Integrand size = 31, antiderivative size = 305 \[ \int \frac {(f+g x)^3 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {b g^3 x \sqrt {1-c^2 x^2}}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {\left (g \left (3 c^2 f^2+g^2\right )+c^2 f \left (c^2 f^2+3 g^2\right ) x\right ) (a+b \arcsin (c x))}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {g^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{c^4 d^2}-\frac {3 f g^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{2 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {b (c f+g)^3 \sqrt {1-c^2 x^2} \log (1-c x)}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {b (c f-g)^3 \sqrt {1-c^2 x^2} \log (1+c x)}{2 c^4 d \sqrt {d-c^2 d x^2}} \] Output:
-b*g^3*x*(-c^2*x^2+1)^(1/2)/c^3/d/(-c^2*d*x^2+d)^(1/2)+(g*(3*c^2*f^2+g^2)+ c^2*f*(c^2*f^2+3*g^2)*x)*(a+b*arcsin(c*x))/c^4/d/(-c^2*d*x^2+d)^(1/2)+g^3* (-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/c^4/d^2-3/2*f*g^2*(-c^2*x^2+1)^(1/2 )*(a+b*arcsin(c*x))^2/b/c^3/d/(-c^2*d*x^2+d)^(1/2)+1/2*b*(c*f+g)^3*(-c^2*x ^2+1)^(1/2)*ln(-c*x+1)/c^4/d/(-c^2*d*x^2+d)^(1/2)+1/2*b*(c*f-g)^3*(-c^2*x^ 2+1)^(1/2)*ln(c*x+1)/c^4/d/(-c^2*d*x^2+d)^(1/2)
Time = 0.70 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.61 \[ \int \frac {(f+g x)^3 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {\sqrt {1-c^2 x^2} \left (-2 b c g^3 x+2 g^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {3 c f g^2 (a+b \arcsin (c x))^2}{b}+(c f-g)^3 \left (-\left ((a+b \arcsin (c x)) \cot \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+b \log (1+c x)\right )+(c f+g)^3 \left (2 b \log \left (\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+(a+b \arcsin (c x)) \tan \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )\right )}{2 c^4 d \sqrt {d-c^2 d x^2}} \] Input:
Integrate[((f + g*x)^3*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^(3/2),x]
Output:
(Sqrt[1 - c^2*x^2]*(-2*b*c*g^3*x + 2*g^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c *x]) - (3*c*f*g^2*(a + b*ArcSin[c*x])^2)/b + (c*f - g)^3*(-((a + b*ArcSin[ c*x])*Cot[(Pi + 2*ArcSin[c*x])/4]) + b*Log[1 + c*x]) + (c*f + g)^3*(2*b*Lo g[Cos[(Pi + 2*ArcSin[c*x])/4]] + (a + b*ArcSin[c*x])*Tan[(Pi + 2*ArcSin[c* x])/4])))/(2*c^4*d*Sqrt[d - c^2*d*x^2])
Time = 0.80 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.70, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {5276, 5274, 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.
