Integrand size = 23, antiderivative size = 121 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x} \, dx=-\frac {1}{4} b c d x \sqrt {1-c^2 x^2}-\frac {1}{4} b d \arcsin (c x)+\frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {i d (a+b \arcsin (c x))^2}{2 b}+d (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )-\frac {1}{2} i b d \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) \]
-1/4*b*d*arcsin(c*x)+1/2*d*(-c^2*x^2+1)*(a+b*arcsin(c*x))-1/2*I*d*(a+b*arc sin(c*x))^2/b+d*(a+b*arcsin(c*x))*ln(1-(I*c*x+(-c^2*x^2+1)^(1/2))^2)-1/2*I *b*d*polylog(2,(I*c*x+(-c^2*x^2+1)^(1/2))^2)-1/4*b*c*d*x*(-c^2*x^2+1)^(1/2 )
Time = 0.11 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.95 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x} \, dx=-\frac {1}{2} a c^2 d x^2-\frac {1}{4} b c d x \sqrt {1-c^2 x^2}+\frac {1}{4} b d \arcsin (c x)-\frac {1}{2} b c^2 d x^2 \arcsin (c x)+b d \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )+a d \log (x)-\frac {1}{2} i b d \left (\arcsin (c x)^2+\operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right ) \]
-1/2*(a*c^2*d*x^2) - (b*c*d*x*Sqrt[1 - c^2*x^2])/4 + (b*d*ArcSin[c*x])/4 - (b*c^2*d*x^2*ArcSin[c*x])/2 + b*d*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x ])] + a*d*Log[x] - (I/2)*b*d*(ArcSin[c*x]^2 + PolyLog[2, E^((2*I)*ArcSin[c *x])])
Time = 0.53 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.12, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {5188, 211, 223, 5136, 3042, 25, 4200, 25, 2620, 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.
| \(\displaystyle \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x} \, dx\) |
| \(\Big \downarrow \) 5188 |
| \(\displaystyle d \int \frac {a+b \arcsin (c x)}{x}dx-\frac {1}{2} b c d \int \sqrt {1-c^2 x^2}dx+\frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle d \int \frac {a+b \arcsin (c x)}{x}dx-\frac {1}{2} b c d \left (\frac {1}{2} \int \frac {1}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle d \int \frac {a+b \arcsin (c x)}{x}dx+\frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {1}{2} b c d \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\) |
| \(\Big \downarrow \) 5136 |
| \(\displaystyle d \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c x}d\arcsin (c x)+\frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {1}{2} b c d \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle d \int -\left ((a+b \arcsin (c x)) \tan \left (\arcsin (c x)+\frac {\pi }{2}\right )\right )d\arcsin (c x)+\frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {1}{2} b c d \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -d \int (a+b \arcsin (c x)) \tan \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)+\frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {1}{2} b c d \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\) |
| \(\Big \downarrow \) 4200 |
| \(\displaystyle d \left (2 i \int -\frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))}{1-e^{2 i \arcsin (c x)}}d\arcsin (c x)-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )+\frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {1}{2} b c d \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle d \left (-2 i \int \frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))}{1-e^{2 i \arcsin (c x)}}d\arcsin (c x)-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )+\frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {1}{2} b c d \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle d \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{2} i b \int \log \left (1-e^{2 i \arcsin (c x)}\right )d\arcsin (c x)\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )+\frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {1}{2} b c d \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle d \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \log \left (1-e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )+\frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {1}{2} b c d \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))+d \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )-\frac {1}{2} b c d \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\) |
(d*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/2 - (b*c*d*((x*Sqrt[1 - c^2*x^2])/2 + ArcSin[c*x]/(2*c)))/2 + d*(((-1/2*I)*(a + b*ArcSin[c*x])^2)/b - (2*I)*(( I/2)*(a + b*ArcSin[c*x])*Log[1 - E^((2*I)*ArcSin[c*x])] + (b*PolyLog[2, E^ ((2*I)*ArcSin[c*x])])/4))
3.1.6.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
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]
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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[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]
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[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol ] :> Simp[I*((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[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]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[( a + b*x)^n*Cot[x], x], x, ArcSin[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]
Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.))/(x_), x_Symbol] :> Simp[(d + e*x^2)^p*((a + b*ArcSin[c*x])/(2*p)), x] + (Simp[d Int[(d + e*x^2)^(p - 1)*((a + b*ArcSin[c*x])/x), x], x] - Simp[b*c*(d^p/(2 *p)) Int[(1 - c^2*x^2)^(p - 1/2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
Time = 0.14 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.30
| method | result | size |
| parts | \(-d a \left (\frac {c^{2} x^{2}}{2}-\ln \left (x \right )\right )-d b \left (\frac {i \arcsin \left (c x \right )^{2}}{2}-\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {\arcsin \left (c x \right ) \cos \left (2 \arcsin \left (c x \right )\right )}{4}+\frac {\sin \left (2 \arcsin \left (c x \right )\right )}{8}\right )\) | \(157\) |
| derivativedivides | \(-d a \left (\frac {c^{2} x^{2}}{2}-\ln \left (c x \right )\right )-d b \left (\frac {i \arcsin \left (c x \right )^{2}}{2}-\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {\arcsin \left (c x \right ) \cos \left (2 \arcsin \left (c x \right )\right )}{4}+\frac {\sin \left (2 \arcsin \left (c x \right )\right )}{8}\right )\) | \(159\) |
| default | \(-d a \left (\frac {c^{2} x^{2}}{2}-\ln \left (c x \right )\right )-d b \left (\frac {i \arcsin \left (c x \right )^{2}}{2}-\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {\arcsin \left (c x \right ) \cos \left (2 \arcsin \left (c x \right )\right )}{4}+\frac {\sin \left (2 \arcsin \left (c x \right )\right )}{8}\right )\) | \(159\) |
-d*a*(1/2*c^2*x^2-ln(x))-d*b*(1/2*I*arcsin(c*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))+I*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))-1/4*arcsi n(c*x)*cos(2*arcsin(c*x))+1/8*sin(2*arcsin(c*x)))
\[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x} \, dx=\int { -\frac {{\left (c^{2} d x^{2} - d\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{x} \,d x } \]
\[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x} \, dx=- d \left (\int \left (- \frac {a}{x}\right )\, dx + \int a c^{2} x\, dx + \int \left (- \frac {b \operatorname {asin}{\left (c x \right )}}{x}\right )\, dx + \int b c^{2} x \operatorname {asin}{\left (c x \right )}\, dx\right ) \]
-d*(Integral(-a/x, x) + Integral(a*c**2*x, x) + Integral(-b*asin(c*x)/x, x ) + Integral(b*c**2*x*asin(c*x), x))
\[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x} \, dx=\int { -\frac {{\left (c^{2} d x^{2} - d\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{x} \,d x } \]
-1/2*a*c^2*d*x^2 + a*d*log(x) - integrate((b*c^2*d*x^2 - b*d)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/x, x)
\[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x} \, dx=\int { -\frac {{\left (c^{2} d x^{2} - d\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{x} \,d x } \]
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x} \, dx=\int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (d-c^2\,d\,x^2\right )}{x} \,d x \]