Integrand size = 12, antiderivative size = 272 \[ \int \frac {\arcsin (a+b x)^2}{x^3} \, dx=-\frac {b \sqrt {1-(a+b x)^2} \arcsin (a+b x)}{\left (1-a^2\right ) x}-\frac {\arcsin (a+b x)^2}{2 x^2}-\frac {i a b^2 \arcsin (a+b x) \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{a-\sqrt {-1+a^2}}\right )}{\left (-1+a^2\right )^{3/2}}+\frac {i a b^2 \arcsin (a+b x) \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{a+\sqrt {-1+a^2}}\right )}{\left (-1+a^2\right )^{3/2}}+\frac {b^2 \log (x)}{1-a^2}-\frac {a b^2 \operatorname {PolyLog}\left (2,-\frac {i e^{i \arcsin (a+b x)}}{a-\sqrt {-1+a^2}}\right )}{\left (-1+a^2\right )^{3/2}}+\frac {a b^2 \operatorname {PolyLog}\left (2,-\frac {i e^{i \arcsin (a+b x)}}{a+\sqrt {-1+a^2}}\right )}{\left (-1+a^2\right )^{3/2}} \]
-1/2*arcsin(b*x+a)^2/x^2+b^2*ln(x)/(-a^2+1)-I*a*b^2*arcsin(b*x+a)*ln(1+I*( I*(b*x+a)+(1-(b*x+a)^2)^(1/2))/(a-(a^2-1)^(1/2)))/(a^2-1)^(3/2)+I*a*b^2*ar csin(b*x+a)*ln(1+I*(I*(b*x+a)+(1-(b*x+a)^2)^(1/2))/(a+(a^2-1)^(1/2)))/(a^2 -1)^(3/2)-a*b^2*polylog(2,-I*(I*(b*x+a)+(1-(b*x+a)^2)^(1/2))/(a-(a^2-1)^(1 /2)))/(a^2-1)^(3/2)+a*b^2*polylog(2,-I*(I*(b*x+a)+(1-(b*x+a)^2)^(1/2))/(a+ (a^2-1)^(1/2)))/(a^2-1)^(3/2)-b*arcsin(b*x+a)*(1-(b*x+a)^2)^(1/2)/(-a^2+1) /x
Time = 0.15 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.15 \[ \int \frac {\arcsin (a+b x)^2}{x^3} \, dx=\frac {2 \sqrt {-1+a^2} b x \sqrt {1-(a+b x)^2} \arcsin (a+b x)+\sqrt {-1+a^2} \arcsin (a+b x)^2-a^2 \sqrt {-1+a^2} \arcsin (a+b x)^2-2 i a b^2 x^2 \arcsin (a+b x) \log \left (\frac {a-\sqrt {-1+a^2}+i e^{i \arcsin (a+b x)}}{a-\sqrt {-1+a^2}}\right )+2 i a b^2 x^2 \arcsin (a+b x) \log \left (\frac {a+\sqrt {-1+a^2}+i e^{i \arcsin (a+b x)}}{a+\sqrt {-1+a^2}}\right )-2 \sqrt {-1+a^2} b^2 x^2 \log (x)-2 a b^2 x^2 \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (a+b x)}}{-a+\sqrt {-1+a^2}}\right )+2 a b^2 x^2 \operatorname {PolyLog}\left (2,-\frac {i e^{i \arcsin (a+b x)}}{a+\sqrt {-1+a^2}}\right )}{2 \left (-1+a^2\right )^{3/2} x^2} \]
(2*Sqrt[-1 + a^2]*b*x*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x] + Sqrt[-1 + a^ 2]*ArcSin[a + b*x]^2 - a^2*Sqrt[-1 + a^2]*ArcSin[a + b*x]^2 - (2*I)*a*b^2* x^2*ArcSin[a + b*x]*Log[(a - Sqrt[-1 + a^2] + I*E^(I*ArcSin[a + b*x]))/(a - Sqrt[-1 + a^2])] + (2*I)*a*b^2*x^2*ArcSin[a + b*x]*Log[(a + Sqrt[-1 + a^ 2] + I*E^(I*ArcSin[a + b*x]))/(a + Sqrt[-1 + a^2])] - 2*Sqrt[-1 + a^2]*b^2 *x^2*Log[x] - 2*a*b^2*x^2*PolyLog[2, (I*E^(I*ArcSin[a + b*x]))/(-a + Sqrt[ -1 + a^2])] + 2*a*b^2*x^2*PolyLog[2, ((-I)*E^(I*ArcSin[a + b*x]))/(a + Sqr t[-1 + a^2])])/(2*(-1 + a^2)^(3/2)*x^2)
Time = 1.04 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.03, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.417, Rules used = {5304, 25, 27, 5242, 5272, 3042, 3805, 3042, 3147, 16, 3804, 25, 2694, 27, 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.
