3.19 \(\int x^6 (a+b \text{sech}^{-1}(c x)) \, dx\)

Optimal. Leaf size=142 \[ \frac{1}{7} x^7 \left (a+b \text{sech}^{-1}(c x)\right )-\frac{b x^5 \sqrt{1-c x}}{42 c^2 \sqrt{\frac{1}{c x+1}}}-\frac{5 b x^3 \sqrt{1-c x}}{168 c^4 \sqrt{\frac{1}{c x+1}}}-\frac{5 b x \sqrt{1-c x}}{112 c^6 \sqrt{\frac{1}{c x+1}}}+\frac{5 b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sin ^{-1}(c x)}{112 c^7} \]

[Out]

(-5*b*x*Sqrt[1 - c*x])/(112*c^6*Sqrt[(1 + c*x)^(-1)]) - (5*b*x^3*Sqrt[1 - c*x])/(168*c^4*Sqrt[(1 + c*x)^(-1)])
- (b*x^5*Sqrt[1 - c*x])/(42*c^2*Sqrt[(1 + c*x)^(-1)]) + (x^7*(a + b*ArcSech[c*x]))/7 + (5*b*Sqrt[(1 + c*x)^(-
1)]*Sqrt[1 + c*x]*ArcSin[c*x])/(112*c^7)

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Rubi [A]  time = 0.0609944, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6283, 100, 12, 90, 41, 216} \[ \frac{1}{7} x^7 \left (a+b \text{sech}^{-1}(c x)\right )-\frac{b x^5 \sqrt{1-c x}}{42 c^2 \sqrt{\frac{1}{c x+1}}}-\frac{5 b x^3 \sqrt{1-c x}}{168 c^4 \sqrt{\frac{1}{c x+1}}}-\frac{5 b x \sqrt{1-c x}}{112 c^6 \sqrt{\frac{1}{c x+1}}}+\frac{5 b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sin ^{-1}(c x)}{112 c^7} \]

Antiderivative was successfully verified.

[In]

Int[x^6*(a + b*ArcSech[c*x]),x]

[Out]

(-5*b*x*Sqrt[1 - c*x])/(112*c^6*Sqrt[(1 + c*x)^(-1)]) - (5*b*x^3*Sqrt[1 - c*x])/(168*c^4*Sqrt[(1 + c*x)^(-1)])
- (b*x^5*Sqrt[1 - c*x])/(42*c^2*Sqrt[(1 + c*x)^(-1)]) + (x^7*(a + b*ArcSech[c*x]))/7 + (5*b*Sqrt[(1 + c*x)^(-
1)]*Sqrt[1 + c*x]*ArcSin[c*x])/(112*c^7)

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 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

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^6 \left (a+b \text{sech}^{-1}(c x)\right ) \, dx &=\frac{1}{7} x^7 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{7} \left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{x^6}{\sqrt{1-c x} \sqrt{1+c x}} \, dx\\ &=-\frac{b x^5 \sqrt{1-c x}}{42 c^2 \sqrt{\frac{1}{1+c x}}}+\frac{1}{7} x^7 \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{5 x^4}{\sqrt{1-c x} \sqrt{1+c x}} \, dx}{42 c^2}\\ &=-\frac{b x^5 \sqrt{1-c x}}{42 c^2 \sqrt{\frac{1}{1+c x}}}+\frac{1}{7} x^7 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{\left (5 b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{x^4}{\sqrt{1-c x} \sqrt{1+c x}} \, dx}{42 c^2}\\ &=-\frac{5 b x^3 \sqrt{1-c x}}{168 c^4 \sqrt{\frac{1}{1+c x}}}-\frac{b x^5 \sqrt{1-c x}}{42 c^2 \sqrt{\frac{1}{1+c x}}}+\frac{1}{7} x^7 \left (a+b \text{sech}^{-1}(c x)\right )-\frac{\left (5 b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int -\frac{3 x^2}{\sqrt{1-c x} \sqrt{1+c x}} \, dx}{168 c^4}\\ &=-\frac{5 b x^3 \sqrt{1-c x}}{168 c^4 \sqrt{\frac{1}{1+c x}}}-\frac{b x^5 \sqrt{1-c x}}{42 c^2 \sqrt{\frac{1}{1+c x}}}+\frac{1}{7} x^7 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{\left (5 b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{x^2}{\sqrt{1-c x} \sqrt{1+c x}} \, dx}{56 c^4}\\ &=-\frac{5 b x \sqrt{1-c x}}{112 c^6 \sqrt{\frac{1}{1+c x}}}-\frac{5 b x^3 \sqrt{1-c x}}{168 c^4 \sqrt{\frac{1}{1+c x}}}-\frac{b x^5 \sqrt{1-c x}}{42 c^2 \sqrt{\frac{1}{1+c x}}}+\frac{1}{7} x^7 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{\left (5 b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{\sqrt{1-c x} \sqrt{1+c x}} \, dx}{112 c^6}\\ &=-\frac{5 b x \sqrt{1-c x}}{112 c^6 \sqrt{\frac{1}{1+c x}}}-\frac{5 b x^3 \sqrt{1-c x}}{168 c^4 \sqrt{\frac{1}{1+c x}}}-\frac{b x^5 \sqrt{1-c x}}{42 c^2 \sqrt{\frac{1}{1+c x}}}+\frac{1}{7} x^7 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{\left (5 b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{112 c^6}\\ &=-\frac{5 b x \sqrt{1-c x}}{112 c^6 \sqrt{\frac{1}{1+c x}}}-\frac{5 b x^3 \sqrt{1-c x}}{168 c^4 \sqrt{\frac{1}{1+c x}}}-\frac{b x^5 \sqrt{1-c x}}{42 c^2 \sqrt{\frac{1}{1+c x}}}+\frac{1}{7} x^7 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{5 b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sin ^{-1}(c x)}{112 c^7}\\ \end{align*}

