, 641, 195, 216} \[ \frac{c^3 (1-a x)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}+\frac{3 c^3 (1-a x) \left (1-a^2 x^2\right )^{5/2}}{14 a}+\frac{3 c^3 \left (1-a^2 x^2\right )^{5/2}}{10 a}+\frac{3}{8} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac{9}{16} c^3 x \sqrt{1-a^2 x^2}+\frac{9 c^3 \sin ^{-1}(a x)}{16 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(9*c^3*x*Sqrt[1 - a^2*x^2])/16 + (3*c^3*x*(1 - a^2*x^2)^(3/2))/8 + (3*c^3*(1 - a^2*x^2)^(5/2))/(10*a) + (3*c^3
*(1 - a*x)*(1 - a^2*x^2)^(5/2))/(14*a) + (c^3*(1 - a*x)^2*(1 - a^2*x^2)^(5/2))/(7*a) + (9*c^3*ArcSin[a*x])/(16
*a)

Rule 6139

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[(1 - a^2*x^2)^(p + n/
2)/(1 - a*x)^n, x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[a^2*c + d, 0] && IntegerQ[p] && ILtQ[(n - 1)/2, 0] &&
!IntegerQ[p - n/2]

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 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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)} \left (c-a^2 c x^2\right )^3 \, dx &=c^3 \int (1-a x)^3 \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=\frac{c^3 (1-a x)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}+\frac{1}{7} \left (9 c^3\right ) \int (1-a x)^2 \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=\frac{3 c^3 (1-a x) \left (1-a^2 x^2\right )^{5/2}}{14 a}+\frac{c^3 (1-a x)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}+\frac{1}{2} \left (3 c^3\right ) \int (1-a x) \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=\frac{3 c^3 \left (1-a^2 x^2\right )^{5/2}}{10 a}+\frac{3 c^3 (1-a x) \left (1-a^2 x^2\right )^{5/2}}{14 a}+\frac{c^3 (1-a x)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}+\frac{1}{2} \left (3 c^3\right ) \int \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=\frac{3}{8} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac{3 c^3 \left (1-a^2 x^2\right )^{5/2}}{10 a}+\frac{3 c^3 (1-a x) \left (1-a^2 x^2\right )^{5/2}}{14 a}+\frac{c^3 (1-a x)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}+\frac{1}{8} \left (9 c^3\right ) \int \sqrt{1-a^2 x^2} \, dx\\ &=\frac{9}{16} c^3 x \sqrt{1-a^2 x^2}+\frac{3}{8} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac{3 c^3 \left (1-a^2 x^2\right )^{5/2}}{10 a}+\frac{3 c^3 (1-a x) \left (1-a^2 x^2\right )^{5/2}}{14 a}+\frac{c^3 (1-a x)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}+\frac{1}{16} \left (9 c^3\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{9}{16} c^3 x \sqrt{1-a^2 x^2}+\frac{3}{8} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac{3 c^3 \left (1-a^2 x^2\right )^{5/2}}{10 a}+\frac{3 c^3 (1-a x) \left (1-a^2 x^2\right )^{5/2}}{14 a}+\frac{c^3 (1-a x)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}+\frac{9 c^3 \sin ^{-1}(a x)}{16 a}\\ \end{align*}

Mathematica [A]  time = 0.119178, size = 91, normalized size = 0.63 \[ \frac{c^3 \left (\sqrt{1-a^2 x^2} \left (80 a^6 x^6-280 a^5 x^5+208 a^4 x^4+350 a^3 x^3-656 a^2 x^2+245 a x+368\right )-630 \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{560 a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^3*(Sqrt[1 - a^2*x^2]*(368 + 245*a*x - 656*a^2*x^2 + 350*a^3*x^3 + 208*a^4*x^4 - 280*a^5*x^5 + 80*a^6*x^6) -
630*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(560*a)

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Maple [A]  time = 0.038, size = 127, normalized size = 0.9 \begin{align*}{\frac{{c}^{3}a{x}^{2}}{7} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{5}{2}}}}+{\frac{23\,{c}^{3}}{35\,a} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{5}{2}}}}-{\frac{{c}^{3}x}{2} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{5}{2}}}}+{\frac{3\,{c}^{3}x}{8} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{\frac{9\,{c}^{3}x}{16}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{9\,{c}^{3}}{16}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*c^3*a*x^2*(-a^2*x^2+1)^(5/2)+23/35*c^3*(-a^2*x^2+1)^(5/2)/a-1/2*c^3*x*(-a^2*x^2+1)^(5/2)+3/8*c^3*x*(-a^2*x
^2+1)^(3/2)+9/16*c^3*x*(-a^2*x^2+1)^(1/2)+9/16*c^3/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))

