3.897 \(\int \frac {e^{\tanh ^{-1}(a x)} x^5}{(c-a^2 c x^2)^2} \, dx\)

Optimal. Leaf size=110 \[ \frac {5 \sin ^{-1}(a x)}{2 a^6 c^2}+\frac {x^4 (a x+1)}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {(15 a x+16) \sqrt {1-a^2 x^2}}{6 a^6 c^2}-\frac {x^2 (5 a x+4)}{3 a^4 c^2 \sqrt {1-a^2 x^2}} \]

[Out]

1/3*x^4*(a*x+1)/a^2/c^2/(-a^2*x^2+1)^(3/2)+5/2*arcsin(a*x)/a^6/c^2-1/3*x^2*(5*a*x+4)/a^4/c^2/(-a^2*x^2+1)^(1/2
)-1/6*(15*a*x+16)*(-a^2*x^2+1)^(1/2)/a^6/c^2

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Rubi [A]  time = 0.14, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6148, 819, 780, 216} \[ \frac {x^4 (a x+1)}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {x^2 (5 a x+4)}{3 a^4 c^2 \sqrt {1-a^2 x^2}}-\frac {(15 a x+16) \sqrt {1-a^2 x^2}}{6 a^6 c^2}+\frac {5 \sin ^{-1}(a x)}{2 a^6 c^2} \]

Antiderivative was successfully verified.

[In]

Int[(E^ArcTanh[a*x]*x^5)/(c - a^2*c*x^2)^2,x]

[Out]

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

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

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Mathematica [A]  time = 0.05, size = 86, normalized size = 0.78 \[ \frac {3 a^4 x^4+3 a^3 x^3-23 a^2 x^2+15 (a x-1) \sqrt {1-a^2 x^2} \sin ^{-1}(a x)-a x+16}{6 a^6 c^2 (a x-1) \sqrt {1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^ArcTanh[a*x]*x^5)/(c - a^2*c*x^2)^2,x]

[Out]

(16 - a*x - 23*a^2*x^2 + 3*a^3*x^3 + 3*a^4*x^4 + 15*(-1 + a*x)*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(6*a^6*c^2*(-1 +
a*x)*Sqrt[1 - a^2*x^2])

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fricas [A]  time = 0.73, size = 152, normalized size = 1.38 \[ -\frac {16 \, a^{3} x^{3} - 16 \, a^{2} x^{2} - 16 \, a x + 30 \, {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (3 \, a^{4} x^{4} + 3 \, a^{3} x^{3} - 23 \, a^{2} x^{2} - a x + 16\right )} \sqrt {-a^{2} x^{2} + 1} + 16}{6 \, {\left (a^{9} c^{2} x^{3} - a^{8} c^{2} x^{2} - a^{7} c^{2} x + a^{6} c^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^5/(-a^2*c*x^2+c)^2,x, algorithm="fricas")

[Out]

-1/6*(16*a^3*x^3 - 16*a^2*x^2 - 16*a*x + 30*(a^3*x^3 - a^2*x^2 - a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x
)) + (3*a^4*x^4 + 3*a^3*x^3 - 23*a^2*x^2 - a*x + 16)*sqrt(-a^2*x^2 + 1) + 16)/(a^9*c^2*x^3 - a^8*c^2*x^2 - a^7
*c^2*x + a^6*c^2)

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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^5/(-a^2*c*x^2+c)^2,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 [B]  time = 0.05, size = 202, normalized size = 1.84 \[ -\frac {x \sqrt {-a^{2} x^{2}+1}}{2 c^{2} a^{5}}+\frac {5 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 c^{2} a^{5} \sqrt {a^{2}}}-\frac {\sqrt {-a^{2} x^{2}+1}}{c^{2} a^{6}}+\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{6 c^{2} a^{8} \left (x -\frac {1}{a}\right )^{2}}+\frac {25 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{12 c^{2} a^{7} \left (x -\frac {1}{a}\right )}+\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 c^{2} a^{7} \left (x +\frac {1}{a}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)/(-a^2*x^2+1)^(1/2)*x^5/(-a^2*c*x^2+c)^2,x)

[Out]

-1/2/c^2/a^5*x*(-a^2*x^2+1)^(1/2)+5/2/c^2/a^5/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))-1/c^2/a^6*(
-a^2*x^2+1)^(1/2)+1/6/c^2/a^8/(x-1/a)^2*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)+25/12/c^2/a^7/(x-1/a)*(-a^2*(x-1/a)
^2-2*a*(x-1/a))^(1/2)+1/4/c^2/a^7/(x+1/a)*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ a \int \frac {x^{6}}{{\left (a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}\,{d x} - \frac {\frac {3 \, \sqrt {-a^{2} x^{2} + 1}}{c^{2}} - \frac {6 \, a^{2} x^{2} - 5}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{2}}}{3 \, a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^5/(-a^2*c*x^2+c)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.90, size = 218, normalized size = 1.98 \[ \frac {\sqrt {1-a^2\,x^2}}{6\,\left (a^8\,c^2\,x^2-2\,a^7\,c^2\,x+a^6\,c^2\right )}-\frac {\sqrt {1-a^2\,x^2}}{4\,\left (a^4\,c^2\,\sqrt {-a^2}+a^5\,c^2\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}+\frac {25\,\sqrt {1-a^2\,x^2}}{12\,\left (a^4\,c^2\,\sqrt {-a^2}-a^5\,c^2\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{a^6\,c^2}-\frac {x\,\sqrt {1-a^2\,x^2}}{2\,a^5\,c^2}+\frac {5\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,a^5\,c^2\,\sqrt {-a^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5*(a*x + 1))/((c - a^2*c*x^2)^2*(1 - a^2*x^2)^(1/2)),x)

[Out]

(1 - a^2*x^2)^(1/2)/(6*(a^6*c^2 - 2*a^7*c^2*x + a^8*c^2*x^2)) - (1 - a^2*x^2)^(1/2)/(4*(a^4*c^2*(-a^2)^(1/2) +
a^5*c^2*x*(-a^2)^(1/2))*(-a^2)^(1/2)) + (25*(1 - a^2*x^2)^(1/2))/(12*(a^4*c^2*(-a^2)^(1/2) - a^5*c^2*x*(-a^2)
^(1/2))*(-a^2)^(1/2)) - (1 - a^2*x^2)^(1/2)/(a^6*c^2) - (x*(1 - a^2*x^2)^(1/2))/(2*a^5*c^2) + (5*asinh(x*(-a^2
)^(1/2)))/(2*a^5*c^2*(-a^2)^(1/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {x^{5}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{6}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**5/(-a**2*c*x**2+c)**2,x)

[Out]

(Integral(x**5/(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**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))/c**2

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