Integrand size = 33, antiderivative size = 520 \[ \int \frac {\left (e f+2 d h x+e h x^2\right ) (a+b \arcsin (c x))^2}{(d+e x)^2} \, dx=-\frac {2 b^2 h x}{e}+\frac {2 a b h \sqrt {1-c^2 x^2}}{c e}+\frac {2 b^2 h \sqrt {1-c^2 x^2} \arcsin (c x)}{c e}+\frac {h x (a+b \arcsin (c x))^2}{e}-\frac {\left (f-\frac {d^2 h}{e^2}\right ) (a+b \arcsin (c x))^2}{d+e x}+\frac {2 a b c \left (e^2 f-d^2 h\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}-\frac {2 i b^2 c \left (e^2 f-d^2 h\right ) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {2 i b^2 c \left (e^2 f-d^2 h\right ) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}-\frac {2 b^2 c \left (e^2 f-d^2 h\right ) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {2 b^2 c \left (e^2 f-d^2 h\right ) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}} \]
-2*b^2*h*x/e+h*x*(a+b*arcsin(c*x))^2/e-(f-d^2*h/e^2)*(a+b*arcsin(c*x))^2/( e*x+d)+2*a*b*c*(-d^2*h+e^2*f)*arctan((c^2*d*x+e)/(c^2*d^2-e^2)^(1/2)/(-c^2 *x^2+1)^(1/2))/e^2/(c^2*d^2-e^2)^(1/2)-2*I*b^2*c*(-d^2*h+e^2*f)*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^2/(c^2* d^2-e^2)^(1/2)+2*I*b^2*c*(-d^2*h+e^2*f)*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^2/(c^2*d^2-e^2)^(1/2)-2*b^2*c*( -d^2*h+e^2*f)*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^2/(c^2*d^2-e^2)^(1/2)+2*b^2*c*(-d^2*h+e^2*f)*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^2/(c^2*d^2-e^2)^(1/2)+2 *a*b*h*(-c^2*x^2+1)^(1/2)/c/e+2*b^2*h*arcsin(c*x)*(-c^2*x^2+1)^(1/2)/c/e
Time = 0.52 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.59 \[ \int \frac {\left (e f+2 d h x+e h x^2\right ) (a+b \arcsin (c x))^2}{(d+e x)^2} \, dx=\frac {h x (a+b \arcsin (c x))^2}{e}-\frac {\left (f-\frac {d^2 h}{e^2}\right ) (a+b \arcsin (c x))^2}{d+e x}-\frac {2 b h \left (b x-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}\right )}{e}+\frac {2 b c \left (e^2 f-d^2 h\right ) \left (-i (a+b \arcsin (c x)) \left (\log \left (1+\frac {i e e^{i \arcsin (c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )-\log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )-b \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )+b \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )}{e^2 \sqrt {c^2 d^2-e^2}} \]
(h*x*(a + b*ArcSin[c*x])^2)/e - ((f - (d^2*h)/e^2)*(a + b*ArcSin[c*x])^2)/ (d + e*x) - (2*b*h*(b*x - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c))/e + (2*b*c*(e^2*f - d^2*h)*((-I)*(a + b*ArcSin[c*x])*(Log[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*ArcSin[c*x]))/(c*d + Sqrt [c^2*d^2 - e^2])]))/(e^2*Sqrt[c^2*d^2 - e^2])
Time = 1.86 (sec) , antiderivative size = 507, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {5256, 5298, 2009}
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.
| \(\displaystyle \int \frac {(a+b \arcsin (c x))^2 \left (2 d h x+e f+e h x^2\right )}{(d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 5256 |
| \(\displaystyle -2 b c \int \frac {\left (\frac {h x}{e}-\frac {f-\frac {d^2 h}{e^2}}{d+e x}\right ) (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx-\frac {\left (f-\frac {d^2 h}{e^2}\right ) (a+b \arcsin (c x))^2}{d+e x}+\frac {h x (a+b \arcsin (c x))^2}{e}\) |
| \(\Big \downarrow \) 5298 |
| \(\displaystyle -2 b c \int \left (\frac {b \arcsin (c x) \left (h d^2+e h x d+e^2 h x^2-e^2 f\right )}{e^2 (d+e x) \sqrt {1-c^2 x^2}}+\frac {a \left (h d^2+e h x d+e^2 h x^2-e^2 f\right )}{e^2 (d+e x) \sqrt {1-c^2 x^2}}\right )dx-\frac {\left (f-\frac {d^2 h}{e^2}\right ) (a+b \arcsin (c x))^2}{d+e x}+\frac {h x (a+b \arcsin (c x))^2}{e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 b c \left (-\frac {a \left (e^2 f-d^2 h\right ) \arctan \left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}-\frac {a h \sqrt {1-c^2 x^2}}{c^2 e}+\frac {b \left (e^2 f-d^2 h\right ) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}-\frac {b \left (e^2 f-d^2 h\right ) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {i b \arcsin (c x) \left (e^2 f-d^2 h\right ) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}-\frac {i b \arcsin (c x) \left (e^2 f-d^2 h\right ) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{e^2 \sqrt {c^2 d^2-e^2}}-\frac {b h \sqrt {1-c^2 x^2} \arcsin (c x)}{c^2 e}+\frac {b h x}{c e}\right )-\frac {\left (f-\frac {d^2 h}{e^2}\right ) (a+b \arcsin (c x))^2}{d+e x}+\frac {h x (a+b \arcsin (c x))^2}{e}\) |
