3.611 \(\int \frac{(d+e x^2)^2 (a+b \sin ^{-1}(c x))}{x^2} \, dx\)
Optimal. Leaf size=126 \[ -\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} e^2 x^3 \left (a+b \sin ^{-1}(c x)\right )-b c d^2 \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )+\frac{b e \sqrt{1-c^2 x^2} \left (6 c^2 d+e\right )}{3 c^3}-\frac{b e^2 \left (1-c^2 x^2\right )^{3/2}}{9 c^3} \]
[Out]
(b*e*(6*c^2*d + e)*Sqrt[1 - c^2*x^2])/(3*c^3) - (b*e^2*(1 - c^2*x^2)^(3/2))/(9*c^3) - (d^2*(a + b*ArcSin[c*x])
)/x + 2*d*e*x*(a + b*ArcSin[c*x]) + (e^2*x^3*(a + b*ArcSin[c*x]))/3 - b*c*d^2*ArcTanh[Sqrt[1 - c^2*x^2]]
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Rubi [A] time = 0.183663, antiderivative size = 126, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used =
{270, 4731, 1251, 897, 1153, 208} \[ -\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} e^2 x^3 \left (a+b \sin ^{-1}(c x)\right )-b c d^2 \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )+\frac{b e \sqrt{1-c^2 x^2} \left (6 c^2 d+e\right )}{3 c^3}-\frac{b e^2 \left (1-c^2 x^2\right )^{3/2}}{9 c^3} \]
Antiderivative was successfully verified.
[In]
Int[((d + e*x^2)^2*(a + b*ArcSin[c*x]))/x^2,x]
[Out]
(b*e*(6*c^2*d + e)*Sqrt[1 - c^2*x^2])/(3*c^3) - (b*e^2*(1 - c^2*x^2)^(3/2))/(9*c^3) - (d^2*(a + b*ArcSin[c*x])
)/x + 2*d*e*x*(a + b*ArcSin[c*x]) + (e^2*x^3*(a + b*ArcSin[c*x]))/3 - b*c*d^2*ArcTanh[Sqrt[1 - c^2*x^2]]
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 1153
Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]
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^2} \, dx &=-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} e^2 x^3 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac{-d^2+2 d e x^2+\frac{e^2 x^4}{3}}{x \sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} e^2 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{2} (b c) \operatorname{Subst}\left (\int \frac{-d^2+2 d e x+\frac{e^2 x^2}{3}}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )\\ &=-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} e^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{b \operatorname{Subst}\left (\int \frac{\frac{-c^4 d^2+2 c^2 d e+\frac{e^2}{3}}{c^4}-\frac{\left (2 c^2 d e+\frac{2 e^2}{3}\right ) x^2}{c^4}+\frac{e^2 x^4}{3 c^4}}{\frac{1}{c^2}-\frac{x^2}{c^2}} \, dx,x,\sqrt{1-c^2 x^2}\right )}{c}\\ &=-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} e^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{b \operatorname{Subst}\left (\int \left (\frac{1}{3} e \left (6 d+\frac{e}{c^2}\right )-\frac{e^2 x^2}{3 c^2}-\frac{d^2}{\frac{1}{c^2}-\frac{x^2}{c^2}}\right ) \, dx,x,\sqrt{1-c^2 x^2}\right )}{c}\\ &=\frac{b e \left (6 c^2 d+e\right ) \sqrt{1-c^2 x^2}}{3 c^3}-\frac{b e^2 \left (1-c^2 x^2\right )^{3/2}}{9 c^3}-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} e^2 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac{\left (b d^2\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 )}{c}\\ &=\frac{b e \left (6 c^2 d+e\right ) \sqrt{1-c^2 x^2}}{3 c^3}-\frac{b e^2 \left (1-c^2 x^2\right )^{3/2}}{9 c^3}-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} e^2 x^3 \left (a+b \sin ^{-1}(c x)\right )-b c d^2 \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.138299, size = 129, normalized size = 1.02 \[ \frac{1}{9} \left (-\frac{9 a d^2}{x}+18 a d e x+3 a e^2 x^3-9 b c d^2 \log \left (\sqrt{1-c^2 x^2}+1\right )+\frac{b e \sqrt{1-c^2 x^2} \left (c^2 \left (18 d+e x^2\right )+2 e\right )}{c^3}+\frac{3 b \sin ^{-1}(c x) \left (-3 d^2+6 d e x^2+e^2 x^4\right )}{x}+9 b c d^2 \log (x)\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[((d + e*x^2)^2*(a + b*ArcSin[c*x]))/x^2,x]
[Out]
((-9*a*d^2)/x + 18*a*d*e*x + 3*a*e^2*x^3 + (b*e*Sqrt[1 - c^2*x^2]*(2*e + c^2*(18*d + e*x^2)))/c^3 + (3*b*(-3*d
^2 + 6*d*e*x^2 + e^2*x^4)*ArcSin[c*x])/x + 9*b*c*d^2*Log[x] - 9*b*c*d^2*Log[1 + Sqrt[1 - c^2*x^2]])/9
