3.43 \(\int x^2 (a+b \text{sech}^{-1}(c x))^3 \, dx\)

Optimal. Leaf size=242 \[ \frac{i b^2 \text{PolyLog}\left (2,-i e^{\text{sech}^{-1}(c x)}\right ) \left (a+b \text{sech}^{-1}(c x)\right )}{c^3}-\frac{i b^2 \text{PolyLog}\left (2,i e^{\text{sech}^{-1}(c x)}\right ) \left (a+b \text{sech}^{-1}(c x)\right )}{c^3}-\frac{i b^3 \text{PolyLog}\left (3,-i e^{\text{sech}^{-1}(c x)}\right )}{c^3}+\frac{i b^3 \text{PolyLog}\left (3,i e^{\text{sech}^{-1}(c x)}\right )}{c^3}-\frac{b^2 x \left (a+b \text{sech}^{-1}(c x)\right )}{c^2}-\frac{b x \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (a+b \text{sech}^{-1}(c x)\right )^2}{2 c^2}-\frac{b \tan ^{-1}\left (e^{\text{sech}^{-1}(c x)}\right ) \left (a+b \text{sech}^{-1}(c x)\right )^2}{c^3}+\frac{1}{3} x^3 \left (a+b \text{sech}^{-1}(c x)\right )^3+\frac{b^3 \tan ^{-1}\left (\frac{\sqrt{\frac{1-c x}{c x+1}} (c x+1)}{c x}\right )}{c^3} \]

[Out]

-((b^2*x*(a + b*ArcSech[c*x]))/c^2) - (b*x*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcSech[c*x])^2)/(2*c^2)
+ (x^3*(a + b*ArcSech[c*x])^3)/3 - (b*(a + b*ArcSech[c*x])^2*ArcTan[E^ArcSech[c*x]])/c^3 + (b^3*ArcTan[(Sqrt[
(1 - c*x)/(1 + c*x)]*(1 + c*x))/(c*x)])/c^3 + (I*b^2*(a + b*ArcSech[c*x])*PolyLog[2, (-I)*E^ArcSech[c*x]])/c^3
- (I*b^2*(a + b*ArcSech[c*x])*PolyLog[2, I*E^ArcSech[c*x]])/c^3 - (I*b^3*PolyLog[3, (-I)*E^ArcSech[c*x]])/c^3
+ (I*b^3*PolyLog[3, I*E^ArcSech[c*x]])/c^3

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Rubi [A]  time = 0.195349, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {6285, 5451, 4186, 3770, 4180, 2531, 2282, 6589} \[ \frac{i b^2 \text{PolyLog}\left (2,-i e^{\text{sech}^{-1}(c x)}\right ) \left (a+b \text{sech}^{-1}(c x)\right )}{c^3}-\frac{i b^2 \text{PolyLog}\left (2,i e^{\text{sech}^{-1}(c x)}\right ) \left (a+b \text{sech}^{-1}(c x)\right )}{c^3}-\frac{i b^3 \text{PolyLog}\left (3,-i e^{\text{sech}^{-1}(c x)}\right )}{c^3}+\frac{i b^3 \text{PolyLog}\left (3,i e^{\text{sech}^{-1}(c x)}\right )}{c^3}-\frac{b^2 x \left (a+b \text{sech}^{-1}(c x)\right )}{c^2}-\frac{b x \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (a+b \text{sech}^{-1}(c x)\right )^2}{2 c^2}-\frac{b \tan ^{-1}\left (e^{\text{sech}^{-1}(c x)}\right ) \left (a+b \text{sech}^{-1}(c x)\right )^2}{c^3}+\frac{1}{3} x^3 \left (a+b \text{sech}^{-1}(c x)\right )^3+\frac{b^3 \tan ^{-1}\left (\frac{\sqrt{\frac{1-c x}{c x+1}} (c x+1)}{c x}\right )}{c^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*ArcSech[c*x])^3,x]

[Out]

-((b^2*x*(a + b*ArcSech[c*x]))/c^2) - (b*x*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcSech[c*x])^2)/(2*c^2)
+ (x^3*(a + b*ArcSech[c*x])^3)/3 - (b*(a + b*ArcSech[c*x])^2*ArcTan[E^ArcSech[c*x]])/c^3 + (b^3*ArcTan[(Sqrt[
(1 - c*x)/(1 + c*x)]*(1 + c*x))/(c*x)])/c^3 + (I*b^2*(a + b*ArcSech[c*x])*PolyLog[2, (-I)*E^ArcSech[c*x]])/c^3
- (I*b^2*(a + b*ArcSech[c*x])*PolyLog[2, I*E^ArcSech[c*x]])/c^3 - (I*b^3*PolyLog[3, (-I)*E^ArcSech[c*x]])/c^3
+ (I*b^3*PolyLog[3, I*E^ArcSech[c*x]])/c^3

