3.20 \(\int x^5 (a+b \text{sech}^{-1}(c x)) \, dx\)

Optimal. Leaf size=109 \[ \frac{1}{6} x^6 \left (a+b \text{sech}^{-1}(c x)\right )-\frac{b x^4 \sqrt{1-c x}}{30 c^2 \sqrt{\frac{1}{c x+1}}}-\frac{2 b x^2 \sqrt{1-c x}}{45 c^4 \sqrt{\frac{1}{c x+1}}}-\frac{4 b \sqrt{1-c x}}{45 c^6 \sqrt{\frac{1}{c x+1}}} \]

[Out]

(-4*b*Sqrt[1 - c*x])/(45*c^6*Sqrt[(1 + c*x)^(-1)]) - (2*b*x^2*Sqrt[1 - c*x])/(45*c^4*Sqrt[(1 + c*x)^(-1)]) - (
b*x^4*Sqrt[1 - c*x])/(30*c^2*Sqrt[(1 + c*x)^(-1)]) + (x^6*(a + b*ArcSech[c*x]))/6

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Rubi [A]  time = 0.0466268, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6283, 100, 12, 74} \[ \frac{1}{6} x^6 \left (a+b \text{sech}^{-1}(c x)\right )-\frac{b x^4 \sqrt{1-c x}}{30 c^2 \sqrt{\frac{1}{c x+1}}}-\frac{2 b x^2 \sqrt{1-c x}}{45 c^4 \sqrt{\frac{1}{c x+1}}}-\frac{4 b \sqrt{1-c x}}{45 c^6 \sqrt{\frac{1}{c x+1}}} \]

Antiderivative was successfully verified.

[In]

Int[x^5*(a + b*ArcSech[c*x]),x]

[Out]

(-4*b*Sqrt[1 - c*x])/(45*c^6*Sqrt[(1 + c*x)^(-1)]) - (2*b*x^2*Sqrt[1 - c*x])/(45*c^4*Sqrt[(1 + c*x)^(-1)]) - (
b*x^4*Sqrt[1 - c*x])/(30*c^2*Sqrt[(1 + c*x)^(-1)]) + (x^6*(a + b*ArcSech[c*x]))/6

Rule 6283

Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcSech[c*
x]))/(d*(m + 1)), x] + Dist[(b*Sqrt[1 + c*x]*Sqrt[1/(1 + c*x)])/(m + 1), Int[(d*x)^m/(Sqrt[1 - c*x]*Sqrt[1 + c
*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]

Rule 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m - 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 1)), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]

Rule 12

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

Rule 74

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rubi steps

\begin{align*} \int x^5 \left (a+b \text{sech}^{-1}(c x)\right ) \, dx &=\frac{1}{6} x^6 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{6} \left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{x^5}{\sqrt{1-c x} \sqrt{1+c x}} \, dx\\ &=-\frac{b x^4 \sqrt{1-c x}}{30 c^2 \sqrt{\frac{1}{1+c x}}}+\frac{1}{6} x^6 \left (a+b \text{sech}^{-1}(c x)\right )-\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int -\frac{4 x^3}{\sqrt{1-c x} \sqrt{1+c x}} \, dx}{30 c^2}\\ &=-\frac{b x^4 \sqrt{1-c x}}{30 c^2 \sqrt{\frac{1}{1+c x}}}+\frac{1}{6} x^6 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{\left (2 b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{x^3}{\sqrt{1-c x} \sqrt{1+c x}} \, dx}{15 c^2}\\ &=-\frac{2 b x^2 \sqrt{1-c x}}{45 c^4 \sqrt{\frac{1}{1+c x}}}-\frac{b x^4 \sqrt{1-c x}}{30 c^2 \sqrt{\frac{1}{1+c x}}}+\frac{1}{6} x^6 \left (a+b \text{sech}^{-1}(c x)\right )-\frac{\left (2 b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int -\frac{2 x}{\sqrt{1-c x} \sqrt{1+c x}} \, dx}{45 c^4}\\ &=-\frac{2 b x^2 \sqrt{1-c x}}{45 c^4 \sqrt{\frac{1}{1+c x}}}-\frac{b x^4 \sqrt{1-c x}}{30 c^2 \sqrt{\frac{1}{1+c x}}}+\frac{1}{6} x^6 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{\left (4 b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{x}{\sqrt{1-c x} \sqrt{1+c x}} \, dx}{45 c^4}\\ &=-\frac{4 b \sqrt{1-c x}}{45 c^6 \sqrt{\frac{1}{1+c x}}}-\frac{2 b x^2 \sqrt{1-c x}}{45 c^4 \sqrt{\frac{1}{1+c x}}}-\frac{b x^4 \sqrt{1-c x}}{30 c^2 \sqrt{\frac{1}{1+c x}}}+\frac{1}{6} x^6 \left (a+b \text{sech}^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.0840902, size = 97, normalized size = 0.89 \[ \frac{a x^6}{6}+b \sqrt{\frac{1-c x}{c x+1}} \left (-\frac{x^4}{30 c^2}-\frac{2 x^3}{45 c^3}-\frac{2 x^2}{45 c^4}-\frac{4 x}{45 c^5}-\frac{4}{45 c^6}-\frac{x^5}{30 c}\right )+\frac{1}{6} b x^6 \text{sech}^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*(a + b*ArcSech[c*x]),x]

