3.908 \(\int \frac{e^{\tanh ^{-1}(a x)} x^6}{(c-a^2 c x^2)^3} \, dx\)

Optimal. Leaf size=133 \[ \frac{x^5 (a x+1)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{x^3 (6 a x+5)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{x (8 a x+5)}{5 a^6 c^3 \sqrt{1-a^2 x^2}}+\frac{16 \sqrt{1-a^2 x^2}}{5 a^7 c^3}-\frac{\sin ^{-1}(a x)}{a^7 c^3} \]

[Out]

(x^5*(1 + a*x))/(5*a^2*c^3*(1 - a^2*x^2)^(5/2)) - (x^3*(5 + 6*a*x))/(15*a^4*c^3*(1 - a^2*x^2)^(3/2)) + (x*(5 +
8*a*x))/(5*a^6*c^3*Sqrt[1 - a^2*x^2]) + (16*Sqrt[1 - a^2*x^2])/(5*a^7*c^3) - ArcSin[a*x]/(a^7*c^3)

________________________________________________________________________________________

Rubi [A]  time = 0.154496, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {6148, 819, 641, 216} \[ \frac{x^5 (a x+1)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{x^3 (6 a x+5)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{x (8 a x+5)}{5 a^6 c^3 \sqrt{1-a^2 x^2}}+\frac{16 \sqrt{1-a^2 x^2}}{5 a^7 c^3}-\frac{\sin ^{-1}(a x)}{a^7 c^3} \]

Antiderivative was successfully verified.

[In]

Int[(E^ArcTanh[a*x]*x^6)/(c - a^2*c*x^2)^3,x]

[Out]

(x^5*(1 + a*x))/(5*a^2*c^3*(1 - a^2*x^2)^(5/2)) - (x^3*(5 + 6*a*x))/(15*a^4*c^3*(1 - a^2*x^2)^(3/2)) + (x*(5 +
8*a*x))/(5*a^6*c^3*Sqrt[1 - a^2*x^2]) + (16*Sqrt[1 - a^2*x^2])/(5*a^7*c^3) - ArcSin[a*x]/(a^7*c^3)

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]

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 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 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]

Rubi steps

\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)} x^6}{\left (c-a^2 c x^2\right )^3} \, dx &=\frac{\int \frac{x^6 (1+a x)}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{c^3}\\ &=\frac{x^5 (1+a x)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{\int \frac{x^4 (5+6 a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{5 a^2 c^3}\\ &=\frac{x^5 (1+a x)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{x^3 (5+6 a x)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{\int \frac{x^2 (15+24 a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{15 a^4 c^3}\\ &=\frac{x^5 (1+a x)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{x^3 (5+6 a x)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{x (5+8 a x)}{5 a^6 c^3 \sqrt{1-a^2 x^2}}-\frac{\int \frac{15+48 a x}{\sqrt{1-a^2 x^2}} \, dx}{15 a^6 c^3}\\ &=\frac{x^5 (1+a x)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{x^3 (5+6 a x)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{x (5+8 a x)}{5 a^6 c^3 \sqrt{1-a^2 x^2}}+\frac{16 \sqrt{1-a^2 x^2}}{5 a^7 c^3}-\frac{\int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{a^6 c^3}\\ &=\frac{x^5 (1+a x)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{x^3 (5+6 a x)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{x (5+8 a x)}{5 a^6 c^3 \sqrt{1-a^2 x^2}}+\frac{16 \sqrt{1-a^2 x^2}}{5 a^7 c^3}-\frac{\sin ^{-1}(a x)}{a^7 c^3}\\ \end{align*}

Mathematica [A]  time = 0.0728295, size = 108, normalized size = 0.81 \[ \frac{-15 a^5 x^5+38 a^4 x^4+52 a^3 x^3-87 a^2 x^2-15 (a x-1)^2 (a x+1) \sqrt{1-a^2 x^2} \sin ^{-1}(a x)-33 a x+48}{15 a^7 c^3 (a x-1)^2 (a x+1) \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^ArcTanh[a*x]*x^6)/(c - a^2*c*x^2)^3,x]

