Integrand size = 25, antiderivative size = 124 \[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d-c^2 d x^2\right )} \, dx=-\frac {b c \sqrt {1-c^2 x^2}}{2 d x}-\frac {a+b \arcsin (c x)}{2 d x^2}-\frac {2 c^2 (a+b \arcsin (c x)) \text {arctanh}\left (e^{2 i \arcsin (c x)}\right )}{d}+\frac {i b c^2 \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{2 d}-\frac {i b c^2 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{2 d} \]
1/2*(-a-b*arcsin(c*x))/d/x^2-2*c^2*(a+b*arcsin(c*x))*arctanh((I*c*x+(-c^2* x^2+1)^(1/2))^2)/d+1/2*I*b*c^2*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)/d- 1/2*I*b*c^2*polylog(2,(I*c*x+(-c^2*x^2+1)^(1/2))^2)/d-1/2*b*c*(-c^2*x^2+1) ^(1/2)/d/x
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(392\) vs. \(2(124)=248\).
Time = 0.23 (sec) , antiderivative size = 392, normalized size of antiderivative = 3.16 \[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d-c^2 d x^2\right )} \, dx=-\frac {a+b c x \sqrt {1-c^2 x^2}+b \arcsin (c x)+2 i b c^2 \pi x^2 \arcsin (c x)+4 b c^2 \pi x^2 \log \left (1+e^{-i \arcsin (c x)}\right )+b c^2 \pi x^2 \log \left (1-i e^{i \arcsin (c x)}\right )+2 b c^2 x^2 \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right )-b c^2 \pi x^2 \log \left (1+i e^{i \arcsin (c x)}\right )+2 b c^2 x^2 \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right )-2 b c^2 x^2 \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )-2 a c^2 x^2 \log (x)+a c^2 x^2 \log \left (1-c^2 x^2\right )-4 b c^2 \pi x^2 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )\right )+b c^2 \pi x^2 \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-b c^2 \pi x^2 \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-2 i b c^2 x^2 \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-2 i b c^2 x^2 \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )+i b c^2 x^2 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{2 d x^2} \]
-1/2*(a + b*c*x*Sqrt[1 - c^2*x^2] + b*ArcSin[c*x] + (2*I)*b*c^2*Pi*x^2*Arc Sin[c*x] + 4*b*c^2*Pi*x^2*Log[1 + E^((-I)*ArcSin[c*x])] + b*c^2*Pi*x^2*Log [1 - I*E^(I*ArcSin[c*x])] + 2*b*c^2*x^2*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[ c*x])] - b*c^2*Pi*x^2*Log[1 + I*E^(I*ArcSin[c*x])] + 2*b*c^2*x^2*ArcSin[c* x]*Log[1 + I*E^(I*ArcSin[c*x])] - 2*b*c^2*x^2*ArcSin[c*x]*Log[1 - E^((2*I) *ArcSin[c*x])] - 2*a*c^2*x^2*Log[x] + a*c^2*x^2*Log[1 - c^2*x^2] - 4*b*c^2 *Pi*x^2*Log[Cos[ArcSin[c*x]/2]] + b*c^2*Pi*x^2*Log[-Cos[(Pi + 2*ArcSin[c*x ])/4]] - b*c^2*Pi*x^2*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (2*I)*b*c^2*x^2*P olyLog[2, (-I)*E^(I*ArcSin[c*x])] - (2*I)*b*c^2*x^2*PolyLog[2, I*E^(I*ArcS in[c*x])] + I*b*c^2*x^2*PolyLog[2, E^((2*I)*ArcSin[c*x])])/(d*x^2)
Time = 0.59 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.93, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {5204, 27, 242, 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.
