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

Optimal. Leaf size=163 \[ -\frac {2 c^3 (1-a x)^5}{a \sqrt {1-a^2 x^2}}-\frac {9 c^3 \sqrt {1-a^2 x^2} (1-a x)^3}{4 a}-\frac {21 c^3 \sqrt {1-a^2 x^2} (1-a x)^2}{4 a}-\frac {105 c^3 \sqrt {1-a^2 x^2} (1-a x)}{8 a}-\frac {315 c^3 \sqrt {1-a^2 x^2}}{8 a}-\frac {315 c^3 \sin ^{-1}(a x)}{8 a} \]

[Out]

-315/8*c^3*arcsin(a*x)/a-2*c^3*(-a*x+1)^5/a/(-a^2*x^2+1)^(1/2)-315/8*c^3*(-a^2*x^2+1)^(1/2)/a-105/8*c^3*(-a*x+
1)*(-a^2*x^2+1)^(1/2)/a-21/4*c^3*(-a*x+1)^2*(-a^2*x^2+1)^(1/2)/a-9/4*c^3*(-a*x+1)^3*(-a^2*x^2+1)^(1/2)/a

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Rubi [A]  time = 0.13, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6127, 669, 671, 641, 216} \[ -\frac {2 c^3 (1-a x)^5}{a \sqrt {1-a^2 x^2}}-\frac {9 c^3 \sqrt {1-a^2 x^2} (1-a x)^3}{4 a}-\frac {21 c^3 \sqrt {1-a^2 x^2} (1-a x)^2}{4 a}-\frac {105 c^3 \sqrt {1-a^2 x^2} (1-a x)}{8 a}-\frac {315 c^3 \sqrt {1-a^2 x^2}}{8 a}-\frac {315 c^3 \sin ^{-1}(a x)}{8 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*c^3*(1 - a*x)^5)/(a*Sqrt[1 - a^2*x^2]) - (315*c^3*Sqrt[1 - a^2*x^2])/(8*a) - (105*c^3*(1 - a*x)*Sqrt[1 - a
^2*x^2])/(8*a) - (21*c^3*(1 - a*x)^2*Sqrt[1 - a^2*x^2])/(4*a) - (9*c^3*(1 - a*x)^3*Sqrt[1 - a^2*x^2])/(4*a) -
(315*c^3*ArcSin[a*x])/(8*a)

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 669

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*(p + 1)), x] - Dist[(e^2*(m + p))/(c*(p + 1)), Int[(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[2*p]

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 6127

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> Dist[c^n, Int[(c + d*x)^(p - n)*(1 -
a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && IntegerQ[2*p]

