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

Optimal. Leaf size=101 \[ \frac {c^2 (1-a x)^2 \sqrt {1-a^2 x^2}}{3 a}+\frac {5 c^2 (1-a x) \sqrt {1-a^2 x^2}}{6 a}+\frac {5 c^2 \sqrt {1-a^2 x^2}}{2 a}+\frac {5 c^2 \sin ^{-1}(a x)}{2 a} \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6127, 671, 641, 216} \[ \frac {c^2 (1-a x)^2 \sqrt {1-a^2 x^2}}{3 a}+\frac {5 c^2 (1-a x) \sqrt {1-a^2 x^2}}{6 a}+\frac {5 c^2 \sqrt {1-a^2 x^2}}{2 a}+\frac {5 c^2 \sin ^{-1}(a x)}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[(c - a*c*x)^2/E^ArcTanh[a*x],x]

[Out]

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

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Mathematica [A]  time = 0.08, size = 72, normalized size = 0.71 \[ \frac {c^2 \left (\frac {\sqrt {a x+1} \left (-2 a^3 x^3+11 a^2 x^2-31 a x+22\right )}{\sqrt {1-a x}}-30 \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{6 a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^2*((Sqrt[1 + a*x]*(22 - 31*a*x + 11*a^2*x^2 - 2*a^3*x^3))/Sqrt[1 - a*x] - 30*ArcSin[Sqrt[1 - a*x]/Sqrt[2]])
)/(6*a)

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fricas [A]  time = 0.52, size = 71, normalized size = 0.70 \[ -\frac {30 \, c^{2} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (2 \, a^{2} c^{2} x^{2} - 9 \, a c^{2} x + 22 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{6 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(30*c^2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (2*a^2*c^2*x^2 - 9*a*c^2*x + 22*c^2)*sqrt(-a^2*x^2 + 1))
/a

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giac [A]  time = 0.76, size = 54, normalized size = 0.53 \[ \frac {5 \, c^{2} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{2 \, {\left | a \right |}} + \frac {1}{6} \, \sqrt {-a^{2} x^{2} + 1} {\left ({\left (2 \, a c^{2} x - 9 \, c^{2}\right )} x + \frac {22 \, c^{2}}{a}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/2*c^2*arcsin(a*x)*sgn(a)/abs(a) + 1/6*sqrt(-a^2*x^2 + 1)*((2*a*c^2*x - 9*c^2)*x + 22*c^2/a)

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maple [A]  time = 0.04, size = 142, normalized size = 1.41 \[ -\frac {c^{2} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3 a}-\frac {3 c^{2} x \sqrt {-a^{2} x^{2}+1}}{2}-\frac {3 c^{2} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}+\frac {4 c^{2} \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{a}+\frac {4 c^{2} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.43, size = 71, normalized size = 0.70 \[ -\frac {3}{2} \, \sqrt {-a^{2} x^{2} + 1} c^{2} x - \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{2}}{3 \, a} + \frac {5 \, c^{2} \arcsin \left (a x\right )}{2 \, a} + \frac {4 \, \sqrt {-a^{2} x^{2} + 1} c^{2}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.04, size = 82, normalized size = 0.81 \[ \frac {5\,c^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,\sqrt {-a^2}}-\frac {3\,c^2\,x\,\sqrt {1-a^2\,x^2}}{2}+\frac {11\,c^2\,\sqrt {1-a^2\,x^2}}{3\,a}+\frac {a\,c^2\,x^2\,\sqrt {1-a^2\,x^2}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(5*c^2*asinh(x*(-a^2)^(1/2)))/(2*(-a^2)^(1/2)) - (3*c^2*x*(1 - a^2*x^2)^(1/2))/2 + (11*c^2*(1 - a^2*x^2)^(1/2)
)/(3*a) + (a*c^2*x^2*(1 - a^2*x^2)^(1/2))/3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**2*(Integral(sqrt(-a**2*x**2 + 1)/(a*x + 1), x) + Integral(-2*a*x*sqrt(-a**2*x**2 + 1)/(a*x + 1), x) + Integ
ral(a**2*x**2*sqrt(-a**2*x**2 + 1)/(a*x + 1), x))

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