3.96 \(\int x^3 (d+e x^2) (a+b \text{sech}^{-1}(c x)) \, dx\)

Optimal. Leaf size=180 \[ \frac{1}{4} d x^4 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{6} e x^6 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (1-c^2 x^2\right )^{3/2} \left (3 c^2 d+4 e\right )}{36 c^6}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (3 c^2 d+2 e\right )}{12 c^6}-\frac{b e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (1-c^2 x^2\right )^{5/2}}{30 c^6} \]

[Out]

-(b*(3*c^2*d + 2*e)*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(12*c^6) + (b*(3*c^2*d + 4*e)*Sqrt[(
1 + c*x)^(-1)]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(3/2))/(36*c^6) - (b*e*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*(1 - c^2*
x^2)^(5/2))/(30*c^6) + (d*x^4*(a + b*ArcSech[c*x]))/4 + (e*x^6*(a + b*ArcSech[c*x]))/6

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Rubi [A]  time = 0.134929, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {14, 6301, 12, 446, 77} \[ \frac{1}{4} d x^4 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{6} e x^6 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (1-c^2 x^2\right )^{3/2} \left (3 c^2 d+4 e\right )}{36 c^6}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (3 c^2 d+2 e\right )}{12 c^6}-\frac{b e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (1-c^2 x^2\right )^{5/2}}{30 c^6} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(d + e*x^2)*(a + b*ArcSech[c*x]),x]

[Out]

-(b*(3*c^2*d + 2*e)*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(12*c^6) + (b*(3*c^2*d + 4*e)*Sqrt[(
1 + c*x)^(-1)]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(3/2))/(36*c^6) - (b*e*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*(1 - c^2*
x^2)^(5/2))/(30*c^6) + (d*x^4*(a + b*ArcSech[c*x]))/4 + (e*x^6*(a + b*ArcSech[c*x]))/6

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 446

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int x^3 \left (d+e x^2\right ) \left (a+b \text{sech}^{-1}(c x)\right ) \, dx &=\frac{1}{4} d x^4 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{6} e x^6 \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^3 \left (3 d+2 e x^2\right )}{12 \sqrt{1-c^2 x^2}} \, dx\\ &=\frac{1}{4} d x^4 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{6} e x^6 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{12} \left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{x^3 \left (3 d+2 e x^2\right )}{\sqrt{1-c^2 x^2}} \, dx\\ &=\frac{1}{4} d x^4 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{6} e x^6 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{24} \left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{x (3 d+2 e x)}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )\\ &=\frac{1}{4} d x^4 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{6} e x^6 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{24} \left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \left (\frac{3 c^2 d+2 e}{c^4 \sqrt{1-c^2 x}}+\frac{\left (-3 c^2 d-4 e\right ) \sqrt{1-c^2 x}}{c^4}+\frac{2 e \left (1-c^2 x\right )^{3/2}}{c^4}\right ) \, dx,x,x^2\right )\\ &=-\frac{b \left (3 c^2 d+2 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{12 c^6}+\frac{b \left (3 c^2 d+4 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \left (1-c^2 x^2\right )^{3/2}}{36 c^6}-\frac{b e \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \left (1-c^2 x^2\right )^{5/2}}{30 c^6}+\frac{1}{4} d x^4 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{6} e x^6 \left (a+b \text{sech}^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.178137, size = 106, normalized size = 0.59 \[ \frac{1}{180} \left (15 a x^4 \left (3 d+2 e x^2\right )-\frac{b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (3 c^4 \left (5 d x^2+2 e x^4\right )+c^2 \left (30 d+8 e x^2\right )+16 e\right )}{c^6}+15 b x^4 \text{sech}^{-1}(c x) \left (3 d+2 e x^2\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(d + e*x^2)*(a + b*ArcSech[c*x]),x]

[Out]

(15*a*x^4*(3*d + 2*e*x^2) - (b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(16*e + c^2*(30*d + 8*e*x^2) + 3*c^4*(5*d*x
^2 + 2*e*x^4)))/c^6 + 15*b*x^4*(3*d + 2*e*x^2)*ArcSech[c*x])/180

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Maple [A]  time = 0.183, size = 132, normalized size = 0.7 \begin{align*}{\frac{1}{{c}^{4}} \left ({\frac{a}{{c}^{2}} \left ({\frac{{c}^{6}{x}^{6}e}{6}}+{\frac{{x}^{4}{c}^{6}d}{4}} \right ) }+{\frac{b}{{c}^{2}} \left ({\frac{{\rm arcsech} \left (cx\right ){c}^{6}{x}^{6}e}{6}}+{\frac{{\rm arcsech} \left (cx\right ){c}^{6}{x}^{4}d}{4}}-{\frac{cx \left ( 6\,{c}^{4}e{x}^{4}+15\,{c}^{4}d{x}^{2}+8\,{c}^{2}{x}^{2}e+30\,{c}^{2}d+16\,e \right ) }{180}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(e*x^2+d)*(a+b*arcsech(c*x)),x)

[Out]