Rule 6285

Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> -Dist[(c^(m + 1))^(-1), Subst[Int[(a + b
*x)^n*Sech[x]^(m + 1)*Tanh[x], x], x, ArcSech[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] &
& (GtQ[n, 0] || LtQ[m, -1])

Rule 5451

Int[((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.)*Tanh[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> -Si
mp[((c + d*x)^m*Sech[a + b*x]^n)/(b*n), x] + Dist[(d*m)/(b*n), Int[(c + d*x)^(m - 1)*Sech[a + b*x]^n, x], x] /
; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]

Rule 4186

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(b^2*(c + d*x)^m*Cot[e
+ f*x]*(b*Csc[e + f*x])^(n - 2))/(f*(n - 1)), x] + (Dist[(b^2*d^2*m*(m - 1))/(f^2*(n - 1)*(n - 2)), Int[(c + d
*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Dist[(b^2*(n - 2))/(n - 1), Int[(c + d*x)^m*(b*Csc[e + f*x])^(n
- 2), x], x] - Simp[(b^2*d*m*(c + d*x)^(m - 1)*(b*Csc[e + f*x])^(n - 2))/(f^2*(n - 1)*(n - 2)), x]) /; FreeQ[
{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 4180

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c
+ d*x)^m*ArcTanh[E^(-(I*e) + f*fz*x)/E^(I*k*Pi)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*
Log[1 - E^(-(I*e) + f*fz*x)/E^(I*k*Pi)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e)
+ f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]

