3.1193 \(\int e^{-\tanh ^{-1}(a x)} (c-a^2 c x^2)^3 \, dx\)

Optimal. Leaf size=105 \[ \frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{7 a}+\frac {1}{6} c^3 x \left (1-a^2 x^2\right )^{5/2}+\frac {5}{24} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac {5}{16} c^3 x \sqrt {1-a^2 x^2}+\frac {5 c^3 \sin ^{-1}(a x)}{16 a} \]

[Out]

5/24*c^3*x*(-a^2*x^2+1)^(3/2)+1/6*c^3*x*(-a^2*x^2+1)^(5/2)+1/7*c^3*(-a^2*x^2+1)^(7/2)/a+5/16*c^3*arcsin(a*x)/a
+5/16*c^3*x*(-a^2*x^2+1)^(1/2)

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Rubi [A]  time = 0.06, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6139, 641, 195, 216} \[ \frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{7 a}+\frac {1}{6} c^3 x \left (1-a^2 x^2\right )^{5/2}+\frac {5}{24} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac {5}{16} c^3 x \sqrt {1-a^2 x^2}+\frac {5 c^3 \sin ^{-1}(a x)}{16 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*c^3*x*Sqrt[1 - a^2*x^2])/16 + (5*c^3*x*(1 - a^2*x^2)^(3/2))/24 + (c^3*x*(1 - a^2*x^2)^(5/2))/6 + (c^3*(1 -
a^2*x^2)^(7/2))/(7*a) + (5*c^3*ArcSin[a*x])/(16*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 )^3 \, dx &=c^3 \int (1-a x) \left (1-a^2 x^2\right )^{5/2} \, dx\\ &=\frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{7 a}+c^3 \int \left (1-a^2 x^2\right )^{5/2} \, dx\\ &=\frac {1}{6} c^3 x \left (1-a^2 x^2\right )^{5/2}+\frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{7 a}+\frac {1}{6} \left (5 c^3\right ) \int \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=\frac {5}{24} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac {1}{6} c^3 x \left (1-a^2 x^2\right )^{5/2}+\frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{7 a}+\frac {1}{8} \left (5 c^3\right ) \int \sqrt {1-a^2 x^2} \, dx\\ &=\frac {5}{16} c^3 x \sqrt {1-a^2 x^2}+\frac {5}{24} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac {1}{6} c^3 x \left (1-a^2 x^2\right )^{5/2}+\frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{7 a}+\frac {1}{16} \left (5 c^3\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {5}{16} c^3 x \sqrt {1-a^2 x^2}+\frac {5}{24} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac {1}{6} c^3 x \left (1-a^2 x^2\right )^{5/2}+\frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{7 a}+\frac {5 c^3 \sin ^{-1}(a x)}{16 a}\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 91, normalized size = 0.87 \[ -\frac {c^3 \left (\sqrt {1-a^2 x^2} \left (48 a^6 x^6-56 a^5 x^5-144 a^4 x^4+182 a^3 x^3+144 a^2 x^2-231 a x-48\right )+210 \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{336 a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/336*(c^3*(Sqrt[1 - a^2*x^2]*(-48 - 231*a*x + 144*a^2*x^2 + 182*a^3*x^3 - 144*a^4*x^4 - 56*a^5*x^5 + 48*a^6*
x^6) + 210*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/a

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fricas [A]  time = 0.72, size = 114, normalized size = 1.09 \[ -\frac {210 \, c^{3} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (48 \, a^{6} c^{3} x^{6} - 56 \, a^{5} c^{3} x^{5} - 144 \, a^{4} c^{3} x^{4} + 182 \, a^{3} c^{3} x^{3} + 144 \, a^{2} c^{3} x^{2} - 231 \, a c^{3} x - 48 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1}}{336 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/336*(210*c^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (48*a^6*c^3*x^6 - 56*a^5*c^3*x^5 - 144*a^4*c^3*x^4 +
182*a^3*c^3*x^3 + 144*a^2*c^3*x^2 - 231*a*c^3*x - 48*c^3)*sqrt(-a^2*x^2 + 1))/a

