3.2.56 \(\int \frac {(a+b \text {ArcCos}(c x))^3}{x} \, dx\) [156]

Optimal. Leaf size=127 \[ -\frac {i (a+b \text {ArcCos}(c x))^4}{4 b}+(a+b \text {ArcCos}(c x))^3 \log \left (1+e^{2 i \text {ArcCos}(c x)}\right )-\frac {3}{2} i b (a+b \text {ArcCos}(c x))^2 \text {PolyLog}\left (2,-e^{2 i \text {ArcCos}(c x)}\right )+\frac {3}{2} b^2 (a+b \text {ArcCos}(c x)) \text {PolyLog}\left (3,-e^{2 i \text {ArcCos}(c x)}\right )+\frac {3}{4} i b^3 \text {PolyLog}\left (4,-e^{2 i \text {ArcCos}(c x)}\right ) \]

[Out]

-1/4*I*(a+b*arccos(c*x))^4/b+(a+b*arccos(c*x))^3*ln(1+(c*x+I*(-c^2*x^2+1)^(1/2))^2)-3/2*I*b*(a+b*arccos(c*x))^
2*polylog(2,-(c*x+I*(-c^2*x^2+1)^(1/2))^2)+3/2*b^2*(a+b*arccos(c*x))*polylog(3,-(c*x+I*(-c^2*x^2+1)^(1/2))^2)+
3/4*I*b^3*polylog(4,-(c*x+I*(-c^2*x^2+1)^(1/2))^2)

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Rubi [A]
time = 0.11, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4722, 3800, 2221, 2611, 6744, 2320, 6724} \begin {gather*} \frac {3}{2} b^2 \text {Li}_3\left (-e^{2 i \text {ArcCos}(c x)}\right ) (a+b \text {ArcCos}(c x))-\frac {3}{2} i b \text {Li}_2\left (-e^{2 i \text {ArcCos}(c x)}\right ) (a+b \text {ArcCos}(c x))^2-\frac {i (a+b \text {ArcCos}(c x))^4}{4 b}+\log \left (1+e^{2 i \text {ArcCos}(c x)}\right ) (a+b \text {ArcCos}(c x))^3+\frac {3}{4} i b^3 \text {Li}_4\left (-e^{2 i \text {ArcCos}(c x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCos[c*x])^3/x,x]

[Out]

((-1/4*I)*(a + b*ArcCos[c*x])^4)/b + (a + b*ArcCos[c*x])^3*Log[1 + E^((2*I)*ArcCos[c*x])] - ((3*I)/2)*b*(a + b
*ArcCos[c*x])^2*PolyLog[2, -E^((2*I)*ArcCos[c*x])] + (3*b^2*(a + b*ArcCos[c*x])*PolyLog[3, -E^((2*I)*ArcCos[c*
x])])/2 + ((3*I)/4)*b^3*PolyLog[4, -E^((2*I)*ArcCos[c*x])]

Rule 2221

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

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 3800

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

Rule 4722

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> -Subst[Int[(a + b*x)^n*Tan[x], x], x, ArcCos[c
*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

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]

