3.622 \(\int \frac {(d+e x^2)^3 (a+b \sin ^{-1}(c x))}{x^4} \, dx\)
Optimal. Leaf size=186 \[ -\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}-\frac {3 d^2 e \left (a+b \sin ^{-1}(c x)\right )}{x}+3 d e^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} e^3 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {b c d^3 \sqrt {1-c^2 x^2}}{6 x^2}-\frac {1}{6} b c d^2 \left (c^2 d+18 e\right ) \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )+\frac {b e^2 \sqrt {1-c^2 x^2} \left (9 c^2 d+e\right )}{3 c^3}-\frac {b e^3 \left (1-c^2 x^2\right )^{3/2}}{9 c^3} \]
[Out]
-1/9*b*e^3*(-c^2*x^2+1)^(3/2)/c^3-1/3*d^3*(a+b*arcsin(c*x))/x^3-3*d^2*e*(a+b*arcsin(c*x))/x+3*d*e^2*x*(a+b*arc
sin(c*x))+1/3*e^3*x^3*(a+b*arcsin(c*x))-1/6*b*c*d^2*(c^2*d+18*e)*arctanh((-c^2*x^2+1)^(1/2))+1/3*b*e^2*(9*c^2*
d+e)*(-c^2*x^2+1)^(1/2)/c^3-1/6*b*c*d^3*(-c^2*x^2+1)^(1/2)/x^2
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Rubi [A] time = 0.32, antiderivative size = 186, normalized size of antiderivative = 1.00,
number of steps used = 8, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used =
{270, 4731, 12, 1799, 1621, 897, 1153, 208} \[ -\frac {3 d^2 e \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}+3 d e^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} e^3 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{6} b c d^2 \left (c^2 d+18 e\right ) \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )-\frac {b c d^3 \sqrt {1-c^2 x^2}}{6 x^2}+\frac {b e^2 \sqrt {1-c^2 x^2} \left (9 c^2 d+e\right )}{3 c^3}-\frac {b e^3 \left (1-c^2 x^2\right )^{3/2}}{9 c^3} \]
Antiderivative was successfully verified.
[In]
Int[((d + e*x^2)^3*(a + b*ArcSin[c*x]))/x^4,x]
[Out]
(b*e^2*(9*c^2*d + e)*Sqrt[1 - c^2*x^2])/(3*c^3) - (b*c*d^3*Sqrt[1 - c^2*x^2])/(6*x^2) - (b*e^3*(1 - c^2*x^2)^(
3/2))/(9*c^3) - (d^3*(a + b*ArcSin[c*x]))/(3*x^3) - (3*d^2*e*(a + b*ArcSin[c*x]))/x + 3*d*e^2*x*(a + b*ArcSin[
c*x]) + (e^3*x^3*(a + b*ArcSin[c*x]))/3 - (b*c*d^2*(c^2*d + 18*e)*ArcTanh[Sqrt[1 - c^2*x^2]])/6
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
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]
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 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 1621
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> With[{Qx = PolynomialQuotient[Px,
a + b*x, x], R = PolynomialRemainder[Px, a + b*x, x]}, Simp[(R*(a + b*x)^(m + 1)*(c + d*x)^(n + 1))/((m + 1)*
(b*c - a*d)), x] + Dist[1/((m + 1)*(b*c - a*d)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*ExpandToSum[(m + 1)*(b*c -
a*d)*Qx - d*R*(m + n + 2), x], x], x]] /; FreeQ[{a, b, c, d, n}, x] && PolyQ[Px, x] && ILtQ[m, -1] && GtQ[Expo
n[Px, x], 2]
Rule 1799
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*SubstFor[x^2,
Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
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]))
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{x^4} \, dx &=-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}-\frac {3 d^2 e \left (a+b \sin ^{-1}(c x)\right )}{x}+3 d e^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} e^3 x^3 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac {-d^3-9 d^2 e x^2+9 d e^2 x^4+e^3 x^6}{3 x^3 \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}-\frac {3 d^2 e \left (a+b \sin ^{-1}(c x)\right )}{x}+3 d e^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} e^3 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{3} (b c) \int \frac {-d^3-9 d^2 e x^2+9 d e^2 x^4+e^3 x^6}{x^3 \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}-\frac {3 d^2 e \left (a+b \sin ^{-1}(c