Integrand size = 23, antiderivative size = 57 \[ \int \frac {x (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {b x}{2 c d^2 \sqrt {1-c^2 x^2}}+\frac {a+b \arccos (c x)}{2 c^2 d^2 \left (1-c^2 x^2\right )} \] Output:
1/2*b*x/c/d^2/(-c^2*x^2+1)^(1/2)+1/2*(a+b*arccos(c*x))/c^2/d^2/(-c^2*x^2+1 )
Time = 0.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.86 \[ \int \frac {x (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {a+b c x \sqrt {1-c^2 x^2}+b \arccos (c x)}{2 c^2 d^2-2 c^4 d^2 x^2} \] Input:
Integrate[(x*(a + b*ArcCos[c*x]))/(d - c^2*d*x^2)^2,x]
Output:
(a + b*c*x*Sqrt[1 - c^2*x^2] + b*ArcCos[c*x])/(2*c^2*d^2 - 2*c^4*d^2*x^2)
Time = 0.22 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {5183, 208}
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.
\(\displaystyle \int \frac {x (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 5183 |
\(\displaystyle \frac {b \int \frac {1}{\left (1-c^2 x^2\right )^{3/2}}dx}{2 c d^2}+\frac {a+b \arccos (c x)}{2 c^2 d^2 \left (1-c^2 x^2\right )}\) |
\(\Big \downarrow \) 208 |
\(\displaystyle \frac {a+b \arccos (c x)}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {b x}{2 c d^2 \sqrt {1-c^2 x^2}}\) |
Input:
Int[(x*(a + b*ArcCos[c*x]))/(d - c^2*d*x^2)^2,x]
Output:
(b*x)/(2*c*d^2*Sqrt[1 - c^2*x^2]) + (a + b*ArcCos[c*x])/(2*c^2*d^2*(1 - c^ 2*x^2))
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
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]
Time = 0.20 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.72
method | result | size |
derivativedivides | \(\frac {-\frac {a}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \left (-\frac {\arccos \left (c x \right )}{2 \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-\left (c x -1\right )^{2}-2 c x +2}}{4 \left (c x -1\right )}-\frac {\sqrt {-\left (c x +1\right )^{2}+2 c x +2}}{4 \left (c x +1\right )}\right )}{d^{2}}}{c^{2}}\) | \(98\) |
default | \(\frac {-\frac {a}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \left (-\frac {\arccos \left (c x \right )}{2 \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-\left (c x -1\right )^{2}-2 c x +2}}{4 \left (c x -1\right )}-\frac {\sqrt {-\left (c x +1\right )^{2}+2 c x +2}}{4 \left (c x +1\right )}\right )}{d^{2}}}{c^{2}}\) | \(98\) |
parts | \(-\frac {a}{2 d^{2} c^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \left (-\frac {\arccos \left (c x \right )}{2 \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-\left (c x -1\right )^{2}-2 c x +2}}{4 \left (c x -1\right )}-\frac {\sqrt {-\left (c x +1\right )^{2}+2 c x +2}}{4 \left (c x +1\right )}\right )}{d^{2} c^{2}}\) | \(100\) |
orering | \(-\frac {\left (c x -1\right ) \left (c x +1\right ) \left (3 c^{2} x^{2}+2\right ) \left (a +b \arccos \left (c x \right )\right )}{2 c^{2} \left (-c^{2} d \,x^{2}+d \right )^{2}}-\frac {\left (c x -1\right )^{2} \left (c x +1\right )^{2} \left (\frac {a +b \arccos \left (c x \right )}{\left (-c^{2} d \,x^{2}+d \right )^{2}}-\frac {x b c}{\sqrt {-c^{2} x^{2}+1}\, \left (-c^{2} d \,x^{2}+d \right )^{2}}+\frac {4 x^{2} \left (a +b \arccos \left (c x \right )\right ) d \,c^{2}}{\left (-c^{2} d \,x^{2}+d \right )^{3}}\right )}{2 c^{2}}\) | \(150\) |
Input:
int(x*(a+b*arccos(c*x))/(-c^2*d*x^2+d)^2,x,method=_RETURNVERBOSE)
Output:
1/c^2*(-1/2*a/d^2/(c^2*x^2-1)+b/d^2*(-1/2/(c^2*x^2-1)*arccos(c*x)-1/4/(c*x -1)*(-(c*x-1)^2-2*c*x+2)^(1/2)-1/4/(c*x+1)*(-(c*x+1)^2+2*c*x+2)^(1/2)))
Time = 0.11 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.95 \[ \int \frac {x (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=-\frac {a c^{2} x^{2} + \sqrt {-c^{2} x^{2} + 1} b c x + b \arccos \left (c x\right )}{2 \, {\left (c^{4} d^{2} x^{2} - c^{2} d^{2}\right )}} \] Input:
integrate(x*(a+b*arccos(c*x))/(-c^2*d*x^2+d)^2,x, algorithm="fricas")
Output:
-1/2*(a*c^2*x^2 + sqrt(-c^2*x^2 + 1)*b*c*x + b*arccos(c*x))/(c^4*d^2*x^2 - c^2*d^2)
\[ \int \frac {x (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a x}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {b x \operatorname {acos}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \] Input:
integrate(x*(a+b*acos(c*x))/(-c**2*d*x**2+d)**2,x)
Output:
(Integral(a*x/(c**4*x**4 - 2*c**2*x**2 + 1), x) + Integral(b*x*acos(c*x)/( c**4*x**4 - 2*c**2*x**2 + 1), x))/d**2
Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (50) = 100\).
