\(\int x^2 (a+b \sec ^{-1}(c x))^3 \, dx\) [25]

Optimal result

Integrand size = 14, antiderivative size = 236 \[ \int x^2 \left (a+b \sec ^{-1}(c x)\right )^3 \, dx=\frac {b^2 x \left (a+b \sec ^{-1}(c x)\right )}{c^2}-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \sec ^{-1}(c x)\right )^2}{2 c}+\frac {1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )^3+\frac {i b \left (a+b \sec ^{-1}(c x)\right )^2 \arctan \left (e^{i \sec ^{-1}(c x)}\right )}{c^3}-\frac {b^3 \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c^3}-\frac {i b^2 \left (a+b \sec ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,-i e^{i \sec ^{-1}(c x)}\right )}{c^3}+\frac {i b^2 \left (a+b \sec ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,i e^{i \sec ^{-1}(c x)}\right )}{c^3}+\frac {b^3 \operatorname {PolyLog}\left (3,-i e^{i \sec ^{-1}(c x)}\right )}{c^3}-\frac {b^3 \operatorname {PolyLog}\left (3,i e^{i \sec ^{-1}(c x)}\right )}{c^3} \]

[Out]

b^2*x*(a+b*arcsec(c*x))/c^2+1/3*x^3*(a+b*arcsec(c*x))^3+I*b*(a+b*arcsec(c*x))^2*arctan(1/c/x+I*(1-1/c^2/x^2)^(
1/2))/c^3-b^3*arctanh((1-1/c^2/x^2)^(1/2))/c^3-I*b^2*(a+b*arcsec(c*x))*polylog(2,-I*(1/c/x+I*(1-1/c^2/x^2)^(1/
2)))/c^3+I*b^2*(a+b*arcsec(c*x))*polylog(2,I*(1/c/x+I*(1-1/c^2/x^2)^(1/2)))/c^3+b^3*polylog(3,-I*(1/c/x+I*(1-1
/c^2/x^2)^(1/2)))/c^3-b^3*polylog(3,I*(1/c/x+I*(1-1/c^2/x^2)^(1/2)))/c^3-1/2*b*x^2*(a+b*arcsec(c*x))^2*(1-1/c^
2/x^2)^(1/2)/c

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5330, 4494, 4271, 3855, 4266, 2611, 2320, 6724} \[ \int x^2 \left (a+b \sec ^{-1}(c x)\right )^3 \, dx=\frac {i b \arctan \left (e^{i \sec ^{-1}(c x)}\right ) \left (a+b \sec ^{-1}(c x)\right )^2}{c^3}-\frac {i b^2 \operatorname {PolyLog}\left (2,-i e^{i \sec ^{-1}(c x)}\right ) \left (a+b \sec ^{-1}(c x)\right )}{c^3}+\frac {i b^2 \operatorname {PolyLog}\left (2,i e^{i \sec ^{-1}(c x)}\right ) \left (a+b \sec ^{-1}(c x)\right )}{c^3}+\frac {b^2 x \left (a+b \sec ^{-1}(c x)\right )}{c^2}-\frac {b x^2 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{2 c}+\frac {1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )^3-\frac {b^3 \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c^3}+\frac {b^3 \operatorname {PolyLog}\left (3,-i e^{i \sec ^{-1}(c x)}\right )}{c^3}-\frac {b^3 \operatorname {PolyLog}\left (3,i e^{i \sec ^{-1}(c x)}\right )}{c^3} \]

[In]

Int[x^2*(a + b*ArcSec[c*x])^3,x]

[Out]

(b^2*x*(a + b*ArcSec[c*x]))/c^2 - (b*Sqrt[1 - 1/(c^2*x^2)]*x^2*(a + b*ArcSec[c*x])^2)/(2*c) + (x^3*(a + b*ArcS
ec[c*x])^3)/3 + (I*b*(a + b*ArcSec[c*x])^2*ArcTan[E^(I*ArcSec[c*x])])/c^3 - (b^3*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]
])/c^3 - (I*b^2*(a + b*ArcSec[c*x])*PolyLog[2, (-I)*E^(I*ArcSec[c*x])])/c^3 + (I*b^2*(a + b*ArcSec[c*x])*PolyL
og[2, I*E^(I*ArcSec[c*x])])/c^3 + (b^3*PolyLog[3, (-I)*E^(I*ArcSec[c*x])])/c^3 - (b^3*PolyLog[3, I*E^(I*ArcSec
[c*x])])/c^3

