3.78 \(\int x (d+e x^2) (a+b \sec ^{-1}(c x)) \, dx\)

Optimal. Leaf size=138 \[ \frac {\left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right )}{4 e}-\frac {b c d^2 x \tan ^{-1}\left (\sqrt {c^2 x^2-1}\right )}{4 e \sqrt {c^2 x^2}}-\frac {b x \sqrt {c^2 x^2-1} \left (2 c^2 d+e\right )}{4 c^3 \sqrt {c^2 x^2}}-\frac {b e x \left (c^2 x^2-1\right )^{3/2}}{12 c^3 \sqrt {c^2 x^2}} \]

[Out]

1/4*(e*x^2+d)^2*(a+b*arcsec(c*x))/e-1/12*b*e*x*(c^2*x^2-1)^(3/2)/c^3/(c^2*x^2)^(1/2)-1/4*b*c*d^2*x*arctan((c^2
*x^2-1)^(1/2))/e/(c^2*x^2)^(1/2)-1/4*b*(2*c^2*d+e)*x*(c^2*x^2-1)^(1/2)/c^3/(c^2*x^2)^(1/2)

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Rubi [A]  time = 0.09, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {5236, 446, 88, 63, 205} \[ \frac {\left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right )}{4 e}-\frac {b c d^2 x \tan ^{-1}\left (\sqrt {c^2 x^2-1}\right )}{4 e \sqrt {c^2 x^2}}-\frac {b x \sqrt {c^2 x^2-1} \left (2 c^2 d+e\right )}{4 c^3 \sqrt {c^2 x^2}}-\frac {b e x \left (c^2 x^2-1\right )^{3/2}}{12 c^3 \sqrt {c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 205

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

Rule 446

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

Rule 5236

Int[((a_.) + ArcSec[(c_.)*(x_)]*(b_.))*(x_)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)^(p +
1)*(a + b*ArcSec[c*x]))/(2*e*(p + 1)), x] - Dist[(b*c*x)/(2*e*(p + 1)*Sqrt[c^2*x^2]), Int[(d + e*x^2)^(p + 1)/
(x*Sqrt[c^2*x^2 - 1]), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]