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Maple [A] time = 0.008, size = 168, normalized size = 1.3 \begin{align*} c \left ({\frac{a}{{c}^{4}} \left ({\frac{{e}^{2}{c}^{3}{x}^{3}}{3}}+2\,{c}^{3}edx-{\frac{{d}^{2}{c}^{3}}{x}} \right ) }+{\frac{b}{{c}^{4}} \left ({\frac{\arcsin \left ( cx \right ){e}^{2}{c}^{3}{x}^{3}}{3}}+2\,\arcsin \left ( cx \right ){c}^{3}edx-{\frac{\arcsin \left ( cx \right ){d}^{2}{c}^{3}}{x}}-{\frac{{e}^{2}}{3} \left ( -{\frac{{c}^{2}{x}^{2}}{3}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{2}{3}\sqrt{-{c}^{2}{x}^{2}+1}} \right ) }+2\,{c}^{2}ed\sqrt{-{c}^{2}{x}^{2}+1}-{d}^{2}{c}^{4}{\it Artanh} \left ({\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}} \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^2,x)
[Out]
c*(a/c^4*(1/3*e^2*c^3*x^3+2*c^3*e*d*x-d^2*c^3/x)+b/c^4*(1/3*arcsin(c*x)*e^2*c^3*x^3+2*arcsin(c*x)*c^3*e*d*x-ar
csin(c*x)*d^2*c^3/x-1/3*e^2*(-1/3*c^2*x^2*(-c^2*x^2+1)^(1/2)-2/3*(-c^2*x^2+1)^(1/2))+2*c^2*e*d*(-c^2*x^2+1)^(1
/2)-d^2*c^4*arctanh(1/(-c^2*x^2+1)^(1/2))))
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Maxima [A] time = 1.46247, size = 204, normalized size = 1.62 \begin{align*} \frac{1}{3} \, a e^{2} x^{3} -{\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^{2} + \frac{1}{9} \,{\left (3 \, x^{3} \arcsin \left (c x\right ) + c{\left (\frac{\sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac{2 \, \sqrt{-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b e^{2} + 2 \, a d e x + \frac{2 \,{\left (c x \arcsin \left (c x\right ) + \sqrt{-c^{2} x^{2} + 1}\right )} b d e}{c} - \frac{a d^{2}}{x} \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^2,x, algorithm="maxima")
[Out]
1/3*a*e^2*x^3 - (c*log(2*sqrt(-c^2*x^2 + 1)/abs(x) + 2/abs(x)) + arcsin(c*x)/x)*b*d^2 + 1/9*(3*x^3*arcsin(c*x)
+ c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4))*b*e^2 + 2*a*d*e*x + 2*(c*x*arcsin(c*x) + sqrt(-c
^2*x^2 + 1))*b*d*e/c - a*d^2/x
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Fricas [A] time = 2.75518, size = 385, normalized size = 3.06 \begin{align*} \frac{6 \, a c^{3} e^{2} x^{4} - 9 \, b c^{4} d^{2} x \log \left (\sqrt{-c^{2} x^{2} + 1} + 1\right ) + 9 \, b c^{4} d^{2} x \log \left (\sqrt{-c^{2} x^{2} + 1} - 1\right ) + 36 \, a c^{3} d e x^{2} - 18 \, a c^{3} d^{2} + 6 \,{\left (b c^{3} e^{2} x^{4} + 6 \, b c^{3} d e x^{2} - 3 \, b c^{3} d^{2}\right )} \arcsin \left (c x\right ) + 2 \,{\left (b c^{2} e^{2} x^{3} + 2 \,{\left (9 \, b c^{2} d e + b e^{2}\right )} x\right )} \sqrt{-c^{2} x^{2} + 1}}{18 \, c^{3} x} \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^2,x, algorithm="fricas")
[Out]
1/18*(6*a*c^3*e^2*x^4 - 9*b*c^4*d^2*x*log(sqrt(-c^2*x^2 + 1) + 1) + 9*b*c^4*d^2*x*log(sqrt(-c^2*x^2 + 1) - 1)
+ 36*a*c^3*d*e*x^2 - 18*a*c^3*d^2 + 6*(b*c^3*e^2*x^4 + 6*b*c^3*d*e*x^2 - 3*b*c^3*d^2)*arcsin(c*x) + 2*(b*c^2*e
^2*x^3 + 2*(9*b*c^2*d*e + b*e^2)*x)*sqrt(-c^2*x^2 + 1))/(c^3*x)
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Sympy [A] time = 5.77377, size = 167, normalized size = 1.33 \begin{align*} - \frac{a d^{2}}{x} + 2 a d e x + \frac{a e^{2} x^{3}}{3} + b c d^{2} \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 c e^{2} \left (\begin{cases} - \frac{x^{2} \sqrt{- c^{2} x^{2} + 1}}{3 c^{2}} - \frac{2 \sqrt{- c^{2} x^{2} + 1}}{3 c^{4}} & \text{for}\: c \neq 0 \\\frac{x^{4}}{4} & \text{otherwise} \end{cases}\right )}{3} - \frac{b d^{2} \operatorname{asin}{\left (c x \right )}}{x} + 2 b d e \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 ) + \frac{b e^{2} x^{3} \operatorname{asin}{\left (c x \right )}}{3} \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**2,x)
[Out]
-a*d**2/x + 2*a*d*e*x + a*e**2*x**3/3 + b*c*d**2*Piecewise((-acosh(1/(c*x)), 1/Abs(c**2*x**2) > 1), (I*asin(1/
(c*x)), True)) - b*c*e**2*Piecewise((-x**2*sqrt(-c**2*x**2 + 1)/(3*c**2) - 2*sqrt(-c**2*x**2 + 1)/(3*c**4), Ne
(c, 0)), (x**4/4, True))/3 - b*d**2*asin(c*x)/x + 2*b*d*e*Piecewise((0, Eq(c, 0)), (x*asin(c*x) + sqrt(-c**2*x