x)\right )}{x}+3 d e^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} e^3 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{6} (b c) \operatorname {Subst}\left (\int \frac {-d^3-9 d^2 e x+9 d e^2 x^2+e^3 x^3}{x^2 \sqrt {1-c^2 x}} \, dx,x,x^2\right )\\ &=-\frac {b c d^3 \sqrt {1-c^2 x^2}}{6 x^2}-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}-\frac {3 d^2 e \left (a+b \sin ^{-1}(c x)\right )}{x}+3 d e^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} e^3 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} (b c) \operatorname {Subst}\left (\int \frac {\frac {1}{2} d^2 \left (c^2 d+18 e\right )-9 d e^2 x-e^3 x^2}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )\\ &=-\frac {b c d^3 \sqrt {1-c^2 x^2}}{6 x^2}-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}-\frac {3 d^2 e \left (a+b \sin ^{-1}(c x)\right )}{x}+3 d e^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} e^3 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {b \operatorname {Subst}\left (\int \frac {\frac {-9 c^2 d e^2-e^3+\frac {1}{2} c^4 d^2 \left (c^2 d+18 e\right )}{c^4}-\frac {\left (-9 c^2 d e^2-2 e^3\right ) x^2}{c^4}-\frac {e^3 x^4}{c^4}}{\frac {1}{c^2}-\frac {x^2}{c^2}} \, dx,x,\sqrt {1-c^2 x^2}\right )}{3 c}\\ &=-\frac {b c d^3 \sqrt {1-c^2 x^2}}{6 x^2}-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}-\frac {3 d^2 e \left (a+b \sin ^{-1}(c x)\right )}{x}+3 d e^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} e^3 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {b \operatorname {Subst}\left (\int \left (-e^2 \left (9 d+\frac {e}{c^2}\right )+\frac {e^3 x^2}{c^2}+\frac {c^2 d^3+18 d^2 e}{2 \left (\frac {1}{c^2}-\frac {x^2}{c^2}\right )}\right ) \, dx,x,\sqrt {1-c^2 x^2}\right )}{3 c}\\ &=\frac {b e^2 \left (9 c^2 d+e\right ) \sqrt {1-c^2 x^2}}{3 c^3}-\frac {b c d^3 \sqrt {1-c^2 x^2}}{6 x^2}-\frac {b e^3 \left (1-c^2 x^2\right )^{3/2}}{9 c^3}-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}-\frac {3 d^2 e \left (a+b \sin ^{-1}(c x)\right )}{x}+3 d e^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} e^3 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {\left (b d^2 \left (c^2 d+18 e\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 )}{6 c}\\ &=\frac {b e^2 \left (9 c^2 d+e\right ) \sqrt {1-c^2 x^2}}{3 c^3}-\frac {b c d^3 \sqrt {1-c^2 x^2}}{6 x^2}-\frac {b e^3 \left (1-c^2 x^2\right )^{3/2}}{9 c^3}-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}-\frac {3 d^2 e \left (a+b \sin ^{-1}(c x)\right )}{x}+3 d e^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} e^3 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{6} b c d^2 \left (c^2 d+18 e\right ) \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.30, size = 194, normalized size = 1.04 \[ \frac {1}{6} \left (-\frac {2 a d^3}{x^3}-\frac {18 a d^2 e}{x}+18 a d e^2 x+2 a e^3 x^3-b c d^2 \left (c^2 d+18 e\right ) \log \left (\sqrt {1-c^2 x^2}+1\right )+b c d^2 \log (x) \left (c^2 d+18 e\right )+\frac {b \sqrt {1-c^2 x^2} \left (-3 c^4 d^3+2 c^2 e^2 x^2 \left (27 d+e x^2\right )+4 e^3 x^2\right )}{3 c^3 x^2}+\frac {2 b \sin ^{-1}(c x) \left (-d^3-9 d^2 e x^2+9 d e^2 x^4+e^3 x^6\right )}{x^3}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[((d + e*x^2)^3*(a + b*ArcSin[c*x]))/x^4,x]
[Out]
((-2*a*d^3)/x^3 - (18*a*d^2*e)/x + 18*a*d*e^2*x + 2*a*e^3*x^3 + (b*Sqrt[1 - c^2*x^2]*(-3*c^4*d^3 + 4*e^3*x^2 +
2*c^2*e^2*x^2*(27*d + e*x^2)))/(3*c^3*x^2) + (2*b*(-d^3 - 9*d^2*e*x^2 + 9*d*e^2*x^4 + e^3*x^6)*ArcSin[c*x])/x
^3 + b*c*d^2*(c^2*d + 18*e)*Log[x] - b*c*d^2*(c^2*d + 18*e)*Log[1 + Sqrt[1 - c^2*x^2]])/6