[Out]

(48 - 33*a*x - 87*a^2*x^2 + 52*a^3*x^3 + 38*a^4*x^4 - 15*a^5*x^5 - 15*(-1 + a*x)^2*(1 + a*x)*Sqrt[1 - a^2*x^2]
*ArcSin[a*x])/(15*a^7*c^3*(-1 + a*x)^2*(1 + a*x)*Sqrt[1 - a^2*x^2])

________________________________________________________________________________________

Maple [B]  time = 0.05, size = 262, normalized size = 2. \begin{align*}{\frac{1}{{a}^{7}{c}^{3}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{1}{{a}^{6}{c}^{3}}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}-{\frac{1}{20\,{c}^{3}{a}^{10}}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-3}}-{\frac{23}{60\,{c}^{3}{a}^{9}}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-2}}-{\frac{493}{240\,{c}^{3}{a}^{8}}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-1}}-{\frac{1}{24\,{c}^{3}{a}^{9} \left ( x+{a}^{-1} \right ) ^{2}}\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}+{\frac{25}{48\,{c}^{3}{a}^{8} \left ( x+{a}^{-1} \right ) }\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)/(-a^2*x^2+1)^(1/2)*x^6/(-a^2*c*x^2+c)^3,x)

[Out]

(-a^2*x^2+1)^(1/2)/a^7/c^3-1/c^3/a^6/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))-1/20/c^3/a^10/(x-1/a
)^3*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)-23/60/c^3/a^9/(x-1/a)^2*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)-493/240/c^3/
a^8/(x-1/a)*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)-1/24/c^3/a^9/(x+1/a)^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)+25/48
/c^3/a^8/(x+1/a)*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{{\left (a x + 1\right )} x^{6}}{{\left (a^{2} c x^{2} - c\right )}^{3} \sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^6/(-a^2*c*x^2+c)^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.58447, size = 462, normalized size = 3.47 \begin{align*} \frac{48 \, a^{5} x^{5} - 48 \, a^{4} x^{4} - 96 \, a^{3} x^{3} + 96 \, a^{2} x^{2} + 48 \, a x + 30 \,{\left (a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) +{\left (15 \, a^{5} x^{5} - 38 \, a^{4} x^{4} - 52 \, a^{3} x^{3} + 87 \, a^{2} x^{2} + 33 \, a x - 48\right )} \sqrt{-a^{2} x^{2} + 1} - 48}{15 \,{\left (a^{12} c^{3} x^{5} - a^{11} c^{3} x^{4} - 2 \, a^{10} c^{3} x^{3} + 2 \, a^{9} c^{3} x^{2} + a^{8} c^{3} x - a^{7} c^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^6/(-a^2*c*x^2+c)^3,x, algorithm="fricas")

[Out]

1/15*(48*a^5*x^5 - 48*a^4*x^4 - 96*a^3*x^3 + 96*a^2*x^2 + 48*a*x + 30*(a^5*x^5 - a^4*x^4 - 2*a^3*x^3 + 2*a^2*x
^2 + a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (15*a^5*x^5 - 38*a^4*x^4 - 52*a^3*x^3 + 87*a^2*x^2 + 33
*a*x - 48)*sqrt(-a^2*x^2 + 1) - 48)/(a^12*c^3*x^5 - a^11*c^3*x^4 - 2*a^10*c^3*x^3 + 2*a^9*c^3*x^2 + a^8*c^3*x
- a^7*c^3)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{x^{6}}{- 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^{7}}{- 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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**6/(-a**2*c*x**2+c)**3,x)

[Out]

(Integral(x**6/(-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**7/(-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., size = 0, normalized size = 0. \begin{align*} \int -\frac{{\left (a x + 1\right )} x^{6}}{{\left (a^{2} c x^{2} - c\right )}^{3} \sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^6/(-a^2*c*x^2+c)^3,x, algorithm="giac")

[Out]

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