Integrand size = 25, antiderivative size = 71 \[ \int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )} \, dx=-\frac {2 (a+b \arcsin (c x)) \text {arctanh}\left (e^{2 i \arcsin (c x)}\right )}{d}+\frac {i b \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{2 d}-\frac {i b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{2 d} \]
-2*(a+b*arcsin(c*x))*arctanh((I*c*x+(-c^2*x^2+1)^(1/2))^2)/d+1/2*I*b*polyl og(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)/d-1/2*I*b*polylog(2,(I*c*x+(-c^2*x^2+1 )^(1/2))^2)/d
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(274\) vs. \(2(71)=142\).
Time = 0.18 (sec) , antiderivative size = 274, normalized size of antiderivative = 3.86 \[ \int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )} \, dx=-\frac {2 i b \pi \arcsin (c x)+4 b \pi \log \left (1+e^{-i \arcsin (c x)}\right )+b \pi \log \left (1-i e^{i \arcsin (c x)}\right )+2 b \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right )-b \pi \log \left (1+i e^{i \arcsin (c x)}\right )+2 b \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right )-2 b \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )-2 a \log (x)+a \log \left (1-c^2 x^2\right )-4 b \pi \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )\right )+b \pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-b \pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-2 i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-2 i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )+i b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{2 d} \]
-1/2*((2*I)*b*Pi*ArcSin[c*x] + 4*b*Pi*Log[1 + E^((-I)*ArcSin[c*x])] + b*Pi *Log[1 - I*E^(I*ArcSin[c*x])] + 2*b*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x] )] - b*Pi*Log[1 + I*E^(I*ArcSin[c*x])] + 2*b*ArcSin[c*x]*Log[1 + I*E^(I*Ar cSin[c*x])] - 2*b*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] - 2*a*Log[x] + a*Log[1 - c^2*x^2] - 4*b*Pi*Log[Cos[ArcSin[c*x]/2]] + b*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] - b*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (2*I)*b*Pol yLog[2, (-I)*E^(I*ArcSin[c*x])] - (2*I)*b*PolyLog[2, I*E^(I*ArcSin[c*x])] + I*b*PolyLog[2, E^((2*I)*ArcSin[c*x])])/d
Time = 0.39 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.94, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {5184, 4919, 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.
| \(\displaystyle \int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )} \, dx\) |
| \(\Big \downarrow \) 5184 |
| \(\displaystyle \frac {\int \frac {a+b \arcsin (c x)}{c x \sqrt {1-c^2 x^2}}d\arcsin (c x)}{d}\) |
| \(\Big \downarrow \) 4919 |
| \(\displaystyle \frac {2 \int (a+b \arcsin (c x)) \csc (2 \arcsin (c x))d\arcsin (c x)}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \int (a+b \arcsin (c x)) \csc (2 \arcsin (c x))d\arcsin (c x)}{d}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle \frac {2 \left (-\frac {1}{2} b \int \log \left (1-e^{2 i \arcsin (c x)}\right )d\arcsin (c x)+\frac {1}{2} b \int \log \left (1+e^{2 i \arcsin (c x)}\right )d\arcsin (c x)-\left (\text {arctanh}\left (e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )\right )}{d}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {2 \left (\frac {1}{4} i b \int e^{-2 i \arcsin (c x)} \log \left (1-e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}-\frac {1}{4} i b \int e^{-2 i \arcsin (c x)} \log \left (1+e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}-\left (\text {arctanh}\left (e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )\right )}{d}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {2 \left (-\left (\text {arctanh}\left (e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )+\frac {1}{4} i b \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )-\frac {1}{4} i b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )}{d}\) |
(2*(-((a + b*ArcSin[c*x])*ArcTanh[E^((2*I)*ArcSin[c*x])]) + (I/4)*b*PolyLo g[2, -E^((2*I)*ArcSin[c*x])] - (I/4)*b*PolyLog[2, E^((2*I)*ArcSin[c*x])])) /d
3.1.33.3.1 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[Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n Int[(c + d*x)^m*Csc[2*a + 2*b*x]^n , x], x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[n] && RationalQ[m]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[1/d Subst[Int[(a + b*x)^n/(Cos[x]*Sin[x]), x], x, ArcSi n[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 192 vs. \(2 (95 ) = 190\).
Time = 0.13 (sec) , antiderivative size = 193, normalized size of antiderivative = 2.72
| method | result | size |
| parts | \(-\frac {a \left (-\ln \left (x \right )+\frac {\ln \left (c x -1\right )}{2}+\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b \left (-\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}\) | \(193\) |
| derivativedivides | \(-\frac {a \left (-\ln \left (c x \right )+\frac {\ln \left (c x -1\right )}{2}+\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b \left (-\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}\) | \(195\) |
| default | \(-\frac {a \left (-\ln \left (c x \right )+\frac {\ln \left (c x -1\right )}{2}+\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b \left (-\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}\) | \(195\) |
-a/d*(-ln(x)+1/2*ln(c*x-1)+1/2*ln(c*x+1))-b/d*(-arcsin(c*x)*ln(1+I*c*x+(-c ^2*x^2+1)^(1/2))+I*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))+arcsin(c*x)*ln(1+( I*c*x+(-c^2*x^2+1)^(1/2))^2)-1/2*I*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2 )-arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))+I*polylog(2,I*c*x+(-c^2*x^2+1 )^(1/2)))
\[ \int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )} \, dx=\int { -\frac {b \arcsin \left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )} x} \,d x } \]
\[ \int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )} \, dx=- \frac {\int \frac {a}{c^{2} x^{3} - x}\, dx + \int \frac {b \operatorname {asin}{\left (c x \right )}}{c^{2} x^{3} - x}\, dx}{d} \]
\[ \int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )} \, dx=\int { -\frac {b \arcsin \left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )} x} \,d x } \]
-1/2*a*(log(c*x + 1)/d + log(c*x - 1)/d - 2*log(x)/d) - b*integrate(arctan 2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/(c^2*d*x^3 - d*x), x)
\[ \int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )} \, dx=\int { -\frac {b \arcsin \left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )} x} \,d x } \]
Timed out. \[ \int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{x\,\left (d-c^2\,d\,x^2\right )} \,d x \]