3.1372 \(\int \frac{e^{-4 \tanh ^{-1}(a x)} x^2}{(c-a^2 c x^2)^9} \, dx\)

Optimal. Leaf size=31 \[ \frac{4 a x+1}{60 a^3 c^9 (1-a x)^6 (a x+1)^{10}} \]

[Out]

(1 + 4*a*x)/(60*a^3*c^9*(1 - a*x)^6*(1 + a*x)^10)

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Rubi [A]  time = 0.0868716, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {6150, 81} \[ \frac{4 a x+1}{60 a^3 c^9 (1-a x)^6 (a x+1)^{10}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(E^(4*ArcTanh[a*x])*(c - a^2*c*x^2)^9),x]

[Out]

(1 + 4*a*x)/(60*a^3*c^9*(1 - a*x)^6*(1 + a*x)^10)

Rule 6150

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

Rule 81

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*
x)^(n + 1)*(e + f*x)^(p + 1)*(2*a*d*f*(n + p + 3) - b*(d*e*(n + 2) + c*f*(p + 2)) + b*d*f*(n + p + 2)*x))/(d^2
*f^2*(n + p + 2)*(n + p + 3)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] && NeQ[n + p + 3,
0] && EqQ[d*f*(n + p + 2)*(a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1)))) - b*(d*e*(n + 1)
+ c*f*(p + 1))*(a*d*f*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2))), 0]

Rubi steps

\begin{align*} \int \frac{e^{-4 \tanh ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^9} \, dx &=\frac{\int \frac{x^2}{(1-a x)^7 (1+a x)^{11}} \, dx}{c^9}\\ &=\frac{1+4 a x}{60 a^3 c^9 (1-a x)^6 (1+a x)^{10}}\\ \end{align*}

Mathematica [A]  time = 0.169063, size = 30, normalized size = 0.97 \[ \frac{4 a x+1}{60 a^3 c^9 (a x-1)^6 (a x+1)^{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(E^(4*ArcTanh[a*x])*(c - a^2*c*x^2)^9),x]

[Out]

(1 + 4*a*x)/(60*a^3*c^9*(-1 + a*x)^6*(1 + a*x)^10)

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Maple [B]  time = 0.04, size = 186, normalized size = 6. \begin{align*}{\frac{1}{{c}^{9}} \left ( -{\frac{1}{1280\,{a}^{3} \left ( ax+1 \right ) ^{10}}}-{\frac{1}{768\,{a}^{3} \left ( ax+1 \right ) ^{9}}}+{\frac{7}{6144\,{a}^{3} \left ( ax+1 \right ) ^{6}}}+{\frac{21}{10240\,{a}^{3} \left ( ax+1 \right ) ^{5}}}+{\frac{21}{8192\,{a}^{3} \left ( ax+1 \right ) ^{4}}}+{\frac{11}{4096\,{a}^{3} \left ( ax+1 \right ) ^{3}}}+{\frac{165}{65536\,{a}^{3} \left ( ax+1 \right ) ^{2}}}+{\frac{143}{65536\,{a}^{3} \left ( ax+1 \right ) }}-{\frac{1}{1024\,{a}^{3} \left ( ax+1 \right ) ^{8}}}+{\frac{1}{12288\,{a}^{3} \left ( ax-1 \right ) ^{6}}}-{\frac{7}{20480\,{a}^{3} \left ( ax-1 \right ) ^{5}}}-{\frac{11}{8192\,{a}^{3} \left ( ax-1 \right ) ^{3}}}+{\frac{121}{65536\,{a}^{3} \left ( ax-1 \right ) ^{2}}}-{\frac{143}{65536\,{a}^{3} \left ( ax-1 \right ) }}+{\frac{13}{16384\,{a}^{3} \left ( ax-1 \right ) ^{4}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a*x+1)^4*(-a^2*x^2+1)^2/(-a^2*c*x^2+c)^9,x)

[Out]

1/c^9*(-1/1280/a^3/(a*x+1)^10-1/768/a^3/(a*x+1)^9+7/6144/a^3/(a*x+1)^6+21/10240/a^3/(a*x+1)^5+21/8192/a^3/(a*x
+1)^4+11/4096/a^3/(a*x+1)^3+165/65536/a^3/(a*x+1)^2+143/65536/a^3/(a*x+1)-1/1024/a^3/(a*x+1)^8+1/12288/a^3/(a*
x-1)^6-7/20480/a^3/(a*x-1)^5-11/8192/a^3/(a*x-1)^3+121/65536/a^3/(a*x-1)^2-143/65536/a^3/(a*x-1)+13/16384/a^3/
(a*x-1)^4)

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Maxima [B]  time = 1.10271, size = 228, normalized size = 7.35 \begin{align*} \frac{4 \, a x + 1}{60 \,{\left (a^{19} c^{9} x^{16} + 4 \, a^{18} c^{9} x^{15} - 20 \, a^{16} c^{9} x^{13} - 20 \, a^{15} c^{9} x^{12} + 36 \, a^{14} c^{9} x^{11} + 64 \, a^{13} c^{9} x^{10} - 20 \, a^{12} c^{9} x^{9} - 90 \, a^{11} c^{9} x^{8} - 20 \, a^{10} c^{9} x^{7} + 64 \, a^{9} c^{9} x^{6} + 36 \, a^{8} c^{9} x^{5} - 20 \, a^{7} c^{9} x^{4} - 20 \, a^{6} c^{9} x^{3} + 4 \, a^{4} c^{9} x + a^{3} c^{9}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a*x+1)^4*(-a^2*x^2+1)^2/(-a^2*c*x^2+c)^9,x, algorithm="maxima")