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fricas [A] time = 0.87, size = 246, normalized size = 1.32 \[ \frac {12 \, a c^{3} e^{3} x^{6} + 108 \, a c^{3} d e^{2} x^{4} - 108 \, a c^{3} d^{2} e x^{2} - 12 \, a c^{3} d^{3} - 3 \, {\left (b c^{6} d^{3} + 18 \, b c^{4} d^{2} e\right )} x^{3} \log \left (\sqrt {-c^{2} x^{2} + 1} + 1\right ) + 3 \, {\left (b c^{6} d^{3} + 18 \, b c^{4} d^{2} e\right )} x^{3} \log \left (\sqrt {-c^{2} x^{2} + 1} - 1\right ) + 12 \, {\left (b c^{3} e^{3} x^{6} + 9 \, b c^{3} d e^{2} x^{4} - 9 \, b c^{3} d^{2} e x^{2} - b c^{3} d^{3}\right )} \arcsin \left (c x\right ) + 2 \, {\left (2 \, b c^{2} e^{3} x^{5} - 3 \, b c^{4} d^{3} x + 2 \, {\left (27 \, b c^{2} d e^{2} + 2 \, b e^{3}\right )} x^{3}\right )} \sqrt {-c^{2} x^{2} + 1}}{36 \, c^{3} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x^2+d)^3*(a+b*arcsin(c*x))/x^4,x, algorithm="fricas")
[Out]
1/36*(12*a*c^3*e^3*x^6 + 108*a*c^3*d*e^2*x^4 - 108*a*c^3*d^2*e*x^2 - 12*a*c^3*d^3 - 3*(b*c^6*d^3 + 18*b*c^4*d^
2*e)*x^3*log(sqrt(-c^2*x^2 + 1) + 1) + 3*(b*c^6*d^3 + 18*b*c^4*d^2*e)*x^3*log(sqrt(-c^2*x^2 + 1) - 1) + 12*(b*
c^3*e^3*x^6 + 9*b*c^3*d*e^2*x^4 - 9*b*c^3*d^2*e*x^2 - b*c^3*d^3)*arcsin(c*x) + 2*(2*b*c^2*e^3*x^5 - 3*b*c^4*d^
3*x + 2*(27*b*c^2*d*e^2 + 2*b*e^3)*x^3)*sqrt(-c^2*x^2 + 1))/(c^3*x^3)
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giac [B] time = 159.58, size = 7973, normalized size = 42.87 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x^2+d)^3*(a+b*arcsin(c*x))/x^4,x, algorithm="giac")
[Out]
-1/24*b*c^18*d^3*x^12*arcsin(c*x)/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^12) -
1/24*a*c^18*d^3*x^12/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^12) + 1/24*b*c^17*d
^3*x^11/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^11) - 1/4*b*c^16*d^3*x^10*arcsin
(c*x)/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^10) - 1/4*a*c^16*d^3*x^10/((c^12*x
^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 +
c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^10) + 1/6*b*c^15*d^3*x^9*log(abs(c)*abs(x))/((c^
12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^9) - 1/6*b*c^15*d^3*x^9*log(sqrt(-c^2*x^2 +
1) + 1)/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^9) + 1/8*b*c^15*d^3*x^9/((c^12*x
^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 +
c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^9) - 5/8*b*c^14*d^3*x^8*arcsin(c*x)/((c^12*x^9/(
sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6
*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^8) - 3/2*b*c^14*d^2*x^10*arcsin(c*x)*e/((c^12*x^9/(s
qrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*
x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^10) - 5/8*a*c^14*d^3*x^8/((c^12*x^9/(sqrt(-c^2*x^2 +
1) + 1)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*
x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^8) - 3/2*a*c^14*d^2*x^10*e/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3
*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)
^3)*(sqrt(-c^2*x^2 + 1) + 1)^10) + 1/2*b*c^13*d^3*x^7*log(abs(c)*abs(x))/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9
+ 3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1)
+ 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^7) + 3*b*c^13*d^2*x^9*e*log(abs(c)*abs(x))/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1
)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 +
1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^9) - 1/2*b*c^13*d^3*x^7*log(sqrt(-c^2*x^2 + 1) + 1)/((c^12*x^9/(sqrt(-c^2*
x^2 + 1) + 1)^9 + 3*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 + c^6*x^3/(sqrt
