\(\int x^2 (d+e x^2)^2 (a+b \sec ^{-1}(c x)) \, dx\) [81]

Optimal result

Integrand size = 21, antiderivative size = 252 \[ \int x^2 \left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right ) \, dx=-\frac {b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) x^2 \sqrt {-1+c^2 x^2}}{1680 c^5 \sqrt {c^2 x^2}}-\frac {b e \left (84 c^2 d+25 e\right ) x^4 \sqrt {-1+c^2 x^2}}{840 c^3 \sqrt {c^2 x^2}}-\frac {b e^2 x^6 \sqrt {-1+c^2 x^2}}{42 c \sqrt {c^2 x^2}}+\frac {1}{3} d^2 x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \sec ^{-1}(c x)\right )-\frac {b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) x \text {arctanh}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{1680 c^6 \sqrt {c^2 x^2}} \]

[Out]

1/3*d^2*x^3*(a+b*arcsec(c*x))+2/5*d*e*x^5*(a+b*arcsec(c*x))+1/7*e^2*x^7*(a+b*arcsec(c*x))-1/1680*b*(280*c^4*d^
2+252*c^2*d*e+75*e^2)*x*arctanh(c*x/(c^2*x^2-1)^(1/2))/c^6/(c^2*x^2)^(1/2)-1/1680*b*(280*c^4*d^2+252*c^2*d*e+7
5*e^2)*x^2*(c^2*x^2-1)^(1/2)/c^5/(c^2*x^2)^(1/2)-1/840*b*e*(84*c^2*d+25*e)*x^4*(c^2*x^2-1)^(1/2)/c^3/(c^2*x^2)
^(1/2)-1/42*b*e^2*x^6*(c^2*x^2-1)^(1/2)/c/(c^2*x^2)^(1/2)

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {276, 5346, 12, 1281, 470, 327, 223, 212} \[ \int x^2 \left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right ) \, dx=\frac {1}{3} d^2 x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \sec ^{-1}(c x)\right )-\frac {b x \text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right ) \left (280 c^4 d^2+252 c^2 d e+75 e^2\right )}{1680 c^6 \sqrt {c^2 x^2}}-\frac {b e^2 x^6 \sqrt {c^2 x^2-1}}{42 c \sqrt {c^2 x^2}}-\frac {b e x^4 \sqrt {c^2 x^2-1} \left (84 c^2 d+25 e\right )}{840 c^3 \sqrt {c^2 x^2}}-\frac {b x^2 \sqrt {c^2 x^2-1} \left (280 c^4 d^2+252 c^2 d e+75 e^2\right )}{1680 c^5 \sqrt {c^2 x^2}} \]

[In]

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

[Out]

-1/1680*(b*(280*c^4*d^2 + 252*c^2*d*e + 75*e^2)*x^2*Sqrt[-1 + c^2*x^2])/(c^5*Sqrt[c^2*x^2]) - (b*e*(84*c^2*d +
25*e)*x^4*Sqrt[-1 + c^2*x^2])/(840*c^3*Sqrt[c^2*x^2]) - (b*e^2*x^6*Sqrt[-1 + c^2*x^2])/(42*c*Sqrt[c^2*x^2]) +
(d^2*x^3*(a + b*ArcSec[c*x]))/3 + (2*d*e*x^5*(a + b*ArcSec[c*x]))/5 + (e^2*x^7*(a + b*ArcSec[c*x]))/7 - (b*(2
80*c^4*d^2 + 252*c^2*d*e + 75*e^2)*x*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(1680*c^6*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 212

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

Rule 223

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 276

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 327

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

Rule 470

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

Rule 1281

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Si
mp[c^p*(f*x)^(m + 4*p - 1)*((d + e*x^2)^(q + 1)/(e*f^(4*p - 1)*(m + 4*p + 2*q + 1))), x] + Dist[1/(e*(m + 4*p
+ 2*q + 1)), Int[(f*x)^m*(d + e*x^2)^q*ExpandToSum[e*(m + 4*p + 2*q + 1)*((a + b*x^2 + c*x^4)^p - c^p*x^(4*p))
- d*c^p*(m + 4*p - 1)*x^(4*p - 2), x], x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] &&
IGtQ[p, 0] &&  !IntegerQ[q] && NeQ[m + 4*p + 2*q + 1, 0]

