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

Optimal. Leaf size=204 \[ d^2 x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \text {sech}^{-1}(c x)\right )-\frac {b e^2 x^3 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{20 c^2}-\frac {b e x \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (40 c^2 d+9 e\right )}{120 c^4}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (120 c^4 d^2+40 c^2 d e+9 e^2\right ) \sin ^{-1}(c x)}{120 c^5} \]

[Out]

d^2*x*(a+b*arcsech(c*x))+2/3*d*e*x^3*(a+b*arcsech(c*x))+1/5*e^2*x^5*(a+b*arcsech(c*x))+1/120*b*(120*c^4*d^2+40
*c^2*d*e+9*e^2)*arcsin(c*x)*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)/c^5-1/120*b*e*(40*c^2*d+9*e)*x*(1/(c*x+1))^(1/2)*(
c*x+1)^(1/2)*(-c^2*x^2+1)^(1/2)/c^4-1/20*b*e^2*x^3*(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 = 0.13, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {194, 6291, 12, 1159, 388, 216} \[ d^2 x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (120 c^4 d^2+40 c^2 d e+9 e^2\right ) \sin ^{-1}(c x)}{120 c^5}-\frac {b e x \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (40 c^2 d+9 e\right )}{120 c^4}-\frac {b e^2 x^3 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{20 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*e*(40*c^2*d + 9*e)*x*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(120*c^4) - (b*e^2*x^3*Sqrt[(1
+ c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(20*c^2) + d^2*x*(a + b*ArcSech[c*x]) + (2*d*e*x^3*(a + b*ArcSec
h[c*x]))/3 + (e^2*x^5*(a + b*ArcSech[c*x]))/5 + (b*(120*c^4*d^2 + 40*c^2*d*e + 9*e^2)*Sqrt[(1 + c*x)^(-1)]*Sqr
t[1 + c*x]*ArcSin[c*x])/(120*c^5)

Rule 12

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

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

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]

Rule 388

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

Rule 1159

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

Rule 6291

Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(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}, x] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])

Rubi steps

\begin {align*} \int \left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx &=d^2 x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {15 d^2+10 d e x^2+3 e^2 x^4}{15 \sqrt {1-c^2 x^2}} \, dx\\ &=d^2 x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{15} \left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {15 d^2+10 d e x^2+3 e^2 x^4}{\sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {b e^2 x^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{20 c^2}+d^2 x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \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 {-60 c^2 d^2-e \left (40 c^2 d+9 e\right ) x^2}{\sqrt {1-c^2 x^2}} \, dx}{60 c^2}\\ &=-\frac {b e \left (40 c^2 d+9 e\right ) x \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{120 c^4}-\frac {b e^2 x^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{20 c^2}+d^2 x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \text {sech}^{-1}(c x)\right )-\frac {\left (b \left (-120 c^4 d^2-e \left (40 c^2 d+9 e\right )\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{120 c^4}\\ &=-\frac {b e \left (40 c^2 d+9 e\right ) x \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{120 c^4}-\frac {b e^2 x^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{20 c^2}+d^2 x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b \left (120 c^4 d^2+40 c^2 d e+9 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sin ^{-1}(c x)}{120 c^5}\\ \end {align*}

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Mathematica [C]  time = 0.35, size = 174, normalized size = 0.85 \[ \frac {8 a c^5 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )+8 b c^5 x \text {sech}^{-1}(c x) \left (15 d^2+10 d e x^2+3 e^2 x^4\right )-b c e x \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (c^2 \left (40 d+6 e x^2\right )+9 e\right )+i b \left (120 c^4 d^2+40 c^2 d e+9 e^2\right ) \log \left (2 \sqrt {\frac {1-c x}{c x+1}} (c x+1)-2 i c x\right )}{120 c^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8*a*c^5*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4) - b*c*e*x*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(9*e + c^2*(40*d +
6*e*x^2)) + 8*b*c^5*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4)*ArcSech[c*x] + I*b*(120*c^4*d^2 + 40*c^2*d*e + 9*e^2)*
Log[(-2*I)*c*x + 2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)])/(120*c^5)

