Integrand size = 22, antiderivative size = 76 \[ \int \frac {a+b \arccos (c x)}{d-c^2 d x^2} \, dx=\frac {2 (a+b \arccos (c x)) \text {arctanh}\left (e^{i \arccos (c x)}\right )}{c d}-\frac {i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{c d}+\frac {i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{c d} \]
2*(a+b*arccos(c*x))*arctanh(c*x+I*(-c^2*x^2+1)^(1/2))/c/d-I*b*polylog(2,-c *x-I*(-c^2*x^2+1)^(1/2))/c/d+I*b*polylog(2,c*x+I*(-c^2*x^2+1)^(1/2))/c/d
Time = 0.18 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.41 \[ \int \frac {a+b \arccos (c x)}{d-c^2 d x^2} \, dx=\frac {-2 b \arccos (c x) \log \left (1-e^{i \arccos (c x)}\right )+2 b \arccos (c x) \log \left (1+e^{i \arccos (c x)}\right )-a \log (1-c x)+a \log (1+c x)-2 i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )+2 i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{2 c d} \]
(-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])])/(2*c*d)
Time = 0.32 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.87, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {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.
| \(\displaystyle \int \frac {a+b \arccos (c x)}{d-c^2 d x^2} \, dx\) |
| \(\Big \downarrow \) 5165 |
| \(\displaystyle -\frac {\int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}d\arccos (c x)}{c d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int (a+b \arccos (c x)) \csc (\arccos (c x))d\arccos (c x)}{c d}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle -\frac {-b \int \log \left (1-e^{i \arccos (c x)}\right )d\arccos (c x)+b \int \log \left (1+e^{i \arccos (c x)}\right )d\arccos (c x)-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))}{c d}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {i b \int e^{-i \arccos (c x)} \log \left (1-e^{i \arccos (c x)}\right )de^{i \arccos (c x)}-i b \int e^{-i \arccos (c x)} \log \left (1+e^{i \arccos (c x)}\right )de^{i \arccos (c x)}-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))}{c d}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{c d}\) |
-((-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*d))
3.1.4.3.1 Defintions of rubi rules used
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]
Time = 1.70 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.84
| method | result | size |
| derivativedivides | \(\frac {\frac {a \,\operatorname {arctanh}\left (c x \right )}{d}-\frac {b \left (-\operatorname {arctanh}\left (c x \right ) \arccos \left (c x \right )-i \operatorname {arctanh}\left (c x \right ) \left (\ln \left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-\ln \left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )\right )+i \operatorname {dilog}\left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )\right )}{d}}{c}\) | \(140\) |
| default | \(\frac {\frac {a \,\operatorname {arctanh}\left (c x \right )}{d}-\frac {b \left (-\operatorname {arctanh}\left (c x \right ) \arccos \left (c x \right )-i \operatorname {arctanh}\left (c x \right ) \left (\ln \left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-\ln \left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )\right )+i \operatorname {dilog}\left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )\right )}{d}}{c}\) | \(140\) |
| parts | \(-\frac {a \ln \left (c x -1\right )}{2 d c}+\frac {a \ln \left (c x +1\right )}{2 d c}-\frac {b \left (-\operatorname {arctanh}\left (c x \right ) \arccos \left (c x \right )-i \operatorname {arctanh}\left (c x \right ) \left (\ln \left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-\ln \left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )\right )+i \operatorname {dilog}\left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )\right )}{d c}\) | \(160\) |
1/c*(a/d*arctanh(c*x)-b/d*(-arctanh(c*x)*arccos(c*x)-I*arctanh(c*x)*(ln(1- I*(c*x+1)/(-c^2*x^2+1)^(1/2))-ln(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2)))+I*dilog( 1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))-I*dilog(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2))))
\[ \int \frac {a+b \arccos (c x)}{d-c^2 d x^2} \, dx=\int { -\frac {b \arccos \left (c x\right ) + a}{c^{2} d x^{2} - d} \,d x } \]
\[ \int \frac {a+b \arccos (c x)}{d-c^2 d x^2} \, dx=- \frac {\int \frac {a}{c^{2} x^{2} - 1}\, dx + \int \frac {b \operatorname {acos}{\left (c x \right )}}{c^{2} x^{2} - 1}\, dx}{d} \]
\[ \int \frac {a+b \arccos (c x)}{d-c^2 d x^2} \, dx=\int { -\frac {b \arccos \left (c x\right ) + a}{c^{2} d x^{2} - d} \,d x } \]
1/2*a*(log(c*x + 1)/(c*d) - log(c*x - 1)/(c*d)) - 1/2*(2*c*d*integrate(1/2 *sqrt(c*x + 1)*sqrt(-c*x + 1)*(log(c*x + 1) - log(-c*x + 1))/(c^2*d*x^2 - d), x) - (log(c*x + 1) - log(-c*x + 1))*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x))*b/(c*d)
\[ \int \frac {a+b \arccos (c x)}{d-c^2 d x^2} \, dx=\int { -\frac {b \arccos \left (c x\right ) + a}{c^{2} d x^{2} - d} \,d x } \]
Timed out. \[ \int \frac {a+b \arccos (c x)}{d-c^2 d x^2} \, dx=\int \frac {a+b\,\mathrm {acos}\left (c\,x\right )}{d-c^2\,d\,x^2} \,d x \]