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

Optimal. Leaf size=252 \[ \frac {1}{3} d^2 x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \sec ^{-1}(c x)\right )-\frac {b e^2 x^6 \sqrt {c^2 x^2-1}}{42 c \sqrt {c^2 x^2}}-\frac {b e x^4 \sqrt {c^2 x^2-1} \left (84 c^2 d+25 e\right )}{840 c^3 \sqrt {c^2 x^2}}-\frac {b x \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) \tanh ^{-1}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{1680 c^6 \sqrt {c^2 x^2}}-\frac {b x^2 \sqrt {c^2 x^2-1} \left (280 c^4 d^2+252 c^2 d e+75 e^2\right )}{1680 c^5 \sqrt {c^2 x^2}} \]

[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)

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Rubi [A]  time = 0.24, antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {270, 5238, 12, 1267, 459, 321, 217, 206} \[ \frac {1}{3} d^2 x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \sec ^{-1}(c x)\right )-\frac {b x^2 \sqrt {c^2 x^2-1} \left (280 c^4 d^2+252 c^2 d e+75 e^2\right )}{1680 c^5 \sqrt {c^2 x^2}}-\frac {b x \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) \tanh ^{-1}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{1680 c^6 \sqrt {c^2 x^2}}-\frac {b e x^4 \sqrt {c^2 x^2-1} \left (84 c^2 d+25 e\right )}{840 c^3 \sqrt {c^2 x^2}}-\frac {b e^2 x^6 \sqrt {c^2 x^2-1}}{42 c \sqrt {c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*(280*c^4*d^2 + 252*c^2*d*e + 75*e^2)*x^2*Sqrt[-1 + c^2*x^2])/(1680*c^5*Sqrt[c^2*x^2]) - (b*e*(84*c^2*d + 2
5*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*(280
*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 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 321

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 459

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 1267

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 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 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 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 ) \operatorname {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 \tanh ^{-1}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{1680 c^6 \sqrt {c^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.37, size = 186, normalized size = 0.74 \[ \frac {c^2 x^2 \left (16 a c^5 x \left (35 d^2+42 d e x^2+15 e^2 x^4\right )-b \sqrt {1-\frac {1}{c^2 x^2}} \left (8 c^4 \left (35 d^2+21 d e x^2+5 e^2 x^4\right )+2 c^2 e \left (126 d+25 e x^2\right )+75 e^2\right )\right )+16 b c^7 x^3 \sec ^{-1}(c x) \left (35 d^2+42 d e x^2+15 e^2 x^4\right )-b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) \log \left (x \left (\sqrt {1-\frac {1}{c^2 x^2}}+1\right )\right )}{1680 c^7} \]

Antiderivative was successfully verified.

[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)

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fricas [A]  time = 4.04, size = 273, normalized size = 1.08 \[ \frac {240 \, a c^{7} e^{2} x^{7} + 672 \, a c^{7} d e x^{5} + 560 \, a c^{7} d^{2} x^{3} + 16 \, {\left (15 \, b c^{7} e^{2} x^{7} + 42 \, b c^{7} d e x^{5} + 35 \, b c^{7} d^{2} x^{3} - 35 \, b c^{7} d^{2} - 42 \, b c^{7} d e - 15 \, b c^{7} e^{2}\right )} \operatorname {arcsec}\left (c x\right ) + 32 \, {\left (35 \, b c^{7} d^{2} + 42 \, b c^{7} d e + 15 \, b c^{7} e^{2}\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (280 \, b c^{4} d^{2} + 252 \, b c^{2} d e + 75 \, b e^{2}\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (40 \, b c^{5} e^{2} x^{5} + 2 \, {\left (84 \, b c^{5} d e + 25 \, b c^{3} e^{2}\right )} x^{3} + {\left (280 \, b c^{5} d^{2} + 252 \, b c^{3} d e + 75 \, b c e^{2}\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{1680 \, c^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [B]  time = 0.05, size = 494, normalized size = 1.96 \[ \frac {a \,e^{2} x^{7}}{7}+\frac {2 a e d \,x^{5}}{5}+\frac {a \,x^{3} d^{2}}{3}+\frac {b \,\mathrm {arcsec}\left (c x \right ) e^{2} x^{7}}{7}+\frac {2 b \,\mathrm {arcsec}\left (c x \right ) e d \,x^{5}}{5}+\frac {b \,\mathrm {arcsec}\left (c x \right ) x^{3} d^{2}}{3}-\frac {b \,x^{6} e^{2}}{42 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \,x^{4} e^{2}}{168 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \,x^{4} e d}{10 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \,x^{2} e d}{20 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \,d^{2} x^{2}}{6 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \,d^{2}}{6 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \sqrt {c^{2} x^{2}-1}\, d^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{6 c^{4} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}-\frac {5 b \,x^{2} e^{2}}{336 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 b e d}{20 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {3 b \sqrt {c^{2} x^{2}-1}\, e d \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{20 c^{6} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {5 b \,e^{2}}{112 c^{7} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {5 b \sqrt {c^{2} x^{2}-1}\, e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{112 c^{8} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*a*e^2*x^7+2/5*a*e*d*x^5+1/3*a*x^3*d^2+1/7*b*arcsec(c*x)*e^2*x^7+2/5*b*arcsec(c*x)*e*d*x^5+1/3*b*arcsec(c*x
)*x^3*d^2-1/42/c*b/((c^2*x^2-1)/c^2/x^2)^(1/2)*x^6*e^2-1/168/c^3*b/((c^2*x^2-1)/c^2/x^2)^(1/2)*x^4*e^2-1/10/c*
b/((c^2*x^2-1)/c^2/x^2)^(1/2)*x^4*e*d-1/20/c^3*b/((c^2*x^2-1)/c^2/x^2)^(1/2)*x^2*e*d-1/6/c*b/((c^2*x^2-1)/c^2/
x^2)^(1/2)*d^2*x^2+1/6/c^3*b/((c^2*x^2-1)/c^2/x^2)^(1/2)*d^2-1/6/c^4*b*(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))-5/336/c^5*b/((c^2*x^2-1)/c^2/x^2)^(1/2)*x^2*e^2+3/20/c^5*b/((c^2*x^2-1)
/c^2/x^2)^(1/2)*e*d-3/20/c^6*b*(c^2*x^2-1)^(1/2)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x*e*d*ln(c*x+(c^2*x^2-1)^(1/2))+5
/112/c^7*b/((c^2*x^2-1)/c^2/x^2)^(1/2)*e^2-5/112/c^8*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 = 0.34, size = 405, normalized size = 1.61 \[ \frac {1}{7} \, a e^{2} x^{7} + \frac {2}{5} \, a d e x^{5} + \frac {1}{3} \, a d^{2} x^{3} + \frac {1}{12} \, {\left (4 \, x^{3} \operatorname {arcsec}\left (c x\right ) - \frac {\frac {2 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{2}} + \frac {\log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} - \frac {\log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{2}}}{c}\right )} b d^{2} + \frac {1}{40} \, {\left (16 \, x^{5} \operatorname {arcsec}\left (c x\right ) + \frac {\frac {2 \, {\left (3 \, {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 5 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1}\right )}}{c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 2 \, c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{4}} - \frac {3 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{4}} + \frac {3 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{4}}}{c}\right )} b d e + \frac {1}{672} \, {\left (96 \, x^{7} \operatorname {arcsec}\left (c x\right ) - \frac {\frac {2 \, {\left (15 \, {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} - 40 \, {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 33 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1}\right )}}{c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{3} + 3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{6}} + \frac {15 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{6}} - \frac {15 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{6}}}{c}\right )} b e^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^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)

