Integrand size = 18, antiderivative size = 309 \[ \int \frac {(a+b \arcsin (c x))^2}{(d+e x)^2} \, dx=-\frac {(a+b \arcsin (c x))^2}{e (d+e x)}-\frac {2 i b c (a+b \arcsin (c x)) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}+\frac {2 i b c (a+b \arcsin (c x)) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {2 b^2 c \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}+\frac {2 b^2 c \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}} \]
-(a+b*arcsin(c*x))^2/e/(e*x+d)-2*I*b*c*(a+b*arcsin(c*x))*ln(1-I*e*(I*c*x+( -c^2*x^2+1)^(1/2))/(c*d-(c^2*d^2-e^2)^(1/2)))/e/(c^2*d^2-e^2)^(1/2)+2*I*b* c*(a+b*arcsin(c*x))*ln(1-I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d+(c^2*d^2-e^2) ^(1/2)))/e/(c^2*d^2-e^2)^(1/2)-2*b^2*c*polylog(2,I*e*(I*c*x+(-c^2*x^2+1)^( 1/2))/(c*d-(c^2*d^2-e^2)^(1/2)))/e/(c^2*d^2-e^2)^(1/2)+2*b^2*c*polylog(2,I *e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d+(c^2*d^2-e^2)^(1/2)))/e/(c^2*d^2-e^2)^( 1/2)
Time = 0.27 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.75 \[ \int \frac {(a+b \arcsin (c x))^2}{(d+e x)^2} \, dx=\frac {-\frac {(a+b \arcsin (c x))^2}{d+e x}+\frac {2 b c \left (-i (a+b \arcsin (c x)) \left (\log \left (1+\frac {i e e^{i \arcsin (c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )-\log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )-b \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )+b \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )}{\sqrt {c^2 d^2-e^2}}}{e} \]
(-((a + b*ArcSin[c*x])^2/(d + e*x)) + (2*b*c*((-I)*(a + b*ArcSin[c*x])*(Lo g[1 + (I*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])] - Log[1 - (I *e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]) - b*PolyLog[2, (I*e*E^ (I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])] + b*PolyLog[2, (I*e*E^(I*Arc Sin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]))/Sqrt[c^2*d^2 - e^2])/e
Time = 0.97 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.93, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5242, 5272, 3042, 3804, 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.
| \(\displaystyle \int \frac {(a+b \arcsin (c x))^2}{(d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 5242 |
| \(\displaystyle \frac {2 b c \int \frac {a+b \arcsin (c x)}{(d+e x) \sqrt {1-c^2 x^2}}dx}{e}-\frac {(a+b \arcsin (c x))^2}{e (d+e x)}\) |
| \(\Big \downarrow \) 5272 |
| \(\displaystyle \frac {2 b c \int \frac {a+b \arcsin (c x)}{c d+c e x}d\arcsin (c x)}{e}-\frac {(a+b \arcsin (c x))^2}{e (d+e x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 b c \int \frac {a+b \arcsin (c x)}{c d+e \sin (\arcsin (c x))}d\arcsin (c x)}{e}-\frac {(a+b \arcsin (c x))^2}{e (d+e x)}\) |
| \(\Big \downarrow \) 3804 |
| \(\displaystyle -\frac {(a+b \arcsin (c x))^2}{e (d+e x)}+\frac {4 b c \int \frac {e^{i \arcsin (c x)} (a+b \arcsin (c x))}{2 c e^{i \arcsin (c x)} d-i e e^{2 i \arcsin (c x)}+i e}d\arcsin (c x)}{e}\) |
| \(\Big \downarrow \) 2694 |
| \(\displaystyle -\frac {(a+b \arcsin (c x))^2}{e (d+e x)}+\frac {4 b c \left (\frac {i e \int \frac {e^{i \arcsin (c x)} (a+b \arcsin (c x))}{2 \left (c d-i e e^{i \arcsin (c x)}+\sqrt {c^2 d^2-e^2}\right )}d\arcsin (c x)}{\sqrt {c^2 d^2-e^2}}-\frac {i e \int \frac {e^{i \arcsin (c x)} (a+b \arcsin (c x))}{2 \left (c d-i e e^{i \arcsin (c x)}-\sqrt {c^2 d^2-e^2}\right )}d\arcsin (c x)}{\sqrt {c^2 d^2-e^2}}\right )}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {(a+b \arcsin (c x))^2}{e (d+e x)}+\frac {4 b c \left (\frac {i e \int \frac {e^{i \arcsin (c x)} (a+b \arcsin (c x))}{c d-i e e^{i \arcsin (c x)}+\sqrt {c^2 d^2-e^2}}d\arcsin (c x)}{2 \sqrt {c^2 d^2-e^2}}-\frac {i e \int \frac {e^{i \arcsin (c x)} (a+b \arcsin (c x))}{c d-i e e^{i \arcsin (c x)}-\sqrt {c^2 d^2-e^2}}d\arcsin (c x)}{2 \sqrt {c^2 d^2-e^2}}\right )}{e}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {(a+b \arcsin (c x))^2}{e (d+e x)}+\frac {4 b c \left (\frac {i e \left (\frac {(a+b \arcsin (c x)) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{e}-\frac {b \int \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )d\arcsin (c x)}{e}\right )}{2 \sqrt {c^2 d^2-e^2}}-\frac {i e \left (\frac {(a+b \arcsin (c x)) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {b \int \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )d\arcsin (c x)}{e}\right )}{2 \sqrt {c^2 d^2-e^2}}\right )}{e}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {(a+b \arcsin (c x))^2}{e (d+e x)}+\frac {4 b c \left (\frac {i e \left (\frac {i b \int e^{-i \arcsin (c x)} \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )de^{i \arcsin (c x)}}{e}+\frac {(a+b \arcsin (c x)) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{e}\right )}{2 \sqrt {c^2 d^2-e^2}}-\frac {i e \left (\frac {i b \int e^{-i \arcsin (c x)} \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )de^{i \arcsin (c x)}}{e}+\frac {(a+b \arcsin (c x)) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}\right )}{2 \sqrt {c^2 d^2-e^2}}\right )}{e}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {(a+b \arcsin (c x))^2}{e (d+e x)}+\frac {4 b c \left (\frac {i e \left (\frac {(a+b \arcsin (c x)) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{e}-\frac {i b \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}\right )}{2 \sqrt {c^2 d^2-e^2}}-\frac {i e \left (\frac {(a+b \arcsin (c x)) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {i b \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}\right )}{2 \sqrt {c^2 d^2-e^2}}\right )}{e}\) |
-((a + b*ArcSin[c*x])^2/(e*(d + e*x))) + (4*b*c*(((-1/2*I)*e*(((a + b*ArcS in[c*x])*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e - (I*b*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e)) /Sqrt[c^2*d^2 - e^2] + ((I/2)*e*(((a + b*ArcSin[c*x])*Log[1 - (I*e*E^(I*Ar cSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e - (I*b*PolyLog[2, (I*e*E^(I*Ar cSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e))/Sqrt[c^2*d^2 - e^2]))/e
3.1.14.3.1 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[(((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[((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[((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])
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 641 vs. \(2 (319 ) = 638\).
Time = 0.78 (sec) , antiderivative size = 642, normalized size of antiderivative = 2.08
| method | result | size |
| derivativedivides | \(\frac {-\frac {a^{2} c^{2}}{\left (c e x +d c \right ) e}+b^{2} c^{2} \left (-\frac {\arcsin \left (c x \right )^{2}}{e \left (c e x +d c \right )}-\frac {2 \sqrt {-c^{2} d^{2}+e^{2}}\, \arcsin \left (c x \right ) \ln \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e -\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e \left (c^{2} d^{2}-e^{2}\right )}+\frac {2 \sqrt {-c^{2} d^{2}+e^{2}}\, \arcsin \left (c x \right ) \ln \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e \left (c^{2} d^{2}-e^{2}\right )}+\frac {2 i \sqrt {-c^{2} d^{2}+e^{2}}\, \operatorname {dilog}\left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e -\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e \left (c^{2} d^{2}-e^{2}\right )}-\frac {2 i \sqrt {-c^{2} d^{2}+e^{2}}\, \operatorname {dilog}\left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e \left (c^{2} d^{2}-e^{2}\right )}\right )+2 a b \,c^{2} \left (-\frac {\arcsin \left (c x \right )}{\left (c e x +d c \right ) e}-\frac {\ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{e^{2} \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{c}\) | \(642\) |
