3.613 \(\int \frac{(d+e x^2)^2 (a+b \sin ^{-1}(c x))}{x^4} \, dx\)

Optimal. Leaf size=126 \[ -\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}-\frac{2 d e \left (a+b \sin ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \sin ^{-1}(c x)\right )-\frac{b c d^2 \sqrt{1-c^2 x^2}}{6 x^2}-\frac{1}{6} b c d \left (c^2 d+12 e\right ) \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )+\frac{b e^2 \sqrt{1-c^2 x^2}}{c} \]

[Out]

(b*e^2*Sqrt[1 - c^2*x^2])/c - (b*c*d^2*Sqrt[1 - c^2*x^2])/(6*x^2) - (d^2*(a + b*ArcSin[c*x]))/(3*x^3) - (2*d*e
*(a + b*ArcSin[c*x]))/x + e^2*x*(a + b*ArcSin[c*x]) - (b*c*d*(c^2*d + 12*e)*ArcTanh[Sqrt[1 - c^2*x^2]])/6

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Rubi [A]  time = 0.201253, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {270, 4731, 1251, 897, 1157, 388, 208} \[ -\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}-\frac{2 d e \left (a+b \sin ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \sin ^{-1}(c x)\right )-\frac{b c d^2 \sqrt{1-c^2 x^2}}{6 x^2}-\frac{1}{6} b c d \left (c^2 d+12 e\right ) \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )+\frac{b e^2 \sqrt{1-c^2 x^2}}{c} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x^2)^2*(a + b*ArcSin[c*x]))/x^4,x]

[Out]

(b*e^2*Sqrt[1 - c^2*x^2])/c - (b*c*d^2*Sqrt[1 - c^2*x^2])/(6*x^2) - (d^2*(a + b*ArcSin[c*x]))/(3*x^3) - (2*d*e
*(a + b*ArcSin[c*x]))/x + e^2*x*(a + b*ArcSin[c*x]) - (b*c*d*(c^2*d + 12*e)*ArcTanh[Sqrt[1 - c^2*x^2]])/6

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 4731

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 -
c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] ||
(IGtQ[(m - 1)/2, 0] && LeQ[m + p, 0]))

Rule 1251

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rule 897

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + (g*x^q)/e)^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - ((2*c*d - b*e)*x^q)/e^2 + (c*x^(2*q))/e^2)^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n,
p] && FractionQ[m]

Rule 1157

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ
uotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2,
x], x, 0]}, -Simp[(R*x*(d + e*x^2)^(q + 1))/(2*d*(q + 1)), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*
ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && N
eQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rule 388

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

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{\left (d+e x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{x^4} \, dx &=-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}-\frac{2 d e \left (a+b \sin ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac{-\frac{d^2}{3}-2 d e x^2+e^2 x^4}{x^3 \sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}-\frac{2 d e \left (a+b \sin ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{2} (b c) \operatorname{Subst}\left (\int \frac{-\frac{d^2}{3}-2 d e x+e^2 x^2}{x^2 \sqrt{1-c^2 x}} \, dx,x,x^2\right )\\ &=-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}-\frac{2 d e \left (a+b \sin ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac{b \operatorname{Subst}\left (\int \frac{\frac{-\frac{1}{3} c^4 d^2-2 c^2 d e+e^2}{c^4}-\frac{\left (-2 c^2 d e+2 e^2\right ) x^2}{c^4}+\frac{e^2 x^4}{c^4}}{\left (\frac{1}{c^2}-\frac{x^2}{c^2}\right )^2} \, dx,x,\sqrt{1-c^2 x^2}\right )}{c}\\ &=-\frac{b c d^2 \sqrt{1-c^2 x^2}}{6 x^2}-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}-\frac{2 d e \left (a+b \sin ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{2} (b c) \operatorname{Subst}\left (\int \frac{\frac{1}{3} \left (d^2+\frac{12 d e}{c^2}-\frac{6 e^2}{c^4}\right )+\frac{2 e^2 x^2}{c^4}}{\frac{1}{c^2}-\frac{x^2}{c^2}} \, dx,x,\sqrt{1-c^2 x^2}\right )\\ &=\frac{b e^2 \sqrt{1-c^2 x^2}}{c}-\frac{b c d^2 \sqrt{1-c^2 x^2}}{6 x^2}-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}-\frac{2 d e \left (a+b \sin ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{6} \left (b c d \left (d+\frac{12 e}{c^2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{c^2}-\frac{x^2}{c^2}} \, dx,x,\sqrt{1-c^2 x^2}\right )\\ &=\frac{b e^2 \sqrt{1-c^2 x^2}}{c}-\frac{b c d^2 \sqrt{1-c^2 x^2}}{6 x^2}-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}-\frac{2 d e \left (a+b \sin ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{6} b c d \left (c^2 d+12 e\right ) \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )\\ \end{align*}

