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

Optimal. Leaf size=158 \[ -\frac {d^2 \left (a+b \sec ^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \sec ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \sec ^{-1}(c x)\right )+\frac {b c d^2 \sqrt {c^2 x^2-1}}{9 x^2 \sqrt {c^2 x^2}}+\frac {2 b c d \sqrt {c^2 x^2-1} \left (c^2 d+9 e\right )}{9 \sqrt {c^2 x^2}}-\frac {b e^2 x \tanh ^{-1}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{\sqrt {c^2 x^2}} \]

[Out]

-1/3*d^2*(a+b*arcsec(c*x))/x^3-2*d*e*(a+b*arcsec(c*x))/x+e^2*x*(a+b*arcsec(c*x))-b*e^2*x*arctanh(c*x/(c^2*x^2-
1)^(1/2))/(c^2*x^2)^(1/2)+2/9*b*c*d*(c^2*d+9*e)*(c^2*x^2-1)^(1/2)/(c^2*x^2)^(1/2)+1/9*b*c*d^2*(c^2*x^2-1)^(1/2
)/x^2/(c^2*x^2)^(1/2)

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Rubi [A]  time = 0.13, antiderivative size = 158, normalized size of antiderivative = 1.00, 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, 5238, 12, 1265, 451, 217, 206} \[ -\frac {d^2 \left (a+b \sec ^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \sec ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \sec ^{-1}(c x)\right )+\frac {b c d^2 \sqrt {c^2 x^2-1}}{9 x^2 \sqrt {c^2 x^2}}+\frac {2 b c d \sqrt {c^2 x^2-1} \left (c^2 d+9 e\right )}{9 \sqrt {c^2 x^2}}-\frac {b e^2 x \tanh ^{-1}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{\sqrt {c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b*c*d*(c^2*d + 9*e)*Sqrt[-1 + c^2*x^2])/(9*Sqrt[c^2*x^2]) + (b*c*d^2*Sqrt[-1 + c^2*x^2])/(9*x^2*Sqrt[c^2*x^
2]) - (d^2*(a + b*ArcSec[c*x]))/(3*x^3) - (2*d*e*(a + b*ArcSec[c*x]))/x + e^2*x*(a + b*ArcSec[c*x]) - (b*e^2*x
*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/Sqrt[c^2*x^2]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

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 451

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

Rule 1265

Int[((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Wit
h[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, f*x, x], R = PolynomialRemainder[(a + b*x^2 + c*x^4)^p, f*x,
x]}, Simp[(R*(f*x)^(m + 1)*(d + e*x^2)^(q + 1))/(d*f*(m + 1)), x] + Dist[1/(d*f^2*(m + 1)), Int[(f*x)^(m + 2)
*(d + e*x^2)^q*ExpandToSum[(d*f*(m + 1)*Qx)/x - e*R*(m + 2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q},
x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && LtQ[m, -1]

Rule 5238

Int[((a_.) + ArcSec[(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*ArcSec[c*x], u, x] - Dist[(b*c*x)/Sqrt[c^2*x^2], Int[SimplifyI
ntegrand[u/(x*Sqrt[c^2*x^2 - 1]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && ((IGtQ[p, 0] &&  !(ILtQ
[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0])) || (IGtQ[(m + 1)/2, 0] &&  !(ILtQ[p, 0] && GtQ[m + 2*p + 3, 0])) || (I
LtQ[(m + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))

