Integrand size = 29, antiderivative size = 206 \[ \int \frac {x^2 (a+b \arccos (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {b^2 x \sqrt {d-c^2 d x^2}}{4 c^2 d}-\frac {b^2 \sqrt {1-c^2 x^2} \arccos (c x)}{4 c^3 \sqrt {d-c^2 d x^2}}+\frac {b x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{2 c \sqrt {d-c^2 d x^2}}-\frac {x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 c^2 d}+\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^3}{6 b c^3 \sqrt {d-c^2 d x^2}} \] Output:
1/4*b^2*x*(-c^2*d*x^2+d)^(1/2)/c^2/d-1/4*b^2*(-c^2*x^2+1)^(1/2)*arccos(c*x )/c^3/(-c^2*d*x^2+d)^(1/2)+1/2*b*x^2*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))/ c/(-c^2*d*x^2+d)^(1/2)-1/2*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^2/c^2/ d+1/6*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))^3/b/c^3/(-c^2*d*x^2+d)^(1/2)
Time = 1.47 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.01 \[ \int \frac {x^2 (a+b \arccos (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {12 a^2 c d x \left (-1+c^2 x^2\right )-12 a^2 \sqrt {d} \sqrt {d-c^2 d x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-b^2 d \sqrt {1-c^2 x^2} \left (4 \arccos (c x)^3+6 \arccos (c x) \cos (2 \arccos (c x))+\left (-3+6 \arccos (c x)^2\right ) \sin (2 \arccos (c x))\right )-6 a b d \sqrt {1-c^2 x^2} (\cos (2 \arccos (c x))+2 \arccos (c x) (\arccos (c x)+\sin (2 \arccos (c x))))}{24 c^3 d \sqrt {d-c^2 d x^2}} \] Input:
Integrate[(x^2*(a + b*ArcCos[c*x])^2)/Sqrt[d - c^2*d*x^2],x]
Output:
(12*a^2*c*d*x*(-1 + c^2*x^2) - 12*a^2*Sqrt[d]*Sqrt[d - c^2*d*x^2]*ArcTan[( c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - b^2*d*Sqrt[1 - c^2*x^ 2]*(4*ArcCos[c*x]^3 + 6*ArcCos[c*x]*Cos[2*ArcCos[c*x]] + (-3 + 6*ArcCos[c* x]^2)*Sin[2*ArcCos[c*x]]) - 6*a*b*d*Sqrt[1 - c^2*x^2]*(Cos[2*ArcCos[c*x]] + 2*ArcCos[c*x]*(ArcCos[c*x] + Sin[2*ArcCos[c*x]])))/(24*c^3*d*Sqrt[d - c^ 2*d*x^2])
Time = 0.65 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.86, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {5211, 5139, 262, 223, 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.
\(\displaystyle \int \frac {x^2 (a+b \arccos (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx\) |
\(\Big \downarrow \) 5211 |
\(\displaystyle -\frac {b \sqrt {1-c^2 x^2} \int x (a+b \arccos (c x))dx}{c \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \arccos (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{2 c^2}-\frac {x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 c^2 d}\) |
\(\Big \downarrow \) 5139 |
\(\displaystyle -\frac {b \sqrt {1-c^2 x^2} \left (\frac {1}{2} b c \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x^2 (a+b \arccos (c x))\right )}{c \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \arccos (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{2 c^2}-\frac {x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 c^2 d}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle -\frac {b \sqrt {1-c^2 x^2} \left (\frac {1}{2} b c \left (\frac {\int \frac {1}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )+\frac {1}{2} x^2 (a+b \arccos (c x))\right )}{c \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \arccos (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{2 c^2}-\frac {x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 c^2 d}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {\int \frac {(a+b \arccos (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{2 c^2}-\frac {b \sqrt {1-c^2 x^2} \left (\frac {1}{2} x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{c \sqrt {d-c^2 d x^2}}-\frac {x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 c^2 d}\) |
\(\Big \downarrow \) 5153 |
\(\displaystyle -\frac {b \sqrt {1-c^2 x^2} \left (\frac {1}{2} x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{c \sqrt {d-c^2 d x^2}}-\frac {x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 c^2 d}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^3}{6 b c^3 \sqrt {d-c^2 d x^2}}\) |
Input:
Int[(x^2*(a + b*ArcCos[c*x])^2)/Sqrt[d - c^2*d*x^2],x]
Output:
-1/2*(x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(c^2*d) - (Sqrt[1 - c^2 *x^2]*(a + b*ArcCos[c*x])^3)/(6*b*c^3*Sqrt[d - c^2*d*x^2]) - (b*Sqrt[1 - c ^2*x^2]*((x^2*(a + b*ArcCos[c*x]))/2 + (b*c*(-1/2*(x*Sqrt[1 - c^2*x^2])/c^ 2 + ArcSin[c*x]/(2*c^3)))/2))/(c*Sqrt[d - c^2*d*x^2])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcCos[c*x])^n/(d*(m + 1))), x] + Simp[b*c*(n /(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2 *x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && 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_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] - S imp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[(f* x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m , 1] && NeQ[m + 2*p + 1, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(516\) vs. \(2(180)=360\).