Rule 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

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

Rubi steps

\begin{align*} \int x^2 \left (a+b \text{sech}^{-1}(c x)\right )^3 \, dx &=-\frac{\operatorname{Subst}\left (\int (a+b x)^3 \text{sech}^3(x) \tanh (x) \, dx,x,\text{sech}^{-1}(c x)\right )}{c^3}\\ &=\frac{1}{3} x^3 \left (a+b \text{sech}^{-1}(c x)\right )^3-\frac{b \operatorname{Subst}\left (\int (a+b x)^2 \text{sech}^3(x) \, dx,x,\text{sech}^{-1}(c x)\right )}{c^3}\\ &=-\frac{b^2 x \left (a+b \text{sech}^{-1}(c x)\right )}{c^2}-\frac{b x \sqrt{\frac{1-c x}{1+c x}} (1+c x) \left (a+b \text{sech}^{-1}(c x)\right )^2}{2 c^2}+\frac{1}{3} x^3 \left (a+b \text{sech}^{-1}(c x)\right )^3-\frac{b \operatorname{Subst}\left (\int (a+b x)^2 \text{sech}(x) \, dx,x,\text{sech}^{-1}(c x)\right )}{2 c^3}+\frac{b^3 \operatorname{Subst}\left (\int \text{sech}(x) \, dx,x,\text{sech}^{-1}(c x)\right )}{c^3}\\ &=-\frac{b^2 x \left (a+b \text{sech}^{-1}(c x)\right )}{c^2}-\frac{b x \sqrt{\frac{1-c x}{1+c x}} (1+c x) \left (a+b \text{sech}^{-1}(c x)\right )^2}{2 c^2}+\frac{1}{3} x^3 \left (a+b \text{sech}^{-1}(c x)\right )^3-\frac{b \left (a+b \text{sech}^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\text{sech}^{-1}(c x)}\right )}{c^3}+\frac{b^3 \tan ^{-1}\left (\frac{\sqrt{\frac{1-c x}{1+c x}} (1+c x)}{c x}\right )}{c^3}+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1-i e^x\right ) \, dx,x,\text{sech}^{-1}(c x)\right )}{c^3}-\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1+i e^x\right ) \, dx,x,\text{sech}^{-1}(c x)\right )}{c^3}\\ &=-\frac{b^2 x \left (a+b \text{sech}^{-1}(c x)\right )}{c^2}-\frac{b x \sqrt{\frac{1-c x}{1+c x}} (1+c x) \left (a+b \text{sech}^{-1}(c x)\right )^2}{2 c^2}+\frac{1}{3} x^3 \left (a+b \text{sech}^{-1}(c x)\right )^3-\frac{b \left (a+b \text{sech}^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\text{sech}^{-1}(c x)}\right )}{c^3}+\frac{b^3 \tan ^{-1}\left (\frac{\sqrt{\frac{1-c x}{1+c x}} (1+c x)}{c x}\right )}{c^3}+\frac{i b^2 \left (a+b \text{sech}^{-1}(c x)\right ) \text{Li}_2\left (-i e^{\text{sech}^{-1}(c x)}\right )}{c^3}-\frac{i b^2 \left (a+b \text{sech}^{-1}(c x)\right ) \text{Li}_2\left (i e^{\text{sech}^{-1}(c x)}\right )}{c^3}-\frac{\left (i b^3\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^x\right ) \, dx,x,\text{sech}^{-1}(c x)\right )}{c^3}+\frac{\left (i b^3\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^x\right ) \, dx,x,\text{sech}^{-1}(c x)\right )}{c^3}\\ &=-\frac{b^2 x \left (a+b \text{sech}^{-1}(c x)\right )}{c^2}-\frac{b x \sqrt{\frac{1-c x}{1+c x}} (1+c x) \left (a+b \text{sech}^{-1}(c x)\right )^2}{2 c^2}+\frac{1}{3} x^3 \left (a+b \text{sech}^{-1}(c x)\right )^3-\frac{b \left (a+b \text{sech}^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\text{sech}^{-1}(c x)}\right )}{c^3}+\frac{b^3 \tan ^{-1}\left (\frac{\sqrt{\frac{1-c x}{1+c x}} (1+c x)}{c x}\right )}{c^3}+\frac{i b^2 \left (a+b \text{sech}^{-1}(c x)\right ) \text{Li}_2\left (-i e^{\text{sech}^{-1}(c x)}\right )}{c^3}-\frac{i b^2 \left (a+b \text{sech}^{-1}(c x)\right ) \text{Li}_2\left (i e^{\text{sech}^{-1}(c x)}\right )}{c^3}-\frac{\left (i b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{\text{sech}^{-1}(c x)}\right )}{c^3}+\frac{\left (i b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{\text{sech}^{-1}(c x)}\right )}{c^3}\\ &=-\frac{b^2 x \left (a+b \text{sech}^{-1}(c x)\right )}{c^2}-\frac{b x \sqrt{\frac{1-c x}{1+c x}} (1+c x) \left (a+b \text{sech}^{-1}(c x)\right )^2}{2 c^2}+\frac{1}{3} x^3 \left (a+b \text{sech}^{-1}(c x)\right )^3-\frac{b \left (a+b \text{sech}^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\text{sech}^{-1}(c x)}\right )}{c^3}+\frac{b^3 \tan ^{-1}\left (\frac{\sqrt{\frac{1-c x}{1+c x}} (1+c x)}{c x}\right )}{c^3}+\frac{i b^2 \left (a+b \text{sech}^{-1}(c x)\right ) \text{Li}_2\left (-i e^{\text{sech}^{-1}(c x)}\right )}{c^3}-\frac{i b^2 \left (a+b \text{sech}^{-1}(c x)\right ) \text{Li}_2\left (i e^{\text{sech}^{-1}(c x)}\right )}{c^3}-\frac{i b^3 \text{Li}_3\left (-i e^{\text{sech}^{-1}(c x)}\right )}{c^3}+\frac{i b^3 \text{Li}_3\left (i e^{\text{sech}^{-1}(c x)}\right )}{c^3}\\ \end{align*}

Mathematica [A]  time = 1.01831, size = 440, normalized size = 1.82 \[ \frac{-6 a b^2 \left (-i \text{PolyLog}\left (2,-i e^{-\text{sech}^{-1}(c x)}\right )+i \text{PolyLog}\left (2,i e^{-\text{sech}^{-1}(c x)}\right )-c^3 x^3 \text{sech}^{-1}(c x)^2+c x+c x \sqrt{\frac{1-c x}{c x+1}} (c x+1) \text{sech}^{-1}(c x)-i \text{sech}^{-1}(c x) \log \left (1-i e^{-\text{sech}^{-1}(c x)}\right )+i \text{sech}^{-1}(c x) \log \left (1+i e^{-\text{sech}^{-1}(c x)}\right )\right )+b^3 \left (-\left (-3 i \left (2 \text{sech}^{-1}(c x) \text{PolyLog}\left (2,-i e^{-\text{sech}^{-1}(c x)}\right )-2 \text{sech}^{-1}(c x) \text{PolyLog}\left (2,i e^{-\text{sech}^{-1}(c x)}\right )+2 \text{PolyLog}\left (3,-i e^{-\text{sech}^{-1}(c x)}\right )-2 \text{PolyLog}\left (3,i e^{-\text{sech}^{-1}(c x)}\right )+\text{sech}^{-1}(c x)^2 \log \left (1-i e^{-\text{sech}^{-1}(c x)}\right )-\text{sech}^{-1}(c x)^2 \log \left (1+i e^{-\text{sech}^{-1}(c x)}\right )-4 i \tan ^{-1}\left (\tanh \left (\frac{1}{2} \text{sech}^{-1}(c x)\right )\right )\right )-2 c^3 x^3 \text{sech}^{-1}(c x)^3+3 c x \sqrt{\frac{1-c x}{c x+1}} (c x+1) \text{sech}^{-1}(c x)^2+6 c x \text{sech}^{-1}(c x)\right )\right )+6 a^2 b c^3 x^3 \text{sech}^{-1}(c x)-3 a^2 b c x \sqrt{\frac{1-c x}{c x+1}} (c x+1)+3 i a^2 b \log \left (2 \sqrt{\frac{1-c x}{c x+1}} (c x+1)-2 i c x\right )+2 a^3 c^3 x^3}{6 c^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^2*(a + b*ArcSech[c*x])^3,x]

