3.88 \(\int x^4 (d+e x^2) (a+b \text{sech}^{-1}(c x)) \, dx\)

Optimal. Leaf size=229 \[ \frac{1}{5} d x^5 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \text{sech}^{-1}(c x)\right )-\frac{b x^3 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (42 c^2 d+25 e\right )}{840 c^4}-\frac{b x \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (42 c^2 d+25 e\right )}{560 c^6}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (42 c^2 d+25 e\right ) \sin ^{-1}(c x)}{560 c^7}-\frac{b e x^5 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{42 c^2} \]

[Out]

-(b*(42*c^2*d + 25*e)*x*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(560*c^6) - (b*(42*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*x^5*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 +
c*x]*Sqrt[1 - c^2*x^2])/(42*c^2) + (d*x^5*(a + b*ArcSech[c*x]))/5 + (e*x^7*(a + b*ArcSech[c*x]))/7 + (b*(42*c^
2*d + 25*e)*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*ArcSin[c*x])/(560*c^7)

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Rubi [A]  time = 0.127182, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {14, 6301, 12, 459, 321, 216} \[ \frac{1}{5} d x^5 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \text{sech}^{-1}(c x)\right )-\frac{b x^3 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (42 c^2 d+25 e\right )}{840 c^4}-\frac{b x \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (42 c^2 d+25 e\right )}{560 c^6}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (42 c^2 d+25 e\right ) \sin ^{-1}(c x)}{560 c^7}-\frac{b e x^5 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{42 c^2} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(d + e*x^2)*(a + b*ArcSech[c*x]),x]

[Out]

-(b*(42*c^2*d + 25*e)*x*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(560*c^6) - (b*(42*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*x^5*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 +
c*x]*Sqrt[1 - c^2*x^2])/(42*c^2) + (d*x^5*(a + b*ArcSech[c*x]))/5 + (e*x^7*(a + b*ArcSech[c*x]))/7 + (b*(42*c^
2*d + 25*e)*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*ArcSin[c*x])/(560*c^7)

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 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^4 \left (d+e x^2\right ) \left (a+b \text{sech}^{-1}(c x)\right ) \, dx &=\frac{1}{5} d x^5 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{7} e 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^4 \left (7 d+5 e x^2\right )}{35 \sqrt{1-c^2 x^2}} \, dx\\ &=\frac{1}{5} d x^5 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{35} \left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{x^4 \left (7 d+5 e x^2\right )}{\sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{b e x^5 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{42 c^2}+\frac{1}{5} d x^5 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{210} \left (b \left (42 d+\frac{25 e}{c^2}\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{x^4}{\sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{b \left (42 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 x^5 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{42 c^2}+\frac{1}{5} d x^5 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{\left (b \left (42 d+\frac{25 e}{c^2}\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{x^2}{\sqrt{1-c^2 x^2}} \, dx}{280 c^2}\\ &=-\frac{b \left (42 c^2 d+25 e\right ) x \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{560 c^6}-\frac{b \left (42 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 x^5 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{42 c^2}+\frac{1}{5} d x^5 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{\left (b \left (42 d+\frac{25 e}{c^2}\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{560 c^4}\\ &=-\frac{b \left (42 c^2 d+25 e\right ) x \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{560 c^6}-\frac{b \left (42 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 x^5 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{42 c^2}+\frac{1}{5} d x^5 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{b \left (42 c^2 d+25 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sin ^{-1}(c x)}{560 c^7}\\ \end{align*}

Mathematica [C]  time = 0.317618, size = 162, normalized size = 0.71 \[ \frac{48 a c^7 x^5 \left (7 d+5 e x^2\right )-b c x \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (c^4 \left (84 d x^2+40 e x^4\right )+2 c^2 \left (63 d+25 e x^2\right )+75 e\right )+48 b c^7 x^5 \text{sech}^{-1}(c x) \left (7 d+5 e x^2\right )+3 i b \left (42 c^2 d+25 e\right ) \log \left (2 \sqrt{\frac{1-c x}{c x+1}} (c x+1)-2 i c x\right )}{1680 c^7} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(d + e*x^2)*(a + b*ArcSech[c*x]),x]

[Out]

