3.1.96 \(\int x^3 (d+e x^2) (a+b \text {sech}^{-1}(c x)) \, dx\) [96]

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

[Out]

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

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Rubi [A]
time = 0.09, antiderivative size = 180, normalized size of antiderivative = 1.00, 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, 6436, 12, 457, 78} \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/12*(b*(3*c^2*d + 2*e)*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/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 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 78

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]))))

Rule 457

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 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^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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.13, size = 106, normalized size = 0.59 \begin {gather*} \frac {1}{180} \left (15 a x^4 \left (3 d+2 e x^2\right )-\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (16 e+c^2 \left (30 d+8 e x^2\right )+3 c^4 \left (5 d x^2+2 e x^4\right )\right )}{c^6}+15 b x^4 \left (3 d+2 e x^2\right ) \text {sech}^{-1}(c x)\right ) \end {gather*}

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 = 0.34, size = 281, normalized size = 1.56

method	result	size
derivativedivides	\(\frac {-\frac {a \left (\frac {c^{2} d \left (e \,c^{2} x^{2}+c^{2} d \right )^{2}}{2}-\frac {\left (e \,c^{2} x^{2}+c^{2} d \right )^{3}}{3}\right )}{2 c^{2} e^{2}}+\frac {b \left (-\frac {\mathrm {arcsech}\left (c x \right ) c^{6} d^{3}}{12 e^{2}}+\frac {\mathrm {arcsech}\left (c x \right ) c^{6} d \,x^{4}}{4}+\frac {e \,\mathrm {arcsech}\left (c x \right ) c^{6} x^{6}}{6}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (-15 c^{6} d^{3} \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )+15 c^{4} d \,e^{2} \sqrt {-c^{2} x^{2}+1}\, x^{2}+6 e^{3} \sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}+30 c^{2} d \,e^{2} \sqrt {-c^{2} x^{2}+1}+8 e^{3} c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}+16 e^{3} \sqrt {-c^{2} x^{2}+1}\right )}{180 e^{2} \sqrt {-c^{2} x^{2}+1}}\right )}{c^{2}}}{c^{4}}\)	\(281\)
default	\(\frac {-\frac {a \left (\frac {c^{2} d \left (e \,c^{2} x^{2}+c^{2} d \right )^{2}}{2}-\frac {\left (e \,c^{2} x^{2}+c^{2} d \right )^{3}}{3}\right )}{2 c^{2} e^{2}}+\frac {b \left (-\frac {\mathrm {arcsech}\left (c x \right ) c^{6} d^{3}}{12 e^{2}}+\frac {\mathrm {arcsech}\left (c x \right ) c^{6} d \,x^{4}}{4}+\frac {e \,\mathrm {arcsech}\left (c x \right ) c^{6} x^{6}}{6}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (-15 c^{6} d^{3} \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )+15 c^{4} d \,e^{2} \sqrt {-c^{2} x^{2}+1}\, x^{2}+6 e^{3} \sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}+30 c^{2} d \,e^{2} \sqrt {-c^{2} x^{2}+1}+8 e^{3} c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}+16 e^{3} \sqrt {-c^{2} x^{2}+1}\right )}{180 e^{2} \sqrt {-c^{2} x^{2}+1}}\right )}{c^{2}}}{c^{4}}\)	\(281\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^4*(-1/2*a/c^2/e^2*(1/2*c^2*d*(c^2*e*x^2+c^2*d)^2-1/3*(c^2*e*x^2+c^2*d)^3)+b/c^2*(-1/12/e^2*arcsech(c*x)*c^
6*d^3+1/4*arcsech(c*x)*c^6*d*x^4+1/6*e*arcsech(c*x)*c^6*x^6-1/180/e^2*(-(c*x-1)/c/x)^(1/2)*c*x*((c*x+1)/c/x)^(
1/2)*(-15*c^6*d^3*arctanh(1/(-c^2*x^2+1)^(1/2))+15*c^4*d*e^2*(-c^2*x^2+1)^(1/2)*x^2+6*e^3*(-c^2*x^2+1)^(1/2)*c
^4*x^4+30*c^2*d*e^2*(-c^2*x^2+1)^(1/2)+8*e^3*c^2*x^2*(-c^2*x^2+1)^(1/2)+16*e^3*(-c^2*x^2+1)^(1/2))/(-c^2*x^2+1
)^(1/2)))

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Maxima [A]
time = 0.25, size = 140, normalized size = 0.78 \begin {gather*} \frac {1}{6} \, a x^{6} e + \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 {gather*}

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*x^6*e + 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 = 0.38, size = 199, normalized size = 1.11 \begin {gather*} \frac {30 \, a c^{5} x^{6} \cosh \left (1\right ) + 30 \, a c^{5} x^{6} \sinh \left (1\right ) + 45 \, a c^{5} d x^{4} + 15 \, {\left (2 \, b c^{5} x^{6} \cosh \left (1\right ) + 2 \, b c^{5} x^{6} \sinh \left (1\right ) + 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 (15 \, b c^{4} d x^{3} + 30 \, b c^{2} d x + 2 \, {\left (3 \, b c^{4} x^{5} + 4 \, b c^{2} x^{3} + 8 \, b x\right )} \cosh \left (1\right ) + 2 \, {\left (3 \, b c^{4} x^{5} + 4 \, b c^{2} x^{3} + 8 \, b x\right )} \sinh \left (1\right )\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{180 \, c^{5}} \end {gather*}

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

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Sympy [A]
time = 1.03, size = 177, normalized size = 0.98 \begin {gather*} \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 {gather*}

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 = 0.00, 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^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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^3\,\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^3*(d + e*x^2)*(a + b*acosh(1/(c*x))),x)

[Out]

int(x^3*(d + e*x^2)*(a + b*acosh(1/(c*x))), x)

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