(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^7) - 3*b*c^13*d^2*x^9*e*log(sqrt(-c^2*x^2 + 1) + 1)/((c^12*x^9
/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c
^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^9) + 1/12*b*c^13*d^3*x^7/((c^12*x^9/(sqrt(-c^2*x^2
+ 1) + 1)^9 + 3*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 + c^6*x^3/(sqrt(-c
^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^7) - 5/6*b*c^12*d^3*x^6*arcsin(c*x)/((c^12*x^9/(sqrt(-c^2*x^2 + 1
) + 1)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x
^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^6) - 6*b*c^12*d^2*x^8*arcsin(c*x)*e/((c^12*x^9/(sqrt(-c^2*x^2 + 1) +
1)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 +
1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^8) - 5/6*a*c^12*d^3*x^6/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqr
t(-c^2*x^2 + 1) + 1)^6) - 6*a*c^12*d^2*x^8*e/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1)
+ 1)^8) + 1/2*b*c^11*d^3*x^5*log(abs(c)*abs(x))/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 +
1) + 1)^5) + 9*b*c^11*d^2*x^7*e*log(abs(c)*abs(x))/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2
+ 1) + 1)^7) - 1/2*b*c^11*d^3*x^5*log(sqrt(-c^2*x^2 + 1) + 1)/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(s
qrt(-c^2*x^2 + 1) + 1)^5) - 9*b*c^11*d^2*x^7*e*log(sqrt(-c^2*x^2 + 1) + 1)/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)
^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1
) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^7) - 1/12*b*c^11*d^3*x^5/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt
(-c^2*x^2 + 1) + 1)^5) - 5/8*b*c^10*d^3*x^4*arcsin(c*x)/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^
2*x^2 + 1) + 1)^4) - 9*b*c^10*d^2*x^6*arcsin(c*x)*e/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^
2 + 1) + 1)^6) - 5/8*a*c^10*d^3*x^4/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^4) -
3*b*c^11*d*x^9*e^2/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^9) + 6*b*c^10*d*x^8*
arcsin(c*x)*e^2/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*c^10*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 3*c^8*x^5/(sqr
t(-c^2*x^2 + 1) + 1)^5 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^8) - 9*a*c^10*d^2*x^6*e/
((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^6) + 1/6*b*c^9*d^3*x^3*log(abs(c)*abs(x)
)/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^3) + 9*b*c^9*d^2*x^5*e*log(abs(c)*abs(
x))/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^5) - 1/6*b*c^9*d^3*x^3*log(sqrt(-c^2
*x^2 + 1) + 1)/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^3) - 9*b*c^9*d^2*x^5*e*lo
g(sqrt(-c^2*x^2 + 1) + 1)/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^5) - 1/8*b*c^9
*d^3*x^3/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^3) - 1/4*b*c^8*d^3*x^2*arcsin(c
*x)/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^2) + 6*a*c^10*d*x^8*e^2/((c^12*x^9/(
sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6
*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^8) - 6*b*c^8*d^2*x^4*arcsin(c*x)*e/((c^12*x^9/(sqrt(
-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*x^3/
(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^4) - 1/4*a*c^8*d^3*x^2/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1
)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 +
1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^2) - 3*b*c^9*d*x^7*e^2/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(
-c^2*x^2 + 1) + 1)^7) + 12*b*c^8*d*x^6*arcsin(c*x)*e^2/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*c^10*x^7/(sqr
