3.1149 \(\int e^{3 \tanh ^{-1}(a x)} (c-a^2 c x^2)^4 \, dx\)

Optimal. Leaf size=165 \[ -\frac{c^4 (a x+1)^2 \left (1-a^2 x^2\right )^{7/2}}{9 a}-\frac{11 c^4 (a x+1) \left (1-a^2 x^2\right )^{7/2}}{72 a}-\frac{11 c^4 \left (1-a^2 x^2\right )^{7/2}}{56 a}+\frac{11}{48} c^4 x \left (1-a^2 x^2\right )^{5/2}+\frac{55}{192} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac{55}{128} c^4 x \sqrt{1-a^2 x^2}+\frac{55 c^4 \sin ^{-1}(a x)}{128 a} \]

[Out]

(55*c^4*x*Sqrt[1 - a^2*x^2])/128 + (55*c^4*x*(1 - a^2*x^2)^(3/2))/192 + (11*c^4*x*(1 - a^2*x^2)^(5/2))/48 - (1
1*c^4*(1 - a^2*x^2)^(7/2))/(56*a) - (11*c^4*(1 + a*x)*(1 - a^2*x^2)^(7/2))/(72*a) - (c^4*(1 + a*x)^2*(1 - a^2*
x^2)^(7/2))/(9*a) + (55*c^4*ArcSin[a*x])/(128*a)

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Rubi [A]  time = 0.0903055, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {6138, 671, 641, 195, 216} \[ -\frac{c^4 (a x+1)^2 \left (1-a^2 x^2\right )^{7/2}}{9 a}-\frac{11 c^4 (a x+1) \left (1-a^2 x^2\right )^{7/2}}{72 a}-\frac{11 c^4 \left (1-a^2 x^2\right )^{7/2}}{56 a}+\frac{11}{48} c^4 x \left (1-a^2 x^2\right )^{5/2}+\frac{55}{192} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac{55}{128} c^4 x \sqrt{1-a^2 x^2}+\frac{55 c^4 \sin ^{-1}(a x)}{128 a} \]

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^4,x]

[Out]

(55*c^4*x*Sqrt[1 - a^2*x^2])/128 + (55*c^4*x*(1 - a^2*x^2)^(3/2))/192 + (11*c^4*x*(1 - a^2*x^2)^(5/2))/48 - (1
1*c^4*(1 - a^2*x^2)^(7/2))/(56*a) - (11*c^4*(1 + a*x)*(1 - a^2*x^2)^(7/2))/(72*a) - (c^4*(1 + a*x)^2*(1 - a^2*
x^2)^(7/2))/(9*a) + (55*c^4*ArcSin[a*x])/(128*a)