Mathematica [A]  time = 0.162501, size = 140, normalized size = 1.11 \[ \frac{1}{6} \left (-\frac{2 a d^2}{x^3}-\frac{12 a d e}{x}+6 a e^2 x+6 b \sqrt{1-c^2 x^2} \left (\frac{e^2}{c}-\frac{c d^2}{6 x^2}\right )-b c d \left (c^2 d+12 e\right ) \log \left (\sqrt{1-c^2 x^2}+1\right )+b c d \log (x) \left (c^2 d+12 e\right )-\frac{2 b \sin ^{-1}(c x) \left (d^2+6 d e x^2-3 e^2 x^4\right )}{x^3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x^2)^2*(a + b*ArcSin[c*x]))/x^4,x]

[Out]

((-2*a*d^2)/x^3 - (12*a*d*e)/x + 6*a*e^2*x + 6*b*(e^2/c - (c*d^2)/(6*x^2))*Sqrt[1 - c^2*x^2] - (2*b*(d^2 + 6*d
*e*x^2 - 3*e^2*x^4)*ArcSin[c*x])/x^3 + b*c*d*(c^2*d + 12*e)*Log[x] - b*c*d*(c^2*d + 12*e)*Log[1 + Sqrt[1 - c^2
*x^2]])/6

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Maple [A]  time = 0.01, size = 156, normalized size = 1.2 \begin{align*}{c}^{3} \left ({\frac{a}{{c}^{4}} \left ( cx{e}^{2}-2\,{\frac{ced}{x}}-{\frac{{d}^{2}c}{3\,{x}^{3}}} \right ) }+{\frac{b}{{c}^{4}} \left ( \arcsin \left ( cx \right ) cx{e}^{2}-2\,{\frac{\arcsin \left ( cx \right ) ced}{x}}-{\frac{\arcsin \left ( cx \right ){d}^{2}c}{3\,{x}^{3}}}+{e}^{2}\sqrt{-{c}^{2}{x}^{2}+1}-2\,{c}^{2}ed{\it Artanh} \left ({\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}} \right ) +{\frac{{d}^{2}{c}^{4}}{3} \left ( -{\frac{1}{2\,{c}^{2}{x}^{2}}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{1}{2}{\it Artanh} \left ({\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}} \right ) } \right ) } \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^2*(a+b*arcsin(c*x))/x^4,x)

[Out]

c^3*(a/c^4*(c*x*e^2-2*c*e*d/x-1/3*d^2*c/x^3)+b/c^4*(arcsin(c*x)*c*x*e^2-2*arcsin(c*x)*c*e*d/x-1/3*arcsin(c*x)*
d^2*c/x^3+e^2*(-c^2*x^2+1)^(1/2)-2*c^2*e*d*arctanh(1/(-c^2*x^2+1)^(1/2))+1/3*d^2*c^4*(-1/2/c^2/x^2*(-c^2*x^2+1
)^(1/2)-1/2*arctanh(1/(-c^2*x^2+1)^(1/2)))))

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Maxima [A]  time = 1.45057, size = 215, normalized size = 1.71 \begin{align*} -\frac{1}{6} \,{\left ({\left (c^{2} \log \left (\frac{2 \, \sqrt{-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) + \frac{\sqrt{-c^{2} x^{2} + 1}}{x^{2}}\right )} c + \frac{2 \, \arcsin \left (c x\right )}{x^{3}}\right )} b d^{2} - 2 \,{\left (c \log \left (\frac{2 \, \sqrt{-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) + \frac{\arcsin \left (c x\right )}{x}\right )} b d e + a e^{2} x + \frac{{\left (c x \arcsin \left (c x\right ) + \sqrt{-c^{2} x^{2} + 1}\right )} b e^{2}}{c} - \frac{2 \, a d e}{x} - \frac{a d^{2}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^2*(a+b*arcsin(c*x))/x^4,x, algorithm="maxima")

[Out]

-1/6*((c^2*log(2*sqrt(-c^2*x^2 + 1)/abs(x) + 2/abs(x)) + sqrt(-c^2*x^2 + 1)/x^2)*c + 2*arcsin(c*x)/x^3)*b*d^2
- 2*(c*log(2*sqrt(-c^2*x^2 + 1)/abs(x) + 2/abs(x)) + arcsin(c*x)/x)*b*d*e + a*e^2*x + (c*x*arcsin(c*x) + sqrt(
-c^2*x^2 + 1))*b*e^2/c - 2*a*d*e/x - 1/3*a*d^2/x^3