| \(\displaystyle \int \frac {a+b \arcsin (c x)}{x^3 \left (d-c^2 d x^2\right )} \, dx\) |
| \(\Big \downarrow \) 5204 |
| \(\displaystyle c^2 \int \frac {a+b \arcsin (c x)}{d x \left (1-c^2 x^2\right )}dx+\frac {b c \int \frac {1}{x^2 \sqrt {1-c^2 x^2}}dx}{2 d}-\frac {a+b \arcsin (c x)}{2 d x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c^2 \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}dx}{d}+\frac {b c \int \frac {1}{x^2 \sqrt {1-c^2 x^2}}dx}{2 d}-\frac {a+b \arcsin (c x)}{2 d x^2}\) |
| \(\Big \downarrow \) 242 |
| \(\displaystyle \frac {c^2 \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}dx}{d}-\frac {a+b \arcsin (c x)}{2 d x^2}-\frac {b c \sqrt {1-c^2 x^2}}{2 d x}\) |
| \(\Big \downarrow \) 5184 |
| \(\displaystyle \frac {c^2 \int \frac {a+b \arcsin (c x)}{c x \sqrt {1-c^2 x^2}}d\arcsin (c x)}{d}-\frac {a+b \arcsin (c x)}{2 d x^2}-\frac {b c \sqrt {1-c^2 x^2}}{2 d x}\) |
| \(\Big \downarrow \) 4919 |
| \(\displaystyle \frac {2 c^2 \int (a+b \arcsin (c x)) \csc (2 \arcsin (c x))d\arcsin (c x)}{d}-\frac {a+b \arcsin (c x)}{2 d x^2}-\frac {b c \sqrt {1-c^2 x^2}}{2 d x}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 c^2 \int (a+b \arcsin (c x)) \csc (2 \arcsin (c x))d\arcsin (c x)}{d}-\frac {a+b \arcsin (c x)}{2 d x^2}-\frac {b c \sqrt {1-c^2 x^2}}{2 d x}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle \frac {2 c^2 \left (-\frac {1}{2} b \int \log \left (1-e^{2 i \arcsin (c x)}\right )d\arcsin (c x)+\frac {1}{2} b \int \log \left (1+e^{2 i \arcsin (c x)}\right )d\arcsin (c x)-\left (\text {arctanh}\left (e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )\right )}{d}-\frac {a+b \arcsin (c x)}{2 d x^2}-\frac {b c \sqrt {1-c^2 x^2}}{2 d x}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {2 c^2 \left (\frac {1}{4} i b \int e^{-2 i \arcsin (c x)} \log \left (1-e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}-\frac {1}{4} i b \int e^{-2 i \arcsin (c x)} \log \left (1+e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}-\left (\text {arctanh}\left (e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )\right )}{d}-\frac {a+b \arcsin (c x)}{2 d x^2}-\frac {b c \sqrt {1-c^2 x^2}}{2 d x}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {2 c^2 \left (-\left (\text {arctanh}\left (e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )+\frac {1}{4} i b \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )-\frac {1}{4} i b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )}{d}-\frac {a+b \arcsin (c x)}{2 d x^2}-\frac {b c \sqrt {1-c^2 x^2}}{2 d x}\) |
-1/2*(b*c*Sqrt[1 - c^2*x^2])/(d*x) - (a + b*ArcSin[c*x])/(2*d*x^2) + (2*c^ 2*(-((a + b*ArcSin[c*x])*ArcTanh[E^((2*I)*ArcSin[c*x])]) + (I/4)*b*PolyLog [2, -E^((2*I)*ArcSin[c*x])] - (I/4)*b*PolyLog[2, E^((2*I)*ArcSin[c*x])]))/ d
3.1.35.3.1 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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x ] && EqQ[m + 2*p + 3, 0] && NeQ[m, -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[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]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b *ArcSin[c*x])^n/(d*f*(m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1)) ) Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] - Simp[b* c*(n/(f*(m + 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*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]
Time = 0.18 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.97
| method | result | size |
| derivativedivides | \(c^{2} \left (-\frac {a \left (\frac {1}{2 c^{2} x^{2}}-\ln \left (c x \right )+\frac {\ln \left (c x -1\right )}{2}+\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b \left (\frac {-i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )}{2 c^{2} x^{2}}-\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}\right )\) | \(244\) |
| default | \(c^{2} \left (-\frac {a \left (\frac {1}{2 c^{2} x^{2}}-\ln \left (c x \right )+\frac {\ln \left (c x -1\right )}{2}+\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b \left (\frac {-i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )}{2 c^{2} x^{2}}-\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}\right )\) | \(244\) |
| parts | \(-\frac {a \left (\frac {1}{2 x^{2}}-c^{2} \ln \left (x \right )+\frac {c^{2} \ln \left (c x -1\right )}{2}+\frac {c^{2} \ln \left (c x +1\right )}{2}\right )}{d}-\frac {b \,c^{2} \left (\frac {-i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )}{2 c^{2} x^{2}}-\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}\) | \(247\) |
c^2*(-a/d*(1/2/c^2/x^2-ln(c*x)+1/2*ln(c*x-1)+1/2*ln(c*x+1))-b/d*(1/2*(-I*c ^2*x^2+c*x*(-c^2*x^2+1)^(1/2)+arcsin(c*x))/c^2/x^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))+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))))
\[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d-c^2 d x^2\right )} \, dx=\int { -\frac {b \arcsin \left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )} x^{3}} \,d x } \]
\[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d-c^2 d x^2\right )} \, dx=- \frac {\int \frac {a}{c^{2} x^{5} - x^{3}}\, dx + \int \frac {b \operatorname {asin}{\left (c x \right )}}{c^{2} x^{5} - x^{3}}\, dx}{d} \]
\[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d-c^2 d x^2\right )} \, dx=\int { -\frac {b \arcsin \left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )} x^{3}} \,d x } \]
-1/2*(c^2*log(c*x + 1)/d + c^2*log(c*x - 1)/d - 2*c^2*log(x)/d + 1/(d*x^2) )*a - b*integrate(arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/(c^2*d*x^5 - d*x^3), x)
\[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d-c^2 d x^2\right )} \, dx=\int { -\frac {b \arcsin \left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )} x^{3}} \,d x } \]
Timed out. \[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d-c^2 d x^2\right )} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{x^3\,\left (d-c^2\,d\,x^2\right )} \,d x \]