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Maxima [A]  time = 1.45898, size = 146, normalized size = 1.01 \begin{align*} \frac{1}{7} \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{5}{2}} a c^{3} x^{2} - \frac{1}{2} \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{5}{2}} c^{3} x + \frac{3}{8} \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} c^{3} x + \frac{23 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{5}{2}} c^{3}}{35 \, a} + \frac{9}{16} \, \sqrt{-a^{2} x^{2} + 1} c^{3} x + \frac{9 \, c^{3} \arcsin \left (a x\right )}{16 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*(-a^2*x^2 + 1)^(5/2)*a*c^3*x^2 - 1/2*(-a^2*x^2 + 1)^(5/2)*c^3*x + 3/8*(-a^2*x^2 + 1)^(3/2)*c^3*x + 23/35*(
-a^2*x^2 + 1)^(5/2)*c^3/a + 9/16*sqrt(-a^2*x^2 + 1)*c^3*x + 9/16*c^3*arcsin(a*x)/a

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Fricas [A]  time = 2.6148, size = 261, normalized size = 1.8 \begin{align*} -\frac{630 \, c^{3} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) -{\left (80 \, a^{6} c^{3} x^{6} - 280 \, a^{5} c^{3} x^{5} + 208 \, a^{4} c^{3} x^{4} + 350 \, a^{3} c^{3} x^{3} - 656 \, a^{2} c^{3} x^{2} + 245 \, a c^{3} x + 368 \, c^{3}\right )} \sqrt{-a^{2} x^{2} + 1}}{560 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/560*(630*c^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (80*a^6*c^3*x^6 - 280*a^5*c^3*x^5 + 208*a^4*c^3*x^4 +
350*a^3*c^3*x^3 - 656*a^2*c^3*x^2 + 245*a*c^3*x + 368*c^3)*sqrt(-a^2*x^2 + 1))/a

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Sympy [C]  time = 34.4053, size = 632, normalized size = 4.36 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**5*c**3*Piecewise((x**6*sqrt(-a**2*x**2 + 1)/7 - x**4*sqrt(-a**2*x**2 + 1)/(35*a**2) - 4*x**2*sqrt(-a**2*x**
2 + 1)/(105*a**4) - 8*sqrt(-a**2*x**2 + 1)/(105*a**6), Ne(a, 0)), (x**6/6, True)) - 3*a**4*c**3*Piecewise((I*a
**2*x**7/(6*sqrt(a**2*x**2 - 1)) - 5*I*x**5/(24*sqrt(a**2*x**2 - 1)) - I*x**3/(48*a**2*sqrt(a**2*x**2 - 1)) +
I*x/(16*a**4*sqrt(a**2*x**2 - 1)) - I*acosh(a*x)/(16*a**5), Abs(a**2*x**2) > 1), (-a**2*x**7/(6*sqrt(-a**2*x**
2 + 1)) + 5*x**5/(24*sqrt(-a**2*x**2 + 1)) + x**3/(48*a**2*sqrt(-a**2*x**2 + 1)) - x/(16*a**4*sqrt(-a**2*x**2
+ 1)) + asin(a*x)/(16*a**5), True)) + 2*a**3*c**3*Piecewise((x**4*sqrt(-a**2*x**2 + 1)/5 - x**2*sqrt(-a**2*x**
2 + 1)/(15*a**2) - 2*sqrt(-a**2*x**2 + 1)/(15*a**4), Ne(a, 0)), (x**4/4, True)) + 2*a**2*c**3*Piecewise((I*a**
2*x**5/(4*sqrt(a**2*x**2 - 1)) - 3*I*x**3/(8*sqrt(a**2*x**2 - 1)) + I*x/(8*a**2*sqrt(a**2*x**2 - 1)) - I*acosh
(a*x)/(8*a**3), Abs(a**2*x**2) > 1), (-a**2*x**5/(4*sqrt(-a**2*x**2 + 1)) + 3*x**3/(8*sqrt(-a**2*x**2 + 1)) -
x/(8*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(8*a**3), True)) - 3*a*c**3*Piecewise((x**2/2, Eq(a**2, 0)), (-(-a
**2*x**2 + 1)**(3/2)/(3*a**2), True)) + c**3*Piecewise((I*a**2*x**3/(2*sqrt(a**2*x**2 - 1)) - I*x/(2*sqrt(a**2
*x**2 - 1)) - I*acosh(a*x)/(2*a), Abs(a**2*x**2) > 1), (x*sqrt(-a**2*x**2 + 1)/2 + asin(a*x)/(2*a), True))

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Giac [A]  time = 1.15974, size = 138, normalized size = 0.95 \begin{align*} \frac{9 \, c^{3} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{16 \,{\left | a \right |}} + \frac{1}{560} \, \sqrt{-a^{2} x^{2} + 1}{\left (\frac{368 \, c^{3}}{a} +{\left (245 \, c^{3} - 2 \,{\left (328 \, a c^{3} -{\left (175 \, a^{2} c^{3} + 4 \,{\left (26 \, a^{3} c^{3} + 5 \,{\left (2 \, a^{5} c^{3} x - 7 \, a^{4} c^{3}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9/16*c^3*arcsin(a*x)*sgn(a)/abs(a) + 1/560*sqrt(-a^2*x^2 + 1)*(368*c^3/a + (245*c^3 - 2*(328*a*c^3 - (175*a^2*
c^3 + 4*(26*a^3*c^3 + 5*(2*a^5*c^3*x - 7*a^4*c^3)*x)*x)*x)*x)*x)