Rule 5346

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*} \text {integral}& = \frac {1}{3} d^2 x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \sec ^{-1}(c x)\right )-\frac {(b c x) \int \frac {x^2 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )}{105 \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}} \\ & = \frac {1}{3} d^2 x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \sec ^{-1}(c x)\right )-\frac {(b c x) \int \frac {x^2 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )}{\sqrt {-1+c^2 x^2}} \, dx}{105 \sqrt {c^2 x^2}} \\ & = -\frac {b e^2 x^6 \sqrt {-1+c^2 x^2}}{42 c \sqrt {c^2 x^2}}+\frac {1}{3} d^2 x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \sec ^{-1}(c x)\right )-\frac {(b x) \int \frac {x^2 \left (210 c^2 d^2+3 e \left (84 c^2 d+25 e\right ) x^2\right )}{\sqrt {-1+c^2 x^2}} \, dx}{630 c \sqrt {c^2 x^2}} \\ & = -\frac {b e \left (84 c^2 d+25 e\right ) x^4 \sqrt {-1+c^2 x^2}}{840 c^3 \sqrt {c^2 x^2}}-\frac {b e^2 x^6 \sqrt {-1+c^2 x^2}}{42 c \sqrt {c^2 x^2}}+\frac {1}{3} d^2 x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \sec ^{-1}(c x)\right )--\frac {\left (b \left (-840 c^4 d^2-9 e \left (84 c^2 d+25 e\right )\right ) x\right ) \int \frac {x^2}{\sqrt {-1+c^2 x^2}} \, dx}{2520 c^3 \sqrt {c^2 x^2}} \\ & = -\frac {b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) x^2 \sqrt {-1+c^2 x^2}}{1680 c^5 \sqrt {c^2 x^2}}-\frac {b e \left (84 c^2 d+25 e\right ) x^4 \sqrt {-1+c^2 x^2}}{840 c^3 \sqrt {c^2 x^2}}-\frac {b e^2 x^6 \sqrt {-1+c^2 x^2}}{42 c \sqrt {c^2 x^2}}+\frac {1}{3} d^2 x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \sec ^{-1}(c x)\right )--\frac {\left (b \left (-840 c^4 d^2-9 e \left (84 c^2 d+25 e\right )\right ) x\right ) \int \frac {1}{\sqrt {-1+c^2 x^2}} \, dx}{5040 c^5 \sqrt {c^2 x^2}} \\ & = -\frac {b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) x^2 \sqrt {-1+c^2 x^2}}{1680 c^5 \sqrt {c^2 x^2}}-\frac {b e \left (84 c^2 d+25 e\right ) x^4 \sqrt {-1+c^2 x^2}}{840 c^3 \sqrt {c^2 x^2}}-\frac {b e^2 x^6 \sqrt {-1+c^2 x^2}}{42 c \sqrt {c^2 x^2}}+\frac {1}{3} d^2 x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \sec ^{-1}(c x)\right )--\frac {\left (b \left (-840 c^4 d^2-9 e \left (84 c^2 d+25 e\right )\right ) x\right ) \text {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\frac {x}{\sqrt {-1+c^2 x^2}}\right )}{5040 c^5 \sqrt {c^2 x^2}} \\ & = -\frac {b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) x^2 \sqrt {-1+c^2 x^2}}{1680 c^5 \sqrt {c^2 x^2}}-\frac {b e \left (84 c^2 d+25 e\right ) x^4 \sqrt {-1+c^2 x^2}}{840 c^3 \sqrt {c^2 x^2}}-\frac {b e^2 x^6 \sqrt {-1+c^2 x^2}}{42 c \sqrt {c^2 x^2}}+\frac {1}{3} d^2 x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \sec ^{-1}(c x)\right )-\frac {b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) x \text {arctanh}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{1680 c^6 \sqrt {c^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.74 \[ \int x^2 \left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right ) \, dx=\frac {c^2 x^2 \left (16 a c^5 x \left (35 d^2+42 d e x^2+15 e^2 x^4\right )-b \sqrt {1-\frac {1}{c^2 x^2}} \left (75 e^2+2 c^2 e \left (126 d+25 e x^2\right )+8 c^4 \left (35 d^2+21 d e x^2+5 e^2 x^4\right )\right )\right )+16 b c^7 x^3 \left (35 d^2+42 d e x^2+15 e^2 x^4\right ) \sec ^{-1}(c x)-b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) \log \left (\left (1+\sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{1680 c^7} \]

[In]

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

[Out]

(c^2*x^2*(16*a*c^5*x*(35*d^2 + 42*d*e*x^2 + 15*e^2*x^4) - b*Sqrt[1 - 1/(c^2*x^2)]*(75*e^2 + 2*c^2*e*(126*d + 2
5*e*x^2) + 8*c^4*(35*d^2 + 21*d*e*x^2 + 5*e^2*x^4))) + 16*b*c^7*x^3*(35*d^2 + 42*d*e*x^2 + 15*e^2*x^4)*ArcSec[
c*x] - b*(280*c^4*d^2 + 252*c^2*d*e + 75*e^2)*Log[(1 + Sqrt[1 - 1/(c^2*x^2)])*x])/(1680*c^7)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(458\) vs. \(2(222)=444\).