Rubi steps

\begin {align*} \int x \left (d+e x^2\right ) \left (a+b \sec ^{-1}(c x)\right ) \, dx &=\frac {\left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right )}{4 e}-\frac {(b c x) \int \frac {\left (d+e x^2\right )^2}{x \sqrt {-1+c^2 x^2}} \, dx}{4 e \sqrt {c^2 x^2}}\\ &=\frac {\left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right )}{4 e}-\frac {(b c x) \operatorname {Subst}\left (\int \frac {(d+e x)^2}{x \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{8 e \sqrt {c^2 x^2}}\\ &=\frac {\left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right )}{4 e}-\frac {(b c x) \operatorname {Subst}\left (\int \left (\frac {e \left (2 c^2 d+e\right )}{c^2 \sqrt {-1+c^2 x}}+\frac {d^2}{x \sqrt {-1+c^2 x}}+\frac {e^2 \sqrt {-1+c^2 x}}{c^2}\right ) \, dx,x,x^2\right )}{8 e \sqrt {c^2 x^2}}\\ &=-\frac {b \left (2 c^2 d+e\right ) x \sqrt {-1+c^2 x^2}}{4 c^3 \sqrt {c^2 x^2}}-\frac {b e x \left (-1+c^2 x^2\right )^{3/2}}{12 c^3 \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right )}{4 e}-\frac {\left (b c d^2 x\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{8 e \sqrt {c^2 x^2}}\\ &=-\frac {b \left (2 c^2 d+e\right ) x \sqrt {-1+c^2 x^2}}{4 c^3 \sqrt {c^2 x^2}}-\frac {b e x \left (-1+c^2 x^2\right )^{3/2}}{12 c^3 \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right )}{4 e}-\frac {\left (b d^2 x\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 )}{4 c e \sqrt {c^2 x^2}}\\ &=-\frac {b \left (2 c^2 d+e\right ) x \sqrt {-1+c^2 x^2}}{4 c^3 \sqrt {c^2 x^2}}-\frac {b e x \left (-1+c^2 x^2\right )^{3/2}}{12 c^3 \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right )}{4 e}-\frac {b c d^2 x \tan ^{-1}\left (\sqrt {-1+c^2 x^2}\right )}{4 e \sqrt {c^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 79, normalized size = 0.57 \[ \frac {x \left (3 a c^3 x \left (2 d+e x^2\right )+3 b c^3 x \sec ^{-1}(c x) \left (2 d+e x^2\right )-b \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 \left (6 d+e x^2\right )+2 e\right )\right )}{12 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.67, size = 86, normalized size = 0.62 \[ \frac {3 \, a c^{4} e x^{4} + 6 \, a c^{4} d x^{2} + 3 \, {\left (b c^{4} e x^{4} + 2 \, b c^{4} d x^{2}\right )} \operatorname {arcsec}\left (c x\right ) - {\left (b c^{2} e x^{2} + 6 \, b c^{2} d + 2 \, b e\right )} \sqrt {c^{2} x^{2} - 1}}{12 \, c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.23, size = 3360, normalized size = 24.35 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(6*b*c^2*d*arccos(1/(c*x))/(c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/
(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8) + 6*a*c^2*
d/(c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x
^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8) - 12*b*c^2*d*sqrt(-1/(c^2*x^2) + 1)/((c^
5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) -
1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)) - 12*b*c^2*d*(1/(c^2*x^2) - 1)
^2*arccos(1/(c*x))/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4
+ 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^4) + 3*b
*arccos(1/(c*x))*e/(c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4
+ 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8) + 36*b*c^2*d*(-1/(c^2*x
^2) + 1)^(3/2)/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4
*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^3) - 12*a*c^
2*d*(1/(c^2*x^2) - 1)^2/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) +
1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^4)
+ 3*a*e/(c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/
(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8) - 12*b*(1/(c^2*x^2) - 1)*arccos(1/
(c*x))*e/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(
1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^2) - 36*b*c^2*d*(1
/(c^2*x^2) - 1)^2*sqrt(-1/(c^2*x^2) + 1)/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2)
- 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*
(1/(c*x) + 1)^5) - 6*b*sqrt(-1/(c^2*x^2) + 1)*e/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^
2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) +
1)^8)*(1/(c*x) + 1)) + 6*b*c^2*d*(1/(c^2*x^2) - 1)^4*arccos(1/(c*x))/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x)
+ 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*
x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^8) - 12*a*(1/(c^2*x^2) - 1)*e/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c
*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c
^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^2) + 18*b*(1/(c^2*x^2) - 1)^2*arccos(1/(c*x))*e/((c^5 + 4*c^5*(1
/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*
x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^4) - 12*b*c^2*d*(1/(c^2*x^2) - 1)^3*sqrt(-1
/(c^2*x^2) + 1)/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 +
4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^7) + 10*b*(
-1/(c^2*x^2) + 1)^(3/2)*e/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x)
+ 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^3
) + 6*a*c^2*d*(1/(c^2*x^2) - 1)^4/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/
(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x
) + 1)^8) + 18*a*(1/(c^2*x^2) - 1)^2*e/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) -
1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1
/(c*x) + 1)^4) - 12*b*(1/(c^2*x^2) - 1)^3*arccos(1/(c*x))*e/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 +
6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^
4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^6) - 10*b*(1/(c^2*x^2) - 1)^2*sqrt(-1/(c^2*x^2) + 1)*e/((c^5 + 4*c^5*(1/(c^2*
x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1
)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^5) - 12*a*(1/(c^2*x^2) - 1)^3*e/((c^5 + 4*c^5*(1/
(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x
) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^6) + 3*b*(1/(c^2*x^2) - 1)^4*arccos(1/(c*x))
*e/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2
*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^8) - 6*b*(1/(c^2*x^2) -
1)^3*sqrt(-1/(c^2*x^2) + 1)*e/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/(1/(
c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) +
1)^7) + 3*a*(1/(c^2*x^2) - 1)^4*e/((c^5 + 4*c^5*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 6*c^5*(1/(c^2*x^2) - 1)^2/
(1/(c*x) + 1)^4 + 4*c^5*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + c^5*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x
) + 1)^8))*c