Rule 6744

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

Rubi steps

\begin {align*} \int \frac {\left (a+b \cos ^{-1}(c x)\right )^3}{x} \, dx &=-\text {Subst}\left (\int (a+b x)^3 \tan (x) \, dx,x,\cos ^{-1}(c x)\right )\\ &=-\frac {i \left (a+b \cos ^{-1}(c x)\right )^4}{4 b}+2 i \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)^3}{1+e^{2 i x}} \, dx,x,\cos ^{-1}(c x)\right )\\ &=-\frac {i \left (a+b \cos ^{-1}(c x)\right )^4}{4 b}+\left (a+b \cos ^{-1}(c x)\right )^3 \log \left (1+e^{2 i \cos ^{-1}(c x)}\right )-(3 b) \text {Subst}\left (\int (a+b x)^2 \log \left (1+e^{2 i x}\right ) \, dx,x,\cos ^{-1}(c x)\right )\\ &=-\frac {i \left (a+b \cos ^{-1}(c x)\right )^4}{4 b}+\left (a+b \cos ^{-1}(c x)\right )^3 \log \left (1+e^{2 i \cos ^{-1}(c x)}\right )-\frac {3}{2} i b \left (a+b \cos ^{-1}(c x)\right )^2 \text {Li}_2\left (-e^{2 i \cos ^{-1}(c x)}\right )+\left (3 i b^2\right ) \text {Subst}\left (\int (a+b x) \text {Li}_2\left (-e^{2 i x}\right ) \, dx,x,\cos ^{-1}(c x)\right )\\ &=-\frac {i \left (a+b \cos ^{-1}(c x)\right )^4}{4 b}+\left (a+b \cos ^{-1}(c x)\right )^3 \log \left (1+e^{2 i \cos ^{-1}(c x)}\right )-\frac {3}{2} i b \left (a+b \cos ^{-1}(c x)\right )^2 \text {Li}_2\left (-e^{2 i \cos ^{-1}(c x)}\right )+\frac {3}{2} b^2 \left (a+b \cos ^{-1}(c x)\right ) \text {Li}_3\left (-e^{2 i \cos ^{-1}(c x)}\right )-\frac {1}{2} \left (3 b^3\right ) \text {Subst}\left (\int \text {Li}_3\left (-e^{2 i x}\right ) \, dx,x,\cos ^{-1}(c x)\right )\\ &=-\frac {i \left (a+b \cos ^{-1}(c x)\right )^4}{4 b}+\left (a+b \cos ^{-1}(c x)\right )^3 \log \left (1+e^{2 i \cos ^{-1}(c x)}\right )-\frac {3}{2} i b \left (a+b \cos ^{-1}(c x)\right )^2 \text {Li}_2\left (-e^{2 i \cos ^{-1}(c x)}\right )+\frac {3}{2} b^2 \left (a+b \cos ^{-1}(c x)\right ) \text {Li}_3\left (-e^{2 i \cos ^{-1}(c x)}\right )+\frac {1}{4} \left (3 i b^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{2 i \cos ^{-1}(c x)}\right )\\ &=-\frac {i \left (a+b \cos ^{-1}(c x)\right )^4}{4 b}+\left (a+b \cos ^{-1}(c x)\right )^3 \log \left (1+e^{2 i \cos ^{-1}(c x)}\right )-\frac {3}{2} i b \left (a+b \cos ^{-1}(c x)\right )^2 \text {Li}_2\left (-e^{2 i \cos ^{-1}(c x)}\right )+\frac {3}{2} b^2 \left (a+b \cos ^{-1}(c x)\right ) \text {Li}_3\left (-e^{2 i \cos ^{-1}(c x)}\right )+\frac {3}{4} i b^3 \text {Li}_4\left (-e^{2 i \cos ^{-1}(c x)}\right )\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 204, normalized size = 1.61 \begin {gather*} \frac {1}{4} \left (-6 i a^2 b \text {ArcCos}(c x)^2-4 i a b^2 \text {ArcCos}(c x)^3-i b^3 \text {ArcCos}(c x)^4+12 a^2 b \text {ArcCos}(c x) \log \left (1+e^{2 i \text {ArcCos}(c x)}\right )+12 a b^2 \text {ArcCos}(c x)^2 \log \left (1+e^{2 i \text {ArcCos}(c x)}\right )+4 b^3 \text {ArcCos}(c x)^3 \log \left (1+e^{2 i \text {ArcCos}(c x)}\right )+4 a^3 \log (c x)-6 i b (a+b \text {ArcCos}(c x))^2 \text {PolyLog}\left (2,-e^{2 i \text {ArcCos}(c x)}\right )+6 b^2 (a+b \text {ArcCos}(c x)) \text {PolyLog}\left (3,-e^{2 i \text {ArcCos}(c x)}\right )+3 i b^3 \text {PolyLog}\left (4,-e^{2 i \text {ArcCos}(c x)}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCos[c*x])^3/x,x]

[Out]

((-6*I)*a^2*b*ArcCos[c*x]^2 - (4*I)*a*b^2*ArcCos[c*x]^3 - I*b^3*ArcCos[c*x]^4 + 12*a^2*b*ArcCos[c*x]*Log[1 + E
^((2*I)*ArcCos[c*x])] + 12*a*b^2*ArcCos[c*x]^2*Log[1 + E^((2*I)*ArcCos[c*x])] + 4*b^3*ArcCos[c*x]^3*Log[1 + E^
((2*I)*ArcCos[c*x])] + 4*a^3*Log[c*x] - (6*I)*b*(a + b*ArcCos[c*x])^2*PolyLog[2, -E^((2*I)*ArcCos[c*x])] + 6*b
^2*(a + b*ArcCos[c*x])*PolyLog[3, -E^((2*I)*ArcCos[c*x])] + (3*I)*b^3*PolyLog[4, -E^((2*I)*ArcCos[c*x])])/4