Time = 0.77 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.82

method	result	size
parts	\(a \left (\frac {1}{7} e^{2} x^{7}+\frac {2}{5} d e \,x^{5}+\frac {1}{3} x^{3} d^{2}\right )+\frac {b \,\operatorname {arcsec}\left (c x \right ) e^{2} x^{7}}{7}+\frac {2 b \,\operatorname {arcsec}\left (c x \right ) d e \,x^{5}}{5}+\frac {b \,\operatorname {arcsec}\left (c x \right ) d^{2} x^{3}}{3}-\frac {b \left (c^{2} x^{2}-1\right ) x^{4} e^{2}}{42 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \left (c^{2} x^{2}-1\right ) x^{2} d e}{10 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {5 b \left (c^{2} x^{2}-1\right ) x^{2} e^{2}}{168 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \left (c^{2} x^{2}-1\right ) d^{2}}{6 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {3 b \left (c^{2} x^{2}-1\right ) d e}{20 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \sqrt {c^{2} x^{2}-1}\, d^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{6 c^{4} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}-\frac {5 b \left (c^{2} x^{2}-1\right ) e^{2}}{112 c^{7} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {3 b \sqrt {c^{2} x^{2}-1}\, d e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{20 c^{6} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}-\frac {5 b \sqrt {c^{2} x^{2}-1}\, e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{112 c^{8} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}\)	\(459\)
derivativedivides	\(\frac {\frac {a \left (\frac {1}{3} d^{2} c^{7} x^{3}+\frac {2}{5} d \,c^{7} e \,x^{5}+\frac {1}{7} e^{2} c^{7} x^{7}\right )}{c^{4}}+\frac {b \,\operatorname {arcsec}\left (c x \right ) d^{2} c^{3} x^{3}}{3}+\frac {2 b \,c^{3} \operatorname {arcsec}\left (c x \right ) d e \,x^{5}}{5}+\frac {b \,c^{3} \operatorname {arcsec}\left (c x \right ) e^{2} x^{7}}{7}-\frac {b \left (c^{2} x^{2}-1\right ) d^{2}}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \left (c^{2} x^{2}-1\right ) x^{2} d e}{10 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \left (c^{2} x^{2}-1\right ) x^{4} e^{2}}{42 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \sqrt {c^{2} x^{2}-1}\, d^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}-\frac {3 b \left (c^{2} x^{2}-1\right ) d e}{20 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {5 b \left (c^{2} x^{2}-1\right ) x^{2} e^{2}}{168 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {3 b \sqrt {c^{2} x^{2}-1}\, d e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{20 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}-\frac {5 b \left (c^{2} x^{2}-1\right ) e^{2}}{112 c^{4} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {5 b \sqrt {c^{2} x^{2}-1}\, e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{112 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c^{3}}\)	\(475\)
default	\(\frac {\frac {a \left (\frac {1}{3} d^{2} c^{7} x^{3}+\frac {2}{5} d \,c^{7} e \,x^{5}+\frac {1}{7} e^{2} c^{7} x^{7}\right )}{c^{4}}+\frac {b \,\operatorname {arcsec}\left (c x \right ) d^{2} c^{3} x^{3}}{3}+\frac {2 b \,c^{3} \operatorname {arcsec}\left (c x \right ) d e \,x^{5}}{5}+\frac {b \,c^{3} \operatorname {arcsec}\left (c x \right ) e^{2} x^{7}}{7}-\frac {b \left (c^{2} x^{2}-1\right ) d^{2}}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \left (c^{2} x^{2}-1\right ) x^{2} d e}{10 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \left (c^{2} x^{2}-1\right ) x^{4} e^{2}}{42 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \sqrt {c^{2} x^{2}-1}\, d^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}-\frac {3 b \left (c^{2} x^{2}-1\right ) d e}{20 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {5 b \left (c^{2} x^{2}-1\right ) x^{2} e^{2}}{168 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {3 b \sqrt {c^{2} x^{2}-1}\, d e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{20 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}-\frac {5 b \left (c^{2} x^{2}-1\right ) e^{2}}{112 c^{4} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {5 b \sqrt {c^{2} x^{2}-1}\, e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{112 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c^{3}}\)	\(475\)

[In]

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

[Out]

a*(1/7*e^2*x^7+2/5*d*e*x^5+1/3*x^3*d^2)+1/7*b*arcsec(c*x)*e^2*x^7+2/5*b*arcsec(c*x)*d*e*x^5+1/3*b*arcsec(c*x)*
d^2*x^3-1/42*b/c^3*(c^2*x^2-1)/((c^2*x^2-1)/c^2/x^2)^(1/2)*x^4*e^2-1/10*b/c^3*(c^2*x^2-1)/((c^2*x^2-1)/c^2/x^2
)^(1/2)*x^2*d*e-5/168*b/c^5*(c^2*x^2-1)/((c^2*x^2-1)/c^2/x^2)^(1/2)*x^2*e^2-1/6*b/c^3*(c^2*x^2-1)/((c^2*x^2-1)
/c^2/x^2)^(1/2)*d^2-3/20*b/c^5*(c^2*x^2-1)/((c^2*x^2-1)/c^2/x^2)^(1/2)*d*e-1/6*b/c^4*(c^2*x^2-1)^(1/2)/((c^2*x
^2-1)/c^2/x^2)^(1/2)/x*d^2*ln(c*x+(c^2*x^2-1)^(1/2))-5/112*b/c^7*(c^2*x^2-1)/((c^2*x^2-1)/c^2/x^2)^(1/2)*e^2-3
/20*b/c^6*(c^2*x^2-1)^(1/2)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x*d*e*ln(c*x+(c^2*x^2-1)^(1/2))-5/112*b/c^8*(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))

