3.1.100 \(\int x^2 (d+e x^2)^2 (a+b \text {sech}^{-1}(c x)) \, dx\) [100]

Optimal. Leaf size=275 \[ -\frac {b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) x \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{1680 c^6}-\frac {b e \left (84 c^2 d+25 e\right ) x^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{840 c^4}-\frac {b e^2 x^5 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{42 c^2}+\frac {1}{3} d^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {ArcSin}(c x)}{1680 c^7} \]

[Out]

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

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Rubi [A]
time = 0.16, antiderivative size = 275, 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 = {276, 6436, 12, 1281, 470, 327, 222} \begin {gather*} \frac {1}{3} d^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \text {ArcSin}(c x) \left (280 c^4 d^2+252 c^2 d e+75 e^2\right )}{1680 c^7}-\frac {b e^2 x^5 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{42 c^2}-\frac {b e x^3 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (84 c^2 d+25 e\right )}{840 c^4}-\frac {b x \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (280 c^4 d^2+252 c^2 d e+75 e^2\right )}{1680 c^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

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

Rule 222

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

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 6436

Int[((a_.) + ArcSech[(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*ArcSech[c*x], u, x] + Dist[b*Sqrt[1 + c*x]*Sqrt[1/(1 + c*x)],
Int[SimplifyIntegrand[u/(x*Sqrt[1 - c*x]*Sqrt[1 + c*x]), 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])) || (ILtQ[(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 \text {sech}^{-1}(c x)\right ) \, dx &=\frac {1}{3} d^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \text {sech}^{-1}(c x)\right )+\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \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\\ &=\frac {1}{3} d^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{105} \left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \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\\ &=-\frac {b e^2 x^5 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{42 c^2}+\frac {1}{3} d^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \text {sech}^{-1}(c x)\right )-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \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^2}\\ &=-\frac {b e \left (84 c^2 d+25 e\right ) x^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{840 c^4}-\frac {b e^2 x^5 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{42 c^2}+\frac {1}{3} d^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \text {sech}^{-1}(c x)\right )+-\frac {\left (b \left (-840 c^4 d^2-9 e \left (84 c^2 d+25 e\right )\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{2520 c^4}\\ &=-\frac {b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) x \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{1680 c^6}-\frac {b e \left (84 c^2 d+25 e\right ) x^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{840 c^4}-\frac {b e^2 x^5 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{42 c^2}+\frac {1}{3} d^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \text {sech}^{-1}(c x)\right )+-\frac {\left (b \left (-840 c^4 d^2-9 e \left (84 c^2 d+25 e\right )\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{5040 c^6}\\ &=-\frac {b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) x \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{1680 c^6}-\frac {b e \left (84 c^2 d+25 e\right ) x^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{840 c^4}-\frac {b e^2 x^5 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{42 c^2}+\frac {1}{3} d^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sin ^{-1}(c x)}{1680 c^7}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.28, size = 207, normalized size = 0.75 \begin {gather*} \frac {16 a c^7 x^3 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )-b c x \sqrt {\frac {1-c x}{1+c x}} (1+c x) \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 )+16 b c^7 x^3 \left (35 d^2+42 d e x^2+15 e^2 x^4\right ) \text {sech}^{-1}(c x)+i b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) \log \left (-2 i c x+2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)\right )}{1680 c^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(16*a*c^7*x^3*(35*d^2 + 42*d*e*x^2 + 15*e^2*x^4) - b*c*x*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(75*e^2 + 2*c^2*e
*(126*d + 25*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)*ArcSech[c*x] + I*b*(280*c^4*d^2 + 252*c^2*d*e + 75*e^2)*Log[(-2*I)*c*x + 2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c
*x)])/(1680*c^7)

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Maple [A]
time = 0.38, size = 300, normalized size = 1.09