Rule 2320

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 2611

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 3855

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

Rule 4266

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

Rule 4271

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]) /; Free
Q[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]

Rule 4494

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

Rule 5330

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

Rule 6724

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*} \text {integral}& = \frac {\text {Subst}\left (\int (a+b x)^3 \sec ^3(x) \tan (x) \, dx,x,\sec ^{-1}(c x)\right )}{c^3} \\ & = \frac {1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )^3-\frac {b \text {Subst}\left (\int (a+b x)^2 \sec ^3(x) \, dx,x,\sec ^{-1}(c x)\right )}{c^3} \\ & = \frac {b^2 x \left (a+b \sec ^{-1}(c x)\right )}{c^2}-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \sec ^{-1}(c x)\right )^2}{2 c}+\frac {1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )^3-\frac {b \text {Subst}\left (\int (a+b x)^2 \sec (x) \, dx,x,\sec ^{-1}(c x)\right )}{2 c^3}-\frac {b^3 \text {Subst}\left (\int \sec (x) \, dx,x,\sec ^{-1}(c x)\right )}{c^3} \\ & = \frac {b^2 x \left (a+b \sec ^{-1}(c x)\right )}{c^2}-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \sec ^{-1}(c x)\right )^2}{2 c}+\frac {1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )^3+\frac {i b \left (a+b \sec ^{-1}(c x)\right )^2 \arctan \left (e^{i \sec ^{-1}(c x)}\right )}{c^3}-\frac {b^3 \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c^3}+\frac {b^2 \text {Subst}\left (\int (a+b x) \log \left (1-i e^{i x}\right ) \, dx,x,\sec ^{-1}(c x)\right )}{c^3}-\frac {b^2 \text {Subst}\left (\int (a+b x) \log \left (1+i e^{i x}\right ) \, dx,x,\sec ^{-1}(c x)\right )}{c^3} \\ & = \frac {b^2 x \left (a+b \sec ^{-1}(c x)\right )}{c^2}-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \sec ^{-1}(c x)\right )^2}{2 c}+\frac {1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )^3+\frac {i b \left (a+b \sec ^{-1}(c x)\right )^2 \arctan \left (e^{i \sec ^{-1}(c x)}\right )}{c^3}-\frac {b^3 \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c^3}-\frac {i b^2 \left (a+b \sec ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,-i e^{i \sec ^{-1}(c x)}\right )}{c^3}+\frac {i b^2 \left (a+b \sec ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,i e^{i \sec ^{-1}(c x)}\right )}{c^3}+\frac {\left (i b^3\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-i e^{i x}\right ) \, dx,x,\sec ^{-1}(c x)\right )}{c^3}-\frac {\left (i b^3\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,i e^{i x}\right ) \, dx,x,\sec ^{-1}(c x)\right )}{c^3} \\ & = \frac {b^2 x \left (a+b \sec ^{-1}(c x)\right )}{c^2}-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \sec ^{-1}(c x)\right )^2}{2 c}+\frac {1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )^3+\frac {i b \left (a+b \sec ^{-1}(c x)\right )^2 \arctan \left (e^{i \sec ^{-1}(c x)}\right )}{c^3}-\frac {b^3 \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c^3}-\frac {i b^2 \left (a+b \sec ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,-i e^{i \sec ^{-1}(c x)}\right )}{c^3}+\frac {i b^2 \left (a+b \sec ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,i e^{i \sec ^{-1}(c x)}\right )}{c^3}+\frac {b^3 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{i \sec ^{-1}(c x)}\right )}{c^3}-\frac {b^3 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{i \sec ^{-1}(c x)}\right )}{c^3} \\ & = \frac {b^2 x \left (a+b \sec ^{-1}(c x)\right )}{c^2}-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \sec ^{-1}(c x)\right )^2}{2 c}+\frac {1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )^3+\frac {i b \left (a+b \sec ^{-1}(c x)\right )^2 \arctan \left (e^{i \sec ^{-1}(c x)}\right )}{c^3}-\frac {b^3 \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c^3}-\frac {i b^2 \left (a+b \sec ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,-i e^{i \sec ^{-1}(c x)}\right )}{c^3}+\frac {i b^2 \left (a+b \sec ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,i e^{i \sec ^{-1}(c x)}\right )}{c^3}+\frac {b^3 \operatorname {PolyLog}\left (3,-i e^{i \sec ^{-1}(c x)}\right )}{c^3}-\frac {b^3 \operatorname {PolyLog}\left (3,i e^{i \sec ^{-1}(c x)}\right )}{c^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.18 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.71 \[ \int x^2 \left (a+b \sec ^{-1}(c x)\right )^3 \, dx=\frac {6 a b^2 c x-3 a^2 b c^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2+2 a^3 c^3 x^3+6 b^3 c x \sec ^{-1}(c x)-6 a b^2 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2 \sec ^{-1}(c x)+6 a^2 b c^3 x^3 \sec ^{-1}(c x)-3 b^3 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2 \sec ^{-1}(c x)^2+6 a b^2 c^3 x^3 \sec ^{-1}(c x)^2+2 b^3 c^3 x^3 \sec ^{-1}(c x)^3+6 i b^3 \sec ^{-1}(c x)^2 \arctan \left (e^{i \sec ^{-1}(c x)}\right )-6 b^3 \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )-6 a b^2 \sec ^{-1}(c x) \log \left (1-i e^{i \sec ^{-1}(c x)}\right )+6 a b^2 \sec ^{-1}(c x) \log \left (1+i e^{i \sec ^{-1}(c x)}\right )-3 a^2 b \log \left (\left (1+\sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )-6 i b^2 \left (a+b \sec ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,-i e^{i \sec ^{-1}(c x)}\right )+6 i b^2 \left (a+b \sec ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,i e^{i \sec ^{-1}(c x)}\right )+6 b^3 \operatorname {PolyLog}\left (3,-i e^{i \sec ^{-1}(c x)}\right )-6 b^3 \operatorname {PolyLog}\left (3,i e^{i \sec ^{-1}(c x)}\right )}{6 c^3} \]

