Integrand size = 25, antiderivative size = 115 \[ \int \frac {x^2 (a+b \arccos (c x))}{d-c^2 d x^2} \, dx=\frac {b \sqrt {1-c^2 x^2}}{c^3 d}-\frac {x (a+b \arccos (c x))}{c^2 d}+\frac {2 (a+b \arccos (c x)) \text {arctanh}\left (e^{i \arccos (c x)}\right )}{c^3 d}-\frac {i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{c^3 d}+\frac {i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{c^3 d} \]
-x*(a+b*arccos(c*x))/c^2/d+2*(a+b*arccos(c*x))*arctanh(c*x+I*(-c^2*x^2+1)^ (1/2))/c^3/d-I*b*polylog(2,-c*x-I*(-c^2*x^2+1)^(1/2))/c^3/d+I*b*polylog(2, c*x+I*(-c^2*x^2+1)^(1/2))/c^3/d+b*(-c^2*x^2+1)^(1/2)/c^3/d
Time = 0.31 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.20 \[ \int \frac {x^2 (a+b \arccos (c x))}{d-c^2 d x^2} \, dx=-\frac {2 a c x-2 b \sqrt {1-c^2 x^2}+2 b c x \arccos (c x)+2 b \arccos (c x) \log \left (1-e^{i \arccos (c x)}\right )-2 b \arccos (c x) \log \left (1+e^{i \arccos (c x)}\right )+a \log (1-c x)-a \log (1+c x)+2 i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-2 i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{2 c^3 d} \]
-1/2*(2*a*c*x - 2*b*Sqrt[1 - c^2*x^2] + 2*b*c*x*ArcCos[c*x] + 2*b*ArcCos[c *x]*Log[1 - E^(I*ArcCos[c*x])] - 2*b*ArcCos[c*x]*Log[1 + E^(I*ArcCos[c*x]) ] + a*Log[1 - c*x] - a*Log[1 + c*x] + (2*I)*b*PolyLog[2, -E^(I*ArcCos[c*x] )] - (2*I)*b*PolyLog[2, E^(I*ArcCos[c*x])])/(c^3*d)
Time = 0.50 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.92, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {5211, 27, 241, 5165, 3042, 4671, 2715, 2838}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^2 (a+b \arccos (c x))}{d-c^2 d x^2} \, dx\) |
| \(\Big \downarrow \) 5211 |
| \(\displaystyle \frac {\int \frac {a+b \arccos (c x)}{d \left (1-c^2 x^2\right )}dx}{c^2}-\frac {b \int \frac {x}{\sqrt {1-c^2 x^2}}dx}{c d}-\frac {x (a+b \arccos (c x))}{c^2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a+b \arccos (c x)}{1-c^2 x^2}dx}{c^2 d}-\frac {b \int \frac {x}{\sqrt {1-c^2 x^2}}dx}{c d}-\frac {x (a+b \arccos (c x))}{c^2 d}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {\int \frac {a+b \arccos (c x)}{1-c^2 x^2}dx}{c^2 d}-\frac {x (a+b \arccos (c x))}{c^2 d}+\frac {b \sqrt {1-c^2 x^2}}{c^3 d}\) |
| \(\Big \downarrow \) 5165 |
| \(\displaystyle -\frac {\int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}d\arccos (c x)}{c^3 d}-\frac {x (a+b \arccos (c x))}{c^2 d}+\frac {b \sqrt {1-c^2 x^2}}{c^3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int (a+b \arccos (c x)) \csc (\arccos (c x))d\arccos (c x)}{c^3 d}-\frac {x (a+b \arccos (c x))}{c^2 d}+\frac {b \sqrt {1-c^2 x^2}}{c^3 d}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle -\frac {-b \int \log \left (1-e^{i \arccos (c x)}\right )d\arccos (c x)+b \int \log \left (1+e^{i \arccos (c x)}\right )d\arccos (c x)-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))}{c^3 d}-\frac {x (a+b \arccos (c x))}{c^2 d}+\frac {b \sqrt {1-c^2 x^2}}{c^3 d}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {i b \int e^{-i \arccos (c x)} \log \left (1-e^{i \arccos (c x)}\right )de^{i \arccos (c x)}-i b \int e^{-i \arccos (c x)} \log \left (1+e^{i \arccos (c x)}\right )de^{i \arccos (c x)}-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))}{c^3 d}-\frac {x (a+b \arccos (c x))}{c^2 d}+\frac {b \sqrt {1-c^2 x^2}}{c^3 d}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{c^3 d}-\frac {x (a+b \arccos (c x))}{c^2 d}+\frac {b \sqrt {1-c^2 x^2}}{c^3 d}\) |
(b*Sqrt[1 - c^2*x^2])/(c^3*d) - (x*(a + b*ArcCos[c*x]))/(c^2*d) - (-2*(a + b*ArcCos[c*x])*ArcTanh[E^(I*ArcCos[c*x])] + I*b*PolyLog[2, -E^(I*ArcCos[c *x])] - I*b*PolyLog[2, E^(I*ArcCos[c*x])])/(c^3*d)
