3.4.56 \(\int \frac {e^{\tanh ^{-1}(a x)} x^3}{(c-a c x)^4} \, dx\) [356]

Optimal. Leaf size=138 \[ -\frac {2 \sqrt {1-a^2 x^2}}{a^4 c^4 (1-a x)}+\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^4 c^4 (1-a x)^5}-\frac {19 \left (1-a^2 x^2\right )^{3/2}}{35 a^4 c^4 (1-a x)^4}+\frac {86 \left (1-a^2 x^2\right )^{3/2}}{105 a^4 c^4 (1-a x)^3}+\frac {\text {ArcSin}(a x)}{a^4 c^4} \]

[Out]

1/7*(-a^2*x^2+1)^(3/2)/a^4/c^4/(-a*x+1)^5-19/35*(-a^2*x^2+1)^(3/2)/a^4/c^4/(-a*x+1)^4+86/105*(-a^2*x^2+1)^(3/2
)/a^4/c^4/(-a*x+1)^3+arcsin(a*x)/a^4/c^4-2*(-a^2*x^2+1)^(1/2)/a^4/c^4/(-a*x+1)

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Rubi [A]
time = 0.19, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6263, 1651, 673, 665, 677, 222} \begin {gather*} \frac {\text {ArcSin}(a x)}{a^4 c^4}+\frac {86 \left (1-a^2 x^2\right )^{3/2}}{105 a^4 c^4 (1-a x)^3}-\frac {19 \left (1-a^2 x^2\right )^{3/2}}{35 a^4 c^4 (1-a x)^4}+\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^4 c^4 (1-a x)^5}-\frac {2 \sqrt {1-a^2 x^2}}{a^4 c^4 (1-a x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^ArcTanh[a*x]*x^3)/(c - a*c*x)^4,x]

[Out]

(-2*Sqrt[1 - a^2*x^2])/(a^4*c^4*(1 - a*x)) + (1 - a^2*x^2)^(3/2)/(7*a^4*c^4*(1 - a*x)^5) - (19*(1 - a^2*x^2)^(
3/2))/(35*a^4*c^4*(1 - a*x)^4) + (86*(1 - a^2*x^2)^(3/2))/(105*a^4*c^4*(1 - a*x)^3) + ArcSin[a*x]/(a^4*c^4)

Rule 222

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 665

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^m*((a + c*x^2)^(p + 1)/
(2*c*d*(p + 1))), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p
+ 2, 0]

Rule 673

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-e)*(d + e*x)^m*((a + c*x^2)^(p +
1)/(2*c*d*(m + p + 1))), x] + Dist[Simplify[m + 2*p + 2]/(2*d*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^
p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[Simplify[m + 2*p +
2], 0]

Rule 677

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

Rule 1651

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + c*x^2)^p,
(d + e*x)^m*Pq, x], x] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && EqQ[c*d^2 + a*e^2, 0] && EqQ[m + Expon[Pq
, x] + 2*p + 1, 0] && ILtQ[m, 0]

Rule 6263

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

Rubi steps

\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x^3}{(c-a c x)^4} \, dx &=c \int \frac {x^3 \sqrt {1-a^2 x^2}}{(c-a c x)^5} \, dx\\ &=c \int \left (-\frac {\sqrt {1-a^2 x^2}}{a^3 c^5 (-1+a x)^5}-\frac {3 \sqrt {1-a^2 x^2}}{a^3 c^5 (-1+a x)^4}-\frac {3 \sqrt {1-a^2 x^2}}{a^3 c^5 (-1+a x)^3}-\frac {\sqrt {1-a^2 x^2}}{a^3 c^5 (-1+a x)^2}\right ) \, dx\\ &=-\frac {\int \frac {\sqrt {1-a^2 x^2}}{(-1+a x)^5} \, dx}{a^3 c^4}-\frac {\int \frac {\sqrt {1-a^2 x^2}}{(-1+a x)^2} \, dx}{a^3 c^4}-\frac {3 \int \frac {\sqrt {1-a^2 x^2}}{(-1+a x)^4} \, dx}{a^3 c^4}-\frac {3 \int \frac {\sqrt {1-a^2 x^2}}{(-1+a x)^3} \, dx}{a^3 c^4}\\ &=-\frac {2 \sqrt {1-a^2 x^2}}{a^4 c^4 (1-a x)}+\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^4 c^4 (1-a x)^5}-\frac {3 \left (1-a^2 x^2\right )^{3/2}}{5 a^4 c^4 (1-a x)^4}+\frac {\left (1-a^2 x^2\right )^{3/2}}{a^4 c^4 (1-a x)^3}+\frac {2 \int \frac {\sqrt {1-a^2 x^2}}{(-1+a x)^4} \, dx}{7 a^3 c^4}+\frac {3 \int \frac {\sqrt {1-a^2 x^2}}{(-1+a x)^3} \, dx}{5 a^3 c^4}+\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a^3 c^4}\\ &=-\frac {2 \sqrt {1-a^2 x^2}}{a^4 c^4 (1-a x)}+\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^4 c^4 (1-a x)^5}-\frac {19 \left (1-a^2 x^2\right )^{3/2}}{35 a^4 c^4 (1-a x)^4}+\frac {4 \left (1-a^2 x^2\right )^{3/2}}{5 a^4 c^4 (1-a x)^3}+\frac {\sin ^{-1}(a x)}{a^4 c^4}-\frac {2 \int \frac {\sqrt {1-a^2 x^2}}{(-1+a x)^3} \, dx}{35 a^3 c^4}\\ &=-\frac {2 \sqrt {1-a^2 x^2}}{a^4 c^4 (1-a x)}+\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^4 c^4 (1-a x)^5}-\frac {19 \left (1-a^2 x^2\right )^{3/2}}{35 a^4 c^4 (1-a x)^4}+\frac {86 \left (1-a^2 x^2\right )^{3/2}}{105 a^4 c^4 (1-a x)^3}+\frac {\sin ^{-1}(a x)}{a^4 c^4}\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 94, normalized size = 0.68 \begin {gather*} \frac {\sqrt {1+a x} \left (\sqrt {1-a^2 x^2} \left (-166+559 a x-659 a^2 x^2+296 a^3 x^3\right )+105 (-1+a x)^4 \text {ArcSin}(a x)\right )}{105 a^4 c^4 (1-a x)^{7/2} \sqrt {1-a^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^ArcTanh[a*x]*x^3)/(c - a*c*x)^4,x]