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Fricas [A]  time = 2.86726, size = 393, normalized size = 3.12 \begin{align*} \frac{12 \, a c e^{2} x^{4} - 24 \, a c d e x^{2} -{\left (b c^{4} d^{2} + 12 \, b c^{2} d e\right )} x^{3} \log \left (\sqrt{-c^{2} x^{2} + 1} + 1\right ) +{\left (b c^{4} d^{2} + 12 \, b c^{2} d e\right )} x^{3} \log \left (\sqrt{-c^{2} x^{2} + 1} - 1\right ) - 4 \, a c d^{2} + 4 \,{\left (3 \, b c e^{2} x^{4} - 6 \, b c d e x^{2} - b c d^{2}\right )} \arcsin \left (c x\right ) - 2 \,{\left (b c^{2} d^{2} x - 6 \, b e^{2} x^{3}\right )} \sqrt{-c^{2} x^{2} + 1}}{12 \, c x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^2*(a+b*arcsin(c*x))/x^4,x, algorithm="fricas")

[Out]

1/12*(12*a*c*e^2*x^4 - 24*a*c*d*e*x^2 - (b*c^4*d^2 + 12*b*c^2*d*e)*x^3*log(sqrt(-c^2*x^2 + 1) + 1) + (b*c^4*d^
2 + 12*b*c^2*d*e)*x^3*log(sqrt(-c^2*x^2 + 1) - 1) - 4*a*c*d^2 + 4*(3*b*c*e^2*x^4 - 6*b*c*d*e*x^2 - b*c*d^2)*ar
csin(c*x) - 2*(b*c^2*d^2*x - 6*b*e^2*x^3)*sqrt(-c^2*x^2 + 1))/(c*x^3)

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Sympy [A]  time = 7.1506, size = 219, normalized size = 1.74 \begin{align*} - \frac{a d^{2}}{3 x^{3}} - \frac{2 a d e}{x} + a e^{2} x + \frac{b c d^{2} \left (\begin{cases} - \frac{c^{2} \operatorname{acosh}{\left (\frac{1}{c x} \right )}}{2} - \frac{c \sqrt{-1 + \frac{1}{c^{2} x^{2}}}}{2 x} & \text{for}\: \frac{1}{\left |{c^{2} x^{2}}\right |} > 1 \\\frac{i c^{2} \operatorname{asin}{\left (\frac{1}{c x} \right )}}{2} - \frac{i c}{2 x \sqrt{1 - \frac{1}{c^{2} x^{2}}}} + \frac{i}{2 c x^{3} \sqrt{1 - \frac{1}{c^{2} x^{2}}}} & \text{otherwise} \end{cases}\right )}{3} + 2 b c d e \left (\begin{cases} - \operatorname{acosh}{\left (\frac{1}{c x} \right )} & \text{for}\: \frac{1}{\left |{c^{2} x^{2}}\right |} > 1 \\i \operatorname{asin}{\left (\frac{1}{c x} \right )} & \text{otherwise} \end{cases}\right ) - \frac{b d^{2} \operatorname{asin}{\left (c x \right )}}{3 x^{3}} - \frac{2 b d e \operatorname{asin}{\left (c x \right )}}{x} + b e^{2} \left (\begin{cases} 0 & \text{for}\: c = 0 \\x \operatorname{asin}{\left (c x \right )} + \frac{\sqrt{- c^{2} x^{2} + 1}}{c} & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)**2*(a+b*asin(c*x))/x**4,x)

[Out]

-a*d**2/(3*x**3) - 2*a*d*e/x + a*e**2*x + b*c*d**2*Piecewise((-c**2*acosh(1/(c*x))/2 - c*sqrt(-1 + 1/(c**2*x**
2))/(2*x), 1/Abs(c**2*x**2) > 1), (I*c**2*asin(1/(c*x))/2 - I*c/(2*x*sqrt(1 - 1/(c**2*x**2))) + I/(2*c*x**3*sq
rt(1 - 1/(c**2*x**2))), True))/3 + 2*b*c*d*e*Piecewise((-acosh(1/(c*x)), 1/Abs(c**2*x**2) > 1), (I*asin(1/(c*x
)), True)) - b*d**2*asin(c*x)/(3*x**3) - 2*b*d*e*asin(c*x)/x + b*e**2*Piecewise((0, Eq(c, 0)), (x*asin(c*x) +
sqrt(-c**2*x**2 + 1)/c, True))