Time = 0.42 (sec) , antiderivative size = 517, normalized size of antiderivative = 2.51
method | result | size |
default | \(-\frac {a^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{2 c^{2} d}+\frac {a^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{2} \sqrt {c^{2} d}}+b^{2} \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{3}}{6 c^{3} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 \arccos \left (c x \right )^{2}-1\right ) x}{16 c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )}{8 c^{3} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 \arccos \left (c x \right )^{2}-1\right ) \cos \left (3 \arccos \left (c x \right )\right )}{16 c^{3} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right ) \sin \left (3 \arccos \left (c x \right )\right )}{8 c^{3} d \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2}}{4 c^{3} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right ) x}{8 c^{2} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-c^{2} x^{2}+1}}{16 c^{3} \sqrt {-d \left (c^{2} x^{2}-1\right )}}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right ) \cos \left (3 \arccos \left (c x \right )\right )}{8 c^{3} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sin \left (3 \arccos \left (c x \right )\right )}{16 c^{3} d \left (c^{2} x^{2}-1\right )}\right )\) | \(517\) |
parts | \(-\frac {a^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{2 c^{2} d}+\frac {a^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{2} \sqrt {c^{2} d}}+b^{2} \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{3}}{6 c^{3} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 \arccos \left (c x \right )^{2}-1\right ) x}{16 c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )}{8 c^{3} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 \arccos \left (c x \right )^{2}-1\right ) \cos \left (3 \arccos \left (c x \right )\right )}{16 c^{3} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right ) \sin \left (3 \arccos \left (c x \right )\right )}{8 c^{3} d \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2}}{4 c^{3} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right ) x}{8 c^{2} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-c^{2} x^{2}+1}}{16 c^{3} \sqrt {-d \left (c^{2} x^{2}-1\right )}}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right ) \cos \left (3 \arccos \left (c x \right )\right )}{8 c^{3} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sin \left (3 \arccos \left (c x \right )\right )}{16 c^{3} d \left (c^{2} x^{2}-1\right )}\right )\) | \(517\) |
Input:
int(x^2*(a+b*arccos(c*x))^2/(-c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/2*a^2*x/c^2/d*(-c^2*d*x^2+d)^(1/2)+1/2*a^2/c^2/(c^2*d)^(1/2)*arctan((c^ 2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+b^2*(1/6*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x ^2+1)^(1/2)/c^3/d/(c^2*x^2-1)*arccos(c*x)^3+1/16*(-d*(c^2*x^2-1))^(1/2)/c^ 2/d/(c^2*x^2-1)*(2*arccos(c*x)^2-1)*x-1/8*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2 +1)^(1/2)/c^3/d/(c^2*x^2-1)*arccos(c*x)-1/16*(-d*(c^2*x^2-1))^(1/2)/c^3/d/ (c^2*x^2-1)*(2*arccos(c*x)^2-1)*cos(3*arccos(c*x))+1/8*(-d*(c^2*x^2-1))^(1 /2)/c^3/d/(c^2*x^2-1)*arccos(c*x)*sin(3*arccos(c*x)))+2*a*b*(1/4*(-d*(c^2* x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/d/(c^2*x^2-1)*arccos(c*x)^2+1/8*(-d*( c^2*x^2-1))^(1/2)/c^2/d/(c^2*x^2-1)*arccos(c*x)*x+1/16/c^3/(-d*(c^2*x^2-1) )^(1/2)*(-c^2*x^2+1)^(1/2)-1/8*(-d*(c^2*x^2-1))^(1/2)/c^3/d/(c^2*x^2-1)*ar ccos(c*x)*cos(3*arccos(c*x))+1/16*(-d*(c^2*x^2-1))^(1/2)/c^3/d/(c^2*x^2-1) *sin(3*arccos(c*x)))
