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

Optimal. Leaf size=127 \[ -\frac{c^4 \left (1-a^2 x^2\right )^{9/2}}{9 a}+\frac{1}{8} c^4 x \left (1-a^2 x^2\right )^{7/2}+\frac{7}{48} c^4 x \left (1-a^2 x^2\right )^{5/2}+\frac{35}{192} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac{35}{128} c^4 x \sqrt{1-a^2 x^2}+\frac{35 c^4 \sin ^{-1}(a x)}{128 a} \]

[Out]

(35*c^4*x*Sqrt[1 - a^2*x^2])/128 + (35*c^4*x*(1 - a^2*x^2)^(3/2))/192 + (7*c^4*x*(1 - a^2*x^2)^(5/2))/48 + (c^
4*x*(1 - a^2*x^2)^(7/2))/8 - (c^4*(1 - a^2*x^2)^(9/2))/(9*a) + (35*c^4*ArcSin[a*x])/(128*a)

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Rubi [A]  time = 0.0640731, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6138, 641, 195, 216} \[ -\frac{c^4 \left (1-a^2 x^2\right )^{9/2}}{9 a}+\frac{1}{8} c^4 x \left (1-a^2 x^2\right )^{7/2}+\frac{7}{48} c^4 x \left (1-a^2 x^2\right )^{5/2}+\frac{35}{192} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac{35}{128} c^4 x \sqrt{1-a^2 x^2}+\frac{35 c^4 \sin ^{-1}(a x)}{128 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(35*c^4*x*Sqrt[1 - a^2*x^2])/128 + (35*c^4*x*(1 - a^2*x^2)^(3/2))/192 + (7*c^4*x*(1 - a^2*x^2)^(5/2))/48 + (c^
4*x*(1 - a^2*x^2)^(7/2))/8 - (c^4*(1 - a^2*x^2)^(9/2))/(9*a) + (35*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 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)} \left (c-a^2 c x^2\right )^4 \, dx &=c^4 \int (1+a x) \left (1-a^2 x^2\right )^{7/2} \, dx\\ &=-\frac{c^4 \left (1-a^2 x^2\right )^{9/2}}{9 a}+c^4 \int \left (1-a^2 x^2\right )^{7/2} \, dx\\ &=\frac{1}{8} c^4 x \left (1-a^2 x^2\right )^{7/2}-\frac{c^4 \left (1-a^2 x^2\right )^{9/2}}{9 a}+\frac{1}{8} \left (7 c^4\right ) \int \left (1-a^2 x^2\right )^{5/2} \, dx\\ &=\frac{7}{48} c^4 x \left (1-a^2 x^2\right )^{5/2}+\frac{1}{8} c^4 x \left (1-a^2 x^2\right )^{7/2}-\frac{c^4 \left (1-a^2 x^2\right )^{9/2}}{9 a}+\frac{1}{48} \left (35 c^4\right ) \int \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=\frac{35}{192} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac{7}{48} c^4 x \left (1-a^2 x^2\right )^{5/2}+\frac{1}{8} c^4 x \left (1-a^2 x^2\right )^{7/2}-\frac{c^4 \left (1-a^2 x^2\right )^{9/2}}{9 a}+\frac{1}{64} \left (35 c^4\right ) \int \sqrt{1-a^2 x^2} \, dx\\ &=\frac{35}{128} c^4 x \sqrt{1-a^2 x^2}+\frac{35}{192} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac{7}{48} c^4 x \left (1-a^2 x^2\right )^{5/2}+\frac{1}{8} c^4 x \left (1-a^2 x^2\right )^{7/2}-\frac{c^4 \left (1-a^2 x^2\right )^{9/2}}{9 a}+\frac{1}{128} \left (35 c^4\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{35}{128} c^4 x \sqrt{1-a^2 x^2}+\frac{35}{192} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac{7}{48} c^4 x \left (1-a^2 x^2\right )^{5/2}+\frac{1}{8} c^4 x \left (1-a^2 x^2\right )^{7/2}-\frac{c^4 \left (1-a^2 x^2\right )^{9/2}}{9 a}+\frac{35 c^4 \sin ^{-1}(a x)}{128 a}\\ \end{align*}