**2 + 1)/c, True)) + b*e**2*x**3*asin(c*x)/3
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Giac [B] time = 3.077, size = 5733, normalized size = 45.5 \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^2,x, algorithm="giac")
[Out]
-1/2*b*c^12*d^2*x^8*arcsin(c*x)/((c^10*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 3*c^8*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 +
3*c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c^4*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^8) - 1/2*a*c^
12*d^2*x^8/((c^10*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 3*c^8*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^6*x^3/(sqrt(-c^2
*x^2 + 1) + 1)^3 + c^4*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^8) + b*c^11*d^2*x^7*log(abs(c)*abs
(x))/((c^10*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 3*c^8*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^6*x^3/(sqrt(-c^2*x^2 +
1) + 1)^3 + c^4*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^7) - b*c^11*d^2*x^7*log(sqrt(-c^2*x^2 +
1) + 1)/((c^10*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 3*c^8*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^6*x^3/(sqrt(-c^2*x^
2 + 1) + 1)^3 + c^4*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^7) - 2*b*c^10*d^2*x^6*arcsin(c*x)/((c
^10*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 3*c^8*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)
^3 + c^4*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^6) - 2*a*c^10*d^2*x^6/((c^10*x^7/(sqrt(-c^2*x^2
+ 1) + 1)^7 + 3*c^8*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c^4*x/(sqrt(-c^2*x
^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^6) + 3*b*c^9*d^2*x^5*log(abs(c)*abs(x))/((c^10*x^7/(sqrt(-c^2*x^2 + 1)
+ 1)^7 + 3*c^8*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c^4*x/(sqrt(-c^2*x^2 +
1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^5) - 3*b*c^9*d^2*x^5*log(sqrt(-c^2*x^2 + 1) + 1)/((c^10*x^7/(sqrt(-c^2*x^2 +
1) + 1)^7 + 3*c^8*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c^4*x/(sqrt(-c^2*x^
2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^5) - 3*b*c^8*d^2*x^4*arcsin(c*x)/((c^10*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 +
3*c^8*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c^4*x/(sqrt(-c^2*x^2 + 1) + 1))
*(sqrt(-c^2*x^2 + 1) + 1)^4) - 2*b*c^9*d*x^7*e/((c^10*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 3*c^8*x^5/(sqrt(-c^2*x^
2 + 1) + 1)^5 + 3*c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c^4*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1
)^7) + 4*b*c^8*d*x^6*arcsin(c*x)*e/((c^10*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 3*c^8*x^5/(sqrt(-c^2*x^2 + 1) + 1)^
5 + 3*c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c^4*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^6) - 3*a*c
^8*d^2*x^4/((c^10*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 3*c^8*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^6*x^3/(sqrt(-c^2
*x^2 + 1) + 1)^3 + c^4*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^4) + 4*a*c^8*d*x^6*e/((c^10*x^7/(s
qrt(-c^2*x^2 + 1) + 1)^7 + 3*c^8*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c^4*x
/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^6) + 3*b*c^7*d^2*x^3*log(abs(c)*abs(x))/((c^10*x^7/(sqrt(-
c^2*x^2 + 1) + 1)^7 + 3*c^8*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c^4*x/(sqr
t(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^3) - 3*b*c^7*d^2*x^3*log(sqrt(-c^2*x^2 + 1) + 1)/((c^10*x^7/(sq
rt(-c^2*x^2 + 1) + 1)^7 + 3*c^8*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c^4*x/
(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^3) - 2*b*c^6*d^2*x^2*arcsin(c*x)/((c^10*x^7/(sqrt(-c^2*x^2
+ 1) + 1)^7 + 3*c^8*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c^4*x/(sqrt(-c^2*x
^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^2) - 2*b*c^7*d*x^5*e/((c^10*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 3*c^8*x^5/
