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

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

[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)

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Rubi [A]  time = 0.126856, antiderivative size = 204, normalized size of antiderivative = 1., 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 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 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])

Rule 12

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

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 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 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 \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*}

Mathematica [C]  time = 0.27851, size = 174, normalized size = 0.85 \[ \frac{8 a c^5 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )+8 b c^5 x \text{sech}^{-1}(c x) \left (15 d^2+10 d e x^2+3 e^2 x^4\right )+i b \left (120 c^4 d^2+40 c^2 d e+9 e^2\right ) \log \left (2 \sqrt{\frac{1-c x}{c x+1}} (c x+1)-2 i c x\right )-b c e x \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (c^2 \left (40 d+6 e x^2\right )+9 e\right )}{120 c^5} \]

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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Maple [A]  time = 0.178, size = 228, normalized size = 1.1 \begin{align*}{\frac{1}{c} \left ({\frac{a}{{c}^{4}} \left ({\frac{{e}^{2}{c}^{5}{x}^{5}}{5}}+{\frac{2\,{c}^{5}de{x}^{3}}{3}}+x{c}^{5}{d}^{2} \right ) }+{\frac{b}{{c}^{4}} \left ({\frac{{\rm arcsech} \left (cx\right ){e}^{2}{c}^{5}{x}^{5}}{5}}+{\frac{2\,{\rm arcsech} \left (cx\right ){c}^{5}{x}^{3}de}{3}}+{\rm arcsech} \left (cx\right ){c}^{5}x{d}^{2}+{\frac{cx}{120}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}} \left ( 120\,{d}^{2}{c}^{4}\arcsin \left ( cx \right ) -6\,{e}^{2}{c}^{3}{x}^{3}\sqrt{-{c}^{2}{x}^{2}+1}-40\,{c}^{3}dex\sqrt{-{c}^{2}{x}^{2}+1}+40\,{c}^{2}de\arcsin \left ( cx \right ) -9\,{e}^{2}cx\sqrt{-{c}^{2}{x}^{2}+1}+9\,{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((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*x^3*d*e+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 = 1.49882, size = 302, normalized size = 1.48 \begin{align*} \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} \end{align*}

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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Fricas [B]  time = 3.00414, size = 679, normalized size = 3.33 \begin{align*} \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}} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \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((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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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 )}\,{d x} \end{align*}

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)