3.1.89 \(\int x^2 (d+e x^2) (a+b \text {sech}^{-1}(c x)) \, dx\) [89]

Optimal. Leaf size=174 \[ -\frac {b \left (20 c^2 d+9 e\right ) x \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{120 c^4}-\frac {b e x^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{20 c^2}+\frac {1}{3} d x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b \left (20 c^2 d+9 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {ArcSin}(c x)}{120 c^5} \]

[Out]

1/3*d*x^3*(a+b*arcsech(c*x))+1/5*e*x^5*(a+b*arcsech(c*x))+1/120*b*(20*c^2*d+9*e)*arcsin(c*x)*(1/(c*x+1))^(1/2)
*(c*x+1)^(1/2)/c^5-1/120*b*(20*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*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 = 0.07, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {14, 6436, 12, 470, 327, 222} \begin {gather*} \frac {1}{3} d x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \text {ArcSin}(c x) \left (20 c^2 d+9 e\right )}{120 c^5}-\frac {b e x^3 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{20 c^2}-\frac {b x \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (20 c^2 d+9 e\right )}{120 c^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/120*(b*(20*c^2*d + 9*e)*x*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/c^4 - (b*e*x^3*Sqrt[(1 + c*
x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(20*c^2) + (d*x^3*(a + b*ArcSech[c*x]))/3 + (e*x^5*(a + b*ArcSech[c*
x]))/5 + (b*(20*c^2*d + 9*e)*Sqrt[(1 + c*x)^(-1)]*Sqrt[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 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 222

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 327

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 470

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 6436

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^2 \left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right ) \, dx &=\frac {1}{3} d x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e 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 {x^2 \left (5 d+3 e x^2\right )}{15 \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {1}{3} d x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e 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 {x^2 \left (5 d+3 e x^2\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {b e x^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{20 c^2}+\frac {1}{3} d x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{60} \left (b \left (20 d+\frac {9 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\\ &=-\frac {b \left (20 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 x^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{20 c^2}+\frac {1}{3} d x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {\left (b \left (20 d+\frac {9 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}{120 c^2}\\ &=-\frac {b \left (20 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 x^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{20 c^2}+\frac {1}{3} d x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b \left (20 c^2 d+9 e\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] Result contains complex when optimal does not.
time = 0.15, size = 144, normalized size = 0.83 \begin {gather*} \frac {8 a c^5 x^3 \left (5 d+3 e x^2\right )-b c x \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (9 e+c^2 \left (20 d+6 e x^2\right )\right )+8 b c^5 x^3 \left (5 d+3 e x^2\right ) \text {sech}^{-1}(c x)+i b \left (20 c^2 d+9 e\right ) \log \left (-2 i c x+2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)\right )}{120 c^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(8*a*c^5*x^3*(5*d + 3*e*x^2) - b*c*x*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(9*e + c^2*(20*d + 6*e*x^2)) + 8*b*c^
5*x^3*(5*d + 3*e*x^2)*ArcSech[c*x] + I*b*(20*c^2*d + 9*e)*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 = 0.31, size = 182, normalized size = 1.05