Rubi steps

\begin {align*} \int \frac {\left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right )}{x^4} \, dx &=-\frac {d^2 \left (a+b \sec ^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \sec ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \sec ^{-1}(c x)\right )-\frac {(b c x) \int \frac {-d^2-6 d e x^2+3 e^2 x^4}{3 x^4 \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}}\\ &=-\frac {d^2 \left (a+b \sec ^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \sec ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \sec ^{-1}(c x)\right )-\frac {(b c x) \int \frac {-d^2-6 d e x^2+3 e^2 x^4}{x^4 \sqrt {-1+c^2 x^2}} \, dx}{3 \sqrt {c^2 x^2}}\\ &=\frac {b c d^2 \sqrt {-1+c^2 x^2}}{9 x^2 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \sec ^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \sec ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \sec ^{-1}(c x)\right )-\frac {(b c x) \int \frac {-2 d \left (c^2 d+9 e\right )+9 e^2 x^2}{x^2 \sqrt {-1+c^2 x^2}} \, dx}{9 \sqrt {c^2 x^2}}\\ &=\frac {2 b c d \left (c^2 d+9 e\right ) \sqrt {-1+c^2 x^2}}{9 \sqrt {c^2 x^2}}+\frac {b c d^2 \sqrt {-1+c^2 x^2}}{9 x^2 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \sec ^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \sec ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \sec ^{-1}(c x)\right )-\frac {\left (b c e^2 x\right ) \int \frac {1}{\sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}}\\ &=\frac {2 b c d \left (c^2 d+9 e\right ) \sqrt {-1+c^2 x^2}}{9 \sqrt {c^2 x^2}}+\frac {b c d^2 \sqrt {-1+c^2 x^2}}{9 x^2 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \sec ^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \sec ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \sec ^{-1}(c x)\right )-\frac {\left (b c e^2 x\right ) \operatorname {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\frac {x}{\sqrt {-1+c^2 x^2}}\right )}{\sqrt {c^2 x^2}}\\ &=\frac {2 b c d \left (c^2 d+9 e\right ) \sqrt {-1+c^2 x^2}}{9 \sqrt {c^2 x^2}}+\frac {b c d^2 \sqrt {-1+c^2 x^2}}{9 x^2 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \sec ^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \sec ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \sec ^{-1}(c x)\right )-\frac {b e^2 x \tanh ^{-1}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{\sqrt {c^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.23, size = 127, normalized size = 0.80 \[ \frac {c \left (b c d x \sqrt {1-\frac {1}{c^2 x^2}} \left (2 c^2 d x^2+d+18 e x^2\right )-3 a \left (d^2+6 d e x^2-3 e^2 x^4\right )\right )-9 b e^2 x^3 \log \left (x \left (\sqrt {1-\frac {1}{c^2 x^2}}+1\right )\right )-3 b c \sec ^{-1}(c x) \left (d^2+6 d e x^2-3 e^2 x^4\right )}{9 c x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*(b*c*d*Sqrt[1 - 1/(c^2*x^2)]*x*(d + 2*c^2*d*x^2 + 18*e*x^2) - 3*a*(d^2 + 6*d*e*x^2 - 3*e^2*x^4)) - 3*b*c*(d
^2 + 6*d*e*x^2 - 3*e^2*x^4)*ArcSec[c*x] - 9*b*e^2*x^3*Log[(1 + Sqrt[1 - 1/(c^2*x^2)])*x])/(9*c*x^3)

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fricas [A]  time = 3.06, size = 221, normalized size = 1.40 \[ \frac {9 \, a c e^{2} x^{4} + 9 \, b e^{2} x^{3} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - 18 \, a c d e x^{2} - 6 \, {\left (b c d^{2} + 6 \, b c d e - 3 \, b c e^{2}\right )} x^{3} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - 3 \, a c d^{2} + 2 \, {\left (b c^{4} d^{2} + 9 \, b c^{2} d e\right )} x^{3} + 3 \, {\left (3 \, b c e^{2} x^{4} - 6 \, b c d e x^{2} - b c d^{2} + {\left (b c d^{2} + 6 \, b c d e - 3 \, b c e^{2}\right )} x^{3}\right )} \operatorname {arcsec}\left (c x\right ) + {\left (b c d^{2} + 2 \, {\left (b c^{3} d^{2} + 9 \, b c d e\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{9 \, c x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(9*a*c*e^2*x^4 + 9*b*e^2*x^3*log(-c*x + sqrt(c^2*x^2 - 1)) - 18*a*c*d*e*x^2 - 6*(b*c*d^2 + 6*b*c*d*e - 3*b
*c*e^2)*x^3*arctan(-c*x + sqrt(c^2*x^2 - 1)) - 3*a*c*d^2 + 2*(b*c^4*d^2 + 9*b*c^2*d*e)*x^3 + 3*(3*b*c*e^2*x^4
- 6*b*c*d*e*x^2 - b*c*d^2 + (b*c*d^2 + 6*b*c*d*e - 3*b*c*e^2)*x^3)*arcsec(c*x) + (b*c*d^2 + 2*(b*c^3*d^2 + 9*b
*c*d*e)*x^2)*sqrt(c^2*x^2 - 1))/(c*x^3)