Fricas [A] (verification not implemented)

none

Time = 0.47 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.08 \[ \int x^2 \left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right ) \, dx=\frac {240 \, a c^{7} e^{2} x^{7} + 672 \, a c^{7} d e x^{5} + 560 \, a c^{7} d^{2} x^{3} + 16 \, {\left (15 \, b c^{7} e^{2} x^{7} + 42 \, b c^{7} d e x^{5} + 35 \, b c^{7} d^{2} x^{3} - 35 \, b c^{7} d^{2} - 42 \, b c^{7} d e - 15 \, b c^{7} e^{2}\right )} \operatorname {arcsec}\left (c x\right ) + 32 \, {\left (35 \, b c^{7} d^{2} + 42 \, b c^{7} d e + 15 \, b c^{7} e^{2}\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (280 \, b c^{4} d^{2} + 252 \, b c^{2} d e + 75 \, b e^{2}\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (40 \, b c^{5} e^{2} x^{5} + 2 \, {\left (84 \, b c^{5} d e + 25 \, b c^{3} e^{2}\right )} x^{3} + {\left (280 \, b c^{5} d^{2} + 252 \, b c^{3} d e + 75 \, b c e^{2}\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{1680 \, c^{7}} \]

[In]

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

[Out]

1/1680*(240*a*c^7*e^2*x^7 + 672*a*c^7*d*e*x^5 + 560*a*c^7*d^2*x^3 + 16*(15*b*c^7*e^2*x^7 + 42*b*c^7*d*e*x^5 +
35*b*c^7*d^2*x^3 - 35*b*c^7*d^2 - 42*b*c^7*d*e - 15*b*c^7*e^2)*arcsec(c*x) + 32*(35*b*c^7*d^2 + 42*b*c^7*d*e +
15*b*c^7*e^2)*arctan(-c*x + sqrt(c^2*x^2 - 1)) + (280*b*c^4*d^2 + 252*b*c^2*d*e + 75*b*e^2)*log(-c*x + sqrt(c
^2*x^2 - 1)) - (40*b*c^5*e^2*x^5 + 2*(84*b*c^5*d*e + 25*b*c^3*e^2)*x^3 + (280*b*c^5*d^2 + 252*b*c^3*d*e + 75*b
*c*e^2)*x)*sqrt(c^2*x^2 - 1))/c^7

Sympy [A] (verification not implemented)

Time = 11.90 (sec) , antiderivative size = 542, normalized size of antiderivative = 2.15 \[ \int x^2 \left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right ) \, dx=\frac {a d^{2} x^{3}}{3} + \frac {2 a d e x^{5}}{5} + \frac {a e^{2} x^{7}}{7} + \frac {b d^{2} x^{3} \operatorname {asec}{\left (c x \right )}}{3} + \frac {2 b d e x^{5} \operatorname {asec}{\left (c x \right )}}{5} + \frac {b e^{2} x^{7} \operatorname {asec}{\left (c x \right )}}{7} - \frac {b d^{2} \left (\begin {cases} \frac {x \sqrt {c^{2} x^{2} - 1}}{2 c} + \frac {\operatorname {acosh}{\left (c x \right )}}{2 c^{2}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {i c x^{3}}{2 \sqrt {- c^{2} x^{2} + 1}} + \frac {i x}{2 c \sqrt {- c^{2} x^{2} + 1}} - \frac {i \operatorname {asin}{\left (c x \right )}}{2 c^{2}} & \text {otherwise} \end {cases}\right )}{3 c} - \frac {2 b d e \left (\begin {cases} \frac {c x^{5}}{4 \sqrt {c^{2} x^{2} - 1}} + \frac {x^{3}}{8 c \sqrt {c^{2} x^{2} - 1}} - \frac {3 x}{8 c^{3} \sqrt {c^{2} x^{2} - 1}} + \frac {3 \operatorname {acosh}{\left (c x \right )}}{8 c^{4}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {i c x^{5}}{4 \sqrt {- c^{2} x^{2} + 1}} - \frac {i x^{3}}{8 c \sqrt {- c^{2} x^{2} + 1}} + \frac {3 i x}{8 c^{3} \sqrt {- c^{2} x^{2} + 1}} - \frac {3 i \operatorname {asin}{\left (c x \right )}}{8 c^{4}} & \text {otherwise} \end {cases}\right )}{5 c} - \frac {b e^{2} \left (\begin {cases} \frac {c x^{7}}{6 \sqrt {c^{2} x^{2} - 1}} + \frac {x^{5}}{24 c \sqrt {c^{2} x^{2} - 1}} + \frac {5 x^{3}}{48 c^{3} \sqrt {c^{2} x^{2} - 1}} - \frac {5 x}{16 c^{5} \sqrt {c^{2} x^{2} - 1}} + \frac {5 \operatorname {acosh}{\left (c x \right )}}{16 c^{6}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {i c x^{7}}{6 \sqrt {- c^{2} x^{2} + 1}} - \frac {i x^{5}}{24 c \sqrt {- c^{2} x^{2} + 1}} - \frac {5 i x^{3}}{48 c^{3} \sqrt {- c^{2} x^{2} + 1}} + \frac {5 i x}{16 c^{5} \sqrt {- c^{2} x^{2} + 1}} - \frac {5 i \operatorname {asin}{\left (c x \right )}}{16 c^{6}} & \text {otherwise} \end {cases}\right )}{7 c} \]

