Integrand size = 20, antiderivative size = 77 \[ \int \left (d-c^2 d x^2\right ) (a+b \arccos (c x)) \, dx=-\frac {2 b d \sqrt {1-c^2 x^2}}{3 c}-\frac {b d \left (1-c^2 x^2\right )^{3/2}}{9 c}+d x (a+b \arccos (c x))-\frac {1}{3} c^2 d x^3 (a+b \arccos (c x)) \] Output:
-2/3*b*d*(-c^2*x^2+1)^(1/2)/c-1/9*b*d*(-c^2*x^2+1)^(3/2)/c+d*x*(a+b*arccos (c*x))-1/3*c^2*d*x^3*(a+b*arccos(c*x))
Time = 0.06 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.87 \[ \int \left (d-c^2 d x^2\right ) (a+b \arccos (c x)) \, dx=\frac {d \left (b \sqrt {1-c^2 x^2} \left (-7+c^2 x^2\right )+a \left (9 c x-3 c^3 x^3\right )-3 b c x \left (-3+c^2 x^2\right ) \arccos (c x)\right )}{9 c} \] Input:
Integrate[(d - c^2*d*x^2)*(a + b*ArcCos[c*x]),x]
Output:
(d*(b*Sqrt[1 - c^2*x^2]*(-7 + c^2*x^2) + a*(9*c*x - 3*c^3*x^3) - 3*b*c*x*( -3 + c^2*x^2)*ArcCos[c*x]))/(9*c)
Time = 0.28 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5155, 27, 353, 53, 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.
\(\displaystyle \int \left (d-c^2 d x^2\right ) (a+b \arccos (c x)) \, dx\) |
\(\Big \downarrow \) 5155 |
\(\displaystyle b c \int \frac {d x \left (3-c^2 x^2\right )}{3 \sqrt {1-c^2 x^2}}dx-\frac {1}{3} c^2 d x^3 (a+b \arccos (c x))+d x (a+b \arccos (c x))\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} b c d \int \frac {x \left (3-c^2 x^2\right )}{\sqrt {1-c^2 x^2}}dx-\frac {1}{3} c^2 d x^3 (a+b \arccos (c x))+d x (a+b \arccos (c x))\) |
\(\Big \downarrow \) 353 |
\(\displaystyle \frac {1}{6} b c d \int \frac {3-c^2 x^2}{\sqrt {1-c^2 x^2}}dx^2-\frac {1}{3} c^2 d x^3 (a+b \arccos (c x))+d x (a+b \arccos (c x))\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \frac {1}{6} b c d \int \left (\sqrt {1-c^2 x^2}+\frac {2}{\sqrt {1-c^2 x^2}}\right )dx^2-\frac {1}{3} c^2 d x^3 (a+b \arccos (c x))+d x (a+b \arccos (c x))\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{3} c^2 d x^3 (a+b \arccos (c x))+d x (a+b \arccos (c x))+\frac {1}{6} b c d \left (-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {4 \sqrt {1-c^2 x^2}}{c^2}\right )\) |
Input:
Int[(d - c^2*d*x^2)*(a + b*ArcCos[c*x]),x]
Output:
(b*c*d*((-4*Sqrt[1 - c^2*x^2])/c^2 - (2*(1 - c^2*x^2)^(3/2))/(3*c^2)))/6 + d*x*(a + b*ArcCos[c*x]) - (c^2*d*x^3*(a + b*ArcCos[c*x]))/3
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[1/2 Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^2], x] /; FreeQ[ {a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbo l] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(a + b*ArcCos[c*x]) u, x ] + Simp[b*c Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; Fr eeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
Time = 0.09 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.04
