Integrand size = 27, antiderivative size = 146 \[ \int \frac {x (a+b \arccos (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {2 a b x \sqrt {1-c^2 x^2}}{c \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \left (1-c^2 x^2\right )}{c^2 \sqrt {d-c^2 d x^2}}+\frac {2 b^2 x \sqrt {1-c^2 x^2} \arccos (c x)}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{c^2 d} \] Output:
2*a*b*x*(-c^2*x^2+1)^(1/2)/c/(-c^2*d*x^2+d)^(1/2)+2*b^2*(-c^2*x^2+1)/c^2/( -c^2*d*x^2+d)^(1/2)+2*b^2*x*(-c^2*x^2+1)^(1/2)*arccos(c*x)/c/(-c^2*d*x^2+d )^(1/2)-(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^2/c^2/d
Time = 0.66 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.95 \[ \int \frac {x (a+b \arccos (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {\sqrt {d-c^2 d x^2} \left (2 a b c x \sqrt {1-c^2 x^2}+a^2 \left (1-c^2 x^2\right )+2 b^2 \left (-1+c^2 x^2\right )+2 b \left (a-a c^2 x^2+b c x \sqrt {1-c^2 x^2}\right ) \arccos (c x)+b^2 \left (1-c^2 x^2\right ) \arccos (c x)^2\right )}{c^2 d \left (-1+c^2 x^2\right )} \] Input:
Integrate[(x*(a + b*ArcCos[c*x])^2)/Sqrt[d - c^2*d*x^2],x]
Output:
(Sqrt[d - c^2*d*x^2]*(2*a*b*c*x*Sqrt[1 - c^2*x^2] + a^2*(1 - c^2*x^2) + 2* b^2*(-1 + c^2*x^2) + 2*b*(a - a*c^2*x^2 + b*c*x*Sqrt[1 - c^2*x^2])*ArcCos[ c*x] + b^2*(1 - c^2*x^2)*ArcCos[c*x]^2))/(c^2*d*(-1 + c^2*x^2))
Time = 0.35 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.68, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {5183, 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.
\(\displaystyle \int \frac {x (a+b \arccos (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx\) |
\(\Big \downarrow \) 5183 |
\(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \int (a+b \arccos (c x))dx}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{c^2 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\) |
Input:
Int[(x*(a + b*ArcCos[c*x])^2)/Sqrt[d - c^2*d*x^2],x]
Output:
-((Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(c^2*d)) - (2*b*Sqrt[1 - c^2 *x^2]*(a*x - (b*Sqrt[1 - c^2*x^2])/c + b*x*ArcCos[c*x]))/(c*Sqrt[d - c^2*d *x^2])
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ .), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCos[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*ArcCos[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]
Result contains complex when optimal does not.
Time = 0.49 (sec) , antiderivative size = 316, normalized size of antiderivative = 2.16
method | result | size |
default | \(-\frac {a^{2} \sqrt {-c^{2} d \,x^{2}+d}}{c^{2} d}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )^{2}-2+2 i \arccos \left (c x \right )\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )^{2}-2-2 i \arccos \left (c x \right )\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )+i\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )-i\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}\right )\) | \(316\) |
parts | \(-\frac {a^{2} \sqrt {-c^{2} d \,x^{2}+d}}{c^{2} d}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )^{2}-2+2 i \arccos \left (c x \right )\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )^{2}-2-2 i \arccos \left (c x \right )\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )+i\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )-i\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}\right )\) | \(316\) |
orering | \(\frac {\left (c^{4} x^{4}-4 c^{2} x^{2}+2\right ) \left (a +b \arccos \left (c x \right )\right )^{2}}{c^{4} x^{2} \sqrt {-c^{2} d \,x^{2}+d}}+\frac {2 \left (c x -1\right ) \left (c x +1\right ) \left (\frac {\left (a +b \arccos \left (c x \right )\right )^{2}}{\sqrt {-c^{2} d \,x^{2}+d}}-\frac {2 x \left (a +b \arccos \left (c x \right )\right ) b c}{\sqrt {-c^{2} d \,x^{2}+d}\, \sqrt {-c^{2} x^{2}+1}}+\frac {x^{2} \left (a +b \arccos \left (c x \right )\right )^{2} d \,c^{2}}{\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}\right )}{c^{4} x^{2}}+\frac {\left (c x -1\right )^{2} \left (c x +1\right )^{2} \left (-\frac {4 \left (a +b \arccos \left (c x \right )\right ) b c}{\sqrt {-c^{2} d \,x^{2}+d}\, \sqrt {-c^{2} x^{2}+1}}+\frac {3 \left (a +b \arccos \left (c x \right )\right )^{2} d \,c^{2} x}{\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {2 x \,b^{2} c^{2}}{\left (-c^{2} x^{2}+1\right ) \sqrt {-c^{2} d \,x^{2}+d}}-\frac {4 x^{2} \left (a +b \arccos \left (c x \right )\right ) b \,c^{3} d}{\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \sqrt {-c^{2} x^{2}+1}}-\frac {2 x^{2} \left (a +b \arccos \left (c x \right )\right ) b \,c^{3}}{\sqrt {-c^{2} d \,x^{2}+d}\, \left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {3 x^{3} \left (a +b \arccos \left (c x \right )\right )^{2} d^{2} c^{4}}{\left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}\right )}{c^{4} x}\) | \(403\) |