[Out]

(Sqrt[1 + a*x]*(Sqrt[1 - a^2*x^2]*(-166 + 559*a*x - 659*a^2*x^2 + 296*a^3*x^3) + 105*(-1 + a*x)^4*ArcSin[a*x])
)/(105*a^4*c^4*(1 - a*x)^(7/2)*Sqrt[1 - a^2*x^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(465\) vs. \(2(124)=248\).
time = 0.81, size = 466, normalized size = 3.38

method	result	size
default	\(\frac {\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{3} \sqrt {a^{2}}}+\frac {\frac {3 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{a \left (x -\frac {1}{a}\right )^{2}}-\frac {3 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{x -\frac {1}{a}}}{a^{5}}+\frac {\frac {7 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {14 a \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}}{a^{6}}+\frac {\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{7 a \left (x -\frac {1}{a}\right )^{4}}-\frac {6 a \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}\right )}{7}}{a^{7}}+\frac {5 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{a^{5} \left (x -\frac {1}{a}\right )}}{c^{4}}\)	\(466\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)/(-a^2*x^2+1)^(1/2)*x^3/(-a*c*x+c)^4,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.47, size = 208, normalized size = 1.51 \begin {gather*} \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{7 \, {\left (a^{8} c^{4} x^{4} - 4 \, a^{7} c^{4} x^{3} + 6 \, a^{6} c^{4} x^{2} - 4 \, a^{5} c^{4} x + a^{4} c^{4}\right )}} + \frac {43 \, \sqrt {-a^{2} x^{2} + 1}}{35 \, {\left (a^{7} c^{4} x^{3} - 3 \, a^{6} c^{4} x^{2} + 3 \, a^{5} c^{4} x - a^{4} c^{4}\right )}} + \frac {229 \, \sqrt {-a^{2} x^{2} + 1}}{105 \, {\left (a^{6} c^{4} x^{2} - 2 \, a^{5} c^{4} x + a^{4} c^{4}\right )}} + \frac {296 \, \sqrt {-a^{2} x^{2} + 1}}{105 \, {\left (a^{5} c^{4} x - a^{4} c^{4}\right )}} + \frac {\arcsin \left (a x\right )}{a^{4} c^{4}} \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^3/(-a*c*x+c)^4,x, algorithm="maxima")

[Out]

2/7*sqrt(-a^2*x^2 + 1)/(a^8*c^4*x^4 - 4*a^7*c^4*x^3 + 6*a^6*c^4*x^2 - 4*a^5*c^4*x + a^4*c^4) + 43/35*sqrt(-a^2
*x^2 + 1)/(a^7*c^4*x^3 - 3*a^6*c^4*x^2 + 3*a^5*c^4*x - a^4*c^4) + 229/105*sqrt(-a^2*x^2 + 1)/(a^6*c^4*x^2 - 2*
a^5*c^4*x + a^4*c^4) + 296/105*sqrt(-a^2*x^2 + 1)/(a^5*c^4*x - a^4*c^4) + arcsin(a*x)/(a^4*c^4)