\(\displaystyle \int \frac {\arcsin (a+b x)^2}{x^3} \, dx\) |
\(\Big \downarrow \) 5304 |
\(\displaystyle \frac {\int \frac {\arcsin (a+b x)^2}{x^3}d(a+b x)}{b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int -\frac {\arcsin (a+b x)^2}{x^3}d(a+b x)}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -b^2 \int -\frac {\arcsin (a+b x)^2}{b^3 x^3}d(a+b x)\) |
\(\Big \downarrow \) 5242 |
\(\displaystyle -b^2 \left (\frac {\arcsin (a+b x)^2}{2 b^2 x^2}-\int \frac {\arcsin (a+b x)}{b^2 x^2 \sqrt {1-(a+b x)^2}}d(a+b x)\right )\) |
\(\Big \downarrow \) 5272 |
\(\displaystyle -b^2 \left (\frac {\arcsin (a+b x)^2}{2 b^2 x^2}-\int \frac {\arcsin (a+b x)}{b^2 x^2}d\arcsin (a+b x)\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -b^2 \left (\frac {\arcsin (a+b x)^2}{2 b^2 x^2}-\int \frac {\arcsin (a+b x)}{(a-\sin (\arcsin (a+b x)))^2}d\arcsin (a+b x)\right )\) |
\(\Big \downarrow \) 3805 |
\(\displaystyle -b^2 \left (\frac {\int -\frac {\sqrt {1-(a+b x)^2}}{b x}d\arcsin (a+b x)}{1-a^2}+\frac {a \int -\frac {\arcsin (a+b x)}{b x}d\arcsin (a+b x)}{1-a^2}+\frac {\sqrt {1-(a+b x)^2} \arcsin (a+b x)}{\left (1-a^2\right ) b x}+\frac {\arcsin (a+b x)^2}{2 b^2 x^2}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -b^2 \left (\frac {a \int \frac {\arcsin (a+b x)}{a-\sin (\arcsin (a+b x))}d\arcsin (a+b x)}{1-a^2}+\frac {\int \frac {\cos (\arcsin (a+b x))}{a-\sin (\arcsin (a+b x))}d\arcsin (a+b x)}{1-a^2}+\frac {\sqrt {1-(a+b x)^2} \arcsin (a+b x)}{\left (1-a^2\right ) b x}+\frac {\arcsin (a+b x)^2}{2 b^2 x^2}\right )\) |
\(\Big \downarrow \) 3147 |
\(\displaystyle -b^2 \left (\frac {a \int \frac {\arcsin (a+b x)}{a-\sin (\arcsin (a+b x))}d\arcsin (a+b x)}{1-a^2}-\frac {\int \frac {1}{2 a+b x}d(-a-b x)}{1-a^2}+\frac {\sqrt {1-(a+b x)^2} \arcsin (a+b x)}{\left (1-a^2\right ) b x}+\frac {\arcsin (a+b x)^2}{2 b^2 x^2}\right )\) |
\(\Big \downarrow \) 16 |
\(\displaystyle -b^2 \left (\frac {a \int \frac {\arcsin (a+b x)}{a-\sin (\arcsin (a+b x))}d\arcsin (a+b x)}{1-a^2}+\frac {\sqrt {1-(a+b x)^2} \arcsin (a+b x)}{\left (1-a^2\right ) b x}-\frac {\log (2 a+b x)}{1-a^2}+\frac {\arcsin (a+b x)^2}{2 b^2 x^2}\right )\) |
\(\Big \downarrow \) 3804 |
\(\displaystyle -b^2 \left (\frac {2 a \int -\frac {e^{i \arcsin (a+b x)} \arcsin (a+b x)}{-2 e^{i \arcsin (a+b x)} a-i e^{2 i \arcsin (a+b x)}+i}d\arcsin (a+b x)}{1-a^2}+\frac {\sqrt {1-(a+b x)^2} \arcsin (a+b x)}{\left (1-a^2\right ) b x}-\frac {\log (2 a+b x)}{1-a^2}+\frac {\arcsin (a+b x)^2}{2 b^2 x^2}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -b^2 \left (-\frac {2 a \int \frac {e^{i \arcsin (a+b x)} \arcsin (a+b x)}{-2 e^{i \arcsin (a+b x)} a-i e^{2 i \arcsin (a+b x)}+i}d\arcsin (a+b x)}{1-a^2}+\frac {\sqrt {1-(a+b x)^2} \arcsin (a+b x)}{\left (1-a^2\right ) b x}-\frac {\log (2 a+b x)}{1-a^2}+\frac {\arcsin (a+b x)^2}{2 b^2 x^2}\right )\) |
\(\Big \downarrow \) 2694 |