1/c^4*(a/c^2*(1/6*c^6*x^6*e+1/4*x^4*c^6*d)+b/c^2*(1/6*arcsech(c*x)*c^6*x^6*e+1/4*arcsech(c*x)*c^6*x^4*d-1/180*
(-(c*x-1)/c/x)^(1/2)*c*x*((c*x+1)/c/x)^(1/2)*(6*c^4*e*x^4+15*c^4*d*x^2+8*c^2*e*x^2+30*c^2*d+16*e)))

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Maxima [A]  time = 1.00878, size = 186, normalized size = 1.03 \begin{align*} \frac{1}{6} \, a e x^{6} + \frac{1}{4} \, a d x^{4} + \frac{1}{12} \,{\left (3 \, x^{4} \operatorname{arsech}\left (c x\right ) + \frac{c^{2} x^{3}{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{\frac{3}{2}} - 3 \, x \sqrt{\frac{1}{c^{2} x^{2}} - 1}}{c^{3}}\right )} b d + \frac{1}{90} \,{\left (15 \, x^{6} \operatorname{arsech}\left (c x\right ) - \frac{3 \, c^{4} x^{5}{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{\frac{5}{2}} - 10 \, c^{2} x^{3}{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{\frac{3}{2}} + 15 \, x \sqrt{\frac{1}{c^{2} x^{2}} - 1}}{c^{5}}\right )} b e \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x^2+d)*(a+b*arcsech(c*x)),x, algorithm="maxima")

[Out]

1/6*a*e*x^6 + 1/4*a*d*x^4 + 1/12*(3*x^4*arcsech(c*x) + (c^2*x^3*(1/(c^2*x^2) - 1)^(3/2) - 3*x*sqrt(1/(c^2*x^2)
- 1))/c^3)*b*d + 1/90*(15*x^6*arcsech(c*x) - (3*c^4*x^5*(1/(c^2*x^2) - 1)^(5/2) - 10*c^2*x^3*(1/(c^2*x^2) - 1
)^(3/2) + 15*x*sqrt(1/(c^2*x^2) - 1))/c^5)*b*e

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Fricas [A]  time = 1.97318, size = 325, normalized size = 1.81 \begin{align*} \frac{30 \, a c^{5} e x^{6} + 45 \, a c^{5} d x^{4} + 15 \,{\left (2 \, b c^{5} e x^{6} + 3 \, b c^{5} d x^{4}\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 x^{5} +{\left (15 \, b c^{4} d + 8 \, b c^{2} e\right )} x^{3} + 2 \,{\left (15 \, b c^{2} d + 8 \, b e\right )} x\right )} \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}}}{180 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x^2+d)*(a+b*arcsech(c*x)),x, algorithm="fricas")

[Out]

1/180*(30*a*c^5*e*x^6 + 45*a*c^5*d*x^4 + 15*(2*b*c^5*e*x^6 + 3*b*c^5*d*x^4)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*
x^2)) + 1)/(c*x)) - (6*b*c^4*e*x^5 + (15*b*c^4*d + 8*b*c^2*e)*x^3 + 2*(15*b*c^2*d + 8*b*e)*x)*sqrt(-(c^2*x^2 -
1)/(c^2*x^2)))/c^5

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Sympy [A]  time = 14.9681, size = 177, normalized size = 0.98 \begin{align*} \begin{cases} \frac{a d x^{4}}{4} + \frac{a e x^{6}}{6} + \frac{b d x^{4} \operatorname{asech}{\left (c x \right )}}{4} + \frac{b e x^{6} \operatorname{asech}{\left (c x \right )}}{6} - \frac{b d x^{2} \sqrt{- c^{2} x^{2} + 1}}{12 c^{2}} - \frac{b e x^{4} \sqrt{- c^{2} x^{2} + 1}}{30 c^{2}} - \frac{b d \sqrt{- c^{2} x^{2} + 1}}{6 c^{4}} - \frac{2 b e x^{2} \sqrt{- c^{2} x^{2} + 1}}{45 c^{4}} - \frac{4 b e \sqrt{- c^{2} x^{2} + 1}}{45 c^{6}} & \text{for}\: c \neq 0 \\\left (a + \infty b\right ) \left (\frac{d x^{4}}{4} + \frac{e x^{6}}{6}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(e*x**2+d)*(a+b*asech(c*x)),x)

[Out]

Piecewise((a*d*x**4/4 + a*e*x**6/6 + b*d*x**4*asech(c*x)/4 + b*e*x**6*asech(c*x)/6 - b*d*x**2*sqrt(-c**2*x**2
+ 1)/(12*c**2) - b*e*x**4*sqrt(-c**2*x**2 + 1)/(30*c**2) - b*d*sqrt(-c**2*x**2 + 1)/(6*c**4) - 2*b*e*x**2*sqrt
(-c**2*x**2 + 1)/(45*c**4) - 4*b*e*sqrt(-c**2*x**2 + 1)/(45*c**6), Ne(c, 0)), ((a + oo*b)*(d*x**4/4 + e*x**6/6
), True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x^{2} + d\right )}{\left (b \operatorname{arsech}\left (c x\right ) + a\right )} x^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x^2+d)*(a+b*arcsech(c*x)),x, algorithm="giac")

[Out]

integrate((e*x^2 + d)*(b*arcsech(c*x) + a)*x^3, x)