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

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

[Out]

1/5*d*x^5*(a+b*arcsech(c*x))+1/7*e*x^7*(a+b*arcsech(c*x))+1/560*b*(42*c^2*d+25*e)*arcsin(c*x)*(1/(c*x+1))^(1/2
)*(c*x+1)^(1/2)/c^7-1/560*b*(42*c^2*d+25*e)*x*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(1/2)/c^6-1/840*b*(
42*c^2*d+25*e)*x^3*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(1/2)/c^4-1/42*b*e*x^5*(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 = 229, normalized size of antiderivative = 1.00, 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 12

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

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 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 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 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 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]))

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*}

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Mathematica [C]  time = 0.34, size = 162, normalized size = 0.71 \[ \frac {48 a c^7 x^5 \left (7 d+5 e x^2\right )+48 b c^7 x^5 \text {sech}^{-1}(c x) \left (7 d+5 e x^2\right )+3 i b \left (42 c^2 d+25 e\right ) \log \left (2 \sqrt {\frac {1-c x}{c x+1}} (c x+1)-2 i c x\right )-b c x \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (c^4 \left (84 d x^2+40 e x^4\right )+2 c^2 \left (63 d+25 e x^2\right )+75 e\right )}{1680 c^7} \]

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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fricas [A]  time = 1.01, size = 259, normalized size = 1.13 \[ \frac {240 \, a c^{7} e x^{7} + 336 \, a c^{7} d x^{5} - 6 \, {\left (42 \, b c^{2} d + 25 \, b e\right )} \arctan \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c x}\right ) - 48 \, {\left (7 \, b c^{7} d + 5 \, b c^{7} e\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) + 48 \, {\left (5 \, b c^{7} e x^{7} + 7 \, b c^{7} d x^{5} - 7 \, b c^{7} d - 5 \, b c^{7} e\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (40 \, b c^{6} e x^{6} + 2 \, {\left (42 \, b c^{6} d + 25 \, b c^{4} e\right )} x^{4} + 3 \, {\left (42 \, b c^{4} d + 25 \, b c^{2} e\right )} x^{2}\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{1680 \, c^{7}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e x^{2} + d\right )} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{4}\,{d x} \]

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)

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maple [A]  time = 0.07, size = 224, normalized size = 0.98 \[ \frac {\frac {a \left (\frac {1}{7} e \,c^{7} x^{7}+\frac {1}{5} c^{7} x^{5} d \right )}{c^{2}}+\frac {b \left (\frac {\mathrm {arcsech}\left (c x \right ) e \,c^{7} x^{7}}{7}+\frac {\mathrm {arcsech}\left (c x \right ) c^{7} x^{5} d}{5}+\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (-40 c^{5} x^{5} e \sqrt {-c^{2} x^{2}+1}-84 c^{5} x^{3} d \sqrt {-c^{2} x^{2}+1}-50 e \,c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-126 \sqrt {-c^{2} x^{2}+1}\, c^{3} x d +126 \arcsin \left (c x \right ) c^{2} d -75 e c x \sqrt {-c^{2} x^{2}+1}+75 e \arcsin \left (c x \right )\right )}{1680 \sqrt {-c^{2} x^{2}+1}}\right )}{c^{2}}}{c^{5}} \]

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 = 0.42, size = 244, normalized size = 1.07 \[ \frac {1}{7} \, a e x^{7} + \frac {1}{5} \, a d x^{5} + \frac {1}{40} \, {\left (8 \, x^{5} \operatorname {arsech}\left (c x\right ) - \frac {\frac {3 \, {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 5 \, \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 2 \, c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{4}} + \frac {3 \, \arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )}{c^{4}}}{c}\right )} b d + \frac {1}{336} \, {\left (48 \, x^{7} \operatorname {arsech}\left (c x\right ) - \frac {\frac {15 \, {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {5}{2}} + 40 \, {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 33 \, \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{3} + 3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{6}} + \frac {15 \, \arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )}{c^{6}}}{c}\right )} b e \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^4\,\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(d + e*x^2)*(a + b*acosh(1/(c*x))),x)

[Out]

int(x^4*(d + e*x^2)*(a + b*acosh(1/(c*x))), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )\, dx \]

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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