3.10.12 \(\int \frac {e^{\tanh ^{-1}(a x)} x^2}{(c-a^2 c x^2)^3} \, dx\) [912]

Optimal. Leaf size=88 \[ \frac {x^2 (1+a x)}{5 a c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {2 (1-a x)}{15 a^3 c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {2 x}{15 a^2 c^3 \sqrt {1-a^2 x^2}} \]

[Out]

1/5*x^2*(a*x+1)/a/c^3/(-a^2*x^2+1)^(5/2)-2/15*(-a*x+1)/a^3/c^3/(-a^2*x^2+1)^(3/2)-2/15*x/a^2/c^3/(-a^2*x^2+1)^
(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.08, antiderivative size = 88, 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 = {6283, 810, 792, 197} \begin {gather*} \frac {x^2 (a x+1)}{5 a c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {2 x}{15 a^2 c^3 \sqrt {1-a^2 x^2}}-\frac {2 (1-a x)}{15 a^3 c^3 \left (1-a^2 x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(1 + a*x))/(5*a*c^3*(1 - a^2*x^2)^(5/2)) - (2*(1 - a*x))/(15*a^3*c^3*(1 - a^2*x^2)^(3/2)) - (2*x)/(15*a^2
*c^3*Sqrt[1 - a^2*x^2])

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 792

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

Rule 810

Int[(x_)^2*((f_) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^2*(a*g - c*f*x)*((a + c*x^2)^(p
+ 1)/(2*a*c*(p + 1))), x] - Dist[1/(2*a*c*(p + 1)), Int[x*Simp[2*a*g - c*f*(2*p + 5)*x, x]*(a + c*x^2)^(p + 1
), x], x] /; FreeQ[{a, c, f, g}, x] && EqQ[a*g^2 + f^2*c, 0] && LtQ[p, -2]

Rule 6283

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^2}{\left (c-a^2 c x^2\right )^3} \, dx &=\frac {\int \frac {x^2 (1+a x)}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{c^3}\\ &=\frac {x^2 (1+a x)}{5 a c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {\int \frac {x \left (2 a-2 a^2 x\right )}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{5 a^2 c^3}\\ &=\frac {x^2 (1+a x)}{5 a c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {2 (1-a x)}{15 a^3 c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {2 \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{15 a^2 c^3}\\ &=\frac {x^2 (1+a x)}{5 a c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {2 (1-a x)}{15 a^3 c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {2 x}{15 a^2 c^3 \sqrt {1-a^2 x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 68, normalized size = 0.77 \begin {gather*} \frac {-2+2 a x+3 a^2 x^2+2 a^3 x^3-2 a^4 x^4}{15 a^3 c^3 (-1+a x)^2 (1+a x) \sqrt {1-a^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2 + 2*a*x + 3*a^2*x^2 + 2*a^3*x^3 - 2*a^4*x^4)/(15*a^3*c^3*(-1 + a*x)^2*(1 + a*x)*Sqrt[1 - a^2*x^2])

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(370\) vs. \(2(76)=152\).
time = 0.06, size = 371, normalized size = 4.22