[In]

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

[Out]

a*d**2*x**3/3 + 2*a*d*e*x**5/5 + a*e**2*x**7/7 + b*d**2*x**3*asec(c*x)/3 + 2*b*d*e*x**5*asec(c*x)/5 + b*e**2*x
**7*asec(c*x)/7 - b*d**2*Piecewise((x*sqrt(c**2*x**2 - 1)/(2*c) + acosh(c*x)/(2*c**2), Abs(c**2*x**2) > 1), (-
I*c*x**3/(2*sqrt(-c**2*x**2 + 1)) + I*x/(2*c*sqrt(-c**2*x**2 + 1)) - I*asin(c*x)/(2*c**2), True))/(3*c) - 2*b*
d*e*Piecewise((c*x**5/(4*sqrt(c**2*x**2 - 1)) + x**3/(8*c*sqrt(c**2*x**2 - 1)) - 3*x/(8*c**3*sqrt(c**2*x**2 -
1)) + 3*acosh(c*x)/(8*c**4), Abs(c**2*x**2) > 1), (-I*c*x**5/(4*sqrt(-c**2*x**2 + 1)) - I*x**3/(8*c*sqrt(-c**2
*x**2 + 1)) + 3*I*x/(8*c**3*sqrt(-c**2*x**2 + 1)) - 3*I*asin(c*x)/(8*c**4), True))/(5*c) - b*e**2*Piecewise((c
*x**7/(6*sqrt(c**2*x**2 - 1)) + x**5/(24*c*sqrt(c**2*x**2 - 1)) + 5*x**3/(48*c**3*sqrt(c**2*x**2 - 1)) - 5*x/(
16*c**5*sqrt(c**2*x**2 - 1)) + 5*acosh(c*x)/(16*c**6), Abs(c**2*x**2) > 1), (-I*c*x**7/(6*sqrt(-c**2*x**2 + 1)
) - I*x**5/(24*c*sqrt(-c**2*x**2 + 1)) - 5*I*x**3/(48*c**3*sqrt(-c**2*x**2 + 1)) + 5*I*x/(16*c**5*sqrt(-c**2*x
**2 + 1)) - 5*I*asin(c*x)/(16*c**6), True))/(7*c)

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.61 \[ \int x^2 \left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right ) \, dx=\frac {1}{7} \, a e^{2} x^{7} + \frac {2}{5} \, a d e x^{5} + \frac {1}{3} \, a d^{2} x^{3} + \frac {1}{12} \, {\left (4 \, x^{3} \operatorname {arcsec}\left (c x\right ) - \frac {\frac {2 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{2}} + \frac {\log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} - \frac {\log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{2}}}{c}\right )} b d^{2} + \frac {1}{40} \, {\left (16 \, x^{5} \operatorname {arcsec}\left (c x\right ) + \frac {\frac {2 \, {\left (3 \, {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 5 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1}\right )}}{c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 2 \, c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{4}} - \frac {3 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{4}} + \frac {3 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{4}}}{c}\right )} b d e + \frac {1}{672} \, {\left (96 \, x^{7} \operatorname {arcsec}\left (c x\right ) - \frac {\frac {2 \, {\left (15 \, {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} - 40 \, {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 33 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1}\right )}}{c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{3} + 3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{6}} + \frac {15 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{6}} - \frac {15 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{6}}}{c}\right )} b e^{2} \]

[In]

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

[Out]

1/7*a*e^2*x^7 + 2/5*a*d*e*x^5 + 1/3*a*d^2*x^3 + 1/12*(4*x^3*arcsec(c*x) - (2*sqrt(-1/(c^2*x^2) + 1)/(c^2*(1/(c
^2*x^2) - 1) + c^2) + log(sqrt(-1/(c^2*x^2) + 1) + 1)/c^2 - log(sqrt(-1/(c^2*x^2) + 1) - 1)/c^2)/c)*b*d^2 + 1/
40*(16*x^5*arcsec(c*x) + (2*(3*(-1/(c^2*x^2) + 1)^(3/2) - 5*sqrt(-1/(c^2*x^2) + 1))/(c^4*(1/(c^2*x^2) - 1)^2 +
2*c^4*(1/(c^2*x^2) - 1) + c^4) - 3*log(sqrt(-1/(c^2*x^2) + 1) + 1)/c^4 + 3*log(sqrt(-1/(c^2*x^2) + 1) - 1)/c^
4)/c)*b*d*e + 1/672*(96*x^7*arcsec(c*x) - (2*(15*(-1/(c^2*x^2) + 1)^(5/2) - 40*(-1/(c^2*x^2) + 1)^(3/2) + 33*s
qrt(-1/(c^2*x^2) + 1))/(c^6*(1/(c^2*x^2) - 1)^3 + 3*c^6*(1/(c^2*x^2) - 1)^2 + 3*c^6*(1/(c^2*x^2) - 1) + c^6) +
15*log(sqrt(-1/(c^2*x^2) + 1) + 1)/c^6 - 15*log(sqrt(-1/(c^2*x^2) + 1) - 1)/c^6)/c)*b*e^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4760 vs. \(2 (222) = 444\).