[Out]

(2*a^3*c^3*x^3 - 3*a^2*b*c*x*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) + 6*a^2*b*c^3*x^3*ArcSech[c*x] + (3*I)*a^2*b*
Log[(-2*I)*c*x + 2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)] - 6*a*b^2*(c*x + c*x*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*
x)*ArcSech[c*x] - c^3*x^3*ArcSech[c*x]^2 - I*ArcSech[c*x]*Log[1 - I/E^ArcSech[c*x]] + I*ArcSech[c*x]*Log[1 + I
/E^ArcSech[c*x]] - I*PolyLog[2, (-I)/E^ArcSech[c*x]] + I*PolyLog[2, I/E^ArcSech[c*x]]) - b^3*(6*c*x*ArcSech[c*
x] + 3*c*x*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*ArcSech[c*x]^2 - 2*c^3*x^3*ArcSech[c*x]^3 - (3*I)*((-4*I)*ArcTa
n[Tanh[ArcSech[c*x]/2]] + ArcSech[c*x]^2*Log[1 - I/E^ArcSech[c*x]] - ArcSech[c*x]^2*Log[1 + I/E^ArcSech[c*x]]
+ 2*ArcSech[c*x]*PolyLog[2, (-I)/E^ArcSech[c*x]] - 2*ArcSech[c*x]*PolyLog[2, I/E^ArcSech[c*x]] + 2*PolyLog[3,
(-I)/E^ArcSech[c*x]] - 2*PolyLog[3, I/E^ArcSech[c*x]])))/(6*c^3)

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Maple [F]  time = 0.49, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( a+b{\rm arcsech} \left (cx\right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*arcsech(c*x))^3,x)

[Out]

int(x^2*(a+b*arcsech(c*x))^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{3} \, a^{3} x^{3} + \frac{1}{2} \,{\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 )} a^{2} b + \int b^{3} x^{2} \log \left (\sqrt{\frac{1}{c x} + 1} \sqrt{\frac{1}{c x} - 1} + \frac{1}{c x}\right )^{3} + 3 \, a b^{2} x^{2} \log \left (\sqrt{\frac{1}{c x} + 1} \sqrt{\frac{1}{c x} - 1} + \frac{1}{c x}\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arcsech(c*x))^3,x, algorithm="maxima")

[Out]

1/3*a^3*x^3 + 1/2*(2*x^3*arcsech(c*x) - (sqrt(1/(c^2*x^2) - 1)/(c^2*(1/(c^2*x^2) - 1) + c^2) + arctan(sqrt(1/(
c^2*x^2) - 1))/c^2)/c)*a^2*b + integrate(b^3*x^2*log(sqrt(1/(c*x) + 1)*sqrt(1/(c*x) - 1) + 1/(c*x))^3 + 3*a*b^
2*x^2*log(sqrt(1/(c*x) + 1)*sqrt(1/(c*x) - 1) + 1/(c*x))^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b^{3} x^{2} \operatorname{arsech}\left (c x\right )^{3} + 3 \, a b^{2} x^{2} \operatorname{arsech}\left (c x\right )^{2} + 3 \, a^{2} b x^{2} \operatorname{arsech}\left (c x\right ) + a^{3} x^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arcsech(c*x))^3,x, algorithm="fricas")

[Out]

integral(b^3*x^2*arcsech(c*x)^3 + 3*a*b^2*x^2*arcsech(c*x)^2 + 3*a^2*b*x^2*arcsech(c*x) + a^3*x^2, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \left (a + b \operatorname{asech}{\left (c x \right )}\right )^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a+b*asech(c*x))**3,x)

[Out]

Integral(x**2*(a + b*asech(c*x))**3, 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 )}^{3} x^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arcsech(c*x))^3,x, algorithm="giac")

[Out]

integrate((b*arcsech(c*x) + a)^3*x^2, x)