(48*a*c^7*x^5*(7*d + 5*e*x^2) - b*c*x*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(75*e + 2*c^2*(63*d + 25*e*x^2) + c^
4*(84*d*x^2 + 40*e*x^4)) + 48*b*c^7*x^5*(7*d + 5*e*x^2)*ArcSech[c*x] + (3*I)*b*(42*c^2*d + 25*e)*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 = 0.185, size = 224, normalized size = 1. \begin{align*}{\frac{1}{{c}^{5}} \left ({\frac{a}{{c}^{2}} \left ({\frac{e{c}^{7}{x}^{7}}{7}}+{\frac{{c}^{7}{x}^{5}d}{5}} \right ) }+{\frac{b}{{c}^{2}} \left ({\frac{{\rm arcsech} \left (cx\right )e{c}^{7}{x}^{7}}{7}}+{\frac{{\rm arcsech} \left (cx\right ){c}^{7}{x}^{5}d}{5}}+{\frac{cx}{1680}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}} \left ( -40\,{c}^{5}{x}^{5}e\sqrt{-{c}^{2}{x}^{2}+1}-84\,{c}^{5}{x}^{3}d\sqrt{-{c}^{2}{x}^{2}+1}-50\,e{c}^{3}{x}^{3}\sqrt{-{c}^{2}{x}^{2}+1}-126\,\sqrt{-{c}^{2}{x}^{2}+1}{c}^{3}xd+126\,\arcsin \left ( cx \right ){c}^{2}d-75\,ecx\sqrt{-{c}^{2}{x}^{2}+1}+75\,e\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^4*(e*x^2+d)*(a+b*arcsech(c*x)),x)

[Out]

1/c^5*(a/c^2*(1/7*e*c^7*x^7+1/5*c^7*x^5*d)+b/c^2*(1/7*arcsech(c*x)*e*c^7*x^7+1/5*arcsech(c*x)*c^7*x^5*d+1/1680
*(-(c*x-1)/c/x)^(1/2)*c*x*((c*x+1)/c/x)^(1/2)*(-40*c^5*x^5*e*(-c^2*x^2+1)^(1/2)-84*c^5*x^3*d*(-c^2*x^2+1)^(1/2
)-50*e*c^3*x^3*(-c^2*x^2+1)^(1/2)-126*(-c^2*x^2+1)^(1/2)*c^3*x*d+126*arcsin(c*x)*c^2*d-75*e*c*x*(-c^2*x^2+1)^(
1/2)+75*e*arcsin(c*x))/(-c^2*x^2+1)^(1/2)))

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Maxima [A]  time = 1.47971, size = 329, normalized size = 1.44 \begin{align*} \frac{1}{7} \, a e x^{7} + \frac{1}{5} \, a d x^{5} + \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 d + \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x^2+d)*(a+b*arcsech(c*x)),x, algorithm="maxima")

[Out]

1/7*a*e*x^7 + 1/5*a*d*x^5 + 1/40*(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 + 1/33
6*(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

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Fricas [A]  time = 2.98418, size = 590, normalized size = 2.58 \begin{align*} \frac{240 \, a c^{7} e x^{7} + 336 \, a c^{7} d x^{5} - 6 \,{\left (42 \, b c^{2} d + 25 \, b e\right )} \arctan \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c x}\right ) - 48 \,{\left (7 \, b c^{7} d + 5 \, b c^{7} e\right )} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) + 48 \,{\left (5 \, b c^{7} e x^{7} + 7 \, b c^{7} d x^{5} - 7 \, b c^{7} d - 5 \, b c^{7} e\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 x^{6} + 2 \,{\left (42 \, b c^{6} d + 25 \, b c^{4} e\right )} x^{4} + 3 \,{\left (42 \, b c^{4} d + 25 \, b c^{2} e\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^4*(e*x^2+d)*(a+b*arcsech(c*x)),x, algorithm="fricas")

[Out]

1/1680*(240*a*c^7*e*x^7 + 336*a*c^7*d*x^5 - 6*(42*b*c^2*d + 25*b*e)*arctan((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2))
- 1)/(c*x)) - 48*(7*b*c^7*d + 5*b*c^7*e)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 1)/x) + 48*(5*b*c^7*e*x^7
+ 7*b*c^7*d*x^5 - 7*b*c^7*d - 5*b*c^7*e)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) - (40*b*c^6*e*x^6
+ 2*(42*b*c^6*d + 25*b*c^4*e)*x^4 + 3*(42*b*c^4*d + 25*b*c^2*e)*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^{4} \left (a + b \operatorname{asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(e*x**2+d)*(a+b*asech(c*x)),x)

[Out]

Integral(x**4*(a + b*asech(c*x))*(d + e*x**2), x)

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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^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x^2+d)*(a+b*arcsech(c*x)),x, algorithm="giac")

[Out]

integrate((e*x^2 + d)*(b*arcsech(c*x) + a)*x^4, x)