\(\displaystyle -b^2 \left (-\frac {2 a \left (\frac {i \int -\frac {e^{i \arcsin (a+b x)} \arcsin (a+b x)}{2 \left (a+i e^{i \arcsin (a+b x)}-\sqrt {a^2-1}\right )}d\arcsin (a+b x)}{\sqrt {a^2-1}}-\frac {i \int -\frac {e^{i \arcsin (a+b x)} \arcsin (a+b x)}{2 \left (a+i e^{i \arcsin (a+b x)}+\sqrt {a^2-1}\right )}d\arcsin (a+b x)}{\sqrt {a^2-1}}\right )}{1-a^2}+\frac {\sqrt {1-(a+b x)^2} \arcsin (a+b x)}{\left (1-a^2\right ) b x}-\frac {\log (2 a+b x)}{1-a^2}+\frac {\arcsin (a+b x)^2}{2 b^2 x^2}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -b^2 \left (-\frac {2 a \left (\frac {i \int \frac {e^{i \arcsin (a+b x)} \arcsin (a+b x)}{a+i e^{i \arcsin (a+b x)}+\sqrt {a^2-1}}d\arcsin (a+b x)}{2 \sqrt {a^2-1}}-\frac {i \int \frac {e^{i \arcsin (a+b x)} \arcsin (a+b x)}{a+i e^{i \arcsin (a+b x)}-\sqrt {a^2-1}}d\arcsin (a+b x)}{2 \sqrt {a^2-1}}\right )}{1-a^2}+\frac {\sqrt {1-(a+b x)^2} \arcsin (a+b x)}{\left (1-a^2\right ) b x}-\frac {\log (2 a+b x)}{1-a^2}+\frac {\arcsin (a+b x)^2}{2 b^2 x^2}\right )\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle -b^2 \left (-\frac {2 a \left (\frac {i \left (\int \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{a+\sqrt {a^2-1}}\right )d\arcsin (a+b x)-\arcsin (a+b x) \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{\sqrt {a^2-1}+a}\right )\right )}{2 \sqrt {a^2-1}}-\frac {i \left (\int \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{a-\sqrt {a^2-1}}\right )d\arcsin (a+b x)-\arcsin (a+b x) \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{a-\sqrt {a^2-1}}\right )\right )}{2 \sqrt {a^2-1}}\right )}{1-a^2}+\frac {\sqrt {1-(a+b x)^2} \arcsin (a+b x)}{\left (1-a^2\right ) b x}-\frac {\log (2 a+b x)}{1-a^2}+\frac {\arcsin (a+b x)^2}{2 b^2 x^2}\right )\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle -b^2 \left (-\frac {2 a \left (\frac {i \left (-i \int e^{-i \arcsin (a+b x)} \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{a+\sqrt {a^2-1}}\right )de^{i \arcsin (a+b x)}-\arcsin (a+b x) \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{\sqrt {a^2-1}+a}\right )\right )}{2 \sqrt {a^2-1}}-\frac {i \left (-i \int e^{-i \arcsin (a+b x)} \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{a-\sqrt {a^2-1}}\right )de^{i \arcsin (a+b x)}-\arcsin (a+b x) \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{a-\sqrt {a^2-1}}\right )\right )}{2 \sqrt {a^2-1}}\right )}{1-a^2}+\frac {\sqrt {1-(a+b x)^2} \arcsin (a+b x)}{\left (1-a^2\right ) b x}-\frac {\log (2 a+b x)}{1-a^2}+\frac {\arcsin (a+b x)^2}{2 b^2 x^2}\right )\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle -b^2 \left (-\frac {2 a \left (\frac {i \left (i \operatorname {PolyLog}\left (2,-\frac {i e^{i \arcsin (a+b x)}}{a+\sqrt {a^2-1}}\right )-\arcsin (a+b x) \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{\sqrt {a^2-1}+a}\right )\right )}{2 \sqrt {a^2-1}}-\frac {i \left (i \operatorname {PolyLog}\left (2,-\frac {i e^{i \arcsin (a+b x)}}{a-\sqrt {a^2-1}}\right )-\arcsin (a+b x) \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{a-\sqrt {a^2-1}}\right )\right )}{2 \sqrt {a^2-1}}\right )}{1-a^2}+\frac {\sqrt {1-(a+b x)^2} \arcsin (a+b x)}{\left (1-a^2\right ) b x}-\frac {\log (2 a+b x)}{1-a^2}+\frac {\arcsin (a+b x)^2}{2 b^2 x^2}\right )\) |