[In]

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

[Out]

(6*a*b^2*c*x - 3*a^2*b*c^2*Sqrt[1 - 1/(c^2*x^2)]*x^2 + 2*a^3*c^3*x^3 + 6*b^3*c*x*ArcSec[c*x] - 6*a*b^2*c^2*Sqr
t[1 - 1/(c^2*x^2)]*x^2*ArcSec[c*x] + 6*a^2*b*c^3*x^3*ArcSec[c*x] - 3*b^3*c^2*Sqrt[1 - 1/(c^2*x^2)]*x^2*ArcSec[
c*x]^2 + 6*a*b^2*c^3*x^3*ArcSec[c*x]^2 + 2*b^3*c^3*x^3*ArcSec[c*x]^3 + (6*I)*b^3*ArcSec[c*x]^2*ArcTan[E^(I*Arc
Sec[c*x])] - 6*b^3*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]] - 6*a*b^2*ArcSec[c*x]*Log[1 - I*E^(I*ArcSec[c*x])] + 6*a*b^2
*ArcSec[c*x]*Log[1 + I*E^(I*ArcSec[c*x])] - 3*a^2*b*Log[(1 + Sqrt[1 - 1/(c^2*x^2)])*x] - (6*I)*b^2*(a + b*ArcS
ec[c*x])*PolyLog[2, (-I)*E^(I*ArcSec[c*x])] + (6*I)*b^2*(a + b*ArcSec[c*x])*PolyLog[2, I*E^(I*ArcSec[c*x])] +
6*b^3*PolyLog[3, (-I)*E^(I*ArcSec[c*x])] - 6*b^3*PolyLog[3, I*E^(I*ArcSec[c*x])])/(6*c^3)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 573 vs. \(2 (286 ) = 572\).