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maple [A]  time = 0.05, size = 115, normalized size = 0.83 \[ \frac {\frac {a \left (\frac {1}{4} c^{4} e \,x^{4}+\frac {1}{2} c^{4} d \,x^{2}\right )}{c^{2}}+\frac {b \left (\frac {\mathrm {arcsec}\left (c x \right ) e \,c^{4} x^{4}}{4}+\frac {\mathrm {arcsec}\left (c x \right ) c^{4} d \,x^{2}}{2}-\frac {\left (c^{2} x^{2}-1\right ) \left (c^{2} e \,x^{2}+6 c^{2} d +2 e \right )}{12 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(e*x^2+d)*(a+b*arcsec(c*x)),x)

[Out]

1/c^2*(a/c^2*(1/4*c^4*e*x^4+1/2*c^4*d*x^2)+b/c^2*(1/4*arcsec(c*x)*e*c^4*x^4+1/2*arcsec(c*x)*c^4*d*x^2-1/12*(c^
2*x^2-1)*(c^2*e*x^2+6*c^2*d+2*e)/((c^2*x^2-1)/c^2/x^2)^(1/2)/c/x))

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maxima [A]  time = 0.32, size = 100, normalized size = 0.72 \[ \frac {1}{4} \, a e x^{4} + \frac {1}{2} \, a d x^{2} + \frac {1}{2} \, {\left (x^{2} \operatorname {arcsec}\left (c x\right ) - \frac {x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c}\right )} b d + \frac {1}{12} \, {\left (3 \, x^{4} \operatorname {arcsec}\left (c x\right ) - \frac {c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 3 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{3}}\right )} b e \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a*e*x^4 + 1/2*a*d*x^2 + 1/2*(x^2*arcsec(c*x) - x*sqrt(-1/(c^2*x^2) + 1)/c)*b*d + 1/12*(3*x^4*arcsec(c*x) -
(c^2*x^3*(-1/(c^2*x^2) + 1)^(3/2) + 3*x*sqrt(-1/(c^2*x^2) + 1))/c^3)*b*e

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(d + e*x^2)*(a + b*acos(1/(c*x))),x)

[Out]

int(x*(d + e*x^2)*(a + b*acos(1/(c*x))), x)

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sympy [A]  time = 3.66, size = 177, normalized size = 1.28 \[ \frac {a d x^{2}}{2} + \frac {a e x^{4}}{4} + \frac {b d x^{2} \operatorname {asec}{\left (c x \right )}}{2} + \frac {b e x^{4} \operatorname {asec}{\left (c x \right )}}{4} - \frac {b d \left (\begin {cases} \frac {\sqrt {c^{2} x^{2} - 1}}{c} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i \sqrt {- c^{2} x^{2} + 1}}{c} & \text {otherwise} \end {cases}\right )}{2 c} - \frac {b e \left (\begin {cases} \frac {x^{2} \sqrt {c^{2} x^{2} - 1}}{3 c} + \frac {2 \sqrt {c^{2} x^{2} - 1}}{3 c^{3}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{2} \sqrt {- c^{2} x^{2} + 1}}{3 c} + \frac {2 i \sqrt {- c^{2} x^{2} + 1}}{3 c^{3}} & \text {otherwise} \end {cases}\right )}{4 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(e*x**2+d)*(a+b*asec(c*x)),x)

[Out]

a*d*x**2/2 + a*e*x**4/4 + b*d*x**2*asec(c*x)/2 + b*e*x**4*asec(c*x)/4 - b*d*Piecewise((sqrt(c**2*x**2 - 1)/c,
Abs(c**2*x**2) > 1), (I*sqrt(-c**2*x**2 + 1)/c, True))/(2*c) - b*e*Piecewise((x**2*sqrt(c**2*x**2 - 1)/(3*c) +
2*sqrt(c**2*x**2 - 1)/(3*c**3), Abs(c**2*x**2) > 1), (I*x**2*sqrt(-c**2*x**2 + 1)/(3*c) + 2*I*sqrt(-c**2*x**2
+ 1)/(3*c**3), True))/(4*c)

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