-(b^2*((Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/((1 - a^2)*b*x) + ArcSin[a + b*x]^2/(2*b^2*x^2) - Log[2*a + b*x]/(1 - a^2) - (2*a*(((-1/2*I)*(-(ArcSi n[a + b*x]*Log[1 + (I*E^(I*ArcSin[a + b*x]))/(a - Sqrt[-1 + a^2])]) + I*Po lyLog[2, ((-I)*E^(I*ArcSin[a + b*x]))/(a - Sqrt[-1 + a^2])]))/Sqrt[-1 + a^ 2] + ((I/2)*(-(ArcSin[a + b*x]*Log[1 + (I*E^(I*ArcSin[a + b*x]))/(a + Sqrt [-1 + a^2])]) + I*PolyLog[2, ((-I)*E^(I*ArcSin[a + b*x]))/(a + Sqrt[-1 + a ^2])]))/Sqrt[-1 + a^2]))/(1 - a^2)))
3.2.37.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) *(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q) Int [(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q) Int[(f + g*x) ^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && 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[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.), x_Symbol] :> Simp[1/(b^p*f) Subst[Int[(a + x)^m*(b^2 - x^2)^((p - 1) /2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Sy mbol] :> Simp[2 Int[(c + d*x)^m*(E^(I*(e + f*x))/(I*b + 2*a*E^(I*(e + f*x )) - I*b*E^(2*I*(e + f*x)))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ [a^2 - b^2, 0] && IGtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_ Symbol] :> Simp[b*(c + d*x)^m*(Cos[e + f*x]/(f*(a^2 - b^2)*(a + b*Sin[e + f *x]))), x] + (Simp[a/(a^2 - b^2) Int[(c + d*x)^m/(a + b*Sin[e + f*x]), x] , x] - Simp[b*d*(m/(f*(a^2 - b^2))) Int[(c + d*x)^(m - 1)*(Cos[e + f*x]/( a + b*Sin[e + f*x])), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(m_.), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcSin[c*x])^n/(e*(m + 1))), x] - Simp[b*c*(n/(e*(m + 1))) Int[(d + e*x)^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.))/Sq rt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[1/(c^(m + 1)*Sqrt[d]) Subst[In t[(a + b*x)^n*(c*f + g*Sin[x])^m, x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c , d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && GtQ[d, 0] && (G tQ[m, 0] || IGtQ[n, 0])
Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m _.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*A rcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Time = 1.29 (sec) , antiderivative size = 521, normalized size of antiderivative = 1.92
method | result | size |
derivativedivides | \(b^{2} \left (-\frac {\arcsin \left (b x +a \right ) \left (-\arcsin \left (b x +a \right )+2 i a^{2}-4 i a \left (b x +a \right )+a^{2} \arcsin \left (b x +a \right )+2 a \sqrt {1-\left (b x +a \right )^{2}}+2 i \left (b x +a \right )^{2}-2 \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}\right )}{2 \left (a^{2}-1\right ) b^{2} x^{2}}-\frac {\ln \left (i \left (i \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )^{2}+2 a \left (i \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )-i\right )}{a^{2}-1}+\frac {2 \ln \left (i \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )}{a^{2}-1}+\frac {\sqrt {-a^{2}+1}\, a \arcsin \left (b x +a \right ) \ln \left (\frac {i a +\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a +\sqrt {-a^{2}+1}}\right )}{\left (a^{2}-1\right )^{2}}-\frac {\sqrt {-a^{2}+1}\, a \arcsin \left (b x +a \right ) \ln \left (\frac {i a -\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a -\sqrt {-a^{2}+1}}\right )}{\left (a^{2}-1\right )^{2}}-\frac {i \sqrt {-a^{2}+1}\, \operatorname {dilog}\left (\frac {i a +\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a +\sqrt {-a^{2}+1}}\right ) a}{\left (a^{2}-1\right )^{2}}+\frac {i \sqrt {-a^{2}+1}\, \operatorname {dilog}\left (\frac {i a -\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a -\sqrt {-a^{2}+1}}\right ) a}{\left (a^{2}-1\right )^{2}}\right )\) | \(521\) |