Time = 6.39 (sec) , antiderivative size = 4760, normalized size of antiderivative = 18.89 \[ \int x^2 \left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/1680*(560*b*c^4*d^2*arccos(1/(c*x)) - 280*b*c^4*d^2*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1)) + 280*b*c
^4*d^2*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1)) + 560*a*c^4*d^2 + 560*b*c^4*d^2*(1/(c^2*x^2) - 1)*arccos
(1/(c*x))/(1/(c*x) + 1)^2 - 1960*b*c^4*d^2*(1/(c^2*x^2) - 1)*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1))/(1
/(c*x) + 1)^2 + 1960*b*c^4*d^2*(1/(c^2*x^2) - 1)*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/(1/(c*x) + 1)^
2 - 560*b*c^4*d^2*sqrt(-1/(c^2*x^2) + 1)/(1/(c*x) + 1) + 560*a*c^4*d^2*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 672
*b*c^2*d*e*arccos(1/(c*x)) - 1680*b*c^4*d^2*(1/(c^2*x^2) - 1)^2*arccos(1/(c*x))/(1/(c*x) + 1)^4 - 252*b*c^2*d*
e*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1)) - 5880*b*c^4*d^2*(1/(c^2*x^2) - 1)^2*log(abs(sqrt(-1/(c^2*x^2
) + 1) + 1/(c*x) + 1))/(1/(c*x) + 1)^4 + 252*b*c^2*d*e*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1)) + 5880*b
*c^4*d^2*(1/(c^2*x^2) - 1)^2*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/(1/(c*x) + 1)^4 + 2240*b*c^4*d^2*(
-1/(c^2*x^2) + 1)^(3/2)/(1/(c*x) + 1)^3 + 672*a*c^2*d*e - 1680*a*c^4*d^2*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 -
2016*b*c^2*d*e*(1/(c^2*x^2) - 1)*arccos(1/(c*x))/(1/(c*x) + 1)^2 - 1680*b*c^4*d^2*(1/(c^2*x^2) - 1)^3*arccos(
1/(c*x))/(1/(c*x) + 1)^6 - 1764*b*c^2*d*e*(1/(c^2*x^2) - 1)*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1))/(1/
(c*x) + 1)^2 - 9800*b*c^4*d^2*(1/(c^2*x^2) - 1)^3*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1))/(1/(c*x) + 1)
^6 + 1764*b*c^2*d*e*(1/(c^2*x^2) - 1)*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/(1/(c*x) + 1)^2 + 9800*b*
c^4*d^2*(1/(c^2*x^2) - 1)^3*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/(1/(c*x) + 1)^6 - 840*b*c^2*d*e*sqr
t(-1/(c^2*x^2) + 1)/(1/(c*x) + 1) - 2800*b*c^4*d^2*(1/(c^2*x^2) - 1)^2*sqrt(-1/(c^2*x^2) + 1)/(1/(c*x) + 1)^5
- 2016*a*c^2*d*e*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 - 1680*a*c^4*d^2*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 240*
b*e^2*arccos(1/(c*x)) + 672*b*c^2*d*e*(1/(c^2*x^2) - 1)^2*arccos(1/(c*x))/(1/(c*x) + 1)^4 + 1680*b*c^4*d^2*(1/
(c^2*x^2) - 1)^4*arccos(1/(c*x))/(1/(c*x) + 1)^8 - 75*b*e^2*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1)) - 5
292*b*c^2*d*e*(1/(c^2*x^2) - 1)^2*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1))/(1/(c*x) + 1)^4 - 9800*b*c^4*
d^2*(1/(c^2*x^2) - 1)^4*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1))/(1/(c*x) + 1)^8 + 75*b*e^2*log(abs(sqrt
(-1/(c^2*x^2) + 1) - 1/(c*x) - 1)) + 5292*b*c^2*d*e*(1/(c^2*x^2) - 1)^2*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*
x) - 1))/(1/(c*x) + 1)^4 + 9800*b*c^4*d^2*(1/(c^2*x^2) - 1)^4*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/(
1/(c*x) + 1)^8 + 2016*b*c^2*d*e*(-1/(c^2*x^2) + 1)^(3/2)/(1/(c*x) + 1)^3 + 240*a*e^2 + 672*a*c^2*d*e*(1/(c^2*x
^2) - 1)^2/(1/(c*x) + 1)^4 + 1680*a*c^4*d^2*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 - 1680*b*e^2*(1/(c^2*x^2) - 1)
*arccos(1/(c*x))/(1/(c*x) + 1)^2 + 3360*b*c^2*d*e*(1/(c^2*x^2) - 1)^3*arccos(1/(c*x))/(1/(c*x) + 1)^6 + 1680*b
