Integrand size = 18, antiderivative size = 249 \[ \int \frac {d+e x^2}{(a+b \arccos (c x))^2} \, dx=-\frac {d \sqrt {1-c^2 x^2}}{b c (a+b \arccos (c x))}-\frac {e x^2 \sqrt {1-c^2 x^2}}{b c (a+b \arccos (c x))}+\frac {d \operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{b^2 c}+\frac {e \operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{4 b^2 c^3}-\frac {3 e \operatorname {CosIntegral}\left (\frac {3 (a+b \arccos (c x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{4 b^2 c^3}-\frac {d \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )}{b^2 c}-\frac {e \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )}{4 b^2 c^3}+\frac {3 e \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arccos (c x))}{b}\right )}{4 b^2 c^3} \] Output:
-d*(-c^2*x^2+1)^(1/2)/b/c/(a+b*arccos(c*x))-e*x^2*(-c^2*x^2+1)^(1/2)/b/c/( a+b*arccos(c*x))+d*Ci((a+b*arccos(c*x))/b)*sin(a/b)/b^2/c+1/4*e*Ci((a+b*ar ccos(c*x))/b)*sin(a/b)/b^2/c^3-3/4*e*Ci(3*(a+b*arccos(c*x))/b)*sin(3*a/b)/ b^2/c^3-d*cos(a/b)*Si((a+b*arccos(c*x))/b)/b^2/c-1/4*e*cos(a/b)*Si((a+b*ar ccos(c*x))/b)/b^2/c^3+3/4*e*cos(3*a/b)*Si(3*(a+b*arccos(c*x))/b)/b^2/c^3
Time = 0.83 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.76 \[ \int \frac {d+e x^2}{(a+b \arccos (c x))^2} \, dx=-\frac {-\frac {4 b c^2 d \sqrt {1-c^2 x^2}}{a+b \arccos (c x)}-\frac {4 b c^2 e x^2 \sqrt {1-c^2 x^2}}{a+b \arccos (c x)}+\left (4 c^2 d+e\right ) \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arccos (c x)\right )+3 e \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arccos (c x)\right )\right )+4 c^2 d \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arccos (c x)\right )+e \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arccos (c x)\right )+3 e \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arccos (c x)\right )\right )}{4 b^2 c^3} \] Input:
Integrate[(d + e*x^2)/(a + b*ArcCos[c*x])^2,x]
Output:
-1/4*((-4*b*c^2*d*Sqrt[1 - c^2*x^2])/(a + b*ArcCos[c*x]) - (4*b*c^2*e*x^2* Sqrt[1 - c^2*x^2])/(a + b*ArcCos[c*x]) + (4*c^2*d + e)*Cos[a/b]*CosIntegra l[a/b + ArcCos[c*x]] + 3*e*Cos[(3*a)/b]*CosIntegral[3*(a/b + ArcCos[c*x])] + 4*c^2*d*Sin[a/b]*SinIntegral[a/b + ArcCos[c*x]] + e*Sin[a/b]*SinIntegra l[a/b + ArcCos[c*x]] + 3*e*Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcCos[c*x])] )/(b^2*c^3)
Time = 0.69 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5173, 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.