(h*x*(a + b*ArcSin[c*x])^2)/e - ((f - (d^2*h)/e^2)*(a + b*ArcSin[c*x])^2)/ (d + e*x) - 2*b*c*((b*h*x)/(c*e) - (a*h*Sqrt[1 - c^2*x^2])/(c^2*e) - (b*h* Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(c^2*e) - (a*(e^2*f - d^2*h)*ArcTan[(e + c^ 2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(e^2*Sqrt[c^2*d^2 - e^2]) + (I*b*(e^2*f - d^2*h)*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/(e^2*Sqrt[c^2*d^2 - e^2]) - (I*b*(e^2*f - d^2*h)*A rcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/( e^2*Sqrt[c^2*d^2 - e^2]) + (b*(e^2*f - d^2*h)*PolyLog[2, (I*e*E^(I*ArcSin[ c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/(e^2*Sqrt[c^2*d^2 - e^2]) - (b*(e^2*f - d^2*h)*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]) /(e^2*Sqrt[c^2*d^2 - e^2]))
3.2.20.3.1 Defintions of rubi rules used
Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((f_.) + (g_.)*(x_) + (h_.)*(x _)^2)^(p_.))/((d_) + (e_.)*(x_))^2, x_Symbol] :> With[{u = IntHide[(f + g*x + h*x^2)^p/(d + e*x)^2, x]}, Simp[(a + b*ArcSin[c*x])^n u, x] - Simp[b*c *n Int[SimplifyIntegrand[u*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2] ), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && IGtQ[n, 0] && IGtQ[ p, 0] && EqQ[e*g - 2*d*h, 0]
Int[(ArcSin[(c_.)*(x_)]*(b_.) + (a_))^(n_.)*(RFx_)*((d_) + (e_.)*(x_)^2)^(p _), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^p, RFx*(a + b*ArcSin[c*x]) ^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && RationalFunctionQ[RFx, x] && IGt Q[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1250 vs. \(2 (520 ) = 1040\).
Time = 3.86 (sec) , antiderivative size = 1251, normalized size of antiderivative = 2.41
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1251\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1277\) |
| default | \(\text {Expression too large to display}\) | \(1277\) |
a^2*(h/e*x-(-d^2*h+e^2*f)/e^2/(e*x+d))+2*b^2*h*arcsin(c*x)*(-c^2*x^2+1)^(1 /2)/c/e+b^2*h/e*arcsin(c*x)^2*x-2*b^2*h*x/e+b^2*c*arcsin(c*x)^2/e^2/(c*e*x +c*d)*d^2*h-b^2*c*arcsin(c*x)^2/(c*e*x+c*d)*f+2*b^2*c*(-c^2*d^2+e^2)^(1/2) /e^2/(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)))*d^2*h-2*b^2*c*(-c^2*d^2+e^ 2)^(1/2)/(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)))*f-2*b^2*c*(-c^2*d^2+e^ 2)^(1/2)/e^2/(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)))*d^2*h+2*b^2*c*(-c^ 2*d^2+e^2)^(1/2)/(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)))*f-2*I*b^2*c*(- c^2*d^2+e^2)^(1/2)/e^2/(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)))*h*d^2+2*I*b^2*c*( -c^2*d^2+e^2)^(1/2)/(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)))*f+2*I*b^2*c*(-c^2*d^ 2+e^2)^(1/2)/e^2/(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)))*h*d^2-2*I*b^2*c*(-c^2*d ^2+e^2)^(1/2)/(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)))*f+2*a*b/c*(arcsin(c*x)*h/e *c*x+arcsin(c*x)*c^2/e^2/(c*e*x+c*d)*d^2*h-arcsin(c*x)*c^2/(c*e*x+c*d)*...
\[ \int \frac {\left (e f+2 d h x+e h x^2\right ) (a+b \arcsin (c x))^2}{(d+e x)^2} \, dx=\int { \frac {{\left (e h x^{2} + 2 \, d h x + e f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{2}} \,d x } \]
integral((a^2*e*h*x^2 + 2*a^2*d*h*x + a^2*e*f + (b^2*e*h*x^2 + 2*b^2*d*h*x + b^2*e*f)*arcsin(c*x)^2 + 2*(a*b*e*h*x^2 + 2*a*b*d*h*x + a*b*e*f)*arcsin (c*x))/(e^2*x^2 + 2*d*e*x + d^2), x)
\[ \int \frac {\left (e f+2 d h x+e h x^2\right ) (a+b \arcsin (c x))^2}{(d+e x)^2} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2} \cdot \left (2 d h x + e f + e h x^{2}\right )}{\left (d + e x\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {\left (e f+2 d h x+e h x^2\right ) (a+b \arcsin (c x))^2}{(d+e x)^2} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {\left (e f+2 d h x+e h x^2\right ) (a+b \arcsin (c x))^2}{(d+e x)^2} \, dx=\int { \frac {{\left (e h x^{2} + 2 \, d h x + e f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (e f+2 d h x+e h x^2\right ) (a+b \arcsin (c x))^2}{(d+e x)^2} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\left (e\,h\,x^2+2\,d\,h\,x+e\,f\right )}{{\left (d+e\,x\right )}^2} \,d x \]