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 352 vs. \(2 (160 ) = 320\).
time = 0.16, size = 353, normalized size = 2.78

method	result	size
derivativedivides	\(a^{3} \ln \left (c x \right )-\frac {i b^{3} \arccos \left (c x \right )^{4}}{4}+b^{3} \arccos \left (c x \right )^{3} \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {3 i b^{3} \arccos \left (c x \right )^{2} \polylog \left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}+\frac {3 b^{3} \arccos \left (c x \right ) \polylog \left (3, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}+\frac {3 i b^{3} \polylog \left (4, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{4}-i a \,b^{2} \arccos \left (c x \right )^{3}+3 a \,b^{2} \arccos \left (c x \right )^{2} \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-3 i a \,b^{2} \arccos \left (c x \right ) \polylog \left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {3 a \,b^{2} \polylog \left (3, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\frac {3 i a^{2} b \arccos \left (c x \right )^{2}}{2}+3 a^{2} b \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {3 i a^{2} b \polylog \left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\)	\(353\)
default	\(a^{3} \ln \left (c x \right )-\frac {i b^{3} \arccos \left (c x \right )^{4}}{4}+b^{3} \arccos \left (c x \right )^{3} \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {3 i b^{3} \arccos \left (c x \right )^{2} \polylog \left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}+\frac {3 b^{3} \arccos \left (c x \right ) \polylog \left (3, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}+\frac {3 i b^{3} \polylog \left (4, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{4}-i a \,b^{2} \arccos \left (c x \right )^{3}+3 a \,b^{2} \arccos \left (c x \right )^{2} \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-3 i a \,b^{2} \arccos \left (c x \right ) \polylog \left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {3 a \,b^{2} \polylog \left (3, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\frac {3 i a^{2} b \arccos \left (c x \right )^{2}}{2}+3 a^{2} b \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {3 i a^{2} b \polylog \left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\)	\(353\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccos(c*x))^3/x,x,method=_RETURNVERBOSE)

[Out]

a^3*ln(c*x)-1/4*I*b^3*arccos(c*x)^4+b^3*arccos(c*x)^3*ln(1+(c*x+I*(-c^2*x^2+1)^(1/2))^2)-3/2*I*b^3*arccos(c*x)
^2*polylog(2,-(c*x+I*(-c^2*x^2+1)^(1/2))^2)+3/2*b^3*arccos(c*x)*polylog(3,-(c*x+I*(-c^2*x^2+1)^(1/2))^2)+3/4*I
*b^3*polylog(4,-(c*x+I*(-c^2*x^2+1)^(1/2))^2)-I*a*b^2*arccos(c*x)^3+3*a*b^2*arccos(c*x)^2*ln(1+(c*x+I*(-c^2*x^
2+1)^(1/2))^2)-3*I*a*b^2*arccos(c*x)*polylog(2,-(c*x+I*(-c^2*x^2+1)^(1/2))^2)+3/2*a*b^2*polylog(3,-(c*x+I*(-c^
2*x^2+1)^(1/2))^2)-3/2*I*a^2*b*arccos(c*x)^2+3*a^2*b*arccos(c*x)*ln(1+(c*x+I*(-c^2*x^2+1)^(1/2))^2)-3/2*I*a^2*
b*polylog(2,-(c*x+I*(-c^2*x^2+1)^(1/2))^2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^3*log(x) + integrate((b^3*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^3 + 3*a*b^2*arctan2(sqrt(c*x + 1)*sqrt(
-c*x + 1), c*x)^2 + 3*a^2*b*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x))/x, x)

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Fricas [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((a+b*arccos(c*x))^3/x,x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {acos}{\left (c x \right )}\right )^{3}}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acos(c*x))**3/x,x)

[Out]

Integral((a + b*acos(c*x))**3/x, x)

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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((a+b*arccos(c*x))^3/x,x, algorithm="giac")

[Out]

integrate((b*arccos(c*x) + a)^3/x, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^3}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acos(c*x))^3/x,x)

[Out]

int((a + b*acos(c*x))^3/x, x)

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