\(\displaystyle \int \frac {d+e x^2}{(a+b \arccos (c x))^2} \, dx\) |
\(\Big \downarrow \) 5173 |
\(\displaystyle \int \left (\frac {d}{(a+b \arccos (c x))^2}+\frac {e x^2}{(a+b \arccos (c x))^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {e \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right )}{4 b^2 c^3}-\frac {3 e \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arccos (c x))}{b}\right )}{4 b^2 c^3}-\frac {e \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )}{4 b^2 c^3}-\frac {3 e \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arccos (c x))}{b}\right )}{4 b^2 c^3}-\frac {d \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right )}{b^2 c}-\frac {d \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )}{b^2 c}+\frac {d \sqrt {1-c^2 x^2}}{b c (a+b \arccos (c x))}+\frac {e x^2 \sqrt {1-c^2 x^2}}{b c (a+b \arccos (c x))}\) |
Input:
Int[(d + e*x^2)/(a + b*ArcCos[c*x])^2,x]
Output:
(d*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcCos[c*x])) + (e*x^2*Sqrt[1 - c^2*x^2] )/(b*c*(a + b*ArcCos[c*x])) - (d*Cos[a/b]*CosIntegral[(a + b*ArcCos[c*x])/ b])/(b^2*c) - (e*Cos[a/b]*CosIntegral[(a + b*ArcCos[c*x])/b])/(4*b^2*c^3) - (3*e*Cos[(3*a)/b]*CosIntegral[(3*(a + b*ArcCos[c*x]))/b])/(4*b^2*c^3) - (d*Sin[a/b]*SinIntegral[(a + b*ArcCos[c*x])/b])/(b^2*c) - (e*Sin[a/b]*SinI ntegral[(a + b*ArcCos[c*x])/b])/(4*b^2*c^3) - (3*e*Sin[(3*a)/b]*SinIntegra l[(3*(a + b*ArcCos[c*x]))/b])/(4*b^2*c^3)
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(a + b*ArcCos[c*x])^n, (d + e*x^2)^p, x], x ] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (G tQ[p, 0] || IGtQ[n, 0])
Time = 0.15 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.51
method | result | size |
derivativedivides | \(\frac {-\frac {d \left (\arccos \left (c x \right ) \operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b +\arccos \left (c x \right ) \operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b +\operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a +\operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a -\sqrt {-c^{2} x^{2}+1}\, b \right )}{\left (a +b \arccos \left (c x \right )\right ) b^{2}}-\frac {e \left (3 \arccos \left (c x \right ) \cos \left (\frac {3 a}{b}\right ) \operatorname {Ci}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) b +3 \arccos \left (c x \right ) \sin \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) b +3 \cos \left (\frac {3 a}{b}\right ) \operatorname {Ci}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) a +3 \sin \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) a -\sin \left (3 \arccos \left (c x \right )\right ) b \right )}{4 c^{2} \left (a +b \arccos \left (c x \right )\right ) b^{2}}-\frac {e \left (\arccos \left (c x \right ) \operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b +\arccos \left (c x \right ) \operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b +\operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a +\operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a -\sqrt {-c^{2} x^{2}+1}\, b \right )}{4 c^{2} \left (a +b \arccos \left (c x \right )\right ) b^{2}}}{c}\) | \(375\) |
