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

Optimal. Leaf size=74 \[ -\frac {3 \sin ^{-1}(a x)}{2 a^4 c}+\frac {x^2 (a x+1)}{a^2 c \sqrt {1-a^2 x^2}}+\frac {(3 a x+4) \sqrt {1-a^2 x^2}}{2 a^4 c} \]

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6148, 819, 780, 216} \[ \frac {x^2 (a x+1)}{a^2 c \sqrt {1-a^2 x^2}}+\frac {(3 a x+4) \sqrt {1-a^2 x^2}}{2 a^4 c}-\frac {3 \sin ^{-1}(a x)}{2 a^4 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(1 + a*x))/(a^2*c*Sqrt[1 - a^2*x^2]) + ((4 + 3*a*x)*Sqrt[1 - a^2*x^2])/(2*a^4*c) - (3*ArcSin[a*x])/(2*a^4
*c)

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 780

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*(a + c*x^2)^(p + 1))/(2*c*(p + 1)*(2*p + 3)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rule 819

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m - 1)*(a + c*x^2)^(p + 1)*(a*(e*f + d*g) - (c*d*f - a*e*g)*x))/(2*a*c*(p + 1)), x] - Dist[1/(2*a*c*(p + 1)),
Int[(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^2*f*(2*p + 3) + e*(a*e*g*m - c*
d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ
[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rule 6148

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

Rubi steps

\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x^3}{c-a^2 c x^2} \, dx &=\frac {\int \frac {x^3 (1+a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c}\\ &=\frac {x^2 (1+a x)}{a^2 c \sqrt {1-a^2 x^2}}-\frac {\int \frac {x (2+3 a x)}{\sqrt {1-a^2 x^2}} \, dx}{a^2 c}\\ &=\frac {x^2 (1+a x)}{a^2 c \sqrt {1-a^2 x^2}}+\frac {(4+3 a x) \sqrt {1-a^2 x^2}}{2 a^4 c}-\frac {3 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{2 a^3 c}\\ &=\frac {x^2 (1+a x)}{a^2 c \sqrt {1-a^2 x^2}}+\frac {(4+3 a x) \sqrt {1-a^2 x^2}}{2 a^4 c}-\frac {3 \sin ^{-1}(a x)}{2 a^4 c}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 65, normalized size = 0.88 \[ -\frac {a^3 x^3+2 a^2 x^2+3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)-3 a x-4}{2 a^4 c \sqrt {1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(-4 - 3*a*x + 2*a^2*x^2 + a^3*x^3 + 3*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(a^4*c*Sqrt[1 - a^2*x^2])

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fricas [A]  time = 0.50, size = 77, normalized size = 1.04 \[ \frac {4 \, a x + 6 \, {\left (a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (a^{2} x^{2} + a x - 4\right )} \sqrt {-a^{2} x^{2} + 1} - 4}{2 \, {\left (a^{5} c x - a^{4} c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(4*a*x + 6*(a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (a^2*x^2 + a*x - 4)*sqrt(-a^2*x^2 + 1) - 4)/
(a^5*c*x - a^4*c)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.04, size = 119, normalized size = 1.61 \[ \frac {x \sqrt {-a^{2} x^{2}+1}}{2 a^{3} c}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 c \,a^{3} \sqrt {a^{2}}}+\frac {\sqrt {-a^{2} x^{2}+1}}{a^{4} c}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{c \,a^{5} \left (x -\frac {1}{a}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.49, size = 178, normalized size = 2.41 \[ -\frac {1}{2} \, a {\left (\frac {\sqrt {-a^{2} x^{2} + 1} c}{a^{6} c^{2} x + a^{5} c^{2}} + \frac {\sqrt {-a^{2} x^{2} + 1} c}{a^{6} c^{2} x - a^{5} c^{2}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{a^{6} c x + a^{5} c} + \frac {\sqrt {-a^{2} x^{2} + 1}}{a^{6} c x - a^{5} c} - \frac {\sqrt {-a^{2} x^{2} + 1} x}{a^{4} c} + \frac {3 \, \arcsin \left (a x\right )}{a^{5} c} - \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a^{5} c}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.93, size = 128, normalized size = 1.73 \[ \frac {\sqrt {1-a^2\,x^2}}{a^3\,c\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}-\frac {3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,a^3\,c\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {1}{a^2\,c\,\sqrt {-a^2}}-\frac {x\,\sqrt {-a^2}}{2\,a^3\,c}\right )}{\sqrt {-a^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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