Input:
int(x*(a+b*arccos(c*x))^2/(-c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE)
Output:
-a^2/c^2/d*(-c^2*d*x^2+d)^(1/2)+b^2*(-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)*(arccos(c*x)^2-2+2*I*arccos(c*x))/c^2/d/(c^2*x ^2-1)-1/2*(-d*(c^2*x^2-1))^(1/2)*(-I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(ar ccos(c*x)^2-2-2*I*arccos(c*x))/c^2/d/(c^2*x^2-1))+2*a*b*(-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)*(arccos(c*x)+I)/c^2/d/(c^2 *x^2-1)-1/2*(-d*(c^2*x^2-1))^(1/2)*(-I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*( arccos(c*x)-I)/c^2/d/(c^2*x^2-1))
Time = 0.12 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.01 \[ \int \frac {x (a+b \arccos (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {2 \, \sqrt {-c^{2} d x^{2} + d} {\left (b^{2} c x \arccos \left (c x\right ) + a b c x\right )} \sqrt {-c^{2} x^{2} + 1} - {\left ({\left (a^{2} - 2 \, b^{2}\right )} c^{2} x^{2} + {\left (b^{2} c^{2} x^{2} - b^{2}\right )} \arccos \left (c x\right )^{2} - a^{2} + 2 \, b^{2} + 2 \, {\left (a b c^{2} x^{2} - a b\right )} \arccos \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{c^{4} d x^{2} - c^{2} d} \] Input:
integrate(x*(a+b*arccos(c*x))^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="fricas" )
Output:
(2*sqrt(-c^2*d*x^2 + d)*(b^2*c*x*arccos(c*x) + a*b*c*x)*sqrt(-c^2*x^2 + 1) - ((a^2 - 2*b^2)*c^2*x^2 + (b^2*c^2*x^2 - b^2)*arccos(c*x)^2 - a^2 + 2*b^ 2 + 2*(a*b*c^2*x^2 - a*b)*arccos(c*x))*sqrt(-c^2*d*x^2 + d))/(c^4*d*x^2 - c^2*d)
Exception generated. \[ \int \frac {x (a+b \arccos (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x*(a+b*acos(c*x))**2/(-c**2*d*x**2+d)**(1/2),x)
Output:
Exception raised: TypeError >> Invalid comparison of non-real zoo
Time = 0.15 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.90 \[ \int \frac {x (a+b \arccos (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=-2 \, b^{2} {\left (\frac {x \arccos \left (c x\right )}{c \sqrt {d}} - \frac {\sqrt {-c^{2} x^{2} + 1}}{c^{2} \sqrt {d}}\right )} - \frac {2 \, a b x}{c \sqrt {d}} - \frac {\sqrt {-c^{2} d x^{2} + d} b^{2} \arccos \left (c x\right )^{2}}{c^{2} d} - \frac {2 \, \sqrt {-c^{2} d x^{2} + d} a b \arccos \left (c x\right )}{c^{2} d} - \frac {\sqrt {-c^{2} d x^{2} + d} a^{2}}{c^{2} d} \] Input:
integrate(x*(a+b*arccos(c*x))^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="maxima" )
Output:
-2*b^2*(x*arccos(c*x)/(c*sqrt(d)) - sqrt(-c^2*x^2 + 1)/(c^2*sqrt(d))) - 2* a*b*x/(c*sqrt(d)) - sqrt(-c^2*d*x^2 + d)*b^2*arccos(c*x)^2/(c^2*d) - 2*sqr t(-c^2*d*x^2 + d)*a*b*arccos(c*x)/(c^2*d) - sqrt(-c^2*d*x^2 + d)*a^2/(c^2* d)
Exception generated. \[ \int \frac {x (a+b \arccos (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(x*(a+b*arccos(c*x))^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="giac")
Output:
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {x (a+b \arccos (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {x\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2}{\sqrt {d-c^2\,d\,x^2}} \,d x \] Input:
int((x*(a + b*acos(c*x))^2)/(d - c^2*d*x^2)^(1/2),x)
Output:
int((x*(a + b*acos(c*x))^2)/(d - c^2*d*x^2)^(1/2), x)
Time = 0.25 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.68 \[ \int \frac {x (a+b \arccos (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {\sqrt {d}\, \left (-\sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )^{2} b^{2}-2 \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) a b -2 \mathit {acos} \left (c x \right ) b^{2} c x -\sqrt {-c^{2} x^{2}+1}\, a^{2}+2 \sqrt {-c^{2} x^{2}+1}\, b^{2}-2 a b c x \right )}{c^{2} d} \] Input:
int(x*(a+b*acos(c*x))^2/(-c^2*d*x^2+d)^(1/2),x)
Output:
(sqrt(d)*( - sqrt( - c**2*x**2 + 1)*acos(c*x)**2*b**2 - 2*sqrt( - c**2*x** 2 + 1)*acos(c*x)*a*b - 2*acos(c*x)*b**2*c*x - sqrt( - c**2*x**2 + 1)*a**2 + 2*sqrt( - c**2*x**2 + 1)*b**2 - 2*a*b*c*x))/(c**2*d)