3.1254 \(\int e^{-3 \tanh ^{-1}(a x)} (c-a^2 c x^2)^4 \, dx\)

Optimal. Leaf size=167 \[ \frac {c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{7/2}}{9 a}+\frac {11 c^4 (1-a x) \left (1-a^2 x^2\right )^{7/2}}{72 a}+\frac {11 c^4 \left (1-a^2 x^2\right )^{7/2}}{56 a}+\frac {11}{48} c^4 x \left (1-a^2 x^2\right )^{5/2}+\frac {55}{192} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac {55}{128} c^4 x \sqrt {1-a^2 x^2}+\frac {55 c^4 \sin ^{-1}(a x)}{128 a} \]

[Out]

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

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Rubi [A]  time = 0.09, antiderivative size = 167, normalized size of antiderivative = 1.00, 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 = {6139, 671, 641, 195, 216} \[ \frac {c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{7/2}}{9 a}+\frac {11 c^4 (1-a x) \left (1-a^2 x^2\right )^{7/2}}{72 a}+\frac {11 c^4 \left (1-a^2 x^2\right )^{7/2}}{56 a}+\frac {11}{48} c^4 x \left (1-a^2 x^2\right )^{5/2}+\frac {55}{192} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac {55}{128} c^4 x \sqrt {1-a^2 x^2}+\frac {55 c^4 \sin ^{-1}(a x)}{128 a} \]

Antiderivative was successfully verified.

[In]

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

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Mathematica [A]  time = 0.18, size = 107, normalized size = 0.64 \[ -\frac {c^4 \left (\sqrt {1-a^2 x^2} \left (896 a^8 x^8-3024 a^7 x^7+1024 a^6 x^6+7224 a^5 x^5-8448 a^4 x^4-3066 a^3 x^3+10240 a^2 x^2-4599 a x-3712\right )+6930 \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{8064 a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/8064*(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]]))/a

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fricas [A]  time = 0.58, size = 136, normalized size = 0.81 \[ -\frac {6930 \, c^{4} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (896 \, a^{8} c^{4} x^{8} - 3024 \, a^{7} c^{4} x^{7} + 1024 \, a^{6} c^{4} x^{6} + 7224 \, a^{5} c^{4} x^{5} - 8448 \, a^{4} c^{4} x^{4} - 3066 \, a^{3} c^{4} x^{3} + 10240 \, a^{2} c^{4} x^{2} - 4599 \, a c^{4} x - 3712 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1}}{8064 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)^4/(a*x+1)^3*(-a^2*x^2+1)^(3/2),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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giac [A]  time = 0.32, size = 126, normalized size = 0.75 \[ \frac {55 \, c^{4} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{128 \, {\left | a \right |}} + \frac {1}{8064} \, \sqrt {-a^{2} x^{2} + 1} {\left (\frac {3712 \, c^{4}}{a} + {\left (4599 \, c^{4} - 2 \, {\left (5120 \, a c^{4} - {\left (1533 \, a^{2} c^{4} + 4 \, {\left (1056 \, a^{3} c^{4} - {\left (903 \, a^{4} c^{4} + 2 \, {\left (64 \, a^{5} c^{4} + 7 \, {\left (8 \, a^{7} c^{4} x - 27 \, a^{6} c^{4}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maple [A]  time = 0.05, size = 173, normalized size = 1.04 \[ -\frac {c^{4} a^{3} x^{4} \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{9}-\frac {22 c^{4} a \,x^{2} \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{63}+\frac {29 c^{4} \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{63 a}+\frac {3 c^{4} a^{2} x^{3} \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{8}-\frac {7 c^{4} x \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{48}+\frac {55 c^{4} x \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{192}+\frac {55 c^{4} x \sqrt {-a^{2} x^{2}+1}}{128}+\frac {55 c^{4} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{128 \sqrt {a^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*c^4*a^3*x^4*(-a^2*x^2+1)^(5/2)-22/63*c^4*a*x^2*(-a^2*x^2+1)^(5/2)+29/63*c^4*(-a^2*x^2+1)^(5/2)/a+3/8*c^4*
a^2*x^3*(-a^2*x^2+1)^(5/2)-7/48*c^4*x*(-a^2*x^2+1)^(5/2)+55/192*c^4*x*(-a^2*x^2+1)^(3/2)+55/128*c^4*x*(-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))

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maxima [A]  time = 0.44, size = 154, normalized size = 0.92 \[ -\frac {1}{9} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} a^{3} c^{4} x^{4} + \frac {3}{8} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} a^{2} c^{4} x^{3} - \frac {22}{63} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} a c^{4} x^{2} - \frac {7}{48} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} c^{4} x + \frac {55}{192} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{4} x + \frac {29 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} c^{4}}{63 \, a} + \frac {55}{128} \, \sqrt {-a^{2} x^{2} + 1} c^{4} x + \frac {55 \, c^{4} \arcsin \left (a x\right )}{128 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*(-a^2*x^2 + 1)^(5/2)*a^3*c^4*x^4 + 3/8*(-a^2*x^2 + 1)^(5/2)*a^2*c^4*x^3 - 22/63*(-a^2*x^2 + 1)^(5/2)*a*c^
4*x^2 - 7/48*(-a^2*x^2 + 1)^(5/2)*c^4*x + 55/192*(-a^2*x^2 + 1)^(3/2)*c^4*x + 29/63*(-a^2*x^2 + 1)^(5/2)*c^4/a
+ 55/128*sqrt(-a^2*x^2 + 1)*c^4*x + 55/128*c^4*arcsin(a*x)/a

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mupad [B]  time = 0.07, size = 220, normalized size = 1.32 \[ \frac {73\,c^4\,x\,\sqrt {1-a^2\,x^2}}{128}+\frac {55\,c^4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{128\,\sqrt {-a^2}}+\frac {29\,c^4\,\sqrt {1-a^2\,x^2}}{63\,a}-\frac {80\,a\,c^4\,x^2\,\sqrt {1-a^2\,x^2}}{63}+\frac {73\,a^2\,c^4\,x^3\,\sqrt {1-a^2\,x^2}}{192}+\frac {22\,a^3\,c^4\,x^4\,\sqrt {1-a^2\,x^2}}{21}-\frac {43\,a^4\,c^4\,x^5\,\sqrt {1-a^2\,x^2}}{48}-\frac {8\,a^5\,c^4\,x^6\,\sqrt {1-a^2\,x^2}}{63}+\frac {3\,a^6\,c^4\,x^7\,\sqrt {1-a^2\,x^2}}{8}-\frac {a^7\,c^4\,x^8\,\sqrt {1-a^2\,x^2}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(73*c^4*x*(1 - a^2*x^2)^(1/2))/128 + (55*c^4*asinh(x*(-a^2)^(1/2)))/(128*(-a^2)^(1/2)) + (29*c^4*(1 - a^2*x^2)
^(1/2))/(63*a) - (80*a*c^4*x^2*(1 - a^2*x^2)^(1/2))/63 + (73*a^2*c^4*x^3*(1 - a^2*x^2)^(1/2))/192 + (22*a^3*c^
4*x^4*(1 - a^2*x^2)^(1/2))/21 - (43*a^4*c^4*x^5*(1 - a^2*x^2)^(1/2))/48 - (8*a^5*c^4*x^6*(1 - a^2*x^2)^(1/2))/
63 + (3*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

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sympy [C]  time = 20.38, size = 996, normalized size = 5.96 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a**2*c*x**2+c)**4/(a*x+1)**3*(-a**2*x**2+1)**(3/2),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)) + 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/(
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
/(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*acos
h(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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