*c^4*d^2*(1/(c^2*x^2) - 1)^5*arccos(1/(c*x))/(1/(c*x) + 1)^10 - 525*b*e^2*(1/(c^2*x^2) - 1)*log(abs(sqrt(-1/(c
^2*x^2) + 1) + 1/(c*x) + 1))/(1/(c*x) + 1)^2 - 8820*b*c^2*d*e*(1/(c^2*x^2) - 1)^3*log(abs(sqrt(-1/(c^2*x^2) +
1) + 1/(c*x) + 1))/(1/(c*x) + 1)^6 - 5880*b*c^4*d^2*(1/(c^2*x^2) - 1)^5*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*
x) + 1))/(1/(c*x) + 1)^10 + 525*b*e^2*(1/(c^2*x^2) - 1)*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/(1/(c*x
) + 1)^2 + 8820*b*c^2*d*e*(1/(c^2*x^2) - 1)^3*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/(1/(c*x) + 1)^6 +
5880*b*c^4*d^2*(1/(c^2*x^2) - 1)^5*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/(1/(c*x) + 1)^10 - 330*b*e^
2*sqrt(-1/(c^2*x^2) + 1)/(1/(c*x) + 1) - 1512*b*c^2*d*e*(1/(c^2*x^2) - 1)^2*sqrt(-1/(c^2*x^2) + 1)/(1/(c*x) +
1)^5 + 2800*b*c^4*d^2*(1/(c^2*x^2) - 1)^4*sqrt(-1/(c^2*x^2) + 1)/(1/(c*x) + 1)^9 - 1680*a*e^2*(1/(c^2*x^2) - 1
)/(1/(c*x) + 1)^2 + 3360*a*c^2*d*e*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 1680*a*c^4*d^2*(1/(c^2*x^2) - 1)^5/(1
/(c*x) + 1)^10 + 5040*b*e^2*(1/(c^2*x^2) - 1)^2*arccos(1/(c*x))/(1/(c*x) + 1)^4 - 3360*b*c^2*d*e*(1/(c^2*x^2)
- 1)^4*arccos(1/(c*x))/(1/(c*x) + 1)^8 - 560*b*c^4*d^2*(1/(c^2*x^2) - 1)^6*arccos(1/(c*x))/(1/(c*x) + 1)^12 -
1575*b*e^2*(1/(c^2*x^2) - 1)^2*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1))/(1/(c*x) + 1)^4 - 8820*b*c^2*d*e
*(1/(c^2*x^2) - 1)^4*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1))/(1/(c*x) + 1)^8 - 1960*b*c^4*d^2*(1/(c^2*x
^2) - 1)^6*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1))/(1/(c*x) + 1)^12 + 1575*b*e^2*(1/(c^2*x^2) - 1)^2*lo
g(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/(1/(c*x) + 1)^4 + 8820*b*c^2*d*e*(1/(c^2*x^2) - 1)^4*log(abs(sqrt
(-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/(1/(c*x) + 1)^8 + 1960*b*c^4*d^2*(1/(c^2*x^2) - 1)^6*log(abs(sqrt(-1/(c^2*x
^2) + 1) - 1/(c*x) - 1))/(1/(c*x) + 1)^12 + 2240*b*c^4*d^2*(1/(c^2*x^2) - 1)^5*sqrt(-1/(c^2*x^2) + 1)/(1/(c*x)
+ 1)^11 + 280*b*e^2*(-1/(c^2*x^2) + 1)^(3/2)/(1/(c*x) + 1)^3 + 5040*a*e^2*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4
- 3360*a*c^2*d*e*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 - 560*a*c^4*d^2*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 - 8
400*b*e^2*(1/(c^2*x^2) - 1)^3*arccos(1/(c*x))/(1/(c*x) + 1)^6 - 672*b*c^2*d*e*(1/(c^2*x^2) - 1)^5*arccos(1/(c*
x))/(1/(c*x) + 1)^10 - 560*b*c^4*d^2*(1/(c^2*x^2) - 1)^7*arccos(1/(c*x))/(1/(c*x) + 1)^14 - 2625*b*e^2*(1/(c^2
*x^2) - 1)^3*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1))/(1/(c*x) + 1)^6 - 5292*b*c^2*d*e*(1/(c^2*x^2) - 1)
^5*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1))/(1/(c*x) + 1)^10 - 280*b*c^4*d^2*(1/(c^2*x^2) - 1)^7*log(abs
(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1))/(1/(c*x) + 1)^14 + 2625*b*e^2*(1/(c^2*x^2) - 1)^3*log(abs(sqrt(-1/(c^2
*x^2) + 1) - 1/(c*x) - 1))/(1/(c*x) + 1)^6 + 5292*b*c^2*d*e*(1/(c^2*x^2) - 1)^5*log(abs(sqrt(-1/(c^2*x^2) + 1)
- 1/(c*x) - 1))/(1/(c*x) + 1)^10 + 280*b*c^4*d^2*(1/(c^2*x^2) - 1)^7*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x)
- 1))/(1/(c*x) + 1)^14 - 850*b*e^2*(1/(c^2*x^2) - 1)^2*sqrt(-1/(c^2*x^2) + 1)/(1/(c*x) + 1)^5 + 1512*b*c^2*d*
e*(1/(c^2*x^2) - 1)^4*sqrt(-1/(c^2*x^2) + 1)/(1/(c*x) + 1)^9 + 560*b*c^4*d^2*(1/(c^2*x^2) - 1)^6*sqrt(-1/(c^2*