| \(\displaystyle \int \frac {(f+g x)^3 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 5276 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \int \frac {(f+g x)^3 (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx}{d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5274 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \int \left (-\frac {x (a+b \arcsin (c x)) g^3}{c^2 \sqrt {1-c^2 x^2}}-\frac {3 f (a+b \arcsin (c x)) g^2}{c^2 \sqrt {1-c^2 x^2}}+\frac {\left (c^2 f^3+3 g^2 f+g \left (3 c^2 f^2+g^2\right ) x\right ) (a+b \arcsin (c x))}{c^2 \left (1-c^2 x^2\right )^{3/2}}\right )dx}{d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (-\frac {3 f g^2 (a+b \arcsin (c x))^2}{2 b c^3}+\frac {\left (c^2 f x \left (c^2 f^2+3 g^2\right )+g \left (3 c^2 f^2+g^2\right )\right ) (a+b \arcsin (c x))}{c^4 \sqrt {1-c^2 x^2}}+\frac {g^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^4}-\frac {b g \text {arctanh}(c x) \left (3 c^2 f^2+g^2\right )}{c^4}-\frac {b g^3 x}{c^3}+\frac {b f \left (c^2 f^2+3 g^2\right ) \log \left (1-c^2 x^2\right )}{2 c^3}\right )}{d \sqrt {d-c^2 d x^2}}\) |
Input:
Int[((f + g*x)^3*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^(3/2),x]
Output:
(Sqrt[1 - c^2*x^2]*(-((b*g^3*x)/c^3) + ((g*(3*c^2*f^2 + g^2) + c^2*f*(c^2* f^2 + 3*g^2)*x)*(a + b*ArcSin[c*x]))/(c^4*Sqrt[1 - c^2*x^2]) + (g^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c^4 - (3*f*g^2*(a + b*ArcSin[c*x])^2)/(2* b*c^3) - (b*g*(3*c^2*f^2 + g^2)*ArcTanh[c*x])/c^4 + (b*f*(c^2*f^2 + 3*g^2) *Log[1 - c^2*x^2])/(2*c^3)))/(d*Sqrt[d - c^2*d*x^2])
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_ ) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcSin[c*x] )^n/Sqrt[d + e*x^2], (f + g*x)^m*(d + e*x^2)^(p + 1/2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && ILtQ[p + 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_ ) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[Simp[(d + e*x^2)^p/(1 - c^2*x^2)^ p] Int[(f + g*x)^m*(1 - c^2*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ [{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && Intege rQ[p - 1/2] && !GtQ[d, 0]
Result contains complex when optimal does not.
Time = 1.29 (sec) , antiderivative size = 724, normalized size of antiderivative = 2.37
| method | result | size |
| default | \(a \left (\frac {f^{3} x}{d \sqrt {-c^{2} d \,x^{2}+d}}+g^{3} \left (-\frac {x^{2}}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {2}{d \,c^{4} \sqrt {-c^{2} d \,x^{2}+d}}\right )+3 f \,g^{2} \left (\frac {x}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {\arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{c^{2} d \sqrt {c^{2} d}}\right )+\frac {3 f^{2} g}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}\right )+b \left (\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} f \,g^{2}}{2 c^{3} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}\, c x -1\right ) g^{3} \left (\arcsin \left (c x \right )+i\right )}{2 c^{4} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) g^{3} \left (\arcsin \left (c x \right )-i\right )}{2 c^{4} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \left (c^{2} f^{2}+3 g^{2}\right ) \arcsin \left (c x \right )}{c^{3} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \left (i \sqrt {-c^{2} x^{2}+1}\, c^{3} f^{3}+c^{4} f^{3} x +3 i \sqrt {-c^{2} x^{2}+1}\, c f \,g^{2}+3 c^{2} f \,g^{2} x +3 f^{2} g \,c^{2}+g^{3}\right )}{c^{4} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (c^{3} f^{3}+3 f^{2} g \,c^{2}+3 c f \,g^{2}+g^{3}\right ) \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )}{c^{4} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (c^{3} f^{3}-3 f^{2} g \,c^{2}+3 c f \,g^{2}-g^{3}\right ) \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right )}{c^{4} d^{2} \left (c^{2} x^{2}-1\right )}\right )\) | \(724\) |