Rubi steps

\begin {align*} \int e^{-3 \tanh ^{-1}(a x)} (c-a c x)^3 \, dx &=\frac {\int \frac {(c-a c x)^6}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=-\frac {2 c^3 (1-a x)^5}{a \sqrt {1-a^2 x^2}}-\frac {9 \int \frac {(c-a c x)^4}{\sqrt {1-a^2 x^2}} \, dx}{c}\\ &=-\frac {2 c^3 (1-a x)^5}{a \sqrt {1-a^2 x^2}}-\frac {9 c^3 (1-a x)^3 \sqrt {1-a^2 x^2}}{4 a}-\frac {63}{4} \int \frac {(c-a c x)^3}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {2 c^3 (1-a x)^5}{a \sqrt {1-a^2 x^2}}-\frac {21 c^3 (1-a x)^2 \sqrt {1-a^2 x^2}}{4 a}-\frac {9 c^3 (1-a x)^3 \sqrt {1-a^2 x^2}}{4 a}-\frac {1}{4} (105 c) \int \frac {(c-a c x)^2}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {2 c^3 (1-a x)^5}{a \sqrt {1-a^2 x^2}}-\frac {105 c^3 (1-a x) \sqrt {1-a^2 x^2}}{8 a}-\frac {21 c^3 (1-a x)^2 \sqrt {1-a^2 x^2}}{4 a}-\frac {9 c^3 (1-a x)^3 \sqrt {1-a^2 x^2}}{4 a}-\frac {1}{8} \left (315 c^2\right ) \int \frac {c-a c x}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {2 c^3 (1-a x)^5}{a \sqrt {1-a^2 x^2}}-\frac {315 c^3 \sqrt {1-a^2 x^2}}{8 a}-\frac {105 c^3 (1-a x) \sqrt {1-a^2 x^2}}{8 a}-\frac {21 c^3 (1-a x)^2 \sqrt {1-a^2 x^2}}{4 a}-\frac {9 c^3 (1-a x)^3 \sqrt {1-a^2 x^2}}{4 a}-\frac {1}{8} \left (315 c^3\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {2 c^3 (1-a x)^5}{a \sqrt {1-a^2 x^2}}-\frac {315 c^3 \sqrt {1-a^2 x^2}}{8 a}-\frac {105 c^3 (1-a x) \sqrt {1-a^2 x^2}}{8 a}-\frac {21 c^3 (1-a x)^2 \sqrt {1-a^2 x^2}}{4 a}-\frac {9 c^3 (1-a x)^3 \sqrt {1-a^2 x^2}}{4 a}-\frac {315 c^3 \sin ^{-1}(a x)}{8 a}\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 45, normalized size = 0.28 \[ -\frac {c^3 (1-a x)^{11/2} \, _2F_1\left (\frac {3}{2},\frac {11}{2};\frac {13}{2};\frac {1}{2} (1-a x)\right )}{11 \sqrt {2} a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/11*(c^3*(1 - a*x)^(11/2)*Hypergeometric2F1[3/2, 11/2, 13/2, (1 - a*x)/2])/(Sqrt[2]*a)

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fricas [A]  time = 0.51, size = 118, normalized size = 0.72 \[ -\frac {496 \, a c^{3} x + 496 \, c^{3} - 630 \, {\left (a c^{3} x + c^{3}\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (2 \, a^{4} c^{3} x^{4} - 14 \, a^{3} c^{3} x^{3} + 51 \, a^{2} c^{3} x^{2} - 173 \, a c^{3} x - 496 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1}}{8 \, {\left (a^{2} x + a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(496*a*c^3*x + 496*c^3 - 630*(a*c^3*x + c^3)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (2*a^4*c^3*x^4 - 14
*a^3*c^3*x^3 + 51*a^2*c^3*x^2 - 173*a*c^3*x - 496*c^3)*sqrt(-a^2*x^2 + 1))/(a^2*x + a)

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giac [A]  time = 0.25, size = 103, normalized size = 0.63 \[ -\frac {315 \, c^{3} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{8 \, {\left | a \right |}} - \frac {1}{8} \, \sqrt {-a^{2} x^{2} + 1} {\left (\frac {240 \, c^{3}}{a} - {\left (67 \, c^{3} + 2 \, {\left (a^{2} c^{3} x - 8 \, a c^{3}\right )} x\right )} x\right )} + \frac {64 \, c^{3}}{{\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )} {\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-315/8*c^3*arcsin(a*x)*sgn(a)/abs(a) - 1/8*sqrt(-a^2*x^2 + 1)*(240*c^3/a - (67*c^3 + 2*(a^2*c^3*x - 8*a*c^3)*x
)*x) + 64*c^3/(((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) + 1)*abs(a))

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maple [A]  time = 0.05, size = 245, normalized size = 1.50 \[ -\frac {c^{3} x \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4}-\frac {3 c^{3} x \sqrt {-a^{2} x^{2}+1}}{8}-\frac {3 c^{3} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 \sqrt {a^{2}}}-\frac {28 c^{3} \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a^{3} \left (x +\frac {1}{a}\right )^{2}}-\frac {26 c^{3} \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{a}-39 c^{3} \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}\, x -\frac {39 c^{3} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}}-\frac {8 c^{3} \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a^{4} \left (x +\frac {1}{a}\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*c^3*x*(-a^2*x^2+1)^(3/2)-3/8*c^3*x*(-a^2*x^2+1)^(1/2)-3/8*c^3/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+
1)^(1/2))-28*c^3/a^3/(x+1/a)^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(5/2)-26*c^3/a*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(3/2)-
39*c^3*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)*x-39*c^3/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*(x+1/a)^2+2*a*(x+1/a
))^(1/2))-8*c^3/a^4/(x+1/a)^3*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(5/2)

