3.9 \(\int \frac{x^3 (a+b \cos ^{-1}(c x))}{(d-c^2 d x^2)^2} \, dx\)

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

[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)

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Rubi [A]  time = 0.18674, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {4704, 4676, 3717, 2190, 2279, 2391, 288, 216} \[ -\frac{i b \text{PolyLog}\left (2,e^{2 i \cos ^{-1}(c x)}\right )}{2 c^4 d^2}+\frac{x^2 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac{i \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c^4 d^2}+\frac{\log \left (1-e^{2 i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right )}{c^4 d^2}+\frac{b x}{2 c^3 d^2 \sqrt{1-c^2 x^2}}-\frac{b \sin ^{-1}(c x)}{2 c^4 d^2} \]

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 4704

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*d^IntPart[p]*(d + e*x^2
)^FracPart[p])/(2*c*(p + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCo
s[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] && Gt
Q[m, 1]

Rule 4676

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 3717

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 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2279

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 2391

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 288

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 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 \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{\operatorname{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) \operatorname{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 \operatorname{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) \operatorname{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*}

Mathematica [A]  time = 0.455747, size = 203, normalized size = 1.31 \[ \frac{-4 i b \text{PolyLog}\left (2,-e^{i \cos ^{-1}(c x)}\right )-4 i b \text{PolyLog}\left (2,e^{i \cos ^{-1}(c x)}\right )-\frac{2 a}{c^2 x^2-1}+2 a \log \left (1-c^2 x^2\right )+\frac{b \sqrt{1-c^2 x^2}}{1-c x}-\frac{b \sqrt{1-c^2 x^2}}{c x+1}-2 i b \cos ^{-1}(c x)^2+\frac{b \cos ^{-1}(c x)}{1-c x}+\frac{b \cos ^{-1}(c x)}{c x+1}+4 b \cos ^{-1}(c x) \log \left (1-e^{i \cos ^{-1}(c x)}\right )+4 b \cos ^{-1}(c x) \log \left (1+e^{i \cos ^{-1}(c x)}\right )}{4 c^4 d^2} \]

Warning: Unable to verify antiderivative.

[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 = 0.25, size = 312, normalized size = 2. \begin{align*} -{\frac{a}{4\,{d}^{2}{c}^{4} \left ( cx-1 \right ) }}+{\frac{a\ln \left ( cx-1 \right ) }{2\,{d}^{2}{c}^{4}}}+{\frac{a}{4\,{d}^{2}{c}^{4} \left ( cx+1 \right ) }}+{\frac{a\ln \left ( cx+1 \right ) }{2\,{d}^{2}{c}^{4}}}-{\frac{{\frac{i}{2}}b \left ( \arccos \left ( cx \right ) \right ) ^{2}}{{d}^{2}{c}^{4}}}-{\frac{{\frac{i}{2}}b{x}^{2}}{{c}^{2}{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{bx}{2\,{c}^{3}{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{b\arccos \left ( cx \right ) }{2\,{d}^{2}{c}^{4} \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{{\frac{i}{2}}b}{{d}^{2}{c}^{4} \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{b\arccos \left ( cx \right ) }{{d}^{2}{c}^{4}}\ln \left ( 1+cx+i\sqrt{-{c}^{2}{x}^{2}+1} \right ) }+{\frac{b\arccos \left ( cx \right ) }{{d}^{2}{c}^{4}}\ln \left ( 1-cx-i\sqrt{-{c}^{2}{x}^{2}+1} \right ) }-{\frac{ib}{{d}^{2}{c}^{4}}{\it polylog} \left ( 2,-cx-i\sqrt{-{c}^{2}{x}^{2}+1} \right ) }-{\frac{ib}{{d}^{2}{c}^{4}}{\it polylog} \left ( 2,cx+i\sqrt{-{c}^{2}{x}^{2}+1} \right ) } \end{align*}

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)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{2} \, a{\left (\frac{1}{c^{6} d^{2} x^{2} - c^{4} d^{2}} - \frac{\log \left (c^{2} x^{2} - 1\right )}{c^{4} d^{2}}\right )} + \frac{{\left ({\left ({\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right ) +{\left (c^{2} x^{2} - 1\right )} \log \left (-c x + 1\right ) - 1\right )} \arctan \left (\sqrt{c x + 1} \sqrt{-c x + 1}, c x\right ) -{\left (c^{6} d^{2} x^{2} - c^{4} d^{2}\right )} \int \frac{{\left (c^{2} x^{2} - 1\right )} e^{\left (\frac{1}{2} \, \log \left (c x + 1\right ) + \frac{1}{2} \, \log \left (-c x + 1\right )\right )} \log \left (c x + 1\right ) +{\left (c^{2} x^{2} - 1\right )} e^{\left (\frac{1}{2} \, \log \left (c x + 1\right ) + \frac{1}{2} \, \log \left (-c x + 1\right )\right )} \log \left (-c x + 1\right ) - e^{\left (\frac{1}{2} \, \log \left (c x + 1\right ) + \frac{1}{2} \, \log \left (-c x + 1\right )\right )}}{c^{9} d^{2} x^{6} - 2 \, c^{7} d^{2} x^{4} + c^{5} d^{2} x^{2} -{\left (c^{7} d^{2} x^{4} - 2 \, c^{5} d^{2} x^{2} + c^{3} d^{2}\right )}{\left (c x + 1\right )}{\left (c x - 1\right )}}\,{d x}\right )} b}{2 \,{\left (c^{6} d^{2} x^{2} - c^{4} d^{2}\right )}} \end{align*}

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., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b x^{3} \arccos \left (c x\right ) + a x^{3}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \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{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \arccos \left (c x\right ) + a\right )} x^{3}}{{\left (c^{2} d x^{2} - d\right )}^{2}}\,{d x} \end{align*}

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)