method	result	size
derivativedivides	\(\frac {\frac {a \left (\frac {1}{3} d^{2} c^{7} x^{3}+\frac {2}{5} d \,c^{7} e \,x^{5}+\frac {1}{7} e^{2} c^{7} x^{7}\right )}{c^{4}}+\frac {b \left (\frac {\mathrm {arcsech}\left (c x \right ) d^{2} c^{7} x^{3}}{3}+\frac {2 \,\mathrm {arcsech}\left (c x \right ) d \,c^{7} e \,x^{5}}{5}+\frac {\mathrm {arcsech}\left (c x \right ) e^{2} c^{7} x^{7}}{7}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (280 d^{2} c^{5} x \sqrt {-c^{2} x^{2}+1}+168 c^{5} d e \sqrt {-c^{2} x^{2}+1}\, x^{3}+40 e^{2} \sqrt {-c^{2} x^{2}+1}\, c^{5} x^{5}-280 d^{2} c^{4} \arcsin \left (c x \right )+252 \sqrt {-c^{2} x^{2}+1}\, c^{3} d e x +50 e^{2} c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-252 \arcsin \left (c x \right ) c^{2} d e +75 e^{2} c x \sqrt {-c^{2} x^{2}+1}-75 e^{2} \arcsin \left (c x \right )\right )}{1680 \sqrt {-c^{2} x^{2}+1}}\right )}{c^{4}}}{c^{3}}\)	\(300\)
default	\(\frac {\frac {a \left (\frac {1}{3} d^{2} c^{7} x^{3}+\frac {2}{5} d \,c^{7} e \,x^{5}+\frac {1}{7} e^{2} c^{7} x^{7}\right )}{c^{4}}+\frac {b \left (\frac {\mathrm {arcsech}\left (c x \right ) d^{2} c^{7} x^{3}}{3}+\frac {2 \,\mathrm {arcsech}\left (c x \right ) d \,c^{7} e \,x^{5}}{5}+\frac {\mathrm {arcsech}\left (c x \right ) e^{2} c^{7} x^{7}}{7}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (280 d^{2} c^{5} x \sqrt {-c^{2} x^{2}+1}+168 c^{5} d e \sqrt {-c^{2} x^{2}+1}\, x^{3}+40 e^{2} \sqrt {-c^{2} x^{2}+1}\, c^{5} x^{5}-280 d^{2} c^{4} \arcsin \left (c x \right )+252 \sqrt {-c^{2} x^{2}+1}\, c^{3} d e x +50 e^{2} c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-252 \arcsin \left (c x \right ) c^{2} d e +75 e^{2} c x \sqrt {-c^{2} x^{2}+1}-75 e^{2} \arcsin \left (c x \right )\right )}{1680 \sqrt {-c^{2} x^{2}+1}}\right )}{c^{4}}}{c^{3}}\)	\(300\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^3*(a/c^4*(1/3*d^2*c^7*x^3+2/5*d*c^7*e*x^5+1/7*e^2*c^7*x^7)+b/c^4*(1/3*arcsech(c*x)*d^2*c^7*x^3+2/5*arcsech
(c*x)*d*c^7*e*x^5+1/7*arcsech(c*x)*e^2*c^7*x^7-1/1680*(-(c*x-1)/c/x)^(1/2)*c*x*((c*x+1)/c/x)^(1/2)*(280*d^2*c^
5*x*(-c^2*x^2+1)^(1/2)+168*c^5*d*e*(-c^2*x^2+1)^(1/2)*x^3+40*e^2*(-c^2*x^2+1)^(1/2)*c^5*x^5-280*d^2*c^4*arcsin
(c*x)+252*(-c^2*x^2+1)^(1/2)*c^3*d*e*x+50*e^2*c^3*x^3*(-c^2*x^2+1)^(1/2)-252*arcsin(c*x)*c^2*d*e+75*e^2*c*x*(-
c^2*x^2+1)^(1/2)-75*e^2*arcsin(c*x))/(-c^2*x^2+1)^(1/2)))

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Maxima [A]
time = 0.47, size = 328, normalized size = 1.19 \begin {gather*} \frac {1}{7} \, a x^{7} e^{2} + \frac {2}{5} \, a d x^{5} e + \frac {1}{3} \, a d^{2} x^{3} + \frac {1}{6} \, {\left (2 \, x^{3} \operatorname {arsech}\left (c x\right ) - \frac {\frac {\sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{2}} + \frac {\arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )}{c^{2}}}{c}\right )} b d^{2} + \frac {1}{20} \, {\left (8 \, x^{5} \operatorname {arsech}\left (c x\right ) - \frac {\frac {3 \, {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 5 \, \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{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 \, \arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )}{c^{4}}}{c}\right )} b d e + \frac {1}{336} \, {\left (48 \, x^{7} \operatorname {arsech}\left (c x\right ) - \frac {\frac {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}}{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 \, \arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )}{c^{6}}}{c}\right )} b e^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*a*x^7*e^2 + 2/5*a*d*x^5*e + 1/3*a*d^2*x^3 + 1/6*(2*x^3*arcsech(c*x) - (sqrt(1/(c^2*x^2) - 1)/(c^2*(1/(c^2*
x^2) - 1) + c^2) + arctan(sqrt(1/(c^2*x^2) - 1))/c^2)/c)*b*d^2 + 1/20*(8*x^5*arcsech(c*x) - ((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*arctan(sqrt
(1/(c^2*x^2) - 1))/c^4)/c)*b*d*e + 1/336*(48*x^7*arcsech(c*x) - ((15*(1/(c^2*x^2) - 1)^(5/2) + 40*(1/(c^2*x^2)
- 1)^(3/2) + 33*sqrt(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*arctan(sqrt(1/(c^2*x^2) - 1))/c^6)/c)*b*e^2