Time = 0.16 (sec) , antiderivative size = 136, normalized size of antiderivative = 2.39 \[ \int \frac {x (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=-\frac {1}{4} \, {\left ({\left (\frac {\sqrt {-c^{2} x^{2} + 1} c^{2} d^{2}}{c^{7} d^{4} x + c^{6} d^{4}} + \frac {\sqrt {-c^{2} x^{2} + 1} c^{2} d^{2}}{c^{7} d^{4} x - c^{6} d^{4}}\right )} c^{2} + \frac {2 \, \arccos \left (c x\right )}{c^{4} d^{2} x^{2} - c^{2} d^{2}}\right )} b - \frac {a}{2 \, {\left (c^{4} d^{2} x^{2} - c^{2} d^{2}\right )}} \] Input:
integrate(x*(a+b*arccos(c*x))/(-c^2*d*x^2+d)^2,x, algorithm="maxima")
Output:
-1/4*((sqrt(-c^2*x^2 + 1)*c^2*d^2/(c^7*d^4*x + c^6*d^4) + sqrt(-c^2*x^2 + 1)*c^2*d^2/(c^7*d^4*x - c^6*d^4))*c^2 + 2*arccos(c*x)/(c^4*d^2*x^2 - c^2*d ^2))*b - 1/2*a/(c^4*d^2*x^2 - c^2*d^2)
Time = 0.15 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.75 \[ \int \frac {x (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=-\frac {b x^{2} \arccos \left (c x\right )}{2 \, {\left (c^{2} x^{2} - 1\right )} d^{2}} - \frac {a x^{2}}{2 \, {\left (c^{2} x^{2} - 1\right )} d^{2}} - \frac {\sqrt {-c^{2} x^{2} + 1} b x}{2 \, {\left (c^{2} x^{2} - 1\right )} c d^{2}} + \frac {b \arccos \left (c x\right )}{2 \, c^{2} d^{2}} + \frac {a}{2 \, c^{2} d^{2}} \] Input:
integrate(x*(a+b*arccos(c*x))/(-c^2*d*x^2+d)^2,x, algorithm="giac")
Output:
-1/2*b*x^2*arccos(c*x)/((c^2*x^2 - 1)*d^2) - 1/2*a*x^2/((c^2*x^2 - 1)*d^2) - 1/2*sqrt(-c^2*x^2 + 1)*b*x/((c^2*x^2 - 1)*c*d^2) + 1/2*b*arccos(c*x)/(c ^2*d^2) + 1/2*a/(c^2*d^2)
Timed out. \[ \int \frac {x (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\int \frac {x\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^2} \,d x \] Input:
int((x*(a + b*acos(c*x)))/(d - c^2*d*x^2)^2,x)
Output:
int((x*(a + b*acos(c*x)))/(d - c^2*d*x^2)^2, x)
\[ \int \frac {x (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {2 \left (\int \frac {\mathit {acos} \left (c x \right ) x}{c^{4} x^{4}-2 c^{2} x^{2}+1}d x \right ) b \,c^{2} x^{2}-2 \left (\int \frac {\mathit {acos} \left (c x \right ) x}{c^{4} x^{4}-2 c^{2} x^{2}+1}d x \right ) b -a \,x^{2}}{2 d^{2} \left (c^{2} x^{2}-1\right )} \] Input:
int(x*(a+b*acos(c*x))/(-c^2*d*x^2+d)^2,x)
Output:
(2*int((acos(c*x)*x)/(c**4*x**4 - 2*c**2*x**2 + 1),x)*b*c**2*x**2 - 2*int( (acos(c*x)*x)/(c**4*x**4 - 2*c**2*x**2 + 1),x)*b - a*x**2)/(2*d**2*(c**2*x **2 - 1))