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sympy [A]  time = 13.68, size = 542, normalized size = 2.15 \[ \frac {a d^{2} x^{3}}{3} + \frac {2 a d e x^{5}}{5} + \frac {a e^{2} x^{7}}{7} + \frac {b d^{2} x^{3} \operatorname {asec}{\left (c x \right )}}{3} + \frac {2 b d e x^{5} \operatorname {asec}{\left (c x \right )}}{5} + \frac {b e^{2} x^{7} \operatorname {asec}{\left (c x \right )}}{7} - \frac {b d^{2} \left (\begin {cases} \frac {x \sqrt {c^{2} x^{2} - 1}}{2 c} + \frac {\operatorname {acosh}{\left (c x \right )}}{2 c^{2}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {i c x^{3}}{2 \sqrt {- c^{2} x^{2} + 1}} + \frac {i x}{2 c \sqrt {- c^{2} x^{2} + 1}} - \frac {i \operatorname {asin}{\left (c x \right )}}{2 c^{2}} & \text {otherwise} \end {cases}\right )}{3 c} - \frac {2 b d e \left (\begin {cases} \frac {c x^{5}}{4 \sqrt {c^{2} x^{2} - 1}} + \frac {x^{3}}{8 c \sqrt {c^{2} x^{2} - 1}} - \frac {3 x}{8 c^{3} \sqrt {c^{2} x^{2} - 1}} + \frac {3 \operatorname {acosh}{\left (c x \right )}}{8 c^{4}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {i c x^{5}}{4 \sqrt {- c^{2} x^{2} + 1}} - \frac {i x^{3}}{8 c \sqrt {- c^{2} x^{2} + 1}} + \frac {3 i x}{8 c^{3} \sqrt {- c^{2} x^{2} + 1}} - \frac {3 i \operatorname {asin}{\left (c x \right )}}{8 c^{4}} & \text {otherwise} \end {cases}\right )}{5 c} - \frac {b e^{2} \left (\begin {cases} \frac {c x^{7}}{6 \sqrt {c^{2} x^{2} - 1}} + \frac {x^{5}}{24 c \sqrt {c^{2} x^{2} - 1}} + \frac {5 x^{3}}{48 c^{3} \sqrt {c^{2} x^{2} - 1}} - \frac {5 x}{16 c^{5} \sqrt {c^{2} x^{2} - 1}} + \frac {5 \operatorname {acosh}{\left (c x \right )}}{16 c^{6}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {i c x^{7}}{6 \sqrt {- c^{2} x^{2} + 1}} - \frac {i x^{5}}{24 c \sqrt {- c^{2} x^{2} + 1}} - \frac {5 i x^{3}}{48 c^{3} \sqrt {- c^{2} x^{2} + 1}} + \frac {5 i x}{16 c^{5} \sqrt {- c^{2} x^{2} + 1}} - \frac {5 i \operatorname {asin}{\left (c x \right )}}{16 c^{6}} & \text {otherwise} \end {cases}\right )}{7 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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