Rule 6138

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] && IGtQ[(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 )^4 \, dx &=c^4 \int (1+a x)^3 \left (1-a^2 x^2\right )^{5/2} \, dx\\ &=-\frac{c^4 (1+a x)^2 \left (1-a^2 x^2\right )^{7/2}}{9 a}+\frac{1}{9} \left (11 c^4\right ) \int (1+a x)^2 \left (1-a^2 x^2\right )^{5/2} \, dx\\ &=-\frac{11 c^4 (1+a x) \left (1-a^2 x^2\right )^{7/2}}{72 a}-\frac{c^4 (1+a x)^2 \left (1-a^2 x^2\right )^{7/2}}{9 a}+\frac{1}{8} \left (11 c^4\right ) \int (1+a x) \left (1-a^2 x^2\right )^{5/2} \, dx\\ &=-\frac{11 c^4 \left (1-a^2 x^2\right )^{7/2}}{56 a}-\frac{11 c^4 (1+a x) \left (1-a^2 x^2\right )^{7/2}}{72 a}-\frac{c^4 (1+a x)^2 \left (1-a^2 x^2\right )^{7/2}}{9 a}+\frac{1}{8} \left (11 c^4\right ) \int \left (1-a^2 x^2\right )^{5/2} \, dx\\ &=\frac{11}{48} c^4 x \left (1-a^2 x^2\right )^{5/2}-\frac{11 c^4 \left (1-a^2 x^2\right )^{7/2}}{56 a}-\frac{11 c^4 (1+a x) \left (1-a^2 x^2\right )^{7/2}}{72 a}-\frac{c^4 (1+a x)^2 \left (1-a^2 x^2\right )^{7/2}}{9 a}+\frac{1}{48} \left (55 c^4\right ) \int \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=\frac{55}{192} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac{11}{48} c^4 x \left (1-a^2 x^2\right )^{5/2}-\frac{11 c^4 \left (1-a^2 x^2\right )^{7/2}}{56 a}-\frac{11 c^4 (1+a x) \left (1-a^2 x^2\right )^{7/2}}{72 a}-\frac{c^4 (1+a x)^2 \left (1-a^2 x^2\right )^{7/2}}{9 a}+\frac{1}{64} \left (55 c^4\right ) \int \sqrt{1-a^2 x^2} \, dx\\ &=\frac{55}{128} c^4 x \sqrt{1-a^2 x^2}+\frac{55}{192} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac{11}{48} c^4 x \left (1-a^2 x^2\right )^{5/2}-\frac{11 c^4 \left (1-a^2 x^2\right )^{7/2}}{56 a}-\frac{11 c^4 (1+a x) \left (1-a^2 x^2\right )^{7/2}}{72 a}-\frac{c^4 (1+a x)^2 \left (1-a^2 x^2\right )^{7/2}}{9 a}+\frac{1}{128} \left (55 c^4\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{55}{128} c^4 x \sqrt{1-a^2 x^2}+\frac{55}{192} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac{11}{48} c^4 x \left (1-a^2 x^2\right )^{5/2}-\frac{11 c^4 \left (1-a^2 x^2\right )^{7/2}}{56 a}-\frac{11 c^4 (1+a x) \left (1-a^2 x^2\right )^{7/2}}{72 a}-\frac{c^4 (1+a x)^2 \left (1-a^2 x^2\right )^{7/2}}{9 a}+\frac{55 c^4 \sin ^{-1}(a x)}{128 a}\\ \end{align*}

Mathematica [A]  time = 0.146806, size = 107, normalized size = 0.65 \[ \frac{c^4 \left (\sqrt{1-a^2 x^2} \left (896 a^8 x^8+3024 a^7 x^7+1024 a^6 x^6-7224 a^5 x^5-8448 a^4 x^4+3066 a^3 x^3+10240 a^2 x^2+4599 a x-3712\right )-6930 \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{8064 a} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^4,x]

[Out]

(c^4*(Sqrt[1 - a^2*x^2]*(-3712 + 4599*a*x + 10240*a^2*x^2 + 3066*a^3*x^3 - 8448*a^4*x^4 - 7224*a^5*x^5 + 1024*
a^6*x^6 + 3024*a^7*x^7 + 896*a^8*x^8) - 6930*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(8064*a)

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Maple [A]  time = 0.24, size = 275, normalized size = 1.7 \begin{align*} -{\frac{3\,{c}^{4}{a}^{8}{x}^{9}}{8}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{61\,{a}^{6}{c}^{4}{x}^{7}}{48}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{245\,{a}^{4}{c}^{4}{x}^{5}}{192}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{73\,{a}^{2}{c}^{4}{x}^{3}}{384}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{{c}^{4}{a}^{9}{x}^{10}}{9}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{{c}^{4}{a}^{7}{x}^{8}}{63}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{74\,{c}^{4}{a}^{5}{x}^{6}}{63}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{146\,{c}^{4}{a}^{3}{x}^{4}}{63}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{109\,{c}^{4}a{x}^{2}}{63}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{29\,{c}^{4}}{63\,a}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{55\,{c}^{4}}{128}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{\frac{73\,{c}^{4}x}{128}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^4,x)

[Out]