[Out]

1/60*(4*a*x + 1)/(a^19*c^9*x^16 + 4*a^18*c^9*x^15 - 20*a^16*c^9*x^13 - 20*a^15*c^9*x^12 + 36*a^14*c^9*x^11 + 6
4*a^13*c^9*x^10 - 20*a^12*c^9*x^9 - 90*a^11*c^9*x^8 - 20*a^10*c^9*x^7 + 64*a^9*c^9*x^6 + 36*a^8*c^9*x^5 - 20*a
^7*c^9*x^4 - 20*a^6*c^9*x^3 + 4*a^4*c^9*x + a^3*c^9)

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Fricas [B]  time = 2.08409, size = 370, normalized size = 11.94 \begin{align*} \frac{4 \, a x + 1}{60 \,{\left (a^{19} c^{9} x^{16} + 4 \, a^{18} c^{9} x^{15} - 20 \, a^{16} c^{9} x^{13} - 20 \, a^{15} c^{9} x^{12} + 36 \, a^{14} c^{9} x^{11} + 64 \, a^{13} c^{9} x^{10} - 20 \, a^{12} c^{9} x^{9} - 90 \, a^{11} c^{9} x^{8} - 20 \, a^{10} c^{9} x^{7} + 64 \, a^{9} c^{9} x^{6} + 36 \, a^{8} c^{9} x^{5} - 20 \, a^{7} c^{9} x^{4} - 20 \, a^{6} c^{9} x^{3} + 4 \, a^{4} c^{9} x + a^{3} c^{9}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a*x+1)^4*(-a^2*x^2+1)^2/(-a^2*c*x^2+c)^9,x, algorithm="fricas")

[Out]

1/60*(4*a*x + 1)/(a^19*c^9*x^16 + 4*a^18*c^9*x^15 - 20*a^16*c^9*x^13 - 20*a^15*c^9*x^12 + 36*a^14*c^9*x^11 + 6
4*a^13*c^9*x^10 - 20*a^12*c^9*x^9 - 90*a^11*c^9*x^8 - 20*a^10*c^9*x^7 + 64*a^9*c^9*x^6 + 36*a^8*c^9*x^5 - 20*a
^7*c^9*x^4 - 20*a^6*c^9*x^3 + 4*a^4*c^9*x + a^3*c^9)

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Sympy [B]  time = 100.454, size = 178, normalized size = 5.74 \begin{align*} \frac{4 a x + 1}{60 a^{19} c^{9} x^{16} + 240 a^{18} c^{9} x^{15} - 1200 a^{16} c^{9} x^{13} - 1200 a^{15} c^{9} x^{12} + 2160 a^{14} c^{9} x^{11} + 3840 a^{13} c^{9} x^{10} - 1200 a^{12} c^{9} x^{9} - 5400 a^{11} c^{9} x^{8} - 1200 a^{10} c^{9} x^{7} + 3840 a^{9} c^{9} x^{6} + 2160 a^{8} c^{9} x^{5} - 1200 a^{7} c^{9} x^{4} - 1200 a^{6} c^{9} x^{3} + 240 a^{4} c^{9} x + 60 a^{3} c^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(a*x+1)**4*(-a**2*x**2+1)**2/(-a**2*c*x**2+c)**9,x)

[Out]

(4*a*x + 1)/(60*a**19*c**9*x**16 + 240*a**18*c**9*x**15 - 1200*a**16*c**9*x**13 - 1200*a**15*c**9*x**12 + 2160
*a**14*c**9*x**11 + 3840*a**13*c**9*x**10 - 1200*a**12*c**9*x**9 - 5400*a**11*c**9*x**8 - 1200*a**10*c**9*x**7
+ 3840*a**9*c**9*x**6 + 2160*a**8*c**9*x**5 - 1200*a**7*c**9*x**4 - 1200*a**6*c**9*x**3 + 240*a**4*c**9*x + 6
0*a**3*c**9)

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Giac [B]  time = 1.14097, size = 188, normalized size = 6.06 \begin{align*} -\frac{2145 \, a^{5} x^{5} - 12540 \, a^{4} x^{4} + 30030 \, a^{3} x^{3} - 37080 \, a^{2} x^{2} + 23841 \, a x - 6476}{983040 \,{\left (a x - 1\right )}^{6} a^{3} c^{9}} + \frac{2145 \, a^{9} x^{9} + 21780 \, a^{8} x^{8} + 99660 \, a^{7} x^{7} + 270480 \, a^{6} x^{6} + 481446 \, a^{5} x^{5} + 584920 \, a^{4} x^{4} + 486220 \, a^{3} x^{3} + 265680 \, a^{2} x^{2} + 84065 \, a x + 9908}{983040 \,{\left (a x + 1\right )}^{10} a^{3} c^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a*x+1)^4*(-a^2*x^2+1)^2/(-a^2*c*x^2+c)^9,x, algorithm="giac")

[Out]

-1/983040*(2145*a^5*x^5 - 12540*a^4*x^4 + 30030*a^3*x^3 - 37080*a^2*x^2 + 23841*a*x - 6476)/((a*x - 1)^6*a^3*c
^9) + 1/983040*(2145*a^9*x^9 + 21780*a^8*x^8 + 99660*a^7*x^7 + 270480*a^6*x^6 + 481446*a^5*x^5 + 584920*a^4*x^
4 + 486220*a^3*x^3 + 265680*a^2*x^2 + 84065*a*x + 9908)/((a*x + 1)^10*a^3*c^9)