3.1.2.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[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]
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]
Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x )^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG tQ[m, 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbo l] :> Simp[-(c*d)^(-1) Subst[Int[(a + b*x)^n*Csc[x], x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
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/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] - S imp[b*f*(n/(c*(m + 2*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, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m , 1] && NeQ[m + 2*p + 1, 0]
Time = 1.86 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.57
| method | result | size |
| derivativedivides | \(\frac {-\frac {a \left (c x +\frac {\ln \left (c x -1\right )}{2}-\frac {\ln \left (c x +1\right )}{2}\right )}{d}+\frac {b \sqrt {-c^{2} x^{2}+1}}{d}+\frac {b \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {b \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {b \arccos \left (c x \right ) c x}{d}-\frac {i b \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {i b \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}}{c^{3}}\) | \(180\) |
| default | \(\frac {-\frac {a \left (c x +\frac {\ln \left (c x -1\right )}{2}-\frac {\ln \left (c x +1\right )}{2}\right )}{d}+\frac {b \sqrt {-c^{2} x^{2}+1}}{d}+\frac {b \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {b \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {b \arccos \left (c x \right ) c x}{d}-\frac {i b \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {i b \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}}{c^{3}}\) | \(180\) |
| parts | \(-\frac {a \left (\frac {x}{c^{2}}+\frac {\ln \left (c x -1\right )}{2 c^{3}}-\frac {\ln \left (c x +1\right )}{2 c^{3}}\right )}{d}+\frac {b \sqrt {-c^{2} x^{2}+1}}{c^{3} d}-\frac {b \arccos \left (c x \right ) x}{d \,c^{2}}-\frac {i b \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{c^{3} d}+\frac {i b \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{c^{3} d}+\frac {b \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d \,c^{3}}-\frac {b \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d \,c^{3}}\) | \(201\) |
1/c^3*(-a/d*(c*x+1/2*ln(c*x-1)-1/2*ln(c*x+1))+b/d*(-c^2*x^2+1)^(1/2)+b/d*a rccos(c*x)*ln(1+c*x+I*(-c^2*x^2+1)^(1/2))-b/d*arccos(c*x)*ln(1-c*x-I*(-c^2 *x^2+1)^(1/2))-b/d*arccos(c*x)*c*x-I*b/d*polylog(2,-c*x-I*(-c^2*x^2+1)^(1/ 2))+I*b/d*polylog(2,c*x+I*(-c^2*x^2+1)^(1/2)))
\[ \int \frac {x^2 (a+b \arccos (c x))}{d-c^2 d x^2} \, dx=\int { -\frac {{\left (b \arccos \left (c x\right ) + a\right )} x^{2}}{c^{2} d x^{2} - d} \,d x } \]
\[ \int \frac {x^2 (a+b \arccos (c x))}{d-c^2 d x^2} \, dx=- \frac {\int \frac {a x^{2}}{c^{2} x^{2} - 1}\, dx + \int \frac {b x^{2} \operatorname {acos}{\left (c x \right )}}{c^{2} x^{2} - 1}\, dx}{d} \]
\[ \int \frac {x^2 (a+b \arccos (c x))}{d-c^2 d x^2} \, dx=\int { -\frac {{\left (b \arccos \left (c x\right ) + a\right )} x^{2}}{c^{2} d x^{2} - d} \,d x } \]
-1/2*a*(2*x/(c^2*d) - log(c*x + 1)/(c^3*d) + log(c*x - 1)/(c^3*d)) - 1/2*( 2*c^3*d*integrate(-1/2*(2*c*x - log(c*x + 1) + log(-c*x + 1))*sqrt(c*x + 1 )*sqrt(-c*x + 1)/(c^4*d*x^2 - c^2*d), x) + (2*c*x - log(c*x + 1) + log(-c* x + 1))*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x))*b/(c^3*d)
\[ \int \frac {x^2 (a+b \arccos (c x))}{d-c^2 d x^2} \, dx=\int { -\frac {{\left (b \arccos \left (c x\right ) + a\right )} x^{2}}{c^{2} d x^{2} - d} \,d x } \]
Timed out. \[ \int \frac {x^2 (a+b \arccos (c x))}{d-c^2 d x^2} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}{d-c^2\,d\,x^2} \,d x \]