3.318 \(\int e^{\tanh ^{-1}(a x)} x (c-a c x)^4 \, dx\)

Optimal. Leaf size=158 \[ -\frac{c^4 (1-a x)^3 \left (1-a^2 x^2\right )^{3/2}}{6 a^2}-\frac{c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{10 a^2}-\frac{7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{40 a^2}-\frac{7 c^4 \left (1-a^2 x^2\right )^{3/2}}{24 a^2}-\frac{7 c^4 x \sqrt{1-a^2 x^2}}{16 a}-\frac{7 c^4 \sin ^{-1}(a x)}{16 a^2} \]

[Out]

(-7*c^4*x*Sqrt[1 - a^2*x^2])/(16*a) - (7*c^4*(1 - a^2*x^2)^(3/2))/(24*a^2) - (7*c^4*(1 - a*x)*(1 - a^2*x^2)^(3
/2))/(40*a^2) - (c^4*(1 - a*x)^2*(1 - a^2*x^2)^(3/2))/(10*a^2) - (c^4*(1 - a*x)^3*(1 - a^2*x^2)^(3/2))/(6*a^2)
- (7*c^4*ArcSin[a*x])/(16*a^2)

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Rubi [A]  time = 0.141219, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {6128, 795, 671, 641, 195, 216} \[ -\frac{c^4 (1-a x)^3 \left (1-a^2 x^2\right )^{3/2}}{6 a^2}-\frac{c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{10 a^2}-\frac{7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{40 a^2}-\frac{7 c^4 \left (1-a^2 x^2\right )^{3/2}}{24 a^2}-\frac{7 c^4 x \sqrt{1-a^2 x^2}}{16 a}-\frac{7 c^4 \sin ^{-1}(a x)}{16 a^2} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]*x*(c - a*c*x)^4,x]

[Out]

(-7*c^4*x*Sqrt[1 - a^2*x^2])/(16*a) - (7*c^4*(1 - a^2*x^2)^(3/2))/(24*a^2) - (7*c^4*(1 - a*x)*(1 - a^2*x^2)^(3
/2))/(40*a^2) - (c^4*(1 - a*x)^2*(1 - a^2*x^2)^(3/2))/(10*a^2) - (c^4*(1 - a*x)^3*(1 - a^2*x^2)^(3/2))/(6*a^2)
- (7*c^4*ArcSin[a*x])/(16*a^2)

Rule 6128

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

Rule 795

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(g*(d + e*x)^m
*(a + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[(m*(d*g + e*f) + 2*e*f*(p + 1))/(e*(m + 2*p + 2)), Int[(d +
e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && NeQ[m + 2*p +
2, 0] && NeQ[m, 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^{\tanh ^{-1}(a x)} x (c-a c x)^4 \, dx &=c \int x (c-a c x)^3 \sqrt{1-a^2 x^2} \, dx\\ &=-\frac{c^4 (1-a x)^3 \left (1-a^2 x^2\right )^{3/2}}{6 a^2}-\frac{c \int (c-a c x)^3 \sqrt{1-a^2 x^2} \, dx}{2 a}\\ &=-\frac{c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{10 a^2}-\frac{c^4 (1-a x)^3 \left (1-a^2 x^2\right )^{3/2}}{6 a^2}-\frac{\left (7 c^2\right ) \int (c-a c x)^2 \sqrt{1-a^2 x^2} \, dx}{10 a}\\ &=-\frac{7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{40 a^2}-\frac{c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{10 a^2}-\frac{c^4 (1-a x)^3 \left (1-a^2 x^2\right )^{3/2}}{6 a^2}-\frac{\left (7 c^3\right ) \int (c-a c x) \sqrt{1-a^2 x^2} \, dx}{8 a}\\ &=-\frac{7 c^4 \left (1-a^2 x^2\right )^{3/2}}{24 a^2}-\frac{7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{40 a^2}-\frac{c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{10 a^2}-\frac{c^4 (1-a x)^3 \left (1-a^2 x^2\right )^{3/2}}{6 a^2}-\frac{\left (7 c^4\right ) \int \sqrt{1-a^2 x^2} \, dx}{8 a}\\ &=-\frac{7 c^4 x \sqrt{1-a^2 x^2}}{16 a}-\frac{7 c^4 \left (1-a^2 x^2\right )^{3/2}}{24 a^2}-\frac{7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{40 a^2}-\frac{c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{10 a^2}-\frac{c^4 (1-a x)^3 \left (1-a^2 x^2\right )^{3/2}}{6 a^2}-\frac{\left (7 c^4\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{16 a}\\ &=-\frac{7 c^4 x \sqrt{1-a^2 x^2}}{16 a}-\frac{7 c^4 \left (1-a^2 x^2\right )^{3/2}}{24 a^2}-\frac{7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{40 a^2}-\frac{c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{10 a^2}-\frac{c^4 (1-a x)^3 \left (1-a^2 x^2\right )^{3/2}}{6 a^2}-\frac{7 c^4 \sin ^{-1}(a x)}{16 a^2}\\ \end{align*}

Mathematica [A]  time = 0.124399, size = 83, normalized size = 0.53 \[ -\frac{c^4 \left (\sqrt{1-a^2 x^2} \left (40 a^5 x^5-144 a^4 x^4+170 a^3 x^3-32 a^2 x^2-105 a x+176\right )-210 \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{240 a^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcTanh[a*x]*x*(c - a*c*x)^4,x]

