3.16 \(\int \frac{(d-c^2 d x^2)^2 (a+b \sin ^{-1}(c x))}{x^2} \, dx\)
Optimal. Leaf size=123 \[ \frac{1}{3} c^4 d^2 x^3 \left (a+b \sin ^{-1}(c x)\right )-2 c^2 d^2 x \left (a+b \sin ^{-1}(c x)\right )-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac{1}{9} b c d^2 \left (1-c^2 x^2\right )^{3/2}-\frac{5}{3} b c d^2 \sqrt{1-c^2 x^2}-b c d^2 \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right ) \]
[Out]
(-5*b*c*d^2*Sqrt[1 - c^2*x^2])/3 - (b*c*d^2*(1 - c^2*x^2)^(3/2))/9 - (d^2*(a + b*ArcSin[c*x]))/x - 2*c^2*d^2*x
*(a + b*ArcSin[c*x]) + (c^4*d^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 = 0.15514, antiderivative size = 123, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used =
{270, 4687, 12, 1251, 897, 1153, 208} \[ \frac{1}{3} c^4 d^2 x^3 \left (a+b \sin ^{-1}(c x)\right )-2 c^2 d^2 x \left (a+b \sin ^{-1}(c x)\right )-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac{1}{9} b c d^2 \left (1-c^2 x^2\right )^{3/2}-\frac{5}{3} b c d^2 \sqrt{1-c^2 x^2}-b c d^2 \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right ) \]
Antiderivative was successfully verified.
[In]
Int[((d - c^2*d*x^2)^2*(a + b*ArcSin[c*x]))/x^2,x]
[Out]
(-5*b*c*d^2*Sqrt[1 - c^2*x^2])/3 - (b*c*d^2*(1 - c^2*x^2)^(3/2))/9 - (d^2*(a + b*ArcSin[c*x]))/x - 2*c^2*d^2*x
*(a + b*ArcSin[c*x]) + (c^4*d^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 4687
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = I
ntHide[(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] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
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-c^2 d 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 c^2 d^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} c^4 d^2 x^3 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac{d^2 \left (-3-6 c^2 x^2+c^4 x^4\right )}{3 x \sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{x}-2 c^2 d^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} c^4 d^2 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{3} \left (b c d^2\right ) \int \frac{-3-6 c^2 x^2+c^4 x^4}{x \sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{x}-2 c^2 d^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} c^4 d^2 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{6} \left (b c d^2\right ) \operatorname{Subst}\left (\int \frac{-3-6 c^2 x+c^4 x^2}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )\\ &=-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{x}-2 c^2 d^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} c^4 d^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{\left (b d^2\right ) \operatorname{Subst}\left (\int \frac{-8+4 x^2+x^4}{\frac{1}{c^2}-\frac{x^2}{c^2}} \, dx,x,\sqrt{1-c^2 x^2}\right )}{3 c}\\ &=-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{x}-2 c^2 d^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} c^4 d^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{\left (b d^2\right ) \operatorname{Subst}\left (\int \left (-5 c^2-c^2 x^2-\frac{3}{\frac{1}{c^2}-\frac{x^2}{c^2}}\right ) \, dx,x,\sqrt{1-c^2 x^2}\right )}{3 c}\\ &=-\frac{5}{3} b c d^2 \sqrt{1-c^2 x^2}-\frac{1}{9} b c d^2 \left (1-c^2 x^2\right )^{3/2}-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{x}-2 c^2 d^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} c^4 d^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{5}{3} b c d^2 \sqrt{1-c^2 x^2}-\frac{1}{9} b c d^2 \left (1-c^2 x^2\right )^{3/2}-\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )}{x}-2 c^2 d^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} c^4 d^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 = 0.0902922, size = 126, normalized size = 1.02 \[ \frac{d^2 \left (3 a c^4 x^4-18 a c^2 x^2-9 a+b c^3 x^3 \sqrt{1-c^2 x^2}-16 b c x \sqrt{1-c^2 x^2}-9 b c x \log \left (\sqrt{1-c^2 x^2}+1\right )+3 b \left (c^4 x^4-6 c^2 x^2-3\right ) \sin ^{-1}(c x)+9 b c x \log (x)\right )}{9 x} \]
Antiderivative was successfully verified.