method	result	size
gosper	\(\frac {2 a^{4} x^{4}-2 a^{3} x^{3}-3 a^{2} x^{2}-2 a x +2}{15 \left (a x -1\right ) c^{3} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} a^{3}}\)	\(58\)
trager	\(\frac {\left (2 a^{4} x^{4}-2 a^{3} x^{3}-3 a^{2} x^{2}-2 a x +2\right ) \sqrt {-a^{2} x^{2}+1}}{15 c^{3} a^{3} \left (a x -1\right )^{3} \left (a x +1\right )^{2}}\)	\(65\)
default	\(-\frac {-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{16 a^{4} \left (x +\frac {1}{a}\right )}-\frac {-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{3 a \left (x +\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{3 \left (x +\frac {1}{a}\right )}}{8 a^{4}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{16 a^{4} \left (x -\frac {1}{a}\right )}+\frac {\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}}{4 a^{5}}+\frac {\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 \left (x -\frac {1}{a}\right )}}{4 a^{4}}}{c^{3}}\)	\(371\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.36, size = 146, normalized size = 1.66 \begin {gather*} -\frac {2 \, a^{5} x^{5} - 2 \, a^{4} x^{4} - 4 \, a^{3} x^{3} + 4 \, a^{2} x^{2} + 2 \, a x - {\left (2 \, a^{4} x^{4} - 2 \, a^{3} x^{3} - 3 \, a^{2} x^{2} - 2 \, a x + 2\right )} \sqrt {-a^{2} x^{2} + 1} - 2}{15 \, {\left (a^{8} c^{3} x^{5} - a^{7} c^{3} x^{4} - 2 \, a^{6} c^{3} x^{3} + 2 \, a^{5} c^{3} x^{2} + a^{4} c^{3} x - a^{3} c^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(2*a^5*x^5 - 2*a^4*x^4 - 4*a^3*x^3 + 4*a^2*x^2 + 2*a*x - (2*a^4*x^4 - 2*a^3*x^3 - 3*a^2*x^2 - 2*a*x + 2)
*sqrt(-a^2*x^2 + 1) - 2)/(a^8*c^3*x^5 - a^7*c^3*x^4 - 2*a^6*c^3*x^3 + 2*a^5*c^3*x^2 + a^4*c^3*x - a^3*c^3)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {x^{2}}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{3}}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(x**2/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x*
*2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**3/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**
2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**3

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 0.91, size = 329, normalized size = 3.74 \begin {gather*} \frac {9\,\sqrt {1-a^2\,x^2}}{80\,\left (a\,c^3\,\sqrt {-a^2}-a^2\,c^3\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{24\,\left (a^5\,c^3\,x^2+2\,a^4\,c^3\,x+a^3\,c^3\right )}-\frac {\sqrt {1-a^2\,x^2}}{48\,\left (a\,c^3\,\sqrt {-a^2}+a^2\,c^3\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{12\,\left (a^5\,c^3\,x^2-2\,a^4\,c^3\,x+a^3\,c^3\right )}+\frac {a^2\,\sqrt {1-a^2\,x^2}}{30\,\left (a^7\,c^3\,x^2-2\,a^6\,c^3\,x+a^5\,c^3\right )}-\frac {\sqrt {1-a^2\,x^2}}{20\,\sqrt {-a^2}\,\left (a\,c^3\,\sqrt {-a^2}+3\,a^3\,c^3\,x^2\,\sqrt {-a^2}-a^4\,c^3\,x^3\,\sqrt {-a^2}-3\,a^2\,c^3\,x\,\sqrt {-a^2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(9*(1 - a^2*x^2)^(1/2))/(80*(a*c^3*(-a^2)^(1/2) - a^2*c^3*x*(-a^2)^(1/2))*(-a^2)^(1/2)) - (1 - a^2*x^2)^(1/2)/
(24*(a^3*c^3 + 2*a^4*c^3*x + a^5*c^3*x^2)) - (1 - a^2*x^2)^(1/2)/(48*(a*c^3*(-a^2)^(1/2) + a^2*c^3*x*(-a^2)^(1
/2))*(-a^2)^(1/2)) - (1 - a^2*x^2)^(1/2)/(12*(a^3*c^3 - 2*a^4*c^3*x + a^5*c^3*x^2)) + (a^2*(1 - a^2*x^2)^(1/2)
)/(30*(a^5*c^3 - 2*a^6*c^3*x + a^7*c^3*x^2)) - (1 - a^2*x^2)^(1/2)/(20*(-a^2)^(1/2)*(a*c^3*(-a^2)^(1/2) + 3*a^
3*c^3*x^2*(-a^2)^(1/2) - a^4*c^3*x^3*(-a^2)^(1/2) - 3*a^2*c^3*x*(-a^2)^(1/2)))

________________________________________________________________________________________