[Out]

-(c^4*(Sqrt[1 - a^2*x^2]*(176 - 105*a*x - 32*a^2*x^2 + 170*a^3*x^3 - 144*a^4*x^4 + 40*a^5*x^5) - 210*ArcSin[Sq
rt[1 - a*x]/Sqrt[2]]))/(240*a^2)

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Maple [A]  time = 0.052, size = 163, normalized size = 1. \begin{align*} -{\frac{{c}^{4}{a}^{3}{x}^{5}}{6}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{17\,{c}^{4}a{x}^{3}}{24}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{7\,{c}^{4}x}{16\,a}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{7\,{c}^{4}}{16\,a}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{\frac{3\,{c}^{4}{a}^{2}{x}^{4}}{5}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{2\,{c}^{4}{x}^{2}}{15}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{11\,{c}^{4}}{15\,{a}^{2}}\sqrt{-{a}^{2}{x}^{2}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)/(-a^2*x^2+1)^(1/2)*x*(-a*c*x+c)^4,x)

[Out]

-1/6*c^4*a^3*x^5*(-a^2*x^2+1)^(1/2)-17/24*c^4*a*x^3*(-a^2*x^2+1)^(1/2)+7/16*c^4*x*(-a^2*x^2+1)^(1/2)/a-7/16*c^
4/a/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+3/5*c^4*a^2*x^4*(-a^2*x^2+1)^(1/2)+2/15*c^4*x^2*(-a^2
*x^2+1)^(1/2)-11/15*c^4/a^2*(-a^2*x^2+1)^(1/2)

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Maxima [A]  time = 1.43336, size = 207, normalized size = 1.31 \begin{align*} -\frac{1}{6} \, \sqrt{-a^{2} x^{2} + 1} a^{3} c^{4} x^{5} + \frac{3}{5} \, \sqrt{-a^{2} x^{2} + 1} a^{2} c^{4} x^{4} - \frac{17}{24} \, \sqrt{-a^{2} x^{2} + 1} a c^{4} x^{3} + \frac{2}{15} \, \sqrt{-a^{2} x^{2} + 1} c^{4} x^{2} + \frac{7 \, \sqrt{-a^{2} x^{2} + 1} c^{4} x}{16 \, a} - \frac{7 \, c^{4} \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{16 \, \sqrt{a^{2}} a} - \frac{11 \, \sqrt{-a^{2} x^{2} + 1} c^{4}}{15 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x*(-a*c*x+c)^4,x, algorithm="maxima")

[Out]

-1/6*sqrt(-a^2*x^2 + 1)*a^3*c^4*x^5 + 3/5*sqrt(-a^2*x^2 + 1)*a^2*c^4*x^4 - 17/24*sqrt(-a^2*x^2 + 1)*a*c^4*x^3
+ 2/15*sqrt(-a^2*x^2 + 1)*c^4*x^2 + 7/16*sqrt(-a^2*x^2 + 1)*c^4*x/a - 7/16*c^4*arcsin(a^2*x/sqrt(a^2))/(sqrt(a
^2)*a) - 11/15*sqrt(-a^2*x^2 + 1)*c^4/a^2

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Fricas [A]  time = 1.6199, size = 236, normalized size = 1.49 \begin{align*} \frac{210 \, c^{4} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) -{\left (40 \, a^{5} c^{4} x^{5} - 144 \, a^{4} c^{4} x^{4} + 170 \, a^{3} c^{4} x^{3} - 32 \, a^{2} c^{4} x^{2} - 105 \, a c^{4} x + 176 \, c^{4}\right )} \sqrt{-a^{2} x^{2} + 1}}{240 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x*(-a*c*x+c)^4,x, algorithm="fricas")

[Out]

1/240*(210*c^4*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (40*a^5*c^4*x^5 - 144*a^4*c^4*x^4 + 170*a^3*c^4*x^3 -
32*a^2*c^4*x^2 - 105*a*c^4*x + 176*c^4)*sqrt(-a^2*x^2 + 1))/a^2

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Sympy [A]  time = 15.6389, size = 617, normalized size = 3.91 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x*(-a*c*x+c)**4,x)

[Out]

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

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Giac [A]  time = 1.22086, size = 127, normalized size = 0.8 \begin{align*} -\frac{7 \, c^{4} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{16 \, a{\left | a \right |}} - \frac{1}{240} \, \sqrt{-a^{2} x^{2} + 1}{\left (\frac{176 \, c^{4}}{a^{2}} -{\left (\frac{105 \, c^{4}}{a} + 2 \,{\left (16 \, c^{4} -{\left (85 \, a c^{4} + 4 \,{\left (5 \, a^{3} c^{4} x - 18 \, a^{2} c^{4}\right )} x\right )} x\right )} x\right )} x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x*(-a*c*x+c)^4,x, algorithm="giac")

[Out]

-7/16*c^4*arcsin(a*x)*sgn(a)/(a*abs(a)) - 1/240*sqrt(-a^2*x^2 + 1)*(176*c^4/a^2 - (105*c^4/a + 2*(16*c^4 - (85
*a*c^4 + 4*(5*a^3*c^4*x - 18*a^2*c^4)*x)*x)*x)*x)