-3/8*c^4*a^8*x^9/(-a^2*x^2+1)^(1/2)+61/48*c^4*a^6*x^7/(-a^2*x^2+1)^(1/2)-245/192*c^4*a^4*x^5/(-a^2*x^2+1)^(1/2
)-73/384*c^4*a^2*x^3/(-a^2*x^2+1)^(1/2)-1/9*c^4*a^9*x^10/(-a^2*x^2+1)^(1/2)-1/63*c^4*a^7*x^8/(-a^2*x^2+1)^(1/2
)+74/63*c^4*a^5*x^6/(-a^2*x^2+1)^(1/2)-146/63*c^4*a^3*x^4/(-a^2*x^2+1)^(1/2)+109/63*c^4*a*x^2/(-a^2*x^2+1)^(1/
2)-29/63*c^4/a/(-a^2*x^2+1)^(1/2)+55/128*c^4/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+73/128*c^4*x
/(-a^2*x^2+1)^(1/2)

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Maxima [A]  time = 1.47877, size = 358, normalized size = 2.17 \begin{align*} -\frac{a^{9} c^{4} x^{10}}{9 \, \sqrt{-a^{2} x^{2} + 1}} - \frac{3 \, a^{8} c^{4} x^{9}}{8 \, \sqrt{-a^{2} x^{2} + 1}} - \frac{a^{7} c^{4} x^{8}}{63 \, \sqrt{-a^{2} x^{2} + 1}} + \frac{61 \, a^{6} c^{4} x^{7}}{48 \, \sqrt{-a^{2} x^{2} + 1}} + \frac{74 \, a^{5} c^{4} x^{6}}{63 \, \sqrt{-a^{2} x^{2} + 1}} - \frac{245 \, a^{4} c^{4} x^{5}}{192 \, \sqrt{-a^{2} x^{2} + 1}} - \frac{146 \, a^{3} c^{4} x^{4}}{63 \, \sqrt{-a^{2} x^{2} + 1}} - \frac{73 \, a^{2} c^{4} x^{3}}{384 \, \sqrt{-a^{2} x^{2} + 1}} + \frac{109 \, a c^{4} x^{2}}{63 \, \sqrt{-a^{2} x^{2} + 1}} + \frac{73 \, c^{4} x}{128 \, \sqrt{-a^{2} x^{2} + 1}} + \frac{55 \, c^{4} \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{128 \, \sqrt{a^{2}}} - \frac{29 \, c^{4}}{63 \, \sqrt{-a^{2} x^{2} + 1} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^4,x, algorithm="maxima")

[Out]

-1/9*a^9*c^4*x^10/sqrt(-a^2*x^2 + 1) - 3/8*a^8*c^4*x^9/sqrt(-a^2*x^2 + 1) - 1/63*a^7*c^4*x^8/sqrt(-a^2*x^2 + 1
) + 61/48*a^6*c^4*x^7/sqrt(-a^2*x^2 + 1) + 74/63*a^5*c^4*x^6/sqrt(-a^2*x^2 + 1) - 245/192*a^4*c^4*x^5/sqrt(-a^
2*x^2 + 1) - 146/63*a^3*c^4*x^4/sqrt(-a^2*x^2 + 1) - 73/384*a^2*c^4*x^3/sqrt(-a^2*x^2 + 1) + 109/63*a*c^4*x^2/
sqrt(-a^2*x^2 + 1) + 73/128*c^4*x/sqrt(-a^2*x^2 + 1) + 55/128*c^4*arcsin(a^2*x/sqrt(a^2))/sqrt(a^2) - 29/63*c^
4/(sqrt(-a^2*x^2 + 1)*a)

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Fricas [A]  time = 2.7294, size = 325, normalized size = 1.97 \begin{align*} -\frac{6930 \, c^{4} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) -{\left (896 \, a^{8} c^{4} x^{8} + 3024 \, a^{7} c^{4} x^{7} + 1024 \, a^{6} c^{4} x^{6} - 7224 \, a^{5} c^{4} x^{5} - 8448 \, a^{4} c^{4} x^{4} + 3066 \, a^{3} c^{4} x^{3} + 10240 \, a^{2} c^{4} x^{2} + 4599 \, a c^{4} x - 3712 \, c^{4}\right )} \sqrt{-a^{2} x^{2} + 1}}{8064 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^4,x, algorithm="fricas")