[Out]

(a*x^6)/6 + b*Sqrt[(1 - c*x)/(1 + c*x)]*(-4/(45*c^6) - (4*x)/(45*c^5) - (2*x^2)/(45*c^4) - (2*x^3)/(45*c^3) -
x^4/(30*c^2) - x^5/(30*c)) + (b*x^6*ArcSech[c*x])/6

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Maple [A]  time = 0.193, size = 81, normalized size = 0.7 \begin{align*}{\frac{1}{{c}^{6}} \left ({\frac{{c}^{6}{x}^{6}a}{6}}+b \left ({\frac{{c}^{6}{x}^{6}{\rm arcsech} \left (cx\right )}{6}}-{\frac{cx \left ( 3\,{c}^{4}{x}^{4}+4\,{c}^{2}{x}^{2}+8 \right ) }{90}\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^5*(a+b*arcsech(c*x)),x)

[Out]

1/c^6*(1/6*c^6*x^6*a+b*(1/6*c^6*x^6*arcsech(c*x)-1/90*(-(c*x-1)/c/x)^(1/2)*c*x*((c*x+1)/c/x)^(1/2)*(3*c^4*x^4+
4*c^2*x^2+8)))

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Maxima [A]  time = 0.989576, size = 105, normalized size = 0.96 \begin{align*} \frac{1}{6} \, a x^{6} + \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arcsech(c*x)),x, algorithm="maxima")

[Out]

1/6*a*x^6 + 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

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Fricas [A]  time = 1.98351, size = 217, normalized size = 1.99 \begin{align*} \frac{15 \, b c^{5} x^{6} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) + 15 \, a c^{5} x^{6} -{\left (3 \, b c^{4} x^{5} + 4 \, b c^{2} x^{3} + 8 \, b x\right )} \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}}}{90 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arcsech(c*x)),x, algorithm="fricas")

[Out]

1/90*(15*b*c^5*x^6*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) + 15*a*c^5*x^6 - (3*b*c^4*x^5 + 4*b*c^2
*x^3 + 8*b*x)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)))/c^5

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Sympy [A]  time = 16.6257, size = 94, normalized size = 0.86 \begin{align*} \begin{cases} \frac{a x^{6}}{6} + \frac{b x^{6} \operatorname{asech}{\left (c x \right )}}{6} - \frac{b x^{4} \sqrt{- c^{2} x^{2} + 1}}{30 c^{2}} - \frac{2 b x^{2} \sqrt{- c^{2} x^{2} + 1}}{45 c^{4}} - \frac{4 b \sqrt{- c^{2} x^{2} + 1}}{45 c^{6}} & \text{for}\: c \neq 0 \\\frac{x^{6} \left (a + \infty b\right )}{6} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(a+b*asech(c*x)),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{arsech}\left (c x\right ) + a\right )} x^{5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arcsech(c*x)),x, algorithm="giac")

[Out]

integrate((b*arcsech(c*x) + a)*x^5, x)