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giac [B]  time = 14.58, size = 4960, normalized size = 31.39 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*(3*b*c^4*d^2*arccos(1/(c*x))/(c^2 - 2*c^2*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 2*c^2*(1/(c^2*x^2) - 1)^3/(
1/(c*x) + 1)^6 - c^2*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8) + 3*a*c^4*d^2/(c^2 - 2*c^2*(1/(c^2*x^2) - 1)/(1/(c*x
) + 1)^2 + 2*c^2*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^2*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8) + 12*b*c^4*d^2
*(1/(c^2*x^2) - 1)*arccos(1/(c*x))/((c^2 - 2*c^2*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 2*c^2*(1/(c^2*x^2) - 1)^3
/(1/(c*x) + 1)^6 - c^2*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^2) - 6*b*c^4*d^2*sqrt(-1/(c^2*x^2) +
1)/((c^2 - 2*c^2*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 2*c^2*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^2*(1/(c^2*
x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)) + 12*a*c^4*d^2*(1/(c^2*x^2) - 1)/((c^2 - 2*c^2*(1/(c^2*x^2) - 1)/(
1/(c*x) + 1)^2 + 2*c^2*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^2*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x)
+ 1)^2) + 18*b*c^4*d^2*(1/(c^2*x^2) - 1)^2*arccos(1/(c*x))/((c^2 - 2*c^2*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 +
2*c^2*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^2*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^4) + 18*b*c
^2*d*arccos(1/(c*x))*e/(c^2 - 2*c^2*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 2*c^2*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1
)^6 - c^2*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8) + 2*b*c^4*d^2*(-1/(c^2*x^2) + 1)^(3/2)/((c^2 - 2*c^2*(1/(c^2*x^
2) - 1)/(1/(c*x) + 1)^2 + 2*c^2*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^2*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)
*(1/(c*x) + 1)^3) + 18*a*c^4*d^2*(1/(c^2*x^2) - 1)^2/((c^2 - 2*c^2*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 2*c^2*(
1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^2*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^4) + 12*b*c^4*d^2*
(1/(c^2*x^2) - 1)^3*arccos(1/(c*x))/((c^2 - 2*c^2*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 2*c^2*(1/(c^2*x^2) - 1)^
3/(1/(c*x) + 1)^6 - c^2*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^6) + 18*a*c^2*d*e/(c^2 - 2*c^2*(1/(
c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 2*c^2*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^2*(1/(c^2*x^2) - 1)^4/(1/(c*x) +
1)^8) - 2*b*c^4*d^2*(1/(c^2*x^2) - 1)^2*sqrt(-1/(c^2*x^2) + 1)/((c^2 - 2*c^2*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^
2 + 2*c^2*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^2*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^5) - 36
*b*c^2*d*sqrt(-1/(c^2*x^2) + 1)*e/((c^2 - 2*c^2*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 2*c^2*(1/(c^2*x^2) - 1)^3/
(1/(c*x) + 1)^6 - c^2*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)) + 12*a*c^4*d^2*(1/(c^2*x^2) - 1)^3/(
(c^2 - 2*c^2*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 2*c^2*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^2*(1/(c^2*x^2)
- 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^6) + 3*b*c^4*d^2*(1/(c^2*x^2) - 1)^4*arccos(1/(c*x))/((c^2 - 2*c^2*(1/(c
^2*x^2) - 1)/(1/(c*x) + 1)^2 + 2*c^2*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^2*(1/(c^2*x^2) - 1)^4/(1/(c*x) +
1)^8)*(1/(c*x) + 1)^8) - 36*b*c^2*d*(1/(c^2*x^2) - 1)^2*arccos(1/(c*x))*e/((c^2 - 2*c^2*(1/(c^2*x^2) - 1)/(1/(
c*x) + 1)^2 + 2*c^2*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^2*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) +
1)^4) - 6*b*c^4*d^2*(1/(c^2*x^2) - 1)^3*sqrt(-1/(c^2*x^2) + 1)/((c^2 - 2*c^2*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2
+ 2*c^2*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^2*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^7) - 36*
b*c^2*d*(-1/(c^2*x^2) + 1)^(3/2)*e/((c^2 - 2*c^2*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 2*c^2*(1/(c^2*x^2) - 1)^3
/(1/(c*x) + 1)^6 - c^2*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^3) + 3*a*c^4*d^2*(1/(c^2*x^2) - 1)^4
/((c^2 - 2*c^2*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 2*c^2*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^2*(1/(c^2*x^2
) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^8) - 9*b*arccos(1/(c*x))*e^2/(c^2 - 2*c^2*(1/(c^2*x^2) - 1)/(1/(c*x) +
1)^2 + 2*c^2*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^2*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8) - 36*a*c^2*d*(1/(
c^2*x^2) - 1)^2*e/((c^2 - 2*c^2*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 2*c^2*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6
- c^2*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^4) + 9*b*e^2*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x)
+ 1))/(c^2 - 2*c^2*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 2*c^2*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^2*(1/(c^
2*x^2) - 1)^4/(1/(c*x) + 1)^8) - 9*b*e^2*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/(c^2 - 2*c^2*(1/(c^2*x
^2) - 1)/(1/(c*x) + 1)^2 + 2*c^2*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^2*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8
) + 36*b*c^2*d*(1/(c^2*x^2) - 1)^2*sqrt(-1/(c^2*x^2) + 1)*e/((c^2 - 2*c^2*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 +
2*c^2*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^2*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^5) - 9*a*e^
2/(c^2 - 2*c^2*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 2*c^2*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^2*(1/(c^2*x^2
) - 1)^4/(1/(c*x) + 1)^8) + 36*b*(1/(c^2*x^2) - 1)*arccos(1/(c*x))*e^2/((c^2 - 2*c^2*(1/(c^2*x^2) - 1)/(1/(c*x
) + 1)^2 + 2*c^2*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^2*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^
2) + 18*b*c^2*d*(1/(c^2*x^2) - 1)^4*arccos(1/(c*x))*e/((c^2 - 2*c^2*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 2*c^2*
(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^2*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^8) - 18*b*(1/(c^2
*x^2) - 1)*e^2*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1))/((c^2 - 2*c^2*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2
+ 2*c^2*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^2*(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)*e^2*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/((c^2 - 2*c^2*(1/(c^2*x^2) - 1)/(1/(c*x)
+ 1)^2 + 2*c^2*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^2*(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)^3*sqrt(-1/(c^2*x^2) + 1)*e/((c^2 - 2*c^2*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 +
2*c^2*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^2*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^7) + 36*a*(
1/(c^2*x^2) - 1)*e^2/((c^2 - 2*c^2*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 2*c^2*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)
^6 - c^2*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^2) - 54*b*(1/(c^2*x^2) - 1)^2*arccos(1/(c*x))*e^2/
((c^2 - 2*c^2*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 2*c^2*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^2*(1/(c^2*x^2)
- 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^4) + 18*a*c^2*d*(1/(c^2*x^2) - 1)^4*e/((c^2 - 2*c^2*(1/(c^2*x^2) - 1)/(
1/(c*x) + 1)^2 + 2*c^2*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^2*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x)
+ 1)^8) - 54*a*(1/(c^2*x^2) - 1)^2*e^2/((c^2 - 2*c^2*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 2*c^2*(1/(c^2*x^2) -
1)^3/(1/(c*x) + 1)^6 - c^2*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^4) + 36*b*(1/(c^2*x^2) - 1)^3*a
rccos(1/(c*x))*e^2/((c^2 - 2*c^2*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 2*c^2*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6
- c^2*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^6) + 18*b*(1/(c^2*x^2) - 1)^3*e^2*log(abs(sqrt(-1/(c
^2*x^2) + 1) + 1/(c*x) + 1))/((c^2 - 2*c^2*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 2*c^2*(1/(c^2*x^2) - 1)^3/(1/(c
*x) + 1)^6 - c^2*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^6) - 18*b*(1/(c^2*x^2) - 1)^3*e^2*log(abs(
sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/((c^2 - 2*c^2*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 2*c^2*(1/(c^2*x^2) -
1)^3/(1/(c*x) + 1)^6 - c^2*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^6) + 36*a*(1/(c^2*x^2) - 1)^3*e^
2/((c^2 - 2*c^2*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 2*c^2*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^2*(1/(c^2*x^
2) - 1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^6) - 9*b*(1/(c^2*x^2) - 1)^4*arccos(1/(c*x))*e^2/((c^2 - 2*c^2*(1/(c^
2*x^2) - 1)/(1/(c*x) + 1)^2 + 2*c^2*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^2*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1
)^8)*(1/(c*x) + 1)^8) - 9*b*(1/(c^2*x^2) - 1)^4*e^2*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1))/((c^2 - 2*c
^2*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 2*c^2*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^2*(1/(c^2*x^2) - 1)^4/(1/
(c*x) + 1)^8)*(1/(c*x) + 1)^8) + 9*b*(1/(c^2*x^2) - 1)^4*e^2*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/((
c^2 - 2*c^2*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 2*c^2*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^2*(1/(c^2*x^2) -
1)^4/(1/(c*x) + 1)^8)*(1/(c*x) + 1)^8) - 9*a*(1/(c^2*x^2) - 1)^4*e^2/((c^2 - 2*c^2*(1/(c^2*x^2) - 1)/(1/(c*x)
+ 1)^2 + 2*c^2*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - c^2*(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 = 0.06, size = 255, normalized size = 1.61 \[ a x \,e^{2}-\frac {a \,d^{2}}{3 x^{3}}-\frac {2 a e d}{x}+b \,\mathrm {arcsec}\left (c x \right ) x \,e^{2}-\frac {b \,\mathrm {arcsec}\left (c x \right ) d^{2}}{3 x^{3}}-\frac {2 b \,\mathrm {arcsec}\left (c x \right ) e d}{x}+\frac {2 c^{3} b \,d^{2}}{9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {c b \,d^{2}}{9 x^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {2 c b e d}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {2 b e d}{c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{2}}-\frac {b \,d^{2}}{9 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{4}}-\frac {b \sqrt {c^{2} x^{2}-1}\, e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x*e^2-1/3*a*d^2/x^3-2*a*e*d/x+b*arcsec(c*x)*x*e^2-1/3*b*arcsec(c*x)*d^2/x^3-2*b*arcsec(c*x)*e*d/x+2/9*c^3*b/
((c^2*x^2-1)/c^2/x^2)^(1/2)*d^2-1/9*c*b/x^2/((c^2*x^2-1)/c^2/x^2)^(1/2)*d^2+2*c*b/((c^2*x^2-1)/c^2/x^2)^(1/2)*
e*d-2/c*b/((c^2*x^2-1)/c^2/x^2)^(1/2)/x^2*e*d-1/9/c*b/((c^2*x^2-1)/c^2/x^2)^(1/2)/x^4*d^2-1/c^2*b*(c^2*x^2-1)^
(1/2)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x*e^2*ln(c*x+(c^2*x^2-1)^(1/2))

