Integrand size = 23, antiderivative size = 139 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x^3} \, dx=-\frac {b c d \sqrt {1-c^2 x^2}}{2 x}-\frac {1}{2} b c^2 d \arcsin (c x)-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}+\frac {i c^2 d (a+b \arcsin (c x))^2}{2 b}-c^2 d (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )+\frac {1}{2} i b c^2 d \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) \] Output:
-1/2*b*c*d*(-c^2*x^2+1)^(1/2)/x-1/2*b*c^2*d*arcsin(c*x)-1/2*d*(-c^2*x^2+1) *(a+b*arcsin(c*x))/x^2+1/2*I*c^2*d*(a+b*arcsin(c*x))^2/b-c^2*d*(a+b*arcsin (c*x))*ln(1-(I*c*x+(-c^2*x^2+1)^(1/2))^2)+1/2*I*b*c^2*d*polylog(2,(I*c*x+( -c^2*x^2+1)^(1/2))^2)
Time = 0.08 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.81 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x^3} \, dx=-\frac {a d}{2 x^2}-\frac {b c d \sqrt {1-c^2 x^2}}{2 x}-\frac {b d \arcsin (c x)}{2 x^2}-b c^2 d \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )-a c^2 d \log (x)+\frac {1}{2} i b c^2 d \left (\arcsin (c x)^2+\operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right ) \] Input:
Integrate[((d - c^2*d*x^2)*(a + b*ArcSin[c*x]))/x^3,x]
Output:
-1/2*(a*d)/x^2 - (b*c*d*Sqrt[1 - c^2*x^2])/(2*x) - (b*d*ArcSin[c*x])/(2*x^ 2) - b*c^2*d*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] - a*c^2*d*Log[x] + (I/2)*b*c^2*d*(ArcSin[c*x]^2 + PolyLog[2, E^((2*I)*ArcSin[c*x])])
Time = 0.59 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.99, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {5190, 247, 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.
\(\displaystyle \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x^3} \, dx\) |
\(\Big \downarrow \) 5190 |
\(\displaystyle c^2 (-d) \int \frac {a+b \arcsin (c x)}{x}dx+\frac {1}{2} b c d \int \frac {\sqrt {1-c^2 x^2}}{x^2}dx-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}\) |
\(\Big \downarrow \) 247 |
\(\displaystyle c^2 (-d) \int \frac {a+b \arcsin (c x)}{x}dx+\frac {1}{2} b c d \left (c^2 \left (-\int \frac {1}{\sqrt {1-c^2 x^2}}dx\right )-\frac {\sqrt {1-c^2 x^2}}{x}\right )-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle c^2 (-d) \int \frac {a+b \arcsin (c x)}{x}dx-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c d \left (-c \arcsin (c x)-\frac {\sqrt {1-c^2 x^2}}{x}\right )\) |
\(\Big \downarrow \) 5136 |
\(\displaystyle c^2 (-d) \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c x}d\arcsin (c x)-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c d \left (-c \arcsin (c x)-\frac {\sqrt {1-c^2 x^2}}{x}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle c^2 (-d) \int -\left ((a+b \arcsin (c x)) \tan \left (\arcsin (c x)+\frac {\pi }{2}\right )\right )d\arcsin (c x)-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c d \left (-c \arcsin (c x)-\frac {\sqrt {1-c^2 x^2}}{x}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle c^2 d \int (a+b \arcsin (c x)) \tan \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c d \left (-c \arcsin (c x)-\frac {\sqrt {1-c^2 x^2}}{x}\right )\) |
\(\Big \downarrow \) 4200 |
\(\displaystyle c^2 (-d) \left (2 i \int -\frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))}{1-e^{2 i \arcsin (c x)}}d\arcsin (c x)-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c d \left (-c \arcsin (c x)-\frac {\sqrt {1-c^2 x^2}}{x}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle c^2 (-d) \left (-2 i \int \frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))}{1-e^{2 i \arcsin (c x)}}d\arcsin (c x)-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c d \left (-c \arcsin (c x)-\frac {\sqrt {1-c^2 x^2}}{x}\right )\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle c^2 (-d) \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{2} i b \int \log \left (1-e^{2 i \arcsin (c x)}\right )d\arcsin (c x)\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c d \left (-c \arcsin (c x)-\frac {\sqrt {1-c^2 x^2}}{x}\right )\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle c^2 (-d) \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \log \left (1-e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c d \left (-c \arcsin (c x)-\frac {\sqrt {1-c^2 x^2}}{x}\right )\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle c^2 (-d) \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{2 x^2}+\frac {1}{2} b c d \left (-c \arcsin (c x)-\frac {\sqrt {1-c^2 x^2}}{x}\right )\) |
Input:
Int[((d - c^2*d*x^2)*(a + b*ArcSin[c*x]))/x^3,x]
Output:
-1/2*(d*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/x^2 + (b*c*d*(-(Sqrt[1 - c^2*x^ 2]/x) - c*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))
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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^p/(c*(m + 1))), x] - Simp[2*b*(p/(c^2*(m + 1))) Int[ (c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
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_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_) ^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*ArcSin[c*x ])/(f*(m + 1))), x] + (-Simp[b*c*(d^p/(f*(m + 1))) Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p - 1/2), x], x] - Simp[2*e*(p/(f^2*(m + 1))) Int[(f*x)^(m + 2 )*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && ILtQ[(m + 1)/2, 0]
Time = 0.26 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.24
method | result | size |
parts | \(-a d \,c^{2} \ln \left (x \right )-\frac {a d}{2 x^{2}}-d b \,c^{2} \left (-\frac {i \arcsin \left (c x \right )^{2}}{2}+\frac {-i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )}{2 c^{2} x^{2}}+\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )\right )\) | \(172\) |
derivativedivides | \(c^{2} \left (-a d \left (\ln \left (c x \right )+\frac {1}{2 c^{2} x^{2}}\right )-d b \left (-\frac {i \arcsin \left (c x \right )^{2}}{2}+\frac {-i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )}{2 c^{2} x^{2}}+\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )\right )\right )\) | \(174\) |
default | \(c^{2} \left (-a d \left (\ln \left (c x \right )+\frac {1}{2 c^{2} x^{2}}\right )-d b \left (-\frac {i \arcsin \left (c x \right )^{2}}{2}+\frac {-i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )}{2 c^{2} x^{2}}+\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )\right )\right )\) | \(174\) |
Input:
int((-c^2*d*x^2+d)*(a+b*arcsin(c*x))/x^3,x,method=_RETURNVERBOSE)
Output:
-a*d*c^2*ln(x)-1/2*a*d/x^2-d*b*c^2*(-1/2*I*arcsin(c*x)^2+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))+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))-I*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2)))
\[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x^3} \, dx=\int { -\frac {{\left (c^{2} d x^{2} - d\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{x^{3}} \,d x } \] Input:
integrate((-c^2*d*x^2+d)*(a+b*arcsin(c*x))/x^3,x, algorithm="fricas")
Output:
integral(-(a*c^2*d*x^2 - a*d + (b*c^2*d*x^2 - b*d)*arcsin(c*x))/x^3, x)
\[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x^3} \, dx=- d \left (\int \left (- \frac {a}{x^{3}}\right )\, dx + \int \frac {a c^{2}}{x}\, dx + \int \left (- \frac {b \operatorname {asin}{\left (c x \right )}}{x^{3}}\right )\, dx + \int \frac {b c^{2} \operatorname {asin}{\left (c x \right )}}{x}\, dx\right ) \] Input:
integrate((-c**2*d*x**2+d)*(a+b*asin(c*x))/x**3,x)
Output:
-d*(Integral(-a/x**3, x) + Integral(a*c**2/x, x) + Integral(-b*asin(c*x)/x **3, x) + Integral(b*c**2*asin(c*x)/x, x))
\[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x^3} \, dx=\int { -\frac {{\left (c^{2} d x^{2} - d\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{x^{3}} \,d x } \] Input:
integrate((-c^2*d*x^2+d)*(a+b*arcsin(c*x))/x^3,x, algorithm="maxima")
Output:
-b*c^2*d*integrate(arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/x, x) - a*c^ 2*d*log(x) - 1/2*b*d*(sqrt(-c^2*x^2 + 1)*c/x + arcsin(c*x)/x^2) - 1/2*a*d/ x^2
Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x^3} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((-c^2*d*x^2+d)*(a+b*arcsin(c*x))/x^3,x, algorithm="giac")
Output:
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x^3} \, dx=\int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (d-c^2\,d\,x^2\right )}{x^3} \,d x \] Input:
int(((a + b*asin(c*x))*(d - c^2*d*x^2))/x^3,x)
Output:
int(((a + b*asin(c*x))*(d - c^2*d*x^2))/x^3, x)
\[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x^3} \, dx=\frac {d \left (-\mathit {asin} \left (c x \right ) b -\sqrt {-c^{2} x^{2}+1}\, b c x -2 \left (\int \frac {\mathit {asin} \left (c x \right )}{x}d x \right ) b \,c^{2} x^{2}-2 \,\mathrm {log}\left (x \right ) a \,c^{2} x^{2}-a \right )}{2 x^{2}} \] Input:
int((-c^2*d*x^2+d)*(a+b*asin(c*x))/x^3,x)
Output:
(d*( - asin(c*x)*b - sqrt( - c**2*x**2 + 1)*b*c*x - 2*int(asin(c*x)/x,x)*b *c**2*x**2 - 2*log(x)*a*c**2*x**2 - a))/(2*x**2)