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maxima [C]  time = 0.45, size = 233, normalized size = 1.43 \[ -\frac {1}{4} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{3} x + 3 \, \sqrt {a^{2} x^{2} + 4 \, a x + 3} c^{3} x - \frac {3}{8} \, \sqrt {-a^{2} x^{2} + 1} c^{3} x + \frac {8 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{3}}{a^{3} x^{2} + 2 \, a^{2} x + a} - \frac {6 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{3}}{a^{2} x + a} + \frac {2 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{3}}{a} - \frac {3 i \, c^{3} \arcsin \left (a x + 2\right )}{a} - \frac {339 \, c^{3} \arcsin \left (a x\right )}{8 \, a} - \frac {48 \, \sqrt {-a^{2} x^{2} + 1} c^{3}}{a^{2} x + a} + \frac {6 \, \sqrt {a^{2} x^{2} + 4 \, a x + 3} c^{3}}{a} - \frac {18 \, \sqrt {-a^{2} x^{2} + 1} c^{3}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(-a^2*x^2 + 1)^(3/2)*c^3*x + 3*sqrt(a^2*x^2 + 4*a*x + 3)*c^3*x - 3/8*sqrt(-a^2*x^2 + 1)*c^3*x + 8*(-a^2*x
^2 + 1)^(3/2)*c^3/(a^3*x^2 + 2*a^2*x + a) - 6*(-a^2*x^2 + 1)^(3/2)*c^3/(a^2*x + a) + 2*(-a^2*x^2 + 1)^(3/2)*c^
3/a - 3*I*c^3*arcsin(a*x + 2)/a - 339/8*c^3*arcsin(a*x)/a - 48*sqrt(-a^2*x^2 + 1)*c^3/(a^2*x + a) + 6*sqrt(a^2
*x^2 + 4*a*x + 3)*c^3/a - 18*sqrt(-a^2*x^2 + 1)*c^3/a

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mupad [B]  time = 0.82, size = 166, normalized size = 1.02 \[ \frac {32\,c^3\,\sqrt {1-a^2\,x^2}}{\left (x\,\sqrt {-a^2}+\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}-\frac {315\,c^3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{8\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {4\,a^3\,c^3}{{\left (-a^2\right )}^{3/2}}-\frac {67\,c^3\,x\,\sqrt {-a^2}}{8}-\frac {26\,a\,c^3}{\sqrt {-a^2}}+\frac {c^3\,x^3\,{\left (-a^2\right )}^{3/2}}{4}+\frac {2\,a^5\,c^3\,x^2}{{\left (-a^2\right )}^{3/2}}\right )}{\sqrt {-a^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(32*c^3*(1 - a^2*x^2)^(1/2))/((x*(-a^2)^(1/2) + (-a^2)^(1/2)/a)*(-a^2)^(1/2)) - (315*c^3*asinh(x*(-a^2)^(1/2))
)/(8*(-a^2)^(1/2)) - ((1 - a^2*x^2)^(1/2)*((4*a^3*c^3)/(-a^2)^(3/2) - (67*c^3*x*(-a^2)^(1/2))/8 - (26*a*c^3)/(
-a^2)^(1/2) + (c^3*x^3*(-a^2)^(3/2))/4 + (2*a^5*c^3*x^2)/(-a^2)^(3/2)))/(-a^2)^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - c^{3} \left (\int \left (- \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\right )\, dx + \int \frac {3 a x \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx + \int \left (- \frac {2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\right )\, dx + \int \left (- \frac {2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\right )\, dx + \int \frac {3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx + \int \left (- \frac {a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\right )\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c**3*(Integral(-sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1), x) + Integral(3*a*x*sqrt(-a**2*x*
*2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1), x) + Integral(-2*a**2*x**2*sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*
a**2*x**2 + 3*a*x + 1), x) + Integral(-2*a**3*x**3*sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1),
x) + Integral(3*a**4*x**4*sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1), x) + Integral(-a**5*x**
5*sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1), x))

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