| default | \(\frac {-\frac {a^{2} c^{2}}{\left (c e x +d c \right ) e}+b^{2} c^{2} \left (-\frac {\arcsin \left (c x \right )^{2}}{e \left (c e x +d c \right )}-\frac {2 \sqrt {-c^{2} d^{2}+e^{2}}\, \arcsin \left (c x \right ) \ln \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e -\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e \left (c^{2} d^{2}-e^{2}\right )}+\frac {2 \sqrt {-c^{2} d^{2}+e^{2}}\, \arcsin \left (c x \right ) \ln \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e \left (c^{2} d^{2}-e^{2}\right )}+\frac {2 i \sqrt {-c^{2} d^{2}+e^{2}}\, \operatorname {dilog}\left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e -\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e \left (c^{2} d^{2}-e^{2}\right )}-\frac {2 i \sqrt {-c^{2} d^{2}+e^{2}}\, \operatorname {dilog}\left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e \left (c^{2} d^{2}-e^{2}\right )}\right )+2 a b \,c^{2} \left (-\frac {\arcsin \left (c x \right )}{\left (c e x +d c \right ) e}-\frac {\ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{e^{2} \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{c}\) | \(642\) |
| parts | \(-\frac {a^{2}}{\left (e x +d \right ) e}+\frac {b^{2} \left (-\frac {c^{2} \arcsin \left (c x \right )^{2}}{e \left (c e x +d c \right )}-\frac {2 \sqrt {-c^{2} d^{2}+e^{2}}\, c^{2} \arcsin \left (c x \right ) \ln \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e -\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e \left (c^{2} d^{2}-e^{2}\right )}+\frac {2 \sqrt {-c^{2} d^{2}+e^{2}}\, c^{2} \arcsin \left (c x \right ) \ln \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e \left (c^{2} d^{2}-e^{2}\right )}+\frac {2 i \sqrt {-c^{2} d^{2}+e^{2}}\, \operatorname {dilog}\left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e -\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right ) c^{2}}{e \left (c^{2} d^{2}-e^{2}\right )}-\frac {2 i \sqrt {-c^{2} d^{2}+e^{2}}\, \operatorname {dilog}\left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right ) c^{2}}{e \left (c^{2} d^{2}-e^{2}\right )}\right )}{c}-\frac {2 a b c \arcsin \left (c x \right )}{\left (c e x +d c \right ) e}-\frac {2 a b c \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{e^{2} \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\) | \(645\) |
1/c*(-a^2*c^2/(c*e*x+c*d)/e+b^2*c^2*(-arcsin(c*x)^2/e/(c*e*x+c*d)-2*(-c^2* d^2+e^2)^(1/2)/e/(c^2*d^2-e^2)*arcsin(c*x)*ln((I*d*c+(I*c*x+(-c^2*x^2+1)^( 1/2))*e-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))+2*(-c^2*d^2+e^ 2)^(1/2)/e/(c^2*d^2-e^2)*arcsin(c*x)*ln((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))* e+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))+2*I*(-c^2*d^2+e^2)^( 1/2)/e/(c^2*d^2-e^2)*dilog((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e-(-c^2*d^2+e ^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))-2*I*(-c^2*d^2+e^2)^(1/2)/e/(c^2*d ^2-e^2)*dilog((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e+(-c^2*d^2+e^2)^(1/2))/(I *d*c+(-c^2*d^2+e^2)^(1/2))))+2*a*b*c^2*(-1/(c*e*x+c*d)/e*arcsin(c*x)-1/e^2 /(-(c^2*d^2-e^2)/e^2)^(1/2)*ln((-2*(c^2*d^2-e^2)/e^2+2*d*c/e*(c*x+d*c/e)+2 *(-(c^2*d^2-e^2)/e^2)^(1/2)*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e ^2)/e^2)^(1/2))/(c*x+d*c/e))))
\[ \int \frac {(a+b \arcsin (c x))^2}{(d+e x)^2} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{2}} \,d x } \]
\[ \int \frac {(a+b \arcsin (c x))^2}{(d+e x)^2} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\left (d + e x\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {(a+b \arcsin (c x))^2}{(d+e x)^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume((e-c*d)*(e+c*d)>0)', see `assume ?` for mor
\[ \int \frac {(a+b \arcsin (c x))^2}{(d+e x)^2} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{(d+e x)^2} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d+e\,x\right )}^2} \,d x \]