Mathematica [A]  time = 0.144968, size = 107, normalized size = 0.84 \[ -\frac{c^4 \left (\sqrt{1-a^2 x^2} \left (128 a^8 x^8+144 a^7 x^7-512 a^6 x^6-600 a^5 x^5+768 a^4 x^4+978 a^3 x^3-512 a^2 x^2-837 a x+128\right )+630 \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{1152 a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(c^4*(Sqrt[1 - a^2*x^2]*(128 - 837*a*x - 512*a^2*x^2 + 978*a^3*x^3 + 768*a^4*x^4 - 600*a^5*x^5 - 512*a^6*x^6
+ 144*a^7*x^7 + 128*a^8*x^8) + 630*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(1152*a)

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Maple [B]  time = 0.088, size = 229, normalized size = 1.8 \begin{align*} -{\frac{{a}^{6}{c}^{4}{x}^{7}}{8}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{25\,{a}^{4}{c}^{4}{x}^{5}}{48}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{163\,{a}^{2}{c}^{4}{x}^{3}}{192}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{{c}^{4}{a}^{7}{x}^{8}}{9}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{4\,{c}^{4}{a}^{5}{x}^{6}}{9}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{2\,{c}^{4}{a}^{3}{x}^{4}}{3}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{4\,{c}^{4}a{x}^{2}}{9}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{35\,{c}^{4}}{128}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{\frac{93\,{c}^{4}x}{128}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{{c}^{4}}{9\,a}\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)*(-a^2*c*x^2+c)^4,x)

[Out]

-1/8*c^4*a^6*x^7*(-a^2*x^2+1)^(1/2)+25/48*c^4*a^4*x^5*(-a^2*x^2+1)^(1/2)-163/192*c^4*a^2*x^3*(-a^2*x^2+1)^(1/2
)-1/9*c^4*a^7*x^8*(-a^2*x^2+1)^(1/2)+4/9*c^4*a^5*x^6*(-a^2*x^2+1)^(1/2)-2/3*c^4*a^3*x^4*(-a^2*x^2+1)^(1/2)+4/9
*c^4*a*x^2*(-a^2*x^2+1)^(1/2)+35/128*c^4/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+93/128*c^4*x*(-a
^2*x^2+1)^(1/2)-1/9*c^4*(-a^2*x^2+1)^(1/2)/a

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Maxima [B]  time = 1.47239, size = 296, normalized size = 2.33 \begin{align*} -\frac{1}{9} \, \sqrt{-a^{2} x^{2} + 1} a^{7} c^{4} x^{8} - \frac{1}{8} \, \sqrt{-a^{2} x^{2} + 1} a^{6} c^{4} x^{7} + \frac{4}{9} \, \sqrt{-a^{2} x^{2} + 1} a^{5} c^{4} x^{6} + \frac{25}{48} \, \sqrt{-a^{2} x^{2} + 1} a^{4} c^{4} x^{5} - \frac{2}{3} \, \sqrt{-a^{2} x^{2} + 1} a^{3} c^{4} x^{4} - \frac{163}{192} \, \sqrt{-a^{2} x^{2} + 1} a^{2} c^{4} x^{3} + \frac{4}{9} \, \sqrt{-a^{2} x^{2} + 1} a c^{4} x^{2} + \frac{93}{128} \, \sqrt{-a^{2} x^{2} + 1} c^{4} x + \frac{35 \, c^{4} \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{128 \, \sqrt{a^{2}}} - \frac{\sqrt{-a^{2} x^{2} + 1} c^{4}}{9 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*sqrt(-a^2*x^2 + 1)*a^7*c^4*x^8 - 1/8*sqrt(-a^2*x^2 + 1)*a^6*c^4*x^7 + 4/9*sqrt(-a^2*x^2 + 1)*a^5*c^4*x^6
+ 25/48*sqrt(-a^2*x^2 + 1)*a^4*c^4*x^5 - 2/3*sqrt(-a^2*x^2 + 1)*a^3*c^4*x^4 - 163/192*sqrt(-a^2*x^2 + 1)*a^2*c
^4*x^3 + 4/9*sqrt(-a^2*x^2 + 1)*a*c^4*x^2 + 93/128*sqrt(-a^2*x^2 + 1)*c^4*x + 35/128*c^4*arcsin(a^2*x/sqrt(a^2
))/sqrt(a^2) - 1/9*sqrt(-a^2*x^2 + 1)*c^4/a