[Out]

-1/8064*(6930*c^4*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (896*a^8*c^4*x^8 + 3024*a^7*c^4*x^7 + 1024*a^6*c^4*
x^6 - 7224*a^5*c^4*x^5 - 8448*a^4*c^4*x^4 + 3066*a^3*c^4*x^3 + 10240*a^2*c^4*x^2 + 4599*a*c^4*x - 3712*c^4)*sq
rt(-a^2*x^2 + 1))/a

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Sympy [C]  time = 39.6861, size = 996, normalized size = 6.04 \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)**3/(-a**2*x**2+1)**(3/2)*(-a**2*c*x**2+c)**4,x)

[Out]

a**7*c**4*Piecewise((x**8*sqrt(-a**2*x**2 + 1)/9 - x**6*sqrt(-a**2*x**2 + 1)/(63*a**2) - 2*x**4*sqrt(-a**2*x**
2 + 1)/(105*a**4) - 8*x**2*sqrt(-a**2*x**2 + 1)/(315*a**6) - 16*sqrt(-a**2*x**2 + 1)/(315*a**8), Ne(a, 0)), (x
**8/8, True)) + 3*a**6*c**4*Piecewise((I*a**2*x**9/(8*sqrt(a**2*x**2 - 1)) - 7*I*x**7/(48*sqrt(a**2*x**2 - 1))
- I*x**5/(192*a**2*sqrt(a**2*x**2 - 1)) - 5*I*x**3/(384*a**4*sqrt(a**2*x**2 - 1)) + 5*I*x/(128*a**6*sqrt(a**2
*x**2 - 1)) - 5*I*acosh(a*x)/(128*a**7), Abs(a**2*x**2) > 1), (-a**2*x**9/(8*sqrt(-a**2*x**2 + 1)) + 7*x**7/(4
8*sqrt(-a**2*x**2 + 1)) + x**5/(192*a**2*sqrt(-a**2*x**2 + 1)) + 5*x**3/(384*a**4*sqrt(-a**2*x**2 + 1)) - 5*x/
(128*a**6*sqrt(-a**2*x**2 + 1)) + 5*asin(a*x)/(128*a**7), True)) + a**5*c**4*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)) - 5*a**4*c**4*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)) - 5*a**
3*c**4*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)) + a**2*c**4*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**4*Piecewise((x**2/2, Eq(a**2, 0)), (-(-a**2*x**2 + 1)**(3/2)/(3*a**2), True)) + c**4*
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 = 1.18579, size = 169, normalized size = 1.02 \begin{align*} \frac{55 \, c^{4} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{128 \,{\left | a \right |}} - \frac{1}{8064} \, \sqrt{-a^{2} x^{2} + 1}{\left (\frac{3712 \, c^{4}}{a} -{\left (4599 \, c^{4} + 2 \,{\left (5120 \, a c^{4} +{\left (1533 \, a^{2} c^{4} - 4 \,{\left (1056 \, a^{3} c^{4} +{\left (903 \, a^{4} c^{4} - 2 \,{\left (64 \, a^{5} c^{4} + 7 \,{\left (8 \, a^{7} c^{4} x + 27 \, a^{6} c^{4}\right )} x\right )} x\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*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^4,x, algorithm="giac")

[Out]

55/128*c^4*arcsin(a*x)*sgn(a)/abs(a) - 1/8064*sqrt(-a^2*x^2 + 1)*(3712*c^4/a - (4599*c^4 + 2*(5120*a*c^4 + (15
33*a^2*c^4 - 4*(1056*a^3*c^4 + (903*a^4*c^4 - 2*(64*a^5*c^4 + 7*(8*a^7*c^4*x + 27*a^6*c^4)*x)*x)*x)*x)*x)*x)*x
)