3.1.9 \(\int \frac {x^3 (a+b \text {ArcCos}(c x))}{(d-c^2 d x^2)^2} \, dx\) [9]

Optimal. Leaf size=155 \[ \frac {b x}{2 c^3 d^2 \sqrt {1-c^2 x^2}}+\frac {x^2 (a+b \text {ArcCos}(c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {i (a+b \text {ArcCos}(c x))^2}{2 b c^4 d^2}-\frac {b \text {ArcSin}(c x)}{2 c^4 d^2}+\frac {(a+b \text {ArcCos}(c x)) \log \left (1-e^{2 i \text {ArcCos}(c x)}\right )}{c^4 d^2}-\frac {i b \text {PolyLog}\left (2,e^{2 i \text {ArcCos}(c x)}\right )}{2 c^4 d^2} \]

[Out]

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

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Rubi [A]
time = 0.13, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {4792, 4766, 3798, 2221, 2317, 2438, 294, 222} \begin {gather*} -\frac {i (a+b \text {ArcCos}(c x))^2}{2 b c^4 d^2}+\frac {\log \left (1-e^{2 i \text {ArcCos}(c x)}\right ) (a+b \text {ArcCos}(c x))}{c^4 d^2}+\frac {x^2 (a+b \text {ArcCos}(c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {i b \text {Li}_2\left (e^{2 i \text {ArcCos}(c x)}\right )}{2 c^4 d^2}-\frac {b \text {ArcSin}(c x)}{2 c^4 d^2}+\frac {b x}{2 c^3 d^2 \sqrt {1-c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^3*(a + b*ArcCos[c*x]))/(d - c^2*d*x^2)^2,x]

[Out]

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

Rule 222

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]

Rule 294

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^
n)^(p + 1)/(b*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, 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 2317

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

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 3798

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x))))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 4766

Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/e, Subst[Int[(a
+ b*x)^n*Cot[x], x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]

Rule 4792

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

Rubi steps

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

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Mathematica [A]
time = 0.26, size = 203, normalized size = 1.31 \begin {gather*} \frac {\frac {b \sqrt {1-c^2 x^2}}{1-c x}-\frac {b \sqrt {1-c^2 x^2}}{1+c x}-\frac {2 a}{-1+c^2 x^2}+\frac {b \text {ArcCos}(c x)}{1-c x}+\frac {b \text {ArcCos}(c x)}{1+c x}-2 i b \text {ArcCos}(c x)^2+4 b \text {ArcCos}(c x) \log \left (1-e^{i \text {ArcCos}(c x)}\right )+4 b \text {ArcCos}(c x) \log \left (1+e^{i \text {ArcCos}(c x)}\right )+2 a \log \left (1-c^2 x^2\right )-4 i b \text {PolyLog}\left (2,-e^{i \text {ArcCos}(c x)}\right )-4 i b \text {PolyLog}\left (2,e^{i \text {ArcCos}(c x)}\right )}{4 c^4 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^3*(a + b*ArcCos[c*x]))/(d - c^2*d*x^2)^2,x]

[Out]

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

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Maple [A]
time = 1.00, size = 281, normalized size = 1.81

method	result	size
derivativedivides	\(\frac {\frac {a}{4 d^{2} \left (c x +1\right )}+\frac {a \ln \left (c x +1\right )}{2 d^{2}}-\frac {a}{4 d^{2} \left (c x -1\right )}+\frac {a \ln \left (c x -1\right )}{2 d^{2}}-\frac {i b \arccos \left (c x \right )^{2}}{2 d^{2}}-\frac {i b \,c^{2} x^{2}}{2 d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b c x \sqrt {-c^{2} x^{2}+1}}{2 d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \arccos \left (c x \right )}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {i b}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {b \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}-\frac {i b \polylog \left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}-\frac {i b \polylog \left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}}{c^{4}}\)	\(281\)
default	\(\frac {\frac {a}{4 d^{2} \left (c x +1\right )}+\frac {a \ln \left (c x +1\right )}{2 d^{2}}-\frac {a}{4 d^{2} \left (c x -1\right )}+\frac {a \ln \left (c x -1\right )}{2 d^{2}}-\frac {i b \arccos \left (c x \right )^{2}}{2 d^{2}}-\frac {i b \,c^{2} x^{2}}{2 d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b c x \sqrt {-c^{2} x^{2}+1}}{2 d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \arccos \left (c x \right )}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {i b}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {b \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}-\frac {i b \polylog \left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}-\frac {i b \polylog \left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}}{c^{4}}\)	\(281\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*arccos(c*x))/(-c^2*d*x^2+d)^2,x,method=_RETURNVERBOSE)

[Out]

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

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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(x^3*(a+b*arccos(c*x))/(-c^2*d*x^2+d)^2,x, algorithm="maxima")

[Out]

-1/2*a*(1/(c^6*d^2*x^2 - c^4*d^2) - log(c^2*x^2 - 1)/(c^4*d^2)) + 1/2*(((c^2*x^2 - 1)*log(c*x + 1) + (c^2*x^2
- 1)*log(-c*x + 1) - 1)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) - 2*(c^6*d^2*x^2 - c^4*d^2)*integrate(1/2*(
(c^2*x^2 - 1)*e^(1/2*log(c*x + 1) + 1/2*log(-c*x + 1))*log(c*x + 1) + (c^2*x^2 - 1)*e^(1/2*log(c*x + 1) + 1/2*
log(-c*x + 1))*log(-c*x + 1) - e^(1/2*log(c*x + 1) + 1/2*log(-c*x + 1)))/(c^9*d^2*x^6 - 2*c^7*d^2*x^4 + c^5*d^
2*x^2 + (c^7*d^2*x^4 - 2*c^5*d^2*x^2 + c^3*d^2)*e^(log(c*x + 1) + log(-c*x + 1))), x))*b/(c^6*d^2*x^2 - c^4*d^
2)

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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(x^3*(a+b*arccos(c*x))/(-c^2*d*x^2+d)^2,x, algorithm="fricas")

[Out]

integral((b*x^3*arccos(c*x) + a*x^3)/(c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {a x^{3}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {b x^{3} \operatorname {acos}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(a+b*acos(c*x))/(-c**2*d*x**2+d)**2,x)

[Out]

(Integral(a*x**3/(c**4*x**4 - 2*c**2*x**2 + 1), x) + Integral(b*x**3*acos(c*x)/(c**4*x**4 - 2*c**2*x**2 + 1),
x))/d**2

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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(x^3*(a+b*arccos(c*x))/(-c^2*d*x^2+d)^2,x, algorithm="giac")

[Out]

integrate((b*arccos(c*x) + a)*x^3/(c^2*d*x^2 - d)^2, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^3\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*(a + b*acos(c*x)))/(d - c^2*d*x^2)^2,x)

[Out]

int((x^3*(a + b*acos(c*x)))/(d - c^2*d*x^2)^2, x)

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