\[ \int \frac {x^2 (a+b \arccos (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2} x^{2}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \] Input:
integrate(x^2*(a+b*arccos(c*x))^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="frica s")
Output:
integral(-(b^2*x^2*arccos(c*x)^2 + 2*a*b*x^2*arccos(c*x) + a^2*x^2)*sqrt(- c^2*d*x^2 + d)/(c^2*d*x^2 - d), x)
\[ \int \frac {x^2 (a+b \arccos (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {x^{2} \left (a + b \operatorname {acos}{\left (c x \right )}\right )^{2}}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \] Input:
integrate(x**2*(a+b*acos(c*x))**2/(-c**2*d*x**2+d)**(1/2),x)
Output:
Integral(x**2*(a + b*acos(c*x))**2/sqrt(-d*(c*x - 1)*(c*x + 1)), x)
\[ \int \frac {x^2 (a+b \arccos (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2} x^{2}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \] Input:
integrate(x^2*(a+b*arccos(c*x))^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="maxim a")
Output:
-1/2*a^2*(sqrt(-c^2*d*x^2 + d)*x/(c^2*d) - arcsin(c*x)/(c^3*sqrt(d))) - sq rt(d)*integrate((b^2*x^2*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^2 + 2* a*b*x^2*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x))*sqrt(c*x + 1)*sqrt(-c* x + 1)/(c^2*d*x^2 - d), x)
Time = 0.64 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.82 \[ \int \frac {x^2 (a+b \arccos (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {6 \, b^{2} c x^{2} \arccos \left (c x\right ) + 6 \, \sqrt {-c^{2} x^{2} + 1} b^{2} x \arccos \left (c x\right )^{2} + 6 \, a b c x^{2} + 12 \, \sqrt {-c^{2} x^{2} + 1} a b x \arccos \left (c x\right ) + \frac {2 \, b^{2} \arccos \left (c x\right )^{3}}{c} + 6 \, \sqrt {-c^{2} x^{2} + 1} a^{2} x - 3 \, \sqrt {-c^{2} x^{2} + 1} b^{2} x + \frac {6 \, a b \arccos \left (c x\right )^{2}}{c} + \frac {6 \, a^{2} \arccos \left (c x\right )}{c} - \frac {3 \, b^{2} \arccos \left (c x\right )}{c} - \frac {3 \, a b}{c}}{12 \, c^{2} \sqrt {d}} \] Input:
integrate(x^2*(a+b*arccos(c*x))^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="giac" )
Output:
-1/12*(6*b^2*c*x^2*arccos(c*x) + 6*sqrt(-c^2*x^2 + 1)*b^2*x*arccos(c*x)^2 + 6*a*b*c*x^2 + 12*sqrt(-c^2*x^2 + 1)*a*b*x*arccos(c*x) + 2*b^2*arccos(c*x )^3/c + 6*sqrt(-c^2*x^2 + 1)*a^2*x - 3*sqrt(-c^2*x^2 + 1)*b^2*x + 6*a*b*ar ccos(c*x)^2/c + 6*a^2*arccos(c*x)/c - 3*b^2*arccos(c*x)/c - 3*a*b/c)/(c^2* sqrt(d))
Timed out. \[ \int \frac {x^2 (a+b \arccos (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {x^2\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2}{\sqrt {d-c^2\,d\,x^2}} \,d x \] Input:
int((x^2*(a + b*acos(c*x))^2)/(d - c^2*d*x^2)^(1/2),x)
Output:
int((x^2*(a + b*acos(c*x))^2)/(d - c^2*d*x^2)^(1/2), x)
\[ \int \frac {x^2 (a+b \arccos (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {\mathit {asin} \left (c x \right ) a^{2}-\sqrt {-c^{2} x^{2}+1}\, a^{2} c x +4 \left (\int \frac {\mathit {acos} \left (c x \right ) x^{2}}{\sqrt {-c^{2} x^{2}+1}}d x \right ) a b \,c^{3}+2 \left (\int \frac {\mathit {acos} \left (c x \right )^{2} x^{2}}{\sqrt {-c^{2} x^{2}+1}}d x \right ) b^{2} c^{3}}{2 \sqrt {d}\, c^{3}} \] Input:
int(x^2*(a+b*acos(c*x))^2/(-c^2*d*x^2+d)^(1/2),x)
Output:
(asin(c*x)*a**2 - sqrt( - c**2*x**2 + 1)*a**2*c*x + 4*int((acos(c*x)*x**2) /sqrt( - c**2*x**2 + 1),x)*a*b*c**3 + 2*int((acos(c*x)**2*x**2)/sqrt( - c* *2*x**2 + 1),x)*b**2*c**3)/(2*sqrt(d)*c**3)