method | result | size |
parts | \(-d a \left (\frac {1}{3} c^{2} x^{3}-x \right )-\frac {d b \left (\frac {c^{3} x^{3} \arccos \left (c x \right )}{3}-c x \arccos \left (c x \right )-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}+\frac {7 \sqrt {-c^{2} x^{2}+1}}{9}\right )}{c}\) | \(80\) |
derivativedivides | \(\frac {-d a \left (\frac {1}{3} c^{3} x^{3}-c x \right )-d b \left (\frac {c^{3} x^{3} \arccos \left (c x \right )}{3}-c x \arccos \left (c x \right )-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}+\frac {7 \sqrt {-c^{2} x^{2}+1}}{9}\right )}{c}\) | \(82\) |
default | \(\frac {-d a \left (\frac {1}{3} c^{3} x^{3}-c x \right )-d b \left (\frac {c^{3} x^{3} \arccos \left (c x \right )}{3}-c x \arccos \left (c x \right )-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}+\frac {7 \sqrt {-c^{2} x^{2}+1}}{9}\right )}{c}\) | \(82\) |
orering | \(\frac {x \left (5 c^{2} x^{2}-23\right ) \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right )}{9 c^{2} x^{2}-9}-\frac {\left (c^{2} x^{2}-7\right ) \left (-2 d \,c^{2} x \left (a +b \arccos \left (c x \right )\right )-\frac {\left (-c^{2} d \,x^{2}+d \right ) b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{9 c^{2}}\) | \(102\) |
Input:
int((-c^2*d*x^2+d)*(a+b*arccos(c*x)),x,method=_RETURNVERBOSE)
Output:
-d*a*(1/3*c^2*x^3-x)-d*b/c*(1/3*c^3*x^3*arccos(c*x)-c*x*arccos(c*x)-1/9*c^ 2*x^2*(-c^2*x^2+1)^(1/2)+7/9*(-c^2*x^2+1)^(1/2))
Time = 0.13 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.94 \[ \int \left (d-c^2 d x^2\right ) (a+b \arccos (c x)) \, dx=-\frac {3 \, a c^{3} d x^{3} - 9 \, a c d x + 3 \, {\left (b c^{3} d x^{3} - 3 \, b c d x\right )} \arccos \left (c x\right ) - {\left (b c^{2} d x^{2} - 7 \, b d\right )} \sqrt {-c^{2} x^{2} + 1}}{9 \, c} \] Input:
integrate((-c^2*d*x^2+d)*(a+b*arccos(c*x)),x, algorithm="fricas")
Output:
-1/9*(3*a*c^3*d*x^3 - 9*a*c*d*x + 3*(b*c^3*d*x^3 - 3*b*c*d*x)*arccos(c*x) - (b*c^2*d*x^2 - 7*b*d)*sqrt(-c^2*x^2 + 1))/c
Time = 0.13 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.23 \[ \int \left (d-c^2 d x^2\right ) (a+b \arccos (c x)) \, dx=\begin {cases} - \frac {a c^{2} d x^{3}}{3} + a d x - \frac {b c^{2} d x^{3} \operatorname {acos}{\left (c x \right )}}{3} + \frac {b c d x^{2} \sqrt {- c^{2} x^{2} + 1}}{9} + b d x \operatorname {acos}{\left (c x \right )} - \frac {7 b d \sqrt {- c^{2} x^{2} + 1}}{9 c} & \text {for}\: c \neq 0 \\d x \left (a + \frac {\pi b}{2}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((-c**2*d*x**2+d)*(a+b*acos(c*x)),x)
Output:
Piecewise((-a*c**2*d*x**3/3 + a*d*x - b*c**2*d*x**3*acos(c*x)/3 + b*c*d*x* *2*sqrt(-c**2*x**2 + 1)/9 + b*d*x*acos(c*x) - 7*b*d*sqrt(-c**2*x**2 + 1)/( 9*c), Ne(c, 0)), (d*x*(a + pi*b/2), True))