Time = 1.44 (sec) , antiderivative size = 574, normalized size of antiderivative = 2.43

method	result	size
derivativedivides	\(\frac {\frac {a^{3} c^{3} x^{3}}{3}+b^{3} \left (\frac {\operatorname {arcsec}\left (c x \right ) \left (2 c^{2} x^{2} \operatorname {arcsec}\left (c x \right )^{2}-3 \,\operatorname {arcsec}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+6\right ) c x}{6}+\frac {\operatorname {arcsec}\left (c x \right )^{2} \ln \left (1+i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{2}-i \operatorname {arcsec}\left (c x \right ) \operatorname {polylog}\left (2, -i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+\operatorname {polylog}\left (3, -i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )-\frac {\operatorname {arcsec}\left (c x \right )^{2} \ln \left (1-i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{2}+i \operatorname {arcsec}\left (c x \right ) \operatorname {polylog}\left (2, i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )-\operatorname {polylog}\left (3, i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+2 i \arctan \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+3 a \,b^{2} \left (\frac {\left (c^{2} x^{2} \operatorname {arcsec}\left (c x \right )^{2}-\operatorname {arcsec}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+1\right ) c x}{3}+\frac {\operatorname {arcsec}\left (c x \right ) \ln \left (1+i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{3}-\frac {\operatorname {arcsec}\left (c x \right ) \ln \left (1-i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{3}-\frac {i \operatorname {dilog}\left (1+i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{3}+\frac {i \operatorname {dilog}\left (1-i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{3}\right )+3 a^{2} b \left (\frac {c^{3} x^{3} \operatorname {arcsec}\left (c x \right )}{3}-\frac {\sqrt {c^{2} x^{2}-1}\, \left (c x \sqrt {c^{2} x^{2}-1}+\ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{3}}\)	\(574\)
default	\(\frac {\frac {a^{3} c^{3} x^{3}}{3}+b^{3} \left (\frac {\operatorname {arcsec}\left (c x \right ) \left (2 c^{2} x^{2} \operatorname {arcsec}\left (c x \right )^{2}-3 \,\operatorname {arcsec}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+6\right ) c x}{6}+\frac {\operatorname {arcsec}\left (c x \right )^{2} \ln \left (1+i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{2}-i \operatorname {arcsec}\left (c x \right ) \operatorname {polylog}\left (2, -i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+\operatorname {polylog}\left (3, -i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )-\frac {\operatorname {arcsec}\left (c x \right )^{2} \ln \left (1-i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{2}+i \operatorname {arcsec}\left (c x \right ) \operatorname {polylog}\left (2, i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )-\operatorname {polylog}\left (3, i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+2 i \arctan \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+3 a \,b^{2} \left (\frac {\left (c^{2} x^{2} \operatorname {arcsec}\left (c x \right )^{2}-\operatorname {arcsec}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+1\right ) c x}{3}+\frac {\operatorname {arcsec}\left (c x \right ) \ln \left (1+i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{3}-\frac {\operatorname {arcsec}\left (c x \right ) \ln \left (1-i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{3}-\frac {i \operatorname {dilog}\left (1+i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{3}+\frac {i \operatorname {dilog}\left (1-i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{3}\right )+3 a^{2} b \left (\frac {c^{3} x^{3} \operatorname {arcsec}\left (c x \right )}{3}-\frac {\sqrt {c^{2} x^{2}-1}\, \left (c x \sqrt {c^{2} x^{2}-1}+\ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{3}}\)	\(574\)
parts	\(\frac {a^{3} x^{3}}{3}+\frac {b^{3} \left (\frac {\operatorname {arcsec}\left (c x \right ) \left (2 c^{2} x^{2} \operatorname {arcsec}\left (c x \right )^{2}-3 \,\operatorname {arcsec}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+6\right ) c x}{6}+\frac {\operatorname {arcsec}\left (c x \right )^{2} \ln \left (1+i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{2}-i \operatorname {arcsec}\left (c x \right ) \operatorname {polylog}\left (2, -i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+\operatorname {polylog}\left (3, -i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )-\frac {\operatorname {arcsec}\left (c x \right )^{2} \ln \left (1-i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{2}+i \operatorname {arcsec}\left (c x \right ) \operatorname {polylog}\left (2, i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )-\operatorname {polylog}\left (3, i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+2 i \arctan \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{c^{3}}+\frac {3 a \,b^{2} \left (\frac {\left (c^{2} x^{2} \operatorname {arcsec}\left (c x \right )^{2}-\operatorname {arcsec}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+1\right ) c x}{3}+\frac {\operatorname {arcsec}\left (c x \right ) \ln \left (1+i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{3}-\frac {\operatorname {arcsec}\left (c x \right ) \ln \left (1-i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{3}-\frac {i \operatorname {dilog}\left (1+i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{3}+\frac {i \operatorname {dilog}\left (1-i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{3}\right )}{c^{3}}+\frac {3 a^{2} b \left (\frac {c^{3} x^{3} \operatorname {arcsec}\left (c x \right )}{3}-\frac {\sqrt {c^{2} x^{2}-1}\, \left (c x \sqrt {c^{2} x^{2}-1}+\ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{3}}\)	\(576\)