[In]
Integrate[((d - c^2*d*x^2)^2*(a + b*ArcSin[c*x]))/x^2,x]
[Out]
(d^2*(-9*a - 18*a*c^2*x^2 + 3*a*c^4*x^4 - 16*b*c*x*Sqrt[1 - c^2*x^2] + b*c^3*x^3*Sqrt[1 - c^2*x^2] + 3*b*(-3 -
6*c^2*x^2 + c^4*x^4)*ArcSin[c*x] + 9*b*c*x*Log[x] - 9*b*c*x*Log[1 + Sqrt[1 - c^2*x^2]]))/(9*x)
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Maple [A] time = 0.007, size = 117, normalized size = 1. \begin{align*} c \left ({d}^{2}a \left ({\frac{{c}^{3}{x}^{3}}{3}}-2\,cx-{\frac{1}{cx}} \right ) +{d}^{2}b \left ({\frac{{c}^{3}{x}^{3}\arcsin \left ( cx \right ) }{3}}-2\,cx\arcsin \left ( cx \right ) -{\frac{\arcsin \left ( cx \right ) }{cx}}+{\frac{{c}^{2}{x}^{2}}{9}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{16}{9}\sqrt{-{c}^{2}{x}^{2}+1}}-{\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((-c^2*d*x^2+d)^2*(a+b*arcsin(c*x))/x^2,x)
[Out]
c*(d^2*a*(1/3*c^3*x^3-2*c*x-1/c/x)+d^2*b*(1/3*c^3*x^3*arcsin(c*x)-2*c*x*arcsin(c*x)-1/c/x*arcsin(c*x)+1/9*c^2*
x^2*(-c^2*x^2+1)^(1/2)-16/9*(-c^2*x^2+1)^(1/2)-arctanh(1/(-c^2*x^2+1)^(1/2))))
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Maxima [A] time = 1.5615, size = 216, normalized size = 1.76 \begin{align*} \frac{1}{3} \, a c^{4} d^{2} x^{3} + \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 c^{4} d^{2} - 2 \, a c^{2} d^{2} x - 2 \,{\left (c x \arcsin \left (c x\right ) + \sqrt{-c^{2} x^{2} + 1}\right )} b c d^{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^{2} - \frac{a d^{2}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-c^2*d*x^2+d)^2*(a+b*arcsin(c*x))/x^2,x, algorithm="maxima")
[Out]
1/3*a*c^4*d^2*x^3 + 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*c^4*
d^2 - 2*a*c^2*d^2*x - 2*(c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*b*c*d^2 - (c*log(2*sqrt(-c^2*x^2 + 1)/abs(x) +
2/abs(x)) + arcsin(c*x)/x)*b*d^2 - a*d^2/x
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Fricas [A] time = 2.46411, size = 343, normalized size = 2.79 \begin{align*} \frac{6 \, a c^{4} d^{2} x^{4} - 36 \, a c^{2} d^{2} x^{2} - 9 \, b c d^{2} x \log \left (\sqrt{-c^{2} x^{2} + 1} + 1\right ) + 9 \, b c d^{2} x \log \left (\sqrt{-c^{2} x^{2} + 1} - 1\right ) - 18 \, a d^{2} + 6 \,{\left (b c^{4} d^{2} x^{4} - 6 \, b c^{2} d^{2} x^{2} - 3 \, b d^{2}\right )} \arcsin \left (c x\right ) + 2 \,{\left (b c^{3} d^{2} x^{3} - 16 \, b c d^{2} x\right )} \sqrt{-c^{2} x^{2} + 1}}{18 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-c^2*d*x^2+d)^2*(a+b*arcsin(c*x))/x^2,x, algorithm="fricas")
[Out]
1/18*(6*a*c^4*d^2*x^4 - 36*a*c^2*d^2*x^2 - 9*b*c*d^2*x*log(sqrt(-c^2*x^2 + 1) + 1) + 9*b*c*d^2*x*log(sqrt(-c^2
*x^2 + 1) - 1) - 18*a*d^2 + 6*(b*c^4*d^2*x^4 - 6*b*c^2*d^2*x^2 - 3*b*d^2)*arcsin(c*x) + 2*(b*c^3*d^2*x^3 - 16*
b*c*d^2*x)*sqrt(-c^2*x^2 + 1))/x
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Sympy [A] time = 10.3568, size = 182, normalized size = 1.48 \begin{align*} \frac{a c^{4} d^{2} x^{3}}{3} - 2 a c^{2} d^{2} x - \frac{a d^{2}}{x} - \frac{b c^{5} d^{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 c^{4} d^{2} x^{3} \operatorname{asin}{\left (c x \right )}}{3} - 2 b c^{2} d^{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 ) + 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 d^{2} \operatorname{asin}{\left (c x \right )}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-c**2*d*x**2+d)**2*(a+b*asin(c*x))/x**2,x)
[Out]
a*c**4*d**2*x**3/3 - 2*a*c**2*d**2*x - a*d**2/x - b*c**5*d**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*c**4*d**2*x**3*asin(c*x)/3 - 2*b*c**2*d**2*
Piecewise((0, Eq(c, 0)), (x*asin(c*x) + sqrt(-c**2*x**2 + 1)/c, True)) + b*c*d**2*Piecewise((-acosh(1/(c*x)),
1/Abs(c**2*x**2) > 1), (I*asin(1/(c*x)), True)) - b*d**2*asin(c*x)/x
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Giac [B] time = 31.3695, size = 3668, normalized size = 29.82 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-c^2*d*x^2+d)^2*(a+b*arcsin(c*x))/x^2,x, algorithm="giac")
[Out]
-1/2*b*c^9*d^2*x^8*arcsin(c*x)/((c^7*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 3*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3
*c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^8) - 1/2*a*c^9*d^
2*x^8/((c^7*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 3*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^3*x^3/(sqrt(-c^2*x^2 +