Time = 0.13 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.30 \[ \int \left (d-c^2 d x^2\right ) (a+b \arccos (c x)) \, dx=-\frac {1}{3} \, a c^{2} d x^{3} - \frac {1}{9} \, {\left (3 \, x^{3} \arccos \left (c x\right ) - c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b c^{2} d + a d x + \frac {{\left (c x \arccos \left (c x\right ) - \sqrt {-c^{2} x^{2} + 1}\right )} b d}{c} \] Input:
integrate((-c^2*d*x^2+d)*(a+b*arccos(c*x)),x, algorithm="maxima")
Output:
-1/3*a*c^2*d*x^3 - 1/9*(3*x^3*arccos(c*x) - c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4))*b*c^2*d + a*d*x + (c*x*arccos(c*x) - sqrt(-c^ 2*x^2 + 1))*b*d/c
Time = 0.14 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.99 \[ \int \left (d-c^2 d x^2\right ) (a+b \arccos (c x)) \, dx=-\frac {1}{3} \, b c^{2} d x^{3} \arccos \left (c x\right ) - \frac {1}{3} \, a c^{2} d x^{3} + \frac {1}{9} \, \sqrt {-c^{2} x^{2} + 1} b c d x^{2} + b d x \arccos \left (c x\right ) + a d x - \frac {7 \, \sqrt {-c^{2} x^{2} + 1} b d}{9 \, c} \] Input:
integrate((-c^2*d*x^2+d)*(a+b*arccos(c*x)),x, algorithm="giac")
Output:
-1/3*b*c^2*d*x^3*arccos(c*x) - 1/3*a*c^2*d*x^3 + 1/9*sqrt(-c^2*x^2 + 1)*b* c*d*x^2 + b*d*x*arccos(c*x) + a*d*x - 7/9*sqrt(-c^2*x^2 + 1)*b*d/c
Timed out. \[ \int \left (d-c^2 d x^2\right ) (a+b \arccos (c x)) \, dx=\left \{\begin {array}{cl} b\,c^2\,d\,\left (\frac {\sqrt {\frac {1}{c^2}-x^2}\,\left (\frac {2}{c^2}+x^2\right )}{9}-\frac {x^3\,\mathrm {acos}\left (c\,x\right )}{3}\right )-\frac {b\,d\,\left (\sqrt {1-c^2\,x^2}-c\,x\,\mathrm {acos}\left (c\,x\right )\right )}{c}-\frac {a\,d\,x\,\left (c^2\,x^2-3\right )}{3} & \text {\ if\ \ }0<c\\ \int \left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,\left (d-c^2\,d\,x^2\right ) \,d x & \text {\ if\ \ }\neg 0<c \end {array}\right . \] Input:
int((a + b*acos(c*x))*(d - c^2*d*x^2),x)
Output:
piecewise(0 < c, b*c^2*d*(((1/c^2 - x^2)^(1/2)*(2/c^2 + x^2))/9 - (x^3*aco s(c*x))/3) - (b*d*((- c^2*x^2 + 1)^(1/2) - c*x*acos(c*x)))/c - (a*d*x*(c^2 *x^2 - 3))/3, ~0 < c, int((a + b*acos(c*x))*(d - c^2*d*x^2), x))
Time = 0.27 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.99 \[ \int \left (d-c^2 d x^2\right ) (a+b \arccos (c x)) \, dx=\frac {d \left (-3 \mathit {acos} \left (c x \right ) b \,c^{3} x^{3}+9 \mathit {acos} \left (c x \right ) b c x +\sqrt {-c^{2} x^{2}+1}\, b \,c^{2} x^{2}-7 \sqrt {-c^{2} x^{2}+1}\, b -3 a \,c^{3} x^{3}+9 a c x \right )}{9 c} \] Input:
int((-c^2*d*x^2+d)*(a+b*acos(c*x)),x)
Output:
(d*( - 3*acos(c*x)*b*c**3*x**3 + 9*acos(c*x)*b*c*x + sqrt( - c**2*x**2 + 1 )*b*c**2*x**2 - 7*sqrt( - c**2*x**2 + 1)*b - 3*a*c**3*x**3 + 9*a*c*x))/(9* c)