[In]

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

[Out]

1/c^3*(1/3*a^3*c^3*x^3+b^3*(1/6*arcsec(c*x)*(2*c^2*x^2*arcsec(c*x)^2-3*arcsec(c*x)*c*x*((c^2*x^2-1)/c^2/x^2)^(
1/2)+6)*c*x+1/2*arcsec(c*x)^2*ln(1+I*(1/c/x+I*(1-1/c^2/x^2)^(1/2)))-I*arcsec(c*x)*polylog(2,-I*(1/c/x+I*(1-1/c
^2/x^2)^(1/2)))+polylog(3,-I*(1/c/x+I*(1-1/c^2/x^2)^(1/2)))-1/2*arcsec(c*x)^2*ln(1-I*(1/c/x+I*(1-1/c^2/x^2)^(1
/2)))+I*arcsec(c*x)*polylog(2,I*(1/c/x+I*(1-1/c^2/x^2)^(1/2)))-polylog(3,I*(1/c/x+I*(1-1/c^2/x^2)^(1/2)))+2*I*
arctan(1/c/x+I*(1-1/c^2/x^2)^(1/2)))+3*a*b^2*(1/3*(c^2*x^2*arcsec(c*x)^2-arcsec(c*x)*c*x*((c^2*x^2-1)/c^2/x^2)
^(1/2)+1)*c*x+1/3*arcsec(c*x)*ln(1+I*(1/c/x+I*(1-1/c^2/x^2)^(1/2)))-1/3*arcsec(c*x)*ln(1-I*(1/c/x+I*(1-1/c^2/x
^2)^(1/2)))-1/3*I*dilog(1+I*(1/c/x+I*(1-1/c^2/x^2)^(1/2)))+1/3*I*dilog(1-I*(1/c/x+I*(1-1/c^2/x^2)^(1/2))))+3*a
^2*b*(1/3*c^3*x^3*arcsec(c*x)-1/6*(c^2*x^2-1)^(1/2)*(c*x*(c^2*x^2-1)^(1/2)+ln(c*x+(c^2*x^2-1)^(1/2)))/((c^2*x^
2-1)/c^2/x^2)^(1/2)/c/x))

Fricas [F]

\[ \int x^2 \left (a+b \sec ^{-1}(c x)\right )^3 \, dx=\int { {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}^{3} x^{2} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int x^2 \left (a+b \sec ^{-1}(c x)\right )^3 \, dx=\int x^{2} \left (a + b \operatorname {asec}{\left (c x \right )}\right )^{3}\, dx \]

[In]

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

[Out]

Integral(x**2*(a + b*asec(c*x))**3, x)

Maxima [F]

\[ \int x^2 \left (a+b \sec ^{-1}(c x)\right )^3 \, dx=\int { {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}^{3} x^{2} \,d x } \]

[In]

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

[Out]