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fricas [B]  time = 0.81, size = 305, normalized size = 1.50 \[ \frac {24 \, a c^{5} e^{2} x^{5} + 80 \, a c^{5} d e x^{3} + 120 \, a c^{5} d^{2} x - 2 \, {\left (120 \, b c^{4} d^{2} + 40 \, b c^{2} d e + 9 \, b e^{2}\right )} \arctan \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c x}\right ) - 8 \, {\left (15 \, b c^{5} d^{2} + 10 \, b c^{5} d e + 3 \, b c^{5} e^{2}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) + 8 \, {\left (3 \, b c^{5} e^{2} x^{5} + 10 \, b c^{5} d e x^{3} + 15 \, b c^{5} d^{2} x - 15 \, b c^{5} d^{2} - 10 \, b c^{5} d e - 3 \, b c^{5} e^{2}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (6 \, b c^{4} e^{2} x^{4} + {\left (40 \, b c^{4} d e + 9 \, b c^{2} e^{2}\right )} x^{2}\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{120 \, c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(24*a*c^5*e^2*x^5 + 80*a*c^5*d*e*x^3 + 120*a*c^5*d^2*x - 2*(120*b*c^4*d^2 + 40*b*c^2*d*e + 9*b*e^2)*arct
an((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 1)/(c*x)) - 8*(15*b*c^5*d^2 + 10*b*c^5*d*e + 3*b*c^5*e^2)*log((c*x*sq
rt(-(c^2*x^2 - 1)/(c^2*x^2)) - 1)/x) + 8*(3*b*c^5*e^2*x^5 + 10*b*c^5*d*e*x^3 + 15*b*c^5*d^2*x - 15*b*c^5*d^2 -
10*b*c^5*d*e - 3*b*c^5*e^2)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) - (6*b*c^4*e^2*x^4 + (40*b*c^
4*d*e + 9*b*c^2*e^2)*x^2)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)))/c^5

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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)

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maple [A]  time = 0.07, size = 228, normalized size = 1.12 \[ \frac {\frac {a \left (\frac {1}{5} e^{2} c^{5} x^{5}+\frac {2}{3} c^{5} d e \,x^{3}+x \,c^{5} d^{2}\right )}{c^{4}}+\frac {b \left (\frac {\mathrm {arcsech}\left (c x \right ) e^{2} c^{5} x^{5}}{5}+\frac {2 \,\mathrm {arcsech}\left (c x \right ) c^{5} d e \,x^{3}}{3}+\mathrm {arcsech}\left (c x \right ) c^{5} x \,d^{2}+\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (120 d^{2} c^{4} \arcsin \left (c x \right )-6 e^{2} c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-40 c^{3} d e x \sqrt {-c^{2} x^{2}+1}+40 c^{2} d e \arcsin \left (c x \right )-9 e^{2} c x \sqrt {-c^{2} x^{2}+1}+9 e^{2} \arcsin \left (c x \right )\right )}{120 \sqrt {-c^{2} x^{2}+1}}\right )}{c^{4}}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(a/c^4*(1/5*e^2*c^5*x^5+2/3*c^5*d*e*x^3+x*c^5*d^2)+b/c^4*(1/5*arcsech(c*x)*e^2*c^5*x^5+2/3*arcsech(c*x)*c^
5*d*e*x^3+arcsech(c*x)*c^5*x*d^2+1/120*(-(c*x-1)/c/x)^(1/2)*c*x*((c*x+1)/c/x)^(1/2)*(120*d^2*c^4*arcsin(c*x)-6
*e^2*c^3*x^3*(-c^2*x^2+1)^(1/2)-40*c^3*d*e*x*(-c^2*x^2+1)^(1/2)+40*c^2*d*e*arcsin(c*x)-9*e^2*c*x*(-c^2*x^2+1)^
(1/2)+9*e^2*arcsin(c*x))/(-c^2*x^2+1)^(1/2)))

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maxima [A]  time = 0.43, size = 224, normalized size = 1.10 \[ \frac {1}{5} \, a e^{2} x^{5} + \frac {2}{3} \, a d e x^{3} + \frac {1}{3} \, {\left (2 \, x^{3} \operatorname {arsech}\left (c x\right ) - \frac {\frac {\sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{2}} + \frac {\arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )}{c^{2}}}{c}\right )} b d e + \frac {1}{40} \, {\left (8 \, x^{5} \operatorname {arsech}\left (c x\right ) - \frac {\frac {3 \, {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 5 \, \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 2 \, c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{4}} + \frac {3 \, \arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )}{c^{4}}}{c}\right )} b e^{2} + a d^{2} x + \frac {{\left (c x \operatorname {arsech}\left (c x\right ) - \arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )\right )} b d^{2}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*a*e^2*x^5 + 2/3*a*d*e*x^3 + 1/3*(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*e + 1/40*(8*x^5*arcsech(c*x) - ((3*(1/(c^2*x^2) - 1)^(3/2) + 5*sq
rt(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*e^2 + a*d^2*x + (c*x*arcsech(c*x) - arctan(sqrt(1/(c^2*x^2) - 1)))*b*d^2/c

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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