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maxima [A]  time = 0.32, size = 159, normalized size = 1.01 \[ 2 \, {\left (c \sqrt {-\frac {1}{c^{2} x^{2}} + 1} - \frac {\operatorname {arcsec}\left (c x\right )}{x}\right )} b d e + a e^{2} x - \frac {1}{9} \, b d^{2} {\left (\frac {c^{4} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, c^{4} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c} + \frac {3 \, \operatorname {arcsec}\left (c x\right )}{x^{3}}\right )} + \frac {{\left (2 \, c x \operatorname {arcsec}\left (c x\right ) - \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right ) + \log \left (-\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )\right )} b e^{2}}{2 \, c} - \frac {2 \, a d e}{x} - \frac {a d^{2}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(c*sqrt(-1/(c^2*x^2) + 1) - arcsec(c*x)/x)*b*d*e + a*e^2*x - 1/9*b*d^2*((c^4*(-1/(c^2*x^2) + 1)^(3/2) - 3*c^
4*sqrt(-1/(c^2*x^2) + 1))/c + 3*arcsec(c*x)/x^3) + 1/2*(2*c*x*arcsec(c*x) - log(sqrt(-1/(c^2*x^2) + 1) + 1) +
log(-sqrt(-1/(c^2*x^2) + 1) + 1))*b*e^2/c - 2*a*d*e/x - 1/3*a*d^2/x^3

