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

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

[Out]

-(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])/(1680*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)

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Rubi [A]  time = 0.234584, antiderivative size = 275, normalized size of antiderivative = 1., 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, 6301, 12, 1267, 459, 321, 216} \[ \frac{1}{3} d^2 x^3 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \text{sech}^{-1}(c x)\right )-\frac{b x \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (280 c^4 d^2+252 c^2 d e+75 e^2\right )}{1680 c^6}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) \sin ^{-1}(c x)}{1680 c^7}-\frac{b e x^3 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (84 c^2 d+25 e\right )}{840 c^4}-\frac{b e^2 x^5 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{42 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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])/(1680*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 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 6301

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]))

Rule 12

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

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 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 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 216

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]

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*}

Mathematica [C]  time = 0.45736, size = 207, normalized size = 0.75 \[ \frac{16 a c^7 x^3 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )-b c x \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (8 c^4 \left (35 d^2+21 d e x^2+5 e^2 x^4\right )+2 c^2 e \left (126 d+25 e x^2\right )+75 e^2\right )+16 b c^7 x^3 \text{sech}^{-1}(c x) \left (35 d^2+42 d e x^2+15 e^2 x^4\right )+i b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) \log \left (2 \sqrt{\frac{1-c x}{c x+1}} (c x+1)-2 i c x\right )}{1680 c^7} \]

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 = 0.212, size = 300, normalized size = 1.1 \begin{align*}{\frac{1}{{c}^{3}} \left ({\frac{a}{{c}^{4}} \left ({\frac{{e}^{2}{c}^{7}{x}^{7}}{7}}+{\frac{2\,{c}^{7}de{x}^{5}}{5}}+{\frac{{x}^{3}{c}^{7}{d}^{2}}{3}} \right ) }+{\frac{b}{{c}^{4}} \left ({\frac{{\rm arcsech} \left (cx\right ){e}^{2}{c}^{7}{x}^{7}}{7}}+{\frac{2\,{\rm arcsech} \left (cx\right ){c}^{7}de{x}^{5}}{5}}+{\frac{{\rm arcsech} \left (cx\right ){c}^{7}{x}^{3}{d}^{2}}{3}}+{\frac{cx}{1680}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}} \left ( -40\,{c}^{5}{x}^{5}{e}^{2}\sqrt{-{c}^{2}{x}^{2}+1}-168\,{c}^{5}{x}^{3}de\sqrt{-{c}^{2}{x}^{2}+1}-280\,{d}^{2}{c}^{5}x\sqrt{-{c}^{2}{x}^{2}+1}+280\,{d}^{2}{c}^{4}\arcsin \left ( cx \right ) -50\,{e}^{2}{c}^{3}{x}^{3}\sqrt{-{c}^{2}{x}^{2}+1}-252\,{c}^{3}dex\sqrt{-{c}^{2}{x}^{2}+1}+252\,{c}^{2}de\arcsin \left ( cx \right ) -75\,{e}^{2}cx\sqrt{-{c}^{2}{x}^{2}+1}+75\,{e}^{2}\arcsin \left ( cx \right ) \right ){\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}}} \right ) } \right ) } \end{align*}

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)

[Out]

1/c^3*(a/c^4*(1/7*e^2*c^7*x^7+2/5*c^7*d*e*x^5+1/3*x^3*c^7*d^2)+b/c^4*(1/7*arcsech(c*x)*e^2*c^7*x^7+2/5*arcsech
(c*x)*c^7*d*e*x^5+1/3*arcsech(c*x)*c^7*x^3*d^2+1/1680*(-(c*x-1)/c/x)^(1/2)*c*x*((c*x+1)/c/x)^(1/2)*(-40*c^5*x^
5*e^2*(-c^2*x^2+1)^(1/2)-168*c^5*x^3*d*e*(-c^2*x^2+1)^(1/2)-280*d^2*c^5*x*(-c^2*x^2+1)^(1/2)+280*d^2*c^4*arcsi
n(c*x)-50*e^2*c^3*x^3*(-c^2*x^2+1)^(1/2)-252*c^3*d*e*x*(-c^2*x^2+1)^(1/2)+252*c^2*d*e*arcsin(c*x)-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 = 1.51487, size = 443, normalized size = 1.61 \begin{align*} \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}{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{align*}

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*e^2*x^7 + 2/5*a*d*e*x^5 + 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 [A]  time = 3.60211, size = 775, normalized size = 2.82 \begin{align*} \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} - 2 \,{\left (280 \, b c^{4} d^{2} + 252 \, b c^{2} d e + 75 \, b e^{2}\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 e + 15 \, b c^{7} e^{2}\right )} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) + 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 )} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) -{\left (40 \, b c^{6} e^{2} x^{6} + 2 \,{\left (84 \, b c^{6} d e + 25 \, b c^{4} e^{2}\right )} x^{4} +{\left (280 \, b c^{6} d^{2} + 252 \, b c^{4} d e + 75 \, b c^{2} e^{2}\right )} x^{2}\right )} \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}}}{1680 \, c^{7}} \end{align*}

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*e^2*x^7 + 672*a*c^7*d*e*x^5 + 560*a*c^7*d^2*x^3 - 2*(280*b*c^4*d^2 + 252*b*c^2*d*e + 75*b*e^
2)*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*e + 15*b*c^7*e^2)*lo
g((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 1)/x) + 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 - 3
5*b*c^7*d^2 - 42*b*c^7*d*e - 15*b*c^7*e^2)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) - (40*b*c^6*e^2
*x^6 + 2*(84*b*c^6*d*e + 25*b*c^4*e^2)*x^4 + (280*b*c^6*d^2 + 252*b*c^4*d*e + 75*b*c^2*e^2)*x^2)*sqrt(-(c^2*x^
2 - 1)/(c^2*x^2)))/c^7

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

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., size = 0, normalized size = 0. \begin{align*} \int{\left (e x^{2} + d\right )}^{2}{\left (b \operatorname{arsech}\left (c x\right ) + a\right )} x^{2}\,{d x} \end{align*}

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)