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

Optimal. Leaf size=137 \[ \frac{x^5 (a x+1)}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac{x^3 (6 a x+5)}{3 a^4 c^2 \sqrt{1-a^2 x^2}}-\frac{8 x^2 \sqrt{1-a^2 x^2}}{3 a^5 c^2}-\frac{(15 a x+32) \sqrt{1-a^2 x^2}}{6 a^7 c^2}+\frac{5 \sin ^{-1}(a x)}{2 a^7 c^2} \]

[Out]

(x^5*(1 + a*x))/(3*a^2*c^2*(1 - a^2*x^2)^(3/2)) - (x^3*(5 + 6*a*x))/(3*a^4*c^2*Sqrt[1 - a^2*x^2]) - (8*x^2*Sqr
t[1 - a^2*x^2])/(3*a^5*c^2) - ((32 + 15*a*x)*Sqrt[1 - a^2*x^2])/(6*a^7*c^2) + (5*ArcSin[a*x])/(2*a^7*c^2)

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Rubi [A]  time = 0.154315, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {6148, 819, 833, 780, 216} \[ \frac{x^5 (a x+1)}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac{x^3 (6 a x+5)}{3 a^4 c^2 \sqrt{1-a^2 x^2}}-\frac{8 x^2 \sqrt{1-a^2 x^2}}{3 a^5 c^2}-\frac{(15 a x+32) \sqrt{1-a^2 x^2}}{6 a^7 c^2}+\frac{5 \sin ^{-1}(a x)}{2 a^7 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^5*(1 + a*x))/(3*a^2*c^2*(1 - a^2*x^2)^(3/2)) - (x^3*(5 + 6*a*x))/(3*a^4*c^2*Sqrt[1 - a^2*x^2]) - (8*x^2*Sqr
t[1 - a^2*x^2])/(3*a^5*c^2) - ((32 + 15*a*x)*Sqrt[1 - a^2*x^2])/(6*a^7*c^2) + (5*ArcSin[a*x])/(2*a^7*c^2)

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 833

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

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 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 )^2} \, dx &=\frac{\int \frac{x^6 (1+a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{c^2}\\ &=\frac{x^5 (1+a x)}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac{\int \frac{x^4 (5+6 a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{3 a^2 c^2}\\ &=\frac{x^5 (1+a x)}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac{x^3 (5+6 a x)}{3 a^4 c^2 \sqrt{1-a^2 x^2}}+\frac{\int \frac{x^2 (15+24 a x)}{\sqrt{1-a^2 x^2}} \, dx}{3 a^4 c^2}\\ &=\frac{x^5 (1+a x)}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac{x^3 (5+6 a x)}{3 a^4 c^2 \sqrt{1-a^2 x^2}}-\frac{8 x^2 \sqrt{1-a^2 x^2}}{3 a^5 c^2}-\frac{\int \frac{x \left (-48 a-45 a^2 x\right )}{\sqrt{1-a^2 x^2}} \, dx}{9 a^6 c^2}\\ &=\frac{x^5 (1+a x)}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac{x^3 (5+6 a x)}{3 a^4 c^2 \sqrt{1-a^2 x^2}}-\frac{8 x^2 \sqrt{1-a^2 x^2}}{3 a^5 c^2}-\frac{(32+15 a x) \sqrt{1-a^2 x^2}}{6 a^7 c^2}+\frac{5 \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{2 a^6 c^2}\\ &=\frac{x^5 (1+a x)}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac{x^3 (5+6 a x)}{3 a^4 c^2 \sqrt{1-a^2 x^2}}-\frac{8 x^2 \sqrt{1-a^2 x^2}}{3 a^5 c^2}-\frac{(32+15 a x) \sqrt{1-a^2 x^2}}{6 a^7 c^2}+\frac{5 \sin ^{-1}(a x)}{2 a^7 c^2}\\ \end{align*}

Mathematica [A]  time = 0.0664661, size = 93, normalized size = 0.68 \[ \frac{2 a^5 x^5+a^4 x^4+11 a^3 x^3-31 a^2 x^2+15 (a x-1) \sqrt{1-a^2 x^2} \sin ^{-1}(a x)-17 a x+32}{6 a^7 c^2 (a x-1) \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(32 - 17*a*x - 31*a^2*x^2 + 11*a^3*x^3 + a^4*x^4 + 2*a^5*x^5 + 15*(-1 + a*x)*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(6
*a^7*c^2*(-1 + a*x)*Sqrt[1 - a^2*x^2])

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Maple [A]  time = 0.047, size = 225, normalized size = 1.6 \begin{align*} -{\frac{{x}^{2}}{3\,{a}^{5}{c}^{2}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{8}{3\,{c}^{2}{a}^{7}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{x}{2\,{c}^{2}{a}^{6}}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{5}{2\,{c}^{2}{a}^{6}}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}-{\frac{1}{4\,{c}^{2}{a}^{8} \left ( x+{a}^{-1} \right ) }\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}+{\frac{1}{6\,{c}^{2}{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{31}{12\,{c}^{2}{a}^{8}}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-1}} \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)^2,x)

[Out]

-1/3*x^2*(-a^2*x^2+1)^(1/2)/a^5/c^2-8/3/c^2/a^7*(-a^2*x^2+1)^(1/2)-1/2/c^2/a^6*x*(-a^2*x^2+1)^(1/2)+5/2/c^2/a^
6/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))-1/4/c^2/a^8/(x+1/a)*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)+
1/6/c^2/a^9/(x-1/a)^2*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)+31/12/c^2/a^8/(x-1/a)*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1
/2)

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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 )}^{2} \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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.64351, size = 343, normalized size = 2.5 \begin{align*} -\frac{32 \, a^{3} x^{3} - 32 \, a^{2} x^{2} - 32 \, a x + 30 \,{\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) +{\left (2 \, a^{5} x^{5} + a^{4} x^{4} + 11 \, a^{3} x^{3} - 31 \, a^{2} x^{2} - 17 \, a x + 32\right )} \sqrt{-a^{2} x^{2} + 1} + 32}{6 \,{\left (a^{10} c^{2} x^{3} - a^{9} c^{2} x^{2} - a^{8} c^{2} x + a^{7} c^{2}\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)^2,x, algorithm="fricas")

[Out]

-1/6*(32*a^3*x^3 - 32*a^2*x^2 - 32*a*x + 30*(a^3*x^3 - a^2*x^2 - a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x
)) + (2*a^5*x^5 + a^4*x^4 + 11*a^3*x^3 - 31*a^2*x^2 - 17*a*x + 32)*sqrt(-a^2*x^2 + 1) + 32)/(a^10*c^2*x^3 - a^
9*c^2*x^2 - a^8*c^2*x + a^7*c^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{x^{6}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{7}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{2}} \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)**2,x)

[Out]

(Integral(x**6/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x)
+ Integral(a*x**7/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)),
x))/c**2

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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 )}^{2} \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)^2,x, algorithm="giac")

[Out]

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