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}{x^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((d + e*x^2)^2*(a + b*acos(1/(c*x))))/x^4,x)

[Out]

int(((d + e*x^2)^2*(a + b*acos(1/(c*x))))/x^4, x)

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sympy [A]  time = 7.99, size = 211, normalized size = 1.34 \[ - \frac {a d^{2}}{3 x^{3}} - \frac {2 a d e}{x} + a e^{2} x + 2 b c d e \sqrt {1 - \frac {1}{c^{2} x^{2}}} - \frac {b d^{2} \operatorname {asec}{\left (c x \right )}}{3 x^{3}} - \frac {2 b d e \operatorname {asec}{\left (c x \right )}}{x} + b e^{2} x \operatorname {asec}{\left (c x \right )} + \frac {b d^{2} \left (\begin {cases} \frac {2 c^{3} \sqrt {c^{2} x^{2} - 1}}{3 x} + \frac {c \sqrt {c^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {2 i c^{3} \sqrt {- c^{2} x^{2} + 1}}{3 x} + \frac {i c \sqrt {- c^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\right )}{3 c} - \frac {b e^{2} \left (\begin {cases} \operatorname {acosh}{\left (c x \right )} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- i \operatorname {asin}{\left (c x \right )} & \text {otherwise} \end {cases}\right )}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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