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Giac [B]  time = 84.8751, size = 3429, normalized size = 27.21 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^2*(a+b*arcsin(c*x))/x^4,x, algorithm="giac")

[Out]

-1/24*b*c^12*d^2*x^8*arcsin(c*x)/((c^6*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^4*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(s
qrt(-c^2*x^2 + 1) + 1)^8) - 1/24*a*c^12*d^2*x^8/((c^6*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^4*x^3/(sqrt(-c^2*x^2
+ 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^8) + 1/24*b*c^11*d^2*x^7/((c^6*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^4*x^3/
(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^7) - 1/6*b*c^10*d^2*x^6*arcsin(c*x)/((c^6*x^5/(sqrt(-c^2*
x^2 + 1) + 1)^5 + c^4*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^6) - 1/6*a*c^10*d^2*x^6/((c^6*x
^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^4*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^6) + 1/6*b*c^9*d^
2*x^5*log(abs(c)*abs(x))/((c^6*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^4*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2
*x^2 + 1) + 1)^5) - 1/6*b*c^9*d^2*x^5*log(sqrt(-c^2*x^2 + 1) + 1)/((c^6*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^4*x
^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^5) + 1/24*b*c^9*d^2*x^5/((c^6*x^5/(sqrt(-c^2*x^2 + 1)
+ 1)^5 + c^4*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^5) - 1/4*b*c^8*d^2*x^4*arcsin(c*x)/((c^6
*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^4*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^4) - b*c^8*d*x^
6*arcsin(c*x)*e/((c^6*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^4*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1)
+ 1)^6) - 1/4*a*c^8*d^2*x^4/((c^6*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^4*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(
-c^2*x^2 + 1) + 1)^4) - a*c^8*d*x^6*e/((c^6*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^4*x^3/(sqrt(-c^2*x^2 + 1) + 1)^
3)*(sqrt(-c^2*x^2 + 1) + 1)^6) + 1/6*b*c^7*d^2*x^3*log(abs(c)*abs(x))/((c^6*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c
^4*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^3) + 2*b*c^7*d*x^5*e*log(abs(c)*abs(x))/((c^6*x^5/
(sqrt(-c^2*x^2 + 1) + 1)^5 + c^4*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^5) - 1/6*b*c^7*d^2*x
^3*log(sqrt(-c^2*x^2 + 1) + 1)/((c^6*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^4*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqr
t(-c^2*x^2 + 1) + 1)^3) - 2*b*c^7*d*x^5*e*log(sqrt(-c^2*x^2 + 1) + 1)/((c^6*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c
^4*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^5) - 1/24*b*c^7*d^2*x^3/((c^6*x^5/(sqrt(-c^2*x^2 +
1) + 1)^5 + c^4*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^3) - 1/6*b*c^6*d^2*x^2*arcsin(c*x)/(
(c^6*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^4*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^2) - 2*b*c^
6*d*x^4*arcsin(c*x)*e/((c^6*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^4*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^
2 + 1) + 1)^4) - 1/6*a*c^6*d^2*x^2/((c^6*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^4*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*
(sqrt(-c^2*x^2 + 1) + 1)^2) - 2*a*c^6*d*x^4*e/((c^6*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^4*x^3/(sqrt(-c^2*x^2 +
1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^4) + 2*b*c^5*d*x^3*e*log(abs(c)*abs(x))/((c^6*x^5/(sqrt(-c^2*x^2 + 1) + 1)
^5 + c^4*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^3) - 2*b*c^5*d*x^3*e*log(sqrt(-c^2*x^2 + 1)
+ 1)/((c^6*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^4*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^3) -
1/24*b*c^5*d^2*x/((c^6*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^4*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1
) + 1)) - 1/24*b*c^4*d^2*arcsin(c*x)/(c^6*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^4*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)
- b*c^4*d*x^2*arcsin(c*x)*e/((c^6*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^4*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(
-c^2*x^2 + 1) + 1)^2) - 1/24*a*c^4*d^2/(c^6*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^4*x^3/(sqrt(-c^2*x^2 + 1) + 1)^
3) - b*c^5*x^5*e^2/((c^6*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^4*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 +
1) + 1)^5) + 2*b*c^4*x^4*arcsin(c*x)*e^2/((c^6*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^4*x^3/(sqrt(-c^2*x^2 + 1) +
1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^4) - a*c^4*d*x^2*e/((c^6*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^4*x^3/(sqrt(-c^2*x
^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^2) + 2*a*c^4*x^4*e^2/((c^6*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^4*x^3/(
sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^4) + b*c^3*x^3*e^2/((c^6*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 +
c^4*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^3)