t(-c^2*x^2 + 1) + 1)^7 + 3*c^8*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2
*x^2 + 1) + 1)^6) - 6*a*c^8*d^2*x^4*e/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^4)
+ 3*b*c^7*d^2*x^3*e*log(abs(c)*abs(x))/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^
3) - 3*b*c^7*d^2*x^3*e*log(sqrt(-c^2*x^2 + 1) + 1)/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2
+ 1) + 1)^3) - 1/24*b*c^7*d^3*x/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)) - 1/24
*b*c^6*d^3*arcsin(c*x)/(c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3) - 2/9*b*c^9*x^9*e^3/((c^12*x^9/(sqrt(-c^2*x
^2 + 1) + 1)^9 + 3*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 + c^6*x^3/(sqrt(
-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^9) + 12*a*c^8*d*x^6*e^2/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 +
3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) +
1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^6) - 3/2*b*c^6*d^2*x^2*arcsin(c*x)*e/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*
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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^
3)*(sqrt(-c^2*x^2 + 1) + 1)^2) - 1/24*a*c^6*d^3/(c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3) + 3*b*c^7*d*x^5*e^
2/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^5) + 6*b*c^6*d*x^4*arcsin(c*x)*e^2/((c
^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^4) - 3/2*a*c^6*d^2*x^2*e/((c^12*x^9/(sqrt(-
c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*x^3/(
sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^2) - 2/3*b*c^7*x^7*e^3/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)
^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1
) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^7) + 8/3*b*c^6*x^6*arcsin(c*x)*e^3/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 +
3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1
)^3)*(sqrt(-c^2*x^2 + 1) + 1)^6) + 6*a*c^6*d*x^4*e^2/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x
^2 + 1) + 1)^4) + 8/3*a*c^6*x^6*e^3/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^6) +
3*b*c^5*d*x^3*e^2/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^3) + 2/3*b*c^5*x^5*e^
3/((c^12*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^5) + 2/9*b*c^3*x^3*e^3/((c^12*x^9/(sqr
t(-c^2*x^2 + 1) + 1)^9 + 3*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 + c^6*x^
3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^3)
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maple [A] time = 0.01, size = 249, normalized size = 1.34 \[ c^{3} \left (\frac {a \left (\frac {e^{3} c^{3} x^{3}}{3}+3 c^{3} x d \,e^{2}-\frac {d^{3} c^{3}}{3 x^{3}}-\frac {3 c^{3} d^{2} e}{x}\right )}{c^{6}}+\frac {b \left (\frac {\arcsin \left (c x \right ) e^{3} c^{3} x^{3}}{3}+3 \arcsin \left (c x \right ) c^{3} x d \,e^{2}-\frac {\arcsin \left (c x \right ) d^{3} c^{3}}{3 x^{3}}-\frac {3 \arcsin \left (c x \right ) c^{3} d^{2} e}{x}-\frac {e^{3} \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3}+3 c^{2} d \,e^{2} \sqrt {-c^{2} x^{2}+1}+\frac {d^{3} c^{6} \left (-\frac {\sqrt {-c^{2} x^{2}+1}}{2 c^{2} x^{2}}-\frac {\arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{2}\right )}{3}-3 c^{4} d^{2} e \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )}{c^{6}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x^2+d)^3*(a+b*arcsin(c*x))/x^4,x)
[Out]
c^3*(a/c^6*(1/3*e^3*c^3*x^3+3*c^3*x*d*e^2-1/3*d^3*c^3/x^3-3*c^3*d^2*e/x)+b/c^6*(1/3*arcsin(c*x)*e^3*c^3*x^3+3*
arcsin(c*x)*c^3*x*d*e^2-1/3*arcsin(c*x)*d^3*c^3/x^3-3*arcsin(c*x)*c^3*d^2*e/x-1/3*e^3*(-1/3*c^2*x^2*(-c^2*x^2+
