3.1195 \(\int e^{-\tanh ^{-1}(a x)} (c-a^2 c x^2) \, dx\)

Optimal. Leaf size=55 \[ \frac {c \left (1-a^2 x^2\right )^{3/2}}{3 a}+\frac {1}{2} c x \sqrt {1-a^2 x^2}+\frac {c \sin ^{-1}(a x)}{2 a} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6139, 641, 195, 216} \[ \frac {c \left (1-a^2 x^2\right )^{3/2}}{3 a}+\frac {1}{2} c x \sqrt {1-a^2 x^2}+\frac {c \sin ^{-1}(a x)}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[(c - a^2*c*x^2)/E^ArcTanh[a*x],x]

[Out]

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

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Mathematica [A]  time = 0.09, size = 57, normalized size = 1.04 \[ \frac {c \left (\left (-2 a^2 x^2+3 a x+2\right ) \sqrt {1-a^2 x^2}-6 \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{6 a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [A]  time = 0.79, size = 62, normalized size = 1.13 \[ -\frac {6 \, c \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (2 \, a^{2} c x^{2} - 3 \, a c x - 2 \, c\right )} \sqrt {-a^{2} x^{2} + 1}}{6 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(6*c*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (2*a^2*c*x^2 - 3*a*c*x - 2*c)*sqrt(-a^2*x^2 + 1))/a

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giac [A]  time = 0.67, size = 46, normalized size = 0.84 \[ \frac {c \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{2 \, {\left | a \right |}} - \frac {1}{6} \, \sqrt {-a^{2} x^{2} + 1} {\left ({\left (2 \, a c x - 3 \, c\right )} x - \frac {2 \, c}{a}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*c*arcsin(a*x)*sgn(a)/abs(a) - 1/6*sqrt(-a^2*x^2 + 1)*((2*a*c*x - 3*c)*x - 2*c/a)

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maple [A]  time = 0.03, size = 64, normalized size = 1.16 \[ \frac {c \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3 a}+\frac {c x \sqrt {-a^{2} x^{2}+1}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.41, size = 45, normalized size = 0.82 \[ \frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} c x + \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c}{3 \, a} + \frac {c \arcsin \left (a x\right )}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.91, size = 80, normalized size = 1.45 \[ \frac {\sqrt {1-a^2\,x^2}\,\left (\frac {c\,x\,\sqrt {-a^2}}{2}-\frac {a\,c}{3\,\sqrt {-a^2}}+\frac {a^3\,c\,x^2}{3\,\sqrt {-a^2}}\right )}{\sqrt {-a^2}}+\frac {c\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,\sqrt {-a^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [C]  time = 4.44, size = 109, normalized size = 1.98 \[ - a c \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\left (- a^{2} x^{2} + 1\right )^{\frac {3}{2}}}{3 a^{2}} & \text {otherwise} \end {cases}\right ) + c \left (\begin {cases} \frac {i a^{2} x^{3}}{2 \sqrt {a^{2} x^{2} - 1}} - \frac {i x}{2 \sqrt {a^{2} x^{2} - 1}} - \frac {i \operatorname {acosh}{\left (a x \right )}}{2 a} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x \sqrt {- a^{2} x^{2} + 1}}{2} + \frac {\operatorname {asin}{\left (a x \right )}}{2 a} & \text {otherwise} \end {cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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