x^2) + 1)/(1/(c*x) + 1)^13 - 8400*a*e^2*(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 - 672*a*c^2*d*e*(1/(c^2*x^2) - 1)^
5/(1/(c*x) + 1)^10 - 560*a*c^4*d^2*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14 + 8400*b*e^2*(1/(c^2*x^2) - 1)^4*arcco
s(1/(c*x))/(1/(c*x) + 1)^8 + 2016*b*c^2*d*e*(1/(c^2*x^2) - 1)^6*arccos(1/(c*x))/(1/(c*x) + 1)^12 - 2625*b*e^2*
(1/(c^2*x^2) - 1)^4*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1))/(1/(c*x) + 1)^8 - 1764*b*c^2*d*e*(1/(c^2*x^
2) - 1)^6*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1))/(1/(c*x) + 1)^12 + 2625*b*e^2*(1/(c^2*x^2) - 1)^4*log
(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/(1/(c*x) + 1)^8 + 1764*b*c^2*d*e*(1/(c^2*x^2) - 1)^6*log(abs(sqrt(
-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/(1/(c*x) + 1)^12 + 2016*b*c^2*d*e*(1/(c^2*x^2) - 1)^5*sqrt(-1/(c^2*x^2) + 1)
/(1/(c*x) + 1)^11 + 8400*a*e^2*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 2016*a*c^2*d*e*(1/(c^2*x^2) - 1)^6/(1/(c*
x) + 1)^12 - 5040*b*e^2*(1/(c^2*x^2) - 1)^5*arccos(1/(c*x))/(1/(c*x) + 1)^10 - 672*b*c^2*d*e*(1/(c^2*x^2) - 1)
^7*arccos(1/(c*x))/(1/(c*x) + 1)^14 - 1575*b*e^2*(1/(c^2*x^2) - 1)^5*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x)
+ 1))/(1/(c*x) + 1)^10 - 252*b*c^2*d*e*(1/(c^2*x^2) - 1)^7*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1))/(1/(
c*x) + 1)^14 + 1575*b*e^2*(1/(c^2*x^2) - 1)^5*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/(1/(c*x) + 1)^10
+ 252*b*c^2*d*e*(1/(c^2*x^2) - 1)^7*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/(1/(c*x) + 1)^14 + 850*b*e^
2*(1/(c^2*x^2) - 1)^4*sqrt(-1/(c^2*x^2) + 1)/(1/(c*x) + 1)^9 + 840*b*c^2*d*e*(1/(c^2*x^2) - 1)^6*sqrt(-1/(c^2*
x^2) + 1)/(1/(c*x) + 1)^13 - 5040*a*e^2*(1/(c^2*x^2) - 1)^5/(1/(c*x) + 1)^10 - 672*a*c^2*d*e*(1/(c^2*x^2) - 1)
^7/(1/(c*x) + 1)^14 + 1680*b*e^2*(1/(c^2*x^2) - 1)^6*arccos(1/(c*x))/(1/(c*x) + 1)^12 - 525*b*e^2*(1/(c^2*x^2)
- 1)^6*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1))/(1/(c*x) + 1)^12 + 525*b*e^2*(1/(c^2*x^2) - 1)^6*log(ab
s(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/(1/(c*x) + 1)^12 + 280*b*e^2*(1/(c^2*x^2) - 1)^5*sqrt(-1/(c^2*x^2) +
1)/(1/(c*x) + 1)^11 + 1680*a*e^2*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 - 240*b*e^2*(1/(c^2*x^2) - 1)^7*arccos(1
/(c*x))/(1/(c*x) + 1)^14 - 75*b*e^2*(1/(c^2*x^2) - 1)^7*log(abs(sqrt(-1/(c^2*x^2) + 1) + 1/(c*x) + 1))/(1/(c*x
) + 1)^14 + 75*b*e^2*(1/(c^2*x^2) - 1)^7*log(abs(sqrt(-1/(c^2*x^2) + 1) - 1/(c*x) - 1))/(1/(c*x) + 1)^14 + 330
*b*e^2*(1/(c^2*x^2) - 1)^6*sqrt(-1/(c^2*x^2) + 1)/(1/(c*x) + 1)^13 - 240*a*e^2*(1/(c^2*x^2) - 1)^7/(1/(c*x) +
1)^14)*c/(c^8 + 7*c^8*(1/(c^2*x^2) - 1)/(1/(c*x) + 1)^2 + 21*c^8*(1/(c^2*x^2) - 1)^2/(1/(c*x) + 1)^4 + 35*c^8*
(1/(c^2*x^2) - 1)^3/(1/(c*x) + 1)^6 + 35*c^8*(1/(c^2*x^2) - 1)^4/(1/(c*x) + 1)^8 + 21*c^8*(1/(c^2*x^2) - 1)^5/
(1/(c*x) + 1)^10 + 7*c^8*(1/(c^2*x^2) - 1)^6/(1/(c*x) + 1)^12 + c^8*(1/(c^2*x^2) - 1)^7/(1/(c*x) + 1)^14)

Mupad [F(-1)]

Timed out. \[ \int x^2 \left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int x^2\,{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right ) \,d x \]

[In]

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

[Out]

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