| parts | \(a \left (\frac {f^{3} x}{d \sqrt {-c^{2} d \,x^{2}+d}}+g^{3} \left (-\frac {x^{2}}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {2}{d \,c^{4} \sqrt {-c^{2} d \,x^{2}+d}}\right )+3 f \,g^{2} \left (\frac {x}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {\arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{c^{2} d \sqrt {c^{2} d}}\right )+\frac {3 f^{2} g}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}\right )+b \left (\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} f \,g^{2}}{2 c^{3} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}\, c x -1\right ) g^{3} \left (\arcsin \left (c x \right )+i\right )}{2 c^{4} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) g^{3} \left (\arcsin \left (c x \right )-i\right )}{2 c^{4} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \left (c^{2} f^{2}+3 g^{2}\right ) \arcsin \left (c x \right )}{c^{3} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \left (i \sqrt {-c^{2} x^{2}+1}\, c^{3} f^{3}+c^{4} f^{3} x +3 i \sqrt {-c^{2} x^{2}+1}\, c f \,g^{2}+3 c^{2} f \,g^{2} x +3 f^{2} g \,c^{2}+g^{3}\right )}{c^{4} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (c^{3} f^{3}+3 f^{2} g \,c^{2}+3 c f \,g^{2}+g^{3}\right ) \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )}{c^{4} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (c^{3} f^{3}-3 f^{2} g \,c^{2}+3 c f \,g^{2}-g^{3}\right ) \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right )}{c^{4} d^{2} \left (c^{2} x^{2}-1\right )}\right )\) | \(724\) |
Input:
int((g*x+f)^3*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(3/2),x,method=_RETURNVERBO SE)
Output:
a*(f^3*x/d/(-c^2*d*x^2+d)^(1/2)+g^3*(-x^2/c^2/d/(-c^2*d*x^2+d)^(1/2)+2/d/c ^4/(-c^2*d*x^2+d)^(1/2))+3*f*g^2*(x/c^2/d/(-c^2*d*x^2+d)^(1/2)-1/c^2/d/(c^ 2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2)))+3*f^2*g/c^2/d/(-c ^2*d*x^2+d)^(1/2))+b*(3/2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/d^ 2/(c^2*x^2-1)*arcsin(c*x)^2*f*g^2+1/2*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*c* x*(-c^2*x^2+1)^(1/2)-1)*g^3*(arcsin(c*x)+I)/c^4/d^2/(c^2*x^2-1)+1/2*(-d*(c ^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*c*x+c^2*x^2-1)*g^3*(arcsin(c*x)-I)/ c^4/d^2/(c^2*x^2-1)+2*I*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/c^3/d^2/ (c^2*x^2-1)*f*(c^2*f^2+3*g^2)*arcsin(c*x)-(-d*(c^2*x^2-1))^(1/2)/c^4/d^2/( c^2*x^2-1)*arcsin(c*x)*(I*(-c^2*x^2+1)^(1/2)*c^3*f^3+c^4*f^3*x+3*I*(-c^2*x ^2+1)^(1/2)*c*f*g^2+3*c^2*f*g^2*x+3*f^2*g*c^2+g^3)-(-d*(c^2*x^2-1))^(1/2)* (-c^2*x^2+1)^(1/2)/c^4/d^2/(c^2*x^2-1)*(c^3*f^3+3*c^2*f^2*g+3*c*f*g^2+g^3) *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^3*f^3-3*c^2*f^2*g+3*c*f*g^2-g^3)*ln(I*c*x+(-c^2*x^2+1)^(1/2)+I)/c^4/d^2 /(c^2*x^2-1))
\[ \int \frac {(f+g x)^3 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (g x + f\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((g*x+f)^3*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(3/2),x, algorithm="f ricas")
Output:
integral((a*g^3*x^3 + 3*a*f*g^2*x^2 + 3*a*f^2*g*x + a*f^3 + (b*g^3*x^3 + 3 *b*f*g^2*x^2 + 3*b*f^2*g*x + b*f^3)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d)/(c^4 *d^2*x^4 - 2*c^2*d^2*x^2 + d^2), x)