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giac [A]  time = 0.21, size = 101, normalized size = 0.96 \[ \frac {5 \, c^{3} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{16 \, {\left | a \right |}} + \frac {1}{336} \, \sqrt {-a^{2} x^{2} + 1} {\left (\frac {48 \, c^{3}}{a} + {\left (231 \, c^{3} - 2 \, {\left (72 \, a c^{3} + {\left (91 \, a^{2} c^{3} - 4 \, {\left (18 \, a^{3} c^{3} - {\left (6 \, a^{5} c^{3} x - 7 \, a^{4} c^{3}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/16*c^3*arcsin(a*x)*sgn(a)/abs(a) + 1/336*sqrt(-a^2*x^2 + 1)*(48*c^3/a + (231*c^3 - 2*(72*a*c^3 + (91*a^2*c^3
- 4*(18*a^3*c^3 - (6*a^5*c^3*x - 7*a^4*c^3)*x)*x)*x)*x)*x)

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maple [A]  time = 0.05, size = 155, normalized size = 1.48 \[ \frac {c^{3} a^{3} x^{4} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{7}-\frac {2 c^{3} a \,x^{2} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{7}+\frac {c^{3} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{7 a}-\frac {c^{3} a^{2} x^{3} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{6}+\frac {3 c^{3} x \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{8}+\frac {5 c^{3} x \sqrt {-a^{2} x^{2}+1}}{16}+\frac {5 c^{3} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{16 \sqrt {a^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*c^3*a^3*x^4*(-a^2*x^2+1)^(3/2)-2/7*c^3*a*x^2*(-a^2*x^2+1)^(3/2)+1/7*c^3*(-a^2*x^2+1)^(3/2)/a-1/6*c^3*a^2*x
^3*(-a^2*x^2+1)^(3/2)+3/8*c^3*x*(-a^2*x^2+1)^(3/2)+5/16*c^3*x*(-a^2*x^2+1)^(1/2)+5/16*c^3/(a^2)^(1/2)*arctan((
a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))

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maxima [A]  time = 0.42, size = 136, normalized size = 1.30 \[ \frac {1}{7} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{3} c^{3} x^{4} - \frac {1}{6} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2} c^{3} x^{3} - \frac {2}{7} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a c^{3} x^{2} + \frac {3}{8} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{3} x + \frac {5}{16} \, \sqrt {-a^{2} x^{2} + 1} c^{3} x + \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{3}}{7 \, a} + \frac {5 \, c^{3} \arcsin \left (a x\right )}{16 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*(-a^2*x^2 + 1)^(3/2)*a^3*c^3*x^4 - 1/6*(-a^2*x^2 + 1)^(3/2)*a^2*c^3*x^3 - 2/7*(-a^2*x^2 + 1)^(3/2)*a*c^3*x
^2 + 3/8*(-a^2*x^2 + 1)^(3/2)*c^3*x + 5/16*sqrt(-a^2*x^2 + 1)*c^3*x + 1/7*(-a^2*x^2 + 1)^(3/2)*c^3/a + 5/16*c^
3*arcsin(a*x)/a

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mupad [B]  time = 0.89, size = 174, normalized size = 1.66 \[ \frac {11\,c^3\,x\,\sqrt {1-a^2\,x^2}}{16}+\frac {5\,c^3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{16\,\sqrt {-a^2}}+\frac {c^3\,\sqrt {1-a^2\,x^2}}{7\,a}-\frac {3\,a\,c^3\,x^2\,\sqrt {1-a^2\,x^2}}{7}-\frac {13\,a^2\,c^3\,x^3\,\sqrt {1-a^2\,x^2}}{24}+\frac {3\,a^3\,c^3\,x^4\,\sqrt {1-a^2\,x^2}}{7}+\frac {a^4\,c^3\,x^5\,\sqrt {1-a^2\,x^2}}{6}-\frac {a^5\,c^3\,x^6\,\sqrt {1-a^2\,x^2}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(11*c^3*x*(1 - a^2*x^2)^(1/2))/16 + (5*c^3*asinh(x*(-a^2)^(1/2)))/(16*(-a^2)^(1/2)) + (c^3*(1 - a^2*x^2)^(1/2)
)/(7*a) - (3*a*c^3*x^2*(1 - a^2*x^2)^(1/2))/7 - (13*a^2*c^3*x^3*(1 - a^2*x^2)^(1/2))/24 + (3*a^3*c^3*x^4*(1 -
a^2*x^2)^(1/2))/7 + (a^4*c^3*x^5*(1 - a^2*x^2)^(1/2))/6 - (a^5*c^3*x^6*(1 - a^2*x^2)^(1/2))/7