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Fricas [A]
time = 0.36, size = 172, normalized size = 1.25 \begin {gather*} -\frac {166 \, a^{4} x^{4} - 664 \, a^{3} x^{3} + 996 \, a^{2} x^{2} - 664 \, a x + 210 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (296 \, a^{3} x^{3} - 659 \, a^{2} x^{2} + 559 \, a x - 166\right )} \sqrt {-a^{2} x^{2} + 1} + 166}{105 \, {\left (a^{8} c^{4} x^{4} - 4 \, a^{7} c^{4} x^{3} + 6 \, a^{6} c^{4} x^{2} - 4 \, a^{5} c^{4} x + a^{4} c^{4}\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^3/(-a*c*x+c)^4,x, algorithm="fricas")

[Out]

-1/105*(166*a^4*x^4 - 664*a^3*x^3 + 996*a^2*x^2 - 664*a*x + 210*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*
arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (296*a^3*x^3 - 659*a^2*x^2 + 559*a*x - 166)*sqrt(-a^2*x^2 + 1) + 166)
/(a^8*c^4*x^4 - 4*a^7*c^4*x^3 + 6*a^6*c^4*x^2 - 4*a^5*c^4*x + a^4*c^4)

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

[Out]

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

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Giac [A]
time = 0.42, size = 220, normalized size = 1.59 \begin {gather*} \frac {\arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{a^{3} c^{4} {\left | a \right |}} + \frac {2 \, {\left (\frac {1057 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{a^{2} x} - \frac {2751 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + \frac {3640 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{6} x^{3}} - \frac {2170 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4}}{a^{8} x^{4}} + \frac {735 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5}}{a^{10} x^{5}} - \frac {105 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{6}}{a^{12} x^{6}} - 166\right )}}{105 \, a^{3} c^{4} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{7} {\left | a \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^3/(-a*c*x+c)^4,x, algorithm="giac")

[Out]

arcsin(a*x)*sgn(a)/(a^3*c^4*abs(a)) + 2/105*(1057*(sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 2751*(sqrt(-a^2*x^
2 + 1)*abs(a) + a)^2/(a^4*x^2) + 3640*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3/(a^6*x^3) - 2170*(sqrt(-a^2*x^2 + 1)*a
bs(a) + a)^4/(a^8*x^4) + 735*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^5/(a^10*x^5) - 105*(sqrt(-a^2*x^2 + 1)*abs(a) + a
)^6/(a^12*x^6) - 166)/(a^3*c^4*((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 1)^7*abs(a))

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Mupad [B]
time = 0.07, size = 281, normalized size = 2.04 \begin {gather*} \frac {2\,\sqrt {1-a^2\,x^2}}{7\,\left (a^8\,c^4\,x^4-4\,a^7\,c^4\,x^3+6\,a^6\,c^4\,x^2-4\,a^5\,c^4\,x+a^4\,c^4\right )}+\frac {229\,\sqrt {1-a^2\,x^2}}{105\,\left (a^6\,c^4\,x^2-2\,a^5\,c^4\,x+a^4\,c^4\right )}+\frac {43\,\sqrt {1-a^2\,x^2}}{35\,\sqrt {-a^2}\,\left (a^2\,c^4\,\sqrt {-a^2}+3\,a^4\,c^4\,x^2\,\sqrt {-a^2}-a^5\,c^4\,x^3\,\sqrt {-a^2}-3\,a^3\,c^4\,x\,\sqrt {-a^2}\right )}+\frac {296\,\sqrt {1-a^2\,x^2}}{105\,\left (a^2\,c^4\,\sqrt {-a^2}-a^3\,c^4\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}+\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{a^3\,c^4\,\sqrt {-a^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*(a*x + 1))/((1 - a^2*x^2)^(1/2)*(c - a*c*x)^4),x)

[Out]

(2*(1 - a^2*x^2)^(1/2))/(7*(a^4*c^4 - 4*a^5*c^4*x + 6*a^6*c^4*x^2 - 4*a^7*c^4*x^3 + a^8*c^4*x^4)) + (229*(1 -
a^2*x^2)^(1/2))/(105*(a^4*c^4 - 2*a^5*c^4*x + a^6*c^4*x^2)) + (43*(1 - a^2*x^2)^(1/2))/(35*(-a^2)^(1/2)*(a^2*c
^4*(-a^2)^(1/2) + 3*a^4*c^4*x^2*(-a^2)^(1/2) - a^5*c^4*x^3*(-a^2)^(1/2) - 3*a^3*c^4*x*(-a^2)^(1/2))) + (296*(1
- a^2*x^2)^(1/2))/(105*(a^2*c^4*(-a^2)^(1/2) - a^3*c^4*x*(-a^2)^(1/2))*(-a^2)^(1/2)) + asinh(x*(-a^2)^(1/2))/
(a^3*c^4*(-a^2)^(1/2))

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