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 600 vs. \(2 (176) = 352\).
time = 0.57, size = 600, normalized size = 2.18 \begin {gather*} \frac {240 \, a c^{7} x^{7} \cosh \left (1\right )^{2} + 240 \, a c^{7} x^{7} \sinh \left (1\right )^{2} + 672 \, a c^{7} d x^{5} \cosh \left (1\right ) + 560 \, a c^{7} d^{2} x^{3} - 2 \, {\left (280 \, b c^{4} d^{2} + 252 \, b c^{2} d \cosh \left (1\right ) + 75 \, b \cosh \left (1\right )^{2} + 75 \, b \sinh \left (1\right )^{2} + 6 \, {\left (42 \, b c^{2} d + 25 \, b \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \arctan \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c x}\right ) - 16 \, {\left (35 \, b c^{7} d^{2} + 42 \, b c^{7} d \cosh \left (1\right ) + 15 \, b c^{7} \cosh \left (1\right )^{2} + 15 \, b c^{7} \sinh \left (1\right )^{2} + 6 \, {\left (7 \, b c^{7} d + 5 \, b c^{7} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) + 16 \, {\left (35 \, b c^{7} d^{2} x^{3} - 35 \, b c^{7} d^{2} + 15 \, {\left (b c^{7} x^{7} - b c^{7}\right )} \cosh \left (1\right )^{2} + 15 \, {\left (b c^{7} x^{7} - b c^{7}\right )} \sinh \left (1\right )^{2} + 42 \, {\left (b c^{7} d x^{5} - b c^{7} d\right )} \cosh \left (1\right ) + 6 \, {\left (7 \, b c^{7} d x^{5} - 7 \, b c^{7} d + 5 \, {\left (b c^{7} x^{7} - b c^{7}\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) + 96 \, {\left (5 \, a c^{7} x^{7} \cosh \left (1\right ) + 7 \, a c^{7} d x^{5}\right )} \sinh \left (1\right ) - {\left (280 \, b c^{6} d^{2} x^{2} + 5 \, {\left (8 \, b c^{6} x^{6} + 10 \, b c^{4} x^{4} + 15 \, b c^{2} x^{2}\right )} \cosh \left (1\right )^{2} + 5 \, {\left (8 \, b c^{6} x^{6} + 10 \, b c^{4} x^{4} + 15 \, b c^{2} x^{2}\right )} \sinh \left (1\right )^{2} + 84 \, {\left (2 \, b c^{6} d x^{4} + 3 \, b c^{4} d x^{2}\right )} \cosh \left (1\right ) + 2 \, {\left (84 \, b c^{6} d x^{4} + 126 \, b c^{4} d x^{2} + 5 \, {\left (8 \, b c^{6} x^{6} + 10 \, b c^{4} x^{4} + 15 \, b c^{2} x^{2}\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{1680 \, c^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1680*(240*a*c^7*x^7*cosh(1)^2 + 240*a*c^7*x^7*sinh(1)^2 + 672*a*c^7*d*x^5*cosh(1) + 560*a*c^7*d^2*x^3 - 2*(2
80*b*c^4*d^2 + 252*b*c^2*d*cosh(1) + 75*b*cosh(1)^2 + 75*b*sinh(1)^2 + 6*(42*b*c^2*d + 25*b*cosh(1))*sinh(1))*
arctan((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 1)/(c*x)) - 16*(35*b*c^7*d^2 + 42*b*c^7*d*cosh(1) + 15*b*c^7*cosh
(1)^2 + 15*b*c^7*sinh(1)^2 + 6*(7*b*c^7*d + 5*b*c^7*cosh(1))*sinh(1))*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2))
- 1)/x) + 16*(35*b*c^7*d^2*x^3 - 35*b*c^7*d^2 + 15*(b*c^7*x^7 - b*c^7)*cosh(1)^2 + 15*(b*c^7*x^7 - b*c^7)*sinh
(1)^2 + 42*(b*c^7*d*x^5 - b*c^7*d)*cosh(1) + 6*(7*b*c^7*d*x^5 - 7*b*c^7*d + 5*(b*c^7*x^7 - b*c^7)*cosh(1))*sin
h(1))*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) + 96*(5*a*c^7*x^7*cosh(1) + 7*a*c^7*d*x^5)*sinh(1) -
(280*b*c^6*d^2*x^2 + 5*(8*b*c^6*x^6 + 10*b*c^4*x^4 + 15*b*c^2*x^2)*cosh(1)^2 + 5*(8*b*c^6*x^6 + 10*b*c^4*x^4
+ 15*b*c^2*x^2)*sinh(1)^2 + 84*(2*b*c^6*d*x^4 + 3*b*c^4*d*x^2)*cosh(1) + 2*(84*b*c^6*d*x^4 + 126*b*c^4*d*x^2 +
5*(8*b*c^6*x^6 + 10*b*c^4*x^4 + 15*b*c^2*x^2)*cosh(1))*sinh(1))*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)))/c^7

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*(a + b*asech(c*x))*(d + e*x**2)**2, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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