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sympy [C]  time = 11.91, size = 629, normalized size = 5.99 \[ - a^{5} c^{3} \left (\begin {cases} \frac {x^{6} \sqrt {- a^{2} x^{2} + 1}}{7} - \frac {x^{4} \sqrt {- a^{2} x^{2} + 1}}{35 a^{2}} - \frac {4 x^{2} \sqrt {- a^{2} x^{2} + 1}}{105 a^{4}} - \frac {8 \sqrt {- a^{2} x^{2} + 1}}{105 a^{6}} & \text {for}\: a \neq 0 \\\frac {x^{6}}{6} & \text {otherwise} \end {cases}\right ) + a^{4} c^{3} \left (\begin {cases} \frac {i a^{2} x^{7}}{6 \sqrt {a^{2} x^{2} - 1}} - \frac {5 i x^{5}}{24 \sqrt {a^{2} x^{2} - 1}} - \frac {i x^{3}}{48 a^{2} \sqrt {a^{2} x^{2} - 1}} + \frac {i x}{16 a^{4} \sqrt {a^{2} x^{2} - 1}} - \frac {i \operatorname {acosh}{\left (a x \right )}}{16 a^{5}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {a^{2} x^{7}}{6 \sqrt {- a^{2} x^{2} + 1}} + \frac {5 x^{5}}{24 \sqrt {- a^{2} x^{2} + 1}} + \frac {x^{3}}{48 a^{2} \sqrt {- a^{2} x^{2} + 1}} - \frac {x}{16 a^{4} \sqrt {- a^{2} x^{2} + 1}} + \frac {\operatorname {asin}{\left (a x \right )}}{16 a^{5}} & \text {otherwise} \end {cases}\right ) + 2 a^{3} c^{3} \left (\begin {cases} \frac {x^{4} \sqrt {- a^{2} x^{2} + 1}}{5} - \frac {x^{2} \sqrt {- a^{2} x^{2} + 1}}{15 a^{2}} - \frac {2 \sqrt {- a^{2} x^{2} + 1}}{15 a^{4}} & \text {for}\: a \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right ) - 2 a^{2} c^{3} \left (\begin {cases} \frac {i a^{2} x^{5}}{4 \sqrt {a^{2} x^{2} - 1}} - \frac {3 i x^{3}}{8 \sqrt {a^{2} x^{2} - 1}} + \frac {i x}{8 a^{2} \sqrt {a^{2} x^{2} - 1}} - \frac {i \operatorname {acosh}{\left (a x \right )}}{8 a^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {a^{2} x^{5}}{4 \sqrt {- a^{2} x^{2} + 1}} + \frac {3 x^{3}}{8 \sqrt {- a^{2} x^{2} + 1}} - \frac {x}{8 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {\operatorname {asin}{\left (a x \right )}}{8 a^{3}} & \text {otherwise} \end {cases}\right ) - a c^{3} \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\left (- a^{2} x^{2} + 1\right )^{\frac {3}{2}}}{3 a^{2}} & \text {otherwise} \end {cases}\right ) + c^{3} \left (\begin {cases} \frac {i a^{2} x^{3}}{2 \sqrt {a^{2} x^{2} - 1}} - \frac {i x}{2 \sqrt {a^{2} x^{2} - 1}} - \frac {i \operatorname {acosh}{\left (a x \right )}}{2 a} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x \sqrt {- a^{2} x^{2} + 1}}{2} + \frac {\operatorname {asin}{\left (a x \right )}}{2 a} & \text {otherwise} \end {cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**5*c**3*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)) + a**4*c**3*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)) + 2*a**3*c**3*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)) - 2*a**2*c**3*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**3*Piecewise((x**2/2, Eq(a**2, 0)), (-(-a**2
*x**2 + 1)**(3/2)/(3*a**2), True)) + c**3*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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