Integrand size = 27, antiderivative size = 208 \[ \int \frac {x (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {(a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {4 i b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}} \]
(a+b*arcsin(c*x))^2/c^2/d/(-c^2*d*x^2+d)^(1/2)+4*I*b*(a+b*arcsin(c*x))*arc tan(I*c*x+(-c^2*x^2+1)^(1/2))*(-c^2*x^2+1)^(1/2)/c^2/d/(-c^2*d*x^2+d)^(1/2 )-2*I*b^2*polylog(2,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))*(-c^2*x^2+1)^(1/2)/c^2/ d/(-c^2*d*x^2+d)^(1/2)+2*I*b^2*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))*(-c ^2*x^2+1)^(1/2)/c^2/d/(-c^2*d*x^2+d)^(1/2)
Time = 1.58 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.33 \[ \int \frac {x (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {a^2+2 a b \arcsin (c x)+b^2 \arcsin (c x)^2-2 b^2 \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right )+2 b^2 \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right )+2 a b \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )-2 a b \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )-2 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )+2 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}} \]
(a^2 + 2*a*b*ArcSin[c*x] + b^2*ArcSin[c*x]^2 - 2*b^2*Sqrt[1 - c^2*x^2]*Arc Sin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] + 2*b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x ]*Log[1 + I*E^(I*ArcSin[c*x])] + 2*a*b*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c* x]/2] - Sin[ArcSin[c*x]/2]] - 2*a*b*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/ 2] + Sin[ArcSin[c*x]/2]] - (2*I)*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^( I*ArcSin[c*x])] + (2*I)*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x ])])/(c^2*d*Sqrt[d - c^2*d*x^2])
Time = 0.55 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.66, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5182, 5164, 3042, 4669, 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.
| \(\displaystyle \int \frac {x (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle \frac {(a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \int \frac {a+b \arcsin (c x)}{1-c^2 x^2}dx}{c d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5164 |
| \(\displaystyle \frac {(a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}d\arcsin (c x)}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \int (a+b \arcsin (c x)) \csc \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle \frac {(a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-b \int \log \left (1-i e^{i \arcsin (c x)}\right )d\arcsin (c x)+b \int \log \left (1+i e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {(a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (i b \int e^{-i \arcsin (c x)} \log \left (1-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-i b \int e^{-i \arcsin (c x)} \log \left (1+i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {(a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )}{c^2 d \sqrt {d-c^2 d x^2}}\) |
(a + b*ArcSin[c*x])^2/(c^2*d*Sqrt[d - c^2*d*x^2]) - (2*b*Sqrt[1 - c^2*x^2] *((-2*I)*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])] + I*b*PolyLog[2, (- I)*E^(I*ArcSin[c*x])] - I*b*PolyLog[2, I*E^(I*ArcSin[c*x])]))/(c^2*d*Sqrt[ d - c^2*d*x^2])
3.3.48.3.1 Defintions of rubi rules used
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[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol ] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Si mp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x ))], x], x]) /; FreeQ[{c, d, e, f}, x] && IntegerQ[2*k] && IGtQ[m, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbo l] :> Simp[1/(c*d) Subst[Int[(a + b*x)^n*Sec[x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ .), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 1))), x] + Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] I nt[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
Time = 0.14 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.93
| method | result | size |
| default | \(\frac {a^{2}}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )^{2}}{c^{2} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {2 \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{c^{2} d^{2} \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )}{c^{2} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right )}{c^{2} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )}{c^{2} d^{2} \left (c^{2} x^{2}-1\right )}\right )\) | \(401\) |
| parts | \(\frac {a^{2}}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )^{2}}{c^{2} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {2 \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{c^{2} d^{2} \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )}{c^{2} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right )}{c^{2} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )}{c^{2} d^{2} \left (c^{2} x^{2}-1\right )}\right )\) | \(401\) |
a^2/c^2/d/(-c^2*d*x^2+d)^(1/2)+b^2*(-(-d*(c^2*x^2-1))^(1/2)/c^2/d^2/(c^2*x ^2-1)*arcsin(c*x)^2-2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*(arcsin(c* x)*ln(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))-arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+ 1)^(1/2)))-I*dilog(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))+I*dilog(1-I*(I*c*x+(-c^ 2*x^2+1)^(1/2))))/c^2/d^2/(c^2*x^2-1))+2*a*b*(-(-d*(c^2*x^2-1))^(1/2)/c^2/ d^2/(c^2*x^2-1)*arcsin(c*x)+(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^2/ d^2/(c^2*x^2-1)*ln(I*c*x+(-c^2*x^2+1)^(1/2)+I)-(-d*(c^2*x^2-1))^(1/2)*(-c^ 2*x^2+1)^(1/2)/c^2/d^2/(c^2*x^2-1)*ln(I*c*x+(-c^2*x^2+1)^(1/2)-I))
\[ \int \frac {x (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
integral(sqrt(-c^2*d*x^2 + d)*(b^2*x*arcsin(c*x)^2 + 2*a*b*x*arcsin(c*x) + a^2*x)/(c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2), x)
\[ \int \frac {x (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {x (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
sqrt(d)*integrate((b^2*x*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2* a*b*x*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1)/(c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2), x) + a^2/(sqrt(-c^2*d*x^2 + d)*c ^2*d)
Exception generated. \[ \int \frac {x (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E rror: Bad Argument Value
Timed out. \[ \int \frac {x (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]