1)^(1/2)-2/3*(-c^2*x^2+1)^(1/2))+3*c^2*d*e^2*(-c^2*x^2+1)^(1/2)+1/3*d^3*c^6*(-1/2/c^2/x^2*(-c^2*x^2+1)^(1/2)-1
/2*arctanh(1/(-c^2*x^2+1)^(1/2)))-3*c^4*d^2*e*arctanh(1/(-c^2*x^2+1)^(1/2))))
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maxima [A] time = 0.66, size = 231, normalized size = 1.24 \[ \frac {1}{3} \, a e^{3} x^{3} - \frac {1}{6} \, {\left ({\left (c^{2} \log \left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {\sqrt {-c^{2} x^{2} + 1}}{x^{2}}\right )} c + \frac {2 \, \arcsin \left (c x\right )}{x^{3}}\right )} b d^{3} - 3 \, {\left (c \log \left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {\arcsin \left (c x\right )}{x}\right )} b d^{2} e + \frac {1}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b e^{3} + 3 \, a d e^{2} x + \frac {3 \, {\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} b d e^{2}}{c} - \frac {3 \, a d^{2} e}{x} - \frac {a d^{3}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x^2+d)^3*(a+b*arcsin(c*x))/x^4,x, algorithm="maxima")
[Out]
1/3*a*e^3*x^3 - 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^3 - 3*(c*log(2*sqrt(-c^2*x^2 + 1)/abs(x) + 2/abs(x)) + arcsin(c*x)/x)*b*d^2*e + 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^3 + 3*a*d*e^2*x + 3*(c*x*arcsin(c*x) +
sqrt(-c^2*x^2 + 1))*b*d*e^2/c - 3*a*d^2*e/x - 1/3*a*d^3/x^3
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^3}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((a + b*asin(c*x))*(d + e*x^2)^3)/x^4,x)
[Out]
int(((a + b*asin(c*x))*(d + e*x^2)^3)/x^4, x)
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sympy [A] time = 8.73, size = 311, normalized size = 1.67 \[ - \frac {a d^{3}}{3 x^{3}} - \frac {3 a d^{2} e}{x} + 3 a d e^{2} x + \frac {a e^{3} x^{3}}{3} + \frac {b c d^{3} \left (\begin {cases} - \frac {c^{2} \operatorname {acosh}{\left (\frac {1}{c x} \right )}}{2} - \frac {c \sqrt {-1 + \frac {1}{c^{2} x^{2}}}}{2 x} & \text {for}\: \frac {1}{\left |{c^{2} x^{2}}\right |} > 1 \\\frac {i c^{2} \operatorname {asin}{\left (\frac {1}{c x} \right )}}{2} - \frac {i c}{2 x \sqrt {1 - \frac {1}{c^{2} x^{2}}}} + \frac {i}{2 c x^{3} \sqrt {1 - \frac {1}{c^{2} x^{2}}}} & \text {otherwise} \end {cases}\right )}{3} + 3 b c d^{2} e \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{c x} \right )} & \text {for}\: \frac {1}{\left |{c^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{c x} \right )} & \text {otherwise} \end {cases}\right ) - \frac {b c e^{3} \left (\begin {cases} - \frac {x^{2} \sqrt {- c^{2} x^{2} + 1}}{3 c^{2}} - \frac {2 \sqrt {- c^{2} x^{2} + 1}}{3 c^{4}} & \text {for}\: c \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right )}{3} - \frac {b d^{3} \operatorname {asin}{\left (c x \right )}}{3 x^{3}} - \frac {3 b d^{2} e \operatorname {asin}{\left (c x \right )}}{x} + 3 b d e^{2} \left (\begin {cases} 0 & \text {for}\: c = 0 \\x \operatorname {asin}{\left (c x \right )} + \frac {\sqrt {- c^{2} x^{2} + 1}}{c} & \text {otherwise} \end {cases}\right ) + \frac {b e^{3} x^{3} \operatorname {asin}{\left (c x \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x**2+d)**3*(a+b*asin(c*x))/x**4,x)
[Out]
-a*d**3/(3*x**3) - 3*a*d**2*e/x + 3*a*d*e**2*x + a*e**3*x**3/3 + b*c*d**3*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*sqrt(1 - 1/(c**2*x**2))), True))/3 + 3*b*c*d**2*e*Piecewise((-acosh(1/(c*x)), 1/Abs(c**2*
x**2) > 1), (I*asin(1/(c*x)), True)) - b*c*e**3*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**3*asin(c*x)/(3*x**3) - 3*b*d**2*e*asin(c*x)/x + 3*b*d*
e**2*Piecewise((0, Eq(c, 0)), (x*asin(c*x) + sqrt(-c**2*x**2 + 1)/c, True)) + b*e**3*x**3*asin(c*x)/3
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