default | \(b^{2} \left (-\frac {\arcsin \left (b x +a \right ) \left (-\arcsin \left (b x +a \right )+2 i a^{2}-4 i a \left (b x +a \right )+a^{2} \arcsin \left (b x +a \right )+2 a \sqrt {1-\left (b x +a \right )^{2}}+2 i \left (b x +a \right )^{2}-2 \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}\right )}{2 \left (a^{2}-1\right ) b^{2} x^{2}}-\frac {\ln \left (i \left (i \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )^{2}+2 a \left (i \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )-i\right )}{a^{2}-1}+\frac {2 \ln \left (i \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )}{a^{2}-1}+\frac {\sqrt {-a^{2}+1}\, a \arcsin \left (b x +a \right ) \ln \left (\frac {i a +\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a +\sqrt {-a^{2}+1}}\right )}{\left (a^{2}-1\right )^{2}}-\frac {\sqrt {-a^{2}+1}\, a \arcsin \left (b x +a \right ) \ln \left (\frac {i a -\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a -\sqrt {-a^{2}+1}}\right )}{\left (a^{2}-1\right )^{2}}-\frac {i \sqrt {-a^{2}+1}\, \operatorname {dilog}\left (\frac {i a +\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a +\sqrt {-a^{2}+1}}\right ) a}{\left (a^{2}-1\right )^{2}}+\frac {i \sqrt {-a^{2}+1}\, \operatorname {dilog}\left (\frac {i a -\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a -\sqrt {-a^{2}+1}}\right ) a}{\left (a^{2}-1\right )^{2}}\right )\) | \(521\) |
b^2*(-1/2*arcsin(b*x+a)*(-arcsin(b*x+a)+2*I*a^2-4*I*a*(b*x+a)+a^2*arcsin(b *x+a)+2*a*(1-(b*x+a)^2)^(1/2)+2*I*(b*x+a)^2-2*(b*x+a)*(1-(b*x+a)^2)^(1/2)) /(a^2-1)/b^2/x^2-1/(a^2-1)*ln(I*(I*(b*x+a)+(1-(b*x+a)^2)^(1/2))^2+2*a*(I*( b*x+a)+(1-(b*x+a)^2)^(1/2))-I)+2/(a^2-1)*ln(I*(b*x+a)+(1-(b*x+a)^2)^(1/2)) +(-a^2+1)^(1/2)/(a^2-1)^2*a*arcsin(b*x+a)*ln((I*a+(-a^2+1)^(1/2)-I*(b*x+a) -(1-(b*x+a)^2)^(1/2))/(I*a+(-a^2+1)^(1/2)))-(-a^2+1)^(1/2)/(a^2-1)^2*a*arc sin(b*x+a)*ln((I*a-(-a^2+1)^(1/2)-I*(b*x+a)-(1-(b*x+a)^2)^(1/2))/(I*a-(-a^ 2+1)^(1/2)))-I*(-a^2+1)^(1/2)/(a^2-1)^2*dilog((I*a+(-a^2+1)^(1/2)-I*(b*x+a )-(1-(b*x+a)^2)^(1/2))/(I*a+(-a^2+1)^(1/2)))*a+I*(-a^2+1)^(1/2)/(a^2-1)^2* dilog((I*a-(-a^2+1)^(1/2)-I*(b*x+a)-(1-(b*x+a)^2)^(1/2))/(I*a-(-a^2+1)^(1/ 2)))*a)
\[ \int \frac {\arcsin (a+b x)^2}{x^3} \, dx=\int { \frac {\arcsin \left (b x + a\right )^{2}}{x^{3}} \,d x } \]
\[ \int \frac {\arcsin (a+b x)^2}{x^3} \, dx=\int \frac {\operatorname {asin}^{2}{\left (a + b x \right )}}{x^{3}}\, dx \]
Exception generated. \[ \int \frac {\arcsin (a+b x)^2}{x^3} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a-1>0)', see `assume?` for more details)Is
\[ \int \frac {\arcsin (a+b x)^2}{x^3} \, dx=\int { \frac {\arcsin \left (b x + a\right )^{2}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\arcsin (a+b x)^2}{x^3} \, dx=\int \frac {{\mathrm {asin}\left (a+b\,x\right )}^2}{x^3} \,d x \]