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Fricas [A]  time = 1.58702, size = 312, normalized size = 2.46 \begin{align*} -\frac{630 \, c^{4} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) +{\left (128 \, a^{8} c^{4} x^{8} + 144 \, a^{7} c^{4} x^{7} - 512 \, a^{6} c^{4} x^{6} - 600 \, a^{5} c^{4} x^{5} + 768 \, a^{4} c^{4} x^{4} + 978 \, a^{3} c^{4} x^{3} - 512 \, a^{2} c^{4} x^{2} - 837 \, a c^{4} x + 128 \, c^{4}\right )} \sqrt{-a^{2} x^{2} + 1}}{1152 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1152*(630*c^4*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (128*a^8*c^4*x^8 + 144*a^7*c^4*x^7 - 512*a^6*c^4*x^6
- 600*a^5*c^4*x^5 + 768*a^4*c^4*x^4 + 978*a^3*c^4*x^3 - 512*a^2*c^4*x^2 - 837*a*c^4*x + 128*c^4)*sqrt(-a^2*x^
2 + 1))/a

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Sympy [A]  time = 20.9683, size = 454, normalized size = 3.57 \begin{align*} \begin{cases} - \frac{\frac{c^{4} \left (- a^{2} x^{2} + 1\right )^{\frac{3}{2}}}{3} - c^{4} \left (\begin{cases} \frac{a x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\operatorname{asin}{\left (a x \right )}}{2} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right ) + 3 c^{4} \left (\begin{cases} - \frac{a x \left (- 2 a^{2} x^{2} + 1\right ) \sqrt{- a^{2} x^{2} + 1}}{8} + \frac{\operatorname{asin}{\left (a x \right )}}{8} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right ) + 3 c^{4} \left (\begin{cases} \frac{\left (- a^{2} x^{2} + 1\right )^{\frac{5}{2}}}{5} - \frac{\left (- a^{2} x^{2} + 1\right )^{\frac{3}{2}}}{3} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right ) - 3 c^{4} \left (\begin{cases} - \frac{a^{3} x^{3} \left (- a^{2} x^{2} + 1\right )^{\frac{3}{2}}}{6} - \frac{a x \left (- 2 a^{2} x^{2} + 1\right ) \sqrt{- a^{2} x^{2} + 1}}{16} + \frac{\operatorname{asin}{\left (a x \right )}}{16} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right ) - 3 c^{4} \left (\begin{cases} - \frac{\left (- a^{2} x^{2} + 1\right )^{\frac{7}{2}}}{7} + \frac{2 \left (- a^{2} x^{2} + 1\right )^{\frac{5}{2}}}{5} - \frac{\left (- a^{2} x^{2} + 1\right )^{\frac{3}{2}}}{3} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right ) + c^{4} \left (\begin{cases} - \frac{a^{3} x^{3} \left (- a^{2} x^{2} + 1\right )^{\frac{3}{2}}}{6} - \frac{a x \left (- 2 a^{2} x^{2} + 1\right ) \sqrt{- a^{2} x^{2} + 1}}{32} - \frac{a x \sqrt{- a^{2} x^{2} + 1} \left (- 16 a^{6} x^{6} + 24 a^{4} x^{4} - 10 a^{2} x^{2} + 