\[ \int \frac {(f+g x)^3 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (f + g x\right )^{3}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((g*x+f)**3*(a+b*asin(c*x))/(-c**2*d*x**2+d)**(3/2),x)
Output:
Integral((a + b*asin(c*x))*(f + g*x)**3/(-d*(c*x - 1)*(c*x + 1))**(3/2), x )
\[ \int \frac {(f+g x)^3 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (g x + f\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((g*x+f)^3*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(3/2),x, algorithm="m axima")
Output:
-a*g^3*(x^2/(sqrt(-c^2*d*x^2 + d)*c^2*d) - 2/(sqrt(-c^2*d*x^2 + d)*c^4*d)) + 3*a*f*g^2*(x/(sqrt(-c^2*d*x^2 + d)*c^2*d) - arcsin(c*x)/(c^3*d^(3/2))) + b*f^3*x*arcsin(c*x)/(sqrt(-c^2*d*x^2 + d)*d) + a*f^3*x/(sqrt(-c^2*d*x^2 + d)*d) - 1/2*b*f^3*log(x^2 - 1/c^2)/(c*d^(3/2)) + 3*a*f^2*g/(sqrt(-c^2*d* x^2 + d)*c^2*d) - integrate((b*g^3*x^3 + 3*b*f*g^2*x^2 + 3*b*f^2*g*x)*arct an2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/((c^2*d*x^2 - d)*sqrt(c*x + 1)*sqrt (-c*x + 1)), x)/sqrt(d)
Exception generated. \[ \int \frac {(f+g x)^3 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((g*x+f)^3*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(3/2),x, algorithm="g iac")
Output:
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. \[ \int \frac {(f+g x)^3 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {{\left (f+g\,x\right )}^3\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \] Input:
int(((f + g*x)^3*(a + b*asin(c*x)))/(d - c^2*d*x^2)^(3/2),x)
Output:
int(((f + g*x)^3*(a + b*asin(c*x)))/(d - c^2*d*x^2)^(3/2), x)
\[ \int \frac {(f+g x)^3 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {-3 \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )^{2} b c f \,g^{2}-6 \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) a c f \,g^{2}-2 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {asin} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-\sqrt {-c^{2} x^{2}+1}}d x \right ) b \,c^{4} f^{3}-6 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {asin} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-\sqrt {-c^{2} x^{2}+1}}d x \right ) b \,c^{2} f \,g^{2}-2 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {asin} \left (c x \right ) x^{3}}{\sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-\sqrt {-c^{2} x^{2}+1}}d x \right ) b \,c^{4} g^{3}-6 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {asin} \left (c x \right ) x}{\sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-\sqrt {-c^{2} x^{2}+1}}d x \right ) b \,c^{4} f^{2} g +2 a \,c^{4} f^{3} x +6 a \,c^{2} f^{2} g +6 a \,c^{2} f \,g^{2} x -2 a \,c^{2} g^{3} x^{2}+4 a \,g^{3}}{2 \sqrt {d}\, \sqrt {-c^{2} x^{2}+1}\, c^{4} d} \] Input:
int((g*x+f)^3*(a+b*asin(c*x))/(-c^2*d*x^2+d)^(3/2),x)
Output:
( - 3*sqrt( - c**2*x**2 + 1)*asin(c*x)**2*b*c*f*g**2 - 6*sqrt( - c**2*x**2 + 1)*asin(c*x)*a*c*f*g**2 - 2*sqrt( - c**2*x**2 + 1)*int(asin(c*x)/(sqrt( - c**2*x**2 + 1)*c**2*x**2 - sqrt( - c**2*x**2 + 1)),x)*b*c**4*f**3 - 6*s qrt( - c**2*x**2 + 1)*int(asin(c*x)/(sqrt( - c**2*x**2 + 1)*c**2*x**2 - sq rt( - c**2*x**2 + 1)),x)*b*c**2*f*g**2 - 2*sqrt( - c**2*x**2 + 1)*int((asi n(c*x)*x**3)/(sqrt( - c**2*x**2 + 1)*c**2*x**2 - sqrt( - c**2*x**2 + 1)),x )*b*c**4*g**3 - 6*sqrt( - c**2*x**2 + 1)*int((asin(c*x)*x)/(sqrt( - c**2*x **2 + 1)*c**2*x**2 - sqrt( - c**2*x**2 + 1)),x)*b*c**4*f**2*g + 2*a*c**4*f **3*x + 6*a*c**2*f**2*g + 6*a*c**2*f*g**2*x - 2*a*c**2*g**3*x**2 + 4*a*g** 3)/(2*sqrt(d)*sqrt( - c**2*x**2 + 1)*c**4*d)