method	result	size
derivativedivides	\(\frac {\frac {a \left (\frac {1}{3} d \,c^{5} x^{3}+\frac {1}{5} e \,c^{5} x^{5}\right )}{c^{2}}+\frac {b \left (\frac {\mathrm {arcsech}\left (c x \right ) d \,c^{5} x^{3}}{3}+\frac {\mathrm {arcsech}\left (c x \right ) e \,c^{5} x^{5}}{5}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (20 \sqrt {-c^{2} x^{2}+1}\, c^{3} d x +6 e \,c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-20 \arcsin \left (c x \right ) c^{2} d +9 e c x \sqrt {-c^{2} x^{2}+1}-9 e \arcsin \left (c x \right )\right )}{120 \sqrt {-c^{2} x^{2}+1}}\right )}{c^{2}}}{c^{3}}\)	\(182\)
default	\(\frac {\frac {a \left (\frac {1}{3} d \,c^{5} x^{3}+\frac {1}{5} e \,c^{5} x^{5}\right )}{c^{2}}+\frac {b \left (\frac {\mathrm {arcsech}\left (c x \right ) d \,c^{5} x^{3}}{3}+\frac {\mathrm {arcsech}\left (c x \right ) e \,c^{5} x^{5}}{5}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (20 \sqrt {-c^{2} x^{2}+1}\, c^{3} d x +6 e \,c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-20 \arcsin \left (c x \right ) c^{2} d +9 e c x \sqrt {-c^{2} x^{2}+1}-9 e \arcsin \left (c x \right )\right )}{120 \sqrt {-c^{2} x^{2}+1}}\right )}{c^{2}}}{c^{3}}\)	\(182\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^3*(a/c^2*(1/3*d*c^5*x^3+1/5*e*c^5*x^5)+b/c^2*(1/3*arcsech(c*x)*d*c^5*x^3+1/5*arcsech(c*x)*e*c^5*x^5-1/120*
(-(c*x-1)/c/x)^(1/2)*c*x*((c*x+1)/c/x)^(1/2)*(20*(-c^2*x^2+1)^(1/2)*c^3*d*x+6*e*c^3*x^3*(-c^2*x^2+1)^(1/2)-20*
arcsin(c*x)*c^2*d+9*e*c*x*(-c^2*x^2+1)^(1/2)-9*e*arcsin(c*x))/(-c^2*x^2+1)^(1/2)))

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Maxima [A]
time = 0.47, size = 184, normalized size = 1.06 \begin {gather*} \frac {1}{5} \, a x^{5} e + \frac {1}{3} \, a d x^{3} + \frac {1}{6} \, {\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 + \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*a*x^5*e + 1/3*a*d*x^3 + 1/6*(2*x^3*arcsech(c*x) - (sqrt(1/(c^2*x^2) - 1)/(c^2*(1/(c^2*x^2) - 1) + c^2) + a
rctan(sqrt(1/(c^2*x^2) - 1))/c^2)/c)*b*d + 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*e

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 313 vs. \(2 (104) = 208\).
time = 0.56, size = 313, normalized size = 1.80 \begin {gather*} \frac {24 \, a c^{5} x^{5} \cosh \left (1\right ) + 24 \, a c^{5} x^{5} \sinh \left (1\right ) + 40 \, a c^{5} d x^{3} - 2 \, {\left (20 \, b c^{2} d + 9 \, b \cosh \left (1\right ) + 9 \, b \sinh \left (1\right )\right )} \arctan \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c x}\right ) - 8 \, {\left (5 \, b c^{5} d + 3 \, b c^{5} \cosh \left (1\right ) + 3 \, b c^{5} \sinh \left (1\right )\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) + 8 \, {\left (5 \, b c^{5} d x^{3} - 5 \, b c^{5} d + 3 \, {\left (b c^{5} x^{5} - b c^{5}\right )} \cosh \left (1\right ) + 3 \, {\left (b c^{5} x^{5} - b c^{5}\right )} \sinh \left (1\right )\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (20 \, b c^{4} d x^{2} + 3 \, {\left (2 \, b c^{4} x^{4} + 3 \, b c^{2} x^{2}\right )} \cosh \left (1\right ) + 3 \, {\left (2 \, b c^{4} x^{4} + 3 \, b c^{2} x^{2}\right )} \sinh \left (1\right )\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{120 \, c^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(24*a*c^5*x^5*cosh(1) + 24*a*c^5*x^5*sinh(1) + 40*a*c^5*d*x^3 - 2*(20*b*c^2*d + 9*b*cosh(1) + 9*b*sinh(1
))*arctan((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 1)/(c*x)) - 8*(5*b*c^5*d + 3*b*c^5*cosh(1) + 3*b*c^5*sinh(1))*
log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 1)/x) + 8*(5*b*c^5*d*x^3 - 5*b*c^5*d + 3*(b*c^5*x^5 - b*c^5)*cosh(1)
+ 3*(b*c^5*x^5 - b*c^5)*sinh(1))*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) - (20*b*c^4*d*x^2 + 3*(2
*b*c^4*x^4 + 3*b*c^2*x^2)*cosh(1) + 3*(2*b*c^4*x^4 + 3*b*c^2*x^2)*sinh(1))*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)))/c^5

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^2\,\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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