1/3*b^3*x^3*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))^3 - 1/4*b^3*x^3*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))*log(c^2*x^
2)^2 - 1/2*a*b^2*c^2*(2*(c^2*x^3 + 3*x)/c^4 - 3*log(c*x + 1)/c^5 + 3*log(c*x - 1)/c^5)*log(c)^2 - 12*b^3*c^2*i
ntegrate(1/4*x^4*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))/(c^2*x^2 - 1), x)*log(c)^2 + 12*b^3*c^2*integrate(1/4*x^4
*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))*log(c^2*x^2)/(c^2*x^2 - 1), x)*log(c) - 24*b^3*c^2*integrate(1/4*x^4*arct
an(sqrt(c*x + 1)*sqrt(c*x - 1))*log(x)/(c^2*x^2 - 1), x)*log(c) + 12*a*b^2*c^2*integrate(1/4*x^4*log(c^2*x^2)/
(c^2*x^2 - 1), x)*log(c) - 24*a*b^2*c^2*integrate(1/4*x^4*log(x)/(c^2*x^2 - 1), x)*log(c) + 1/3*a^3*x^3 + 12*b
^3*c^2*integrate(1/4*x^4*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))*log(c^2*x^2)*log(x)/(c^2*x^2 - 1), x) - 12*b^3*c^
2*integrate(1/4*x^4*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))*log(x)^2/(c^2*x^2 - 1), x) + 12*a*b^2*c^2*integrate(1/
4*x^4*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))^2/(c^2*x^2 - 1), x) + 4*b^3*c^2*integrate(1/4*x^4*arctan(sqrt(c*x +
1)*sqrt(c*x - 1))*log(c^2*x^2)/(c^2*x^2 - 1), x) - 3*a*b^2*c^2*integrate(1/4*x^4*log(c^2*x^2)^2/(c^2*x^2 - 1),
x) + 12*a*b^2*c^2*integrate(1/4*x^4*log(c^2*x^2)*log(x)/(c^2*x^2 - 1), x) - 12*a*b^2*c^2*integrate(1/4*x^4*lo
g(x)^2/(c^2*x^2 - 1), x) + 3/2*a*b^2*(2*x/c^2 - log(c*x + 1)/c^3 + log(c*x - 1)/c^3)*log(c)^2 + 12*b^3*integra
te(1/4*x^2*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))/(c^2*x^2 - 1), x)*log(c)^2 - 12*b^3*integrate(1/4*x^2*arctan(sq
rt(c*x + 1)*sqrt(c*x - 1))*log(c^2*x^2)/(c^2*x^2 - 1), x)*log(c) + 24*b^3*integrate(1/4*x^2*arctan(sqrt(c*x +
1)*sqrt(c*x - 1))*log(x)/(c^2*x^2 - 1), x)*log(c) - 12*a*b^2*integrate(1/4*x^2*log(c^2*x^2)/(c^2*x^2 - 1), x)*
log(c) + 24*a*b^2*integrate(1/4*x^2*log(x)/(c^2*x^2 - 1), x)*log(c) + 1/4*(4*x^3*arcsec(c*x) - (2*sqrt(-1/(c^2
*x^2) + 1)/(c^2*(1/(c^2*x^2) - 1) + c^2) + log(sqrt(-1/(c^2*x^2) + 1) + 1)/c^2 - log(sqrt(-1/(c^2*x^2) + 1) -
1)/c^2)/c)*a^2*b - 4*b^3*integrate(1/4*sqrt(c*x + 1)*sqrt(c*x - 1)*x^2*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))^2/(
c^2*x^2 - 1), x) + b^3*integrate(1/4*sqrt(c*x + 1)*sqrt(c*x - 1)*x^2*log(c^2*x^2)^2/(c^2*x^2 - 1), x) - 12*b^3
*integrate(1/4*x^2*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))*log(c^2*x^2)*log(x)/(c^2*x^2 - 1), x) + 12*b^3*integrat
e(1/4*x^2*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))*log(x)^2/(c^2*x^2 - 1), x) - 12*a*b^2*integrate(1/4*x^2*arctan(s
qrt(c*x + 1)*sqrt(c*x - 1))^2/(c^2*x^2 - 1), x) - 4*b^3*integrate(1/4*x^2*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))*
log(c^2*x^2)/(c^2*x^2 - 1), x) + 3*a*b^2*integrate(1/4*x^2*log(c^2*x^2)^2/(c^2*x^2 - 1), x) - 12*a*b^2*integra
te(1/4*x^2*log(c^2*x^2)*log(x)/(c^2*x^2 - 1), x) + 12*a*b^2*integrate(1/4*x^2*log(x)^2/(c^2*x^2 - 1), x)

Giac [F]

\[ \int x^2 \left (a+b \sec ^{-1}(c x)\right )^3 \, dx=\int { {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}^{3} x^{2} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int x^2 \left (a+b \sec ^{-1}(c x)\right )^3 \, dx=\int x^2\,{\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}^3 \,d x \]

[In]

int(x^2*(a + b*acos(1/(c*x)))^3,x)

[Out]

int(x^2*(a + b*acos(1/(c*x)))^3, x)