1\right )}{128} + \frac{5 \operatorname{asin}{\left (a x \right )}}{128} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right ) + c^{4} \left (\begin{cases} \frac{\left (- a^{2} x^{2} + 1\right )^{\frac{9}{2}}}{9} - \frac{3 \left (- a^{2} x^{2} + 1\right )^{\frac{7}{2}}}{7} + \frac{3 \left (- a^{2} x^{2} + 1\right )^{\frac{5}{2}}}{5} - \frac{\left (- a^{2} x^{2} + 1\right )^{\frac{3}{2}}}{3} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right )}{a} & \text{for}\: a \neq 0 \\c^{4} x & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-(c**4*(-a**2*x**2 + 1)**(3/2)/3 - c**4*Piecewise((a*x*sqrt(-a**2*x**2 + 1)/2 + asin(a*x)/2, (a*x >
-1) & (a*x < 1))) + 3*c**4*Piecewise((-a*x*(-2*a**2*x**2 + 1)*sqrt(-a**2*x**2 + 1)/8 + asin(a*x)/8, (a*x > -1
) & (a*x < 1))) + 3*c**4*Piecewise(((-a**2*x**2 + 1)**(5/2)/5 - (-a**2*x**2 + 1)**(3/2)/3, (a*x > -1) & (a*x <
1))) - 3*c**4*Piecewise((-a**3*x**3*(-a**2*x**2 + 1)**(3/2)/6 - a*x*(-2*a**2*x**2 + 1)*sqrt(-a**2*x**2 + 1)/1
6 + asin(a*x)/16, (a*x > -1) & (a*x < 1))) - 3*c**4*Piecewise((-(-a**2*x**2 + 1)**(7/2)/7 + 2*(-a**2*x**2 + 1)
**(5/2)/5 - (-a**2*x**2 + 1)**(3/2)/3, (a*x > -1) & (a*x < 1))) + c**4*Piecewise((-a**3*x**3*(-a**2*x**2 + 1)*
*(3/2)/6 - a*x*(-2*a**2*x**2 + 1)*sqrt(-a**2*x**2 + 1)/32 - a*x*sqrt(-a**2*x**2 + 1)*(-16*a**6*x**6 + 24*a**4*
x**4 - 10*a**2*x**2 + 1)/128 + 5*asin(a*x)/128, (a*x > -1) & (a*x < 1))) + c**4*Piecewise(((-a**2*x**2 + 1)**(
9/2)/9 - 3*(-a**2*x**2 + 1)**(7/2)/7 + 3*(-a**2*x**2 + 1)**(5/2)/5 - (-a**2*x**2 + 1)**(3/2)/3, (a*x > -1) & (
a*x < 1))))/a, Ne(a, 0)), (c**4*x, True))

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Giac [A]  time = 1.19874, size = 171, normalized size = 1.35 \begin{align*} \frac{35 \, c^{4} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{128 \,{\left | a \right |}} - \frac{1}{1152} \, \sqrt{-a^{2} x^{2} + 1}{\left (\frac{128 \, c^{4}}{a} -{\left (837 \, c^{4} + 2 \,{\left (256 \, a c^{4} -{\left (489 \, a^{2} c^{4} + 4 \,{\left (96 \, a^{3} c^{4} -{\left (75 \, a^{4} c^{4} + 2 \,{\left (32 \, a^{5} c^{4} -{\left (8 \, a^{7} c^{4} x + 9 \, 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)/(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^4,x, algorithm="giac")

[Out]

35/128*c^4*arcsin(a*x)*sgn(a)/abs(a) - 1/1152*sqrt(-a^2*x^2 + 1)*(128*c^4/a - (837*c^4 + 2*(256*a*c^4 - (489*a
^2*c^4 + 4*(96*a^3*c^4 - (75*a^4*c^4 + 2*(32*a^5*c^4 - (8*a^7*c^4*x + 9*a^6*c^4)*x)*x)*x)*x)*x)*x)*x)