1) + 1)^3 + c*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^8) + b*c^8*d^2*x^7*log(abs(c)*abs(x))/((c^
7*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 3*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3
+ c*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^7) - b*c^8*d^2*x^7*log(sqrt(-c^2*x^2 + 1) + 1)/((c^7
*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 3*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3
+ c*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^7) + 16/9*b*c^8*d^2*x^7/((c^7*x^7/(sqrt(-c^2*x^2 + 1)
+ 1)^7 + 3*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c*x/(sqrt(-c^2*x^2 + 1
) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^7) - 6*b*c^7*d^2*x^6*arcsin(c*x)/((c^7*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 3*c^5
*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-
c^2*x^2 + 1) + 1)^6) - 6*a*c^7*d^2*x^6/((c^7*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 3*c^5*x^5/(sqrt(-c^2*x^2 + 1) +
1)^5 + 3*c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^6) + 3*b*
c^6*d^2*x^5*log(abs(c)*abs(x))/((c^7*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 3*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3
*c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^5) - 3*b*c^6*d^2*
x^5*log(sqrt(-c^2*x^2 + 1) + 1)/((c^7*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 3*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 +
3*c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^5) + 4/3*b*c^6*d
^2*x^5/((c^7*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 3*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^3*x^3/(sqrt(-c^2*x^2
+ 1) + 1)^3 + c*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^5) - 25/3*b*c^5*d^2*x^4*arcsin(c*x)/((c^7
*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 3*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3
+ c*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^4) - 25/3*a*c^5*d^2*x^4/((c^7*x^7/(sqrt(-c^2*x^2 + 1)
+ 1)^7 + 3*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c*x/(sqrt(-c^2*x^2 + 1
) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^4) + 3*b*c^4*d^2*x^3*log(abs(c)*abs(x))/((c^7*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7
+ 3*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c*x/(sqrt(-c^2*x^2 + 1) + 1))*
(sqrt(-c^2*x^2 + 1) + 1)^3) - 3*b*c^4*d^2*x^3*log(sqrt(-c^2*x^2 + 1) + 1)/((c^7*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7
+ 3*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c*x/(sqrt(-c^2*x^2 + 1) + 1))
*(sqrt(-c^2*x^2 + 1) + 1)^3) - 4/3*b*c^4*d^2*x^3/((c^7*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 3*c^5*x^5/(sqrt(-c^2*x
^2 + 1) + 1)^5 + 3*c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)
^3) - 6*b*c^3*d^2*x^2*arcsin(c*x)/((c^7*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 3*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5
+ 3*c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^2) - 6*a*c^3*d
^2*x^2/((c^7*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 3*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^3*x^3/(sqrt(-c^2*x^2
+ 1) + 1)^3 + c*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^2) + b*c^2*d^2*x*log(abs(c)*abs(x))/((c^7
*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 3*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3
+ c*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)) - b*c^2*d^2*x*log(sqrt(-c^2*x^2 + 1) + 1)/((c^7*x^7/
(sqrt(-c^2*x^2 + 1) + 1)^7 + 3*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c*x
/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)) - 16/9*b*c^2*d^2*x/((c^7*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 +
3*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c*x/(sqrt(-c^2*x^2 + 1) + 1))*(
sqrt(-c^2*x^2 + 1) + 1)) - 1/2*b*c*d^2*arcsin(c*x)/(c^7*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 3*c^5*x^5/(sqrt(-c^2*
x^2 + 1) + 1)^5 + 3*c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c*x/(sqrt(-c^2*x^2 + 1) + 1)) - 1/2*a*c*d^2/(c^7*x^7/
(sqrt(-c^2*x^2 + 1) + 1)^7 + 3*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 3*c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c*x
/(sqrt(-c^2*x^2 + 1) + 1))