(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c^4*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2
*x^2 + 1) + 1)^5) + 8*b*c^6*d*x^4*arcsin(c*x)*e/((c^10*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 3*c^8*x^5/(sqrt(-c^2*x
^2 + 1) + 1)^5 + 3*c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c^4*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) +
1)^4) - 2*a*c^6*d^2*x^2/((c^10*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 3*c^8*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^6*x
^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c^4*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^2) - 2/9*b*c^7*x^7*e^
2/((c^10*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 3*c^8*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^6*x^3/(sqrt(-c^2*x^2 + 1)
+ 1)^3 + c^4*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^7) + 8*a*c^6*d*x^4*e/((c^10*x^7/(sqrt(-c^2*
x^2 + 1) + 1)^7 + 3*c^8*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c^4*x/(sqrt(-c
^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^4) + b*c^5*d^2*x*log(abs(c)*abs(x))/((c^10*x^7/(sqrt(-c^2*x^2 + 1)
+ 1)^7 + 3*c^8*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c^4*x/(sqrt(-c^2*x^2 +
1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)) - b*c^5*d^2*x*log(sqrt(-c^2*x^2 + 1) + 1)/((c^10*x^7/(sqrt(-c^2*x^2 + 1) +
1)^7 + 3*c^8*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c^4*x/(sqrt(-c^2*x^2 + 1)
+ 1))*(sqrt(-c^2*x^2 + 1) + 1)) - 1/2*b*c^4*d^2*arcsin(c*x)/(c^10*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 3*c^8*x^5/
(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c^4*x/(sqrt(-c^2*x^2 + 1) + 1)) + 2*b*c^5*
d*x^3*e/((c^10*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 3*c^8*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^6*x^3/(sqrt(-c^2*x^
2 + 1) + 1)^3 + c^4*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^3) + 4*b*c^4*d*x^2*arcsin(c*x)*e/((c^
10*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 3*c^8*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^
3 + c^4*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^2) - 1/2*a*c^4*d^2/(c^10*x^7/(sqrt(-c^2*x^2 + 1)
+ 1)^7 + 3*c^8*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c^4*x/(sqrt(-c^2*x^2 +
1) + 1)) - 2/3*b*c^5*x^5*e^2/((c^10*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 3*c^8*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*
c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c^4*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^5) + 8/3*b*c^4*x
^4*arcsin(c*x)*e^2/((c^10*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 3*c^8*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^6*x^3/(s
qrt(-c^2*x^2 + 1) + 1)^3 + c^4*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^4) + 4*a*c^4*d*x^2*e/((c^1
0*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 3*c^8*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3
+ c^4*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^2) + 8/3*a*c^4*x^4*e^2/((c^10*x^7/(sqrt(-c^2*x^2 +
1) + 1)^7 + 3*c^8*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c^4*x/(sqrt(-c^2*x^
2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^4) + 2*b*c^3*d*x*e/((c^10*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 3*c^8*x^5/(sq
rt(-c^2*x^2 + 1) + 1)^5 + 3*c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c^4*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^
2 + 1) + 1)) + 2/3*b*c^3*x^3*e^2/((c^10*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 3*c^8*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5
+ 3*c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c^4*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^3) + 2/9*b*c
*x*e^2/((c^10*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 3*c^8*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^6*x^3/(sqrt(-c^2*x^2
+ 1) + 1)^3 + c^4*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1))