Integrand size = 24, antiderivative size = 116 \[ \int \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) \, dx=\frac {b c x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{4 b c \sqrt {1-c^2 x^2}} \] Output:
1/4*b*c*x^2*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+1/2*x*(-c^2*d*x^2+d)^( 1/2)*(a+b*arccos(c*x))-1/4*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^2/b/c/(- c^2*x^2+1)^(1/2)
Time = 0.45 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.15 \[ \int \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) \, dx=\frac {1}{8} \left (4 a x \sqrt {d-c^2 d x^2}-\frac {4 a \sqrt {d} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )}{c}+\frac {b \sqrt {d-c^2 d x^2} \left (-2 \arccos (c x)^2+\cos (2 \arccos (c x))+2 \arccos (c x) \sin (2 \arccos (c x))\right )}{c \sqrt {1-c^2 x^2}}\right ) \] Input:
Integrate[Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]),x]
Output:
(4*a*x*Sqrt[d - c^2*d*x^2] - (4*a*Sqrt[d]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2]) /(Sqrt[d]*(-1 + c^2*x^2))])/c + (b*Sqrt[d - c^2*d*x^2]*(-2*ArcCos[c*x]^2 + Cos[2*ArcCos[c*x]] + 2*ArcCos[c*x]*Sin[2*ArcCos[c*x]]))/(c*Sqrt[1 - c^2*x ^2]))/8
Time = 0.34 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5157, 15, 5153}
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.
\(\displaystyle \int \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) \, dx\) |
\(\Big \downarrow \) 5157 |
\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}dx}{2 \sqrt {1-c^2 x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \int xdx}{2 \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}dx}{2 \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {b c x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {1-c^2 x^2}}\) |
\(\Big \downarrow \) 5153 |
\(\displaystyle \frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{4 b c \sqrt {1-c^2 x^2}}+\frac {b c x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {1-c^2 x^2}}\) |
Input:
Int[Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]),x]
Output:
(b*c*x^2*Sqrt[d - c^2*d*x^2])/(4*Sqrt[1 - c^2*x^2]) + (x*Sqrt[d - c^2*d*x^ 2]*(a + b*ArcCos[c*x]))/2 - (Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(4 *b*c*Sqrt[1 - c^2*x^2])
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[(-(b*c*(n + 1))^(-1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2] ]*(a + b*ArcCos[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^ 2*d + e, 0] && NeQ[n, -1]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcCos[c*x])^n/2), x] + (Simp[(1/2 )*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int[(a + b*ArcCos[c*x])^n/Sqrt[ 1 - c^2*x^2], x], x] + Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2 ]] Int[x*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x ] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 280, normalized size of antiderivative = 2.41
method | result | size |
default | \(\frac {a x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {a d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2}}{4 \left (c^{2} x^{2}-1\right ) c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-i \sqrt {-c^{2} x^{2}+1}\right ) \left (i+2 \arccos \left (c x \right )\right )}{16 \left (c^{2} x^{2}-1\right ) c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) \left (-i+2 \arccos \left (c x \right )\right )}{16 \left (c^{2} x^{2}-1\right ) c}\right )\) | \(280\) |
parts | \(\frac {a x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {a d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2}}{4 \left (c^{2} x^{2}-1\right ) c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-i \sqrt {-c^{2} x^{2}+1}\right ) \left (i+2 \arccos \left (c x \right )\right )}{16 \left (c^{2} x^{2}-1\right ) c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) \left (-i+2 \arccos \left (c x \right )\right )}{16 \left (c^{2} x^{2}-1\right ) c}\right )\) | \(280\) |
Input:
int((-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x)),x,method=_RETURNVERBOSE)
Output:
1/2*a*x*(-c^2*d*x^2+d)^(1/2)+1/2*a*d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/ (-c^2*d*x^2+d)^(1/2))+b*(1/4*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^ 2*x^2-1)/c*arccos(c*x)^2+1/16*(-d*(c^2*x^2-1))^(1/2)*(2*c^3*x^3-2*c*x+2*I* (-c^2*x^2+1)^(1/2)*x^2*c^2-I*(-c^2*x^2+1)^(1/2))*(I+2*arccos(c*x))/(c^2*x^ 2-1)/c+1/16*(-d*(c^2*x^2-1))^(1/2)*(-2*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+2*c^3* x^3+I*(-c^2*x^2+1)^(1/2)-2*c*x)*(-I+2*arccos(c*x))/(c^2*x^2-1)/c)
\[ \int \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (b \arccos \left (c x\right ) + a\right )} \,d x } \] Input:
integrate((-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x)),x, algorithm="fricas")
Output:
integral(sqrt(-c^2*d*x^2 + d)*(b*arccos(c*x) + a), x)
\[ \int \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) \, dx=\int \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acos}{\left (c x \right )}\right )\, dx \] Input:
integrate((-c**2*d*x**2+d)**(1/2)*(a+b*acos(c*x)),x)
Output:
Integral(sqrt(-d*(c*x - 1)*(c*x + 1))*(a + b*acos(c*x)), x)
\[ \int \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (b \arccos \left (c x\right ) + a\right )} \,d x } \] Input:
integrate((-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x)),x, algorithm="maxima")
Output:
b*sqrt(d)*integrate(sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(sqrt(c*x + 1)*sqr t(-c*x + 1), c*x), x) + 1/2*(sqrt(-c^2*d*x^2 + d)*x + sqrt(d)*arcsin(c*x)/ c)*a
Exception generated. \[ \int \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x)),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) \, dx=\int \left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2} \,d x \] Input:
int((a + b*acos(c*x))*(d - c^2*d*x^2)^(1/2),x)
Output:
int((a + b*acos(c*x))*(d - c^2*d*x^2)^(1/2), x)
\[ \int \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) \, dx=\frac {\sqrt {d}\, \left (\mathit {asin} \left (c x \right ) a +\sqrt {-c^{2} x^{2}+1}\, a c x +2 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )d x \right ) b c \right )}{2 c} \] Input:
int((-c^2*d*x^2+d)^(1/2)*(a+b*acos(c*x)),x)
Output:
(sqrt(d)*(asin(c*x)*a + sqrt( - c**2*x**2 + 1)*a*c*x + 2*int(sqrt( - c**2* x**2 + 1)*acos(c*x),x)*b*c))/(2*c)