3.1192 \(\int e^{-\tanh ^{-1}(a x)} (c-a^2 c x^2)^4 \, dx\)

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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6139, 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[(c - a^2*c*x^2)^4/E^ArcTanh[a*x],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 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]

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 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]

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 = 0.15, 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[(c - a^2*c*x^2)^4/E^ArcTanh[a*x],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)

________________________________________________________________________________________

fricas [A]  time = 0.66, size = 137, normalized size = 1.08 \[ -\frac {630 \, c^{4} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (128 \, a^{8} c^{4} x^{8} - 144 \, a^{7} c^{4} x^{7} - 512 \, a^{6} c^{4} x^{6} + 600 \, a^{5} c^{4} x^{5} + 768 \, a^{4} c^{4} x^{4} - 978 \, a^{3} c^{4} x^{3} - 512 \, a^{2} c^{4} x^{2} + 837 \, a c^{4} x + 128 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1}}{1152 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

giac [A]  time = 0.20, size = 124, normalized size = 0.98 \[ \frac {35 \, c^{4} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{128 \, {\left | a \right |}} + \frac {1}{1152} \, \sqrt {-a^{2} x^{2} + 1} {\left (\frac {128 \, c^{4}}{a} + {\left (837 \, c^{4} - 2 \, {\left (256 \, a c^{4} + {\left (489 \, a^{2} c^{4} - 4 \, {\left (96 \, a^{3} c^{4} + {\left (75 \, a^{4} c^{4} - 2 \, {\left (32 \, a^{5} c^{4} - {\left (8 \, a^{7} c^{4} x - 9 \, a^{6} c^{4}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

maple [A]  time = 0.07, size = 201, normalized size = 1.58 \[ -\frac {c^{4} a^{5} x^{6} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{9}+\frac {c^{4} a^{3} x^{4} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3}-\frac {c^{4} a \,x^{2} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3}+\frac {c^{4} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{9 a}+\frac {c^{4} a^{4} x^{5} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{8}-\frac {19 c^{4} a^{2} x^{3} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{48}+\frac {29 c^{4} x \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{64}+\frac {35 c^{4} x \sqrt {-a^{2} x^{2}+1}}{128}+\frac {35 c^{4} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{128 \sqrt {a^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*c^4*a^5*x^6*(-a^2*x^2+1)^(3/2)+1/3*c^4*a^3*x^4*(-a^2*x^2+1)^(3/2)-1/3*c^4*a*x^2*(-a^2*x^2+1)^(3/2)+1/9*c^
4*(-a^2*x^2+1)^(3/2)/a+1/8*c^4*a^4*x^5*(-a^2*x^2+1)^(3/2)-19/48*c^4*a^2*x^3*(-a^2*x^2+1)^(3/2)+29/64*c^4*x*(-a
^2*x^2+1)^(3/2)+35/128*c^4*x*(-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)
)

________________________________________________________________________________________

maxima [A]  time = 0.43, size = 182, normalized size = 1.43 \[ -\frac {1}{9} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{5} c^{4} x^{6} + \frac {1}{8} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{4} c^{4} x^{5} + \frac {1}{3} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{3} c^{4} x^{4} - \frac {19}{48} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2} c^{4} x^{3} - \frac {1}{3} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a c^{4} x^{2} + \frac {29}{64} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{4} x + \frac {35}{128} \, \sqrt {-a^{2} x^{2} + 1} c^{4} x + \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{4}}{9 \, a} + \frac {35 \, c^{4} \arcsin \left (a x\right )}{128 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*(-a^2*x^2 + 1)^(3/2)*a^5*c^4*x^6 + 1/8*(-a^2*x^2 + 1)^(3/2)*a^4*c^4*x^5 + 1/3*(-a^2*x^2 + 1)^(3/2)*a^3*c^
4*x^4 - 19/48*(-a^2*x^2 + 1)^(3/2)*a^2*c^4*x^3 - 1/3*(-a^2*x^2 + 1)^(3/2)*a*c^4*x^2 + 29/64*(-a^2*x^2 + 1)^(3/
2)*c^4*x + 35/128*sqrt(-a^2*x^2 + 1)*c^4*x + 1/9*(-a^2*x^2 + 1)^(3/2)*c^4/a + 35/128*c^4*arcsin(a*x)/a

________________________________________________________________________________________

mupad [B]  time = 0.93, size = 220, normalized size = 1.73 \[ \frac {93\,c^4\,x\,\sqrt {1-a^2\,x^2}}{128}+\frac {35\,c^4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{128\,\sqrt {-a^2}}+\frac {c^4\,\sqrt {1-a^2\,x^2}}{9\,a}-\frac {4\,a\,c^4\,x^2\,\sqrt {1-a^2\,x^2}}{9}-\frac {163\,a^2\,c^4\,x^3\,\sqrt {1-a^2\,x^2}}{192}+\frac {2\,a^3\,c^4\,x^4\,\sqrt {1-a^2\,x^2}}{3}+\frac {25\,a^4\,c^4\,x^5\,\sqrt {1-a^2\,x^2}}{48}-\frac {4\,a^5\,c^4\,x^6\,\sqrt {1-a^2\,x^2}}{9}-\frac {a^6\,c^4\,x^7\,\sqrt {1-a^2\,x^2}}{8}+\frac {a^7\,c^4\,x^8\,\sqrt {1-a^2\,x^2}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [C]  time = 20.60, size = 996, normalized size = 7.84 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a**2*c*x**2+c)**4/(a*x+1)*(-a**2*x**2+1)**(1/2),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)) - 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/(48*
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/(1
28*a**6*sqrt(-a**2*x**2 + 1)) + 5*asin(a*x)/(128*a**7), True)) - 3*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)) + 3*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)) + 3*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)) - 3*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)) - 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))

________________________________________________________________________________________