Mathematica [C]  time = 0.187581, size = 143, normalized size = 1.01 \[ \frac{a x^7}{7}+b \sqrt{\frac{1-c x}{c x+1}} \left (-\frac{x^5}{42 c^2}-\frac{5 x^4}{168 c^3}-\frac{5 x^3}{168 c^4}-\frac{5 x^2}{112 c^5}-\frac{5 x}{112 c^6}-\frac{x^6}{42 c}\right )+\frac{5 i b \log \left (2 \sqrt{\frac{1-c x}{c x+1}} (c x+1)-2 i c x\right )}{112 c^7}+\frac{1}{7} b x^7 \text{sech}^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

Integrate[x^6*(a + b*ArcSech[c*x]),x]

[Out]

(a*x^7)/7 + b*Sqrt[(1 - c*x)/(1 + c*x)]*((-5*x)/(112*c^6) - (5*x^2)/(112*c^5) - (5*x^3)/(168*c^4) - (5*x^4)/(1
68*c^3) - x^5/(42*c^2) - x^6/(42*c)) + (b*x^7*ArcSech[c*x])/7 + (((5*I)/112)*b*Log[(-2*I)*c*x + 2*Sqrt[(1 - c*
x)/(1 + c*x)]*(1 + c*x)])/c^7

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Maple [A]  time = 0.217, size = 138, normalized size = 1. \begin{align*}{\frac{1}{{c}^{7}} \left ({\frac{{c}^{7}{x}^{7}a}{7}}+b \left ({\frac{{c}^{7}{x}^{7}{\rm arcsech} \left (cx\right )}{7}}+{\frac{cx}{336}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}} \left ( -8\,{c}^{5}{x}^{5}\sqrt{-{c}^{2}{x}^{2}+1}-10\,{c}^{3}{x}^{3}\sqrt{-{c}^{2}{x}^{2}+1}-15\,cx\sqrt{-{c}^{2}{x}^{2}+1}+15\,\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^6*(a+b*arcsech(c*x)),x)

[Out]

1/c^7*(1/7*c^7*x^7*a+b*(1/7*c^7*x^7*arcsech(c*x)+1/336*(-(c*x-1)/c/x)^(1/2)*c*x*((c*x+1)/c/x)^(1/2)*(-8*c^5*x^
5*(-c^2*x^2+1)^(1/2)-10*c^3*x^3*(-c^2*x^2+1)^(1/2)-15*c*x*(-c^2*x^2+1)^(1/2)+15*arcsin(c*x))/(-c^2*x^2+1)^(1/2
)))

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Maxima [A]  time = 1.49201, size = 182, normalized size = 1.28 \begin{align*} \frac{1}{7} \, a x^{7} + \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(a+b*arcsech(c*x)),x, algorithm="maxima")

[Out]

1/7*a*x^7 + 1/336*(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*ar
ctan(sqrt(1/(c^2*x^2) - 1))/c^6)/c)*b

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Fricas [A]  time = 2.33069, size = 406, normalized size = 2.86 \begin{align*} \frac{48 \, a c^{7} x^{7} - 48 \, b c^{7} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) - 30 \, b \arctan \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c x}\right ) + 48 \,{\left (b c^{7} x^{7} - b c^{7}\right )} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) -{\left (8 \, b c^{6} x^{6} + 10 \, b c^{4} x^{4} + 15 \, b c^{2} x^{2}\right )} \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}}}{336 \, c^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(a+b*arcsech(c*x)),x, algorithm="fricas")

[Out]

1/336*(48*a*c^7*x^7 - 48*b*c^7*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 1)/x) - 30*b*arctan((c*x*sqrt(-(c^2*x
^2 - 1)/(c^2*x^2)) - 1)/(c*x)) + 48*(b*c^7*x^7 - b*c^7)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) -
(8*b*c^6*x^6 + 10*b*c^4*x^4 + 15*b*c^2*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^{6} \left (a + b \operatorname{asech}{\left (c x \right )}\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(a+b*asech(c*x)),x)

[Out]

Integral(x**6*(a + b*asech(c*x)), x)

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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^{6}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(a+b*arcsech(c*x)),x, algorithm="giac")

[Out]

integrate((b*arcsech(c*x) + a)*x^6, x)