default | \(\frac {-\frac {d \left (\arccos \left (c x \right ) \operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b +\arccos \left (c x \right ) \operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b +\operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a +\operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a -\sqrt {-c^{2} x^{2}+1}\, b \right )}{\left (a +b \arccos \left (c x \right )\right ) b^{2}}-\frac {e \left (3 \arccos \left (c x \right ) \cos \left (\frac {3 a}{b}\right ) \operatorname {Ci}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) b +3 \arccos \left (c x \right ) \sin \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) b +3 \cos \left (\frac {3 a}{b}\right ) \operatorname {Ci}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) a +3 \sin \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) a -\sin \left (3 \arccos \left (c x \right )\right ) b \right )}{4 c^{2} \left (a +b \arccos \left (c x \right )\right ) b^{2}}-\frac {e \left (\arccos \left (c x \right ) \operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b +\arccos \left (c x \right ) \operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b +\operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a +\operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a -\sqrt {-c^{2} x^{2}+1}\, b \right )}{4 c^{2} \left (a +b \arccos \left (c x \right )\right ) b^{2}}}{c}\) | \(375\) |
Input:
int((e*x^2+d)/(a+b*arccos(c*x))^2,x,method=_RETURNVERBOSE)
Output:
1/c*(-d*(arccos(c*x)*Si(arccos(c*x)+a/b)*sin(a/b)*b+arccos(c*x)*Ci(arccos( c*x)+a/b)*cos(a/b)*b+Si(arccos(c*x)+a/b)*sin(a/b)*a+Ci(arccos(c*x)+a/b)*co s(a/b)*a-(-c^2*x^2+1)^(1/2)*b)/(a+b*arccos(c*x))/b^2-1/4*e/c^2*(3*arccos(c *x)*cos(3*a/b)*Ci(3*arccos(c*x)+3*a/b)*b+3*arccos(c*x)*sin(3*a/b)*Si(3*arc cos(c*x)+3*a/b)*b+3*cos(3*a/b)*Ci(3*arccos(c*x)+3*a/b)*a+3*sin(3*a/b)*Si(3 *arccos(c*x)+3*a/b)*a-sin(3*arccos(c*x))*b)/(a+b*arccos(c*x))/b^2-1/4*e/c^ 2*(arccos(c*x)*Si(arccos(c*x)+a/b)*sin(a/b)*b+arccos(c*x)*Ci(arccos(c*x)+a /b)*cos(a/b)*b+Si(arccos(c*x)+a/b)*sin(a/b)*a+Ci(arccos(c*x)+a/b)*cos(a/b) *a-(-c^2*x^2+1)^(1/2)*b)/(a+b*arccos(c*x))/b^2)
\[ \int \frac {d+e x^2}{(a+b \arccos (c x))^2} \, dx=\int { \frac {e x^{2} + d}{{\left (b \arccos \left (c x\right ) + a\right )}^{2}} \,d x } \] Input:
integrate((e*x^2+d)/(a+b*arccos(c*x))^2,x, algorithm="fricas")
Output:
integral((e*x^2 + d)/(b^2*arccos(c*x)^2 + 2*a*b*arccos(c*x) + a^2), x)
\[ \int \frac {d+e x^2}{(a+b \arccos (c x))^2} \, dx=\int \frac {d + e x^{2}}{\left (a + b \operatorname {acos}{\left (c x \right )}\right )^{2}}\, dx \] Input:
integrate((e*x**2+d)/(a+b*acos(c*x))**2,x)
Output:
Integral((d + e*x**2)/(a + b*acos(c*x))**2, x)
\[ \int \frac {d+e x^2}{(a+b \arccos (c x))^2} \, dx=\int { \frac {e x^{2} + d}{{\left (b \arccos \left (c x\right ) + a\right )}^{2}} \,d x } \] Input:
integrate((e*x^2+d)/(a+b*arccos(c*x))^2,x, algorithm="maxima")
Output:
((e*x^2 + d)*sqrt(c*x + 1)*sqrt(-c*x + 1) - (b^2*c*arctan2(sqrt(c*x + 1)*s qrt(-c*x + 1), c*x) + a*b*c)*integrate((3*c^2*e*x^3 + (c^2*d - 2*e)*x)*sqr t(c*x + 1)*sqrt(-c*x + 1)/(a*b*c^3*x^2 - a*b*c + (b^2*c^3*x^2 - b^2*c)*arc tan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)), x))/(b^2*c*arctan2(sqrt(c*x + 1) *sqrt(-c*x + 1), c*x) + a*b*c)
Leaf count of result is larger than twice the leaf count of optimal. 860 vs. \(2 (237) = 474\).
Time = 0.20 (sec) , antiderivative size = 860, normalized size of antiderivative = 3.45 \[ \int \frac {d+e x^2}{(a+b \arccos (c x))^2} \, dx=\text {Too large to display} \] Input:
integrate((e*x^2+d)/(a+b*arccos(c*x))^2,x, algorithm="giac")
Output:
-3*b*e*arccos(c*x)*cos(a/b)^3*cos_integral(3*a/b + 3*arccos(c*x))/(b^3*c^3 *arccos(c*x) + a*b^2*c^3) - b*c^2*d*arccos(c*x)*cos(a/b)*cos_integral(a/b + arccos(c*x))/(b^3*c^3*arccos(c*x) + a*b^2*c^3) - 3*b*e*arccos(c*x)*cos(a /b)^2*sin(a/b)*sin_integral(3*a/b + 3*arccos(c*x))/(b^3*c^3*arccos(c*x) + a*b^2*c^3) - b*c^2*d*arccos(c*x)*sin(a/b)*sin_integral(a/b + arccos(c*x))/ (b^3*c^3*arccos(c*x) + a*b^2*c^3) + sqrt(-c^2*x^2 + 1)*b*c^2*e*x^2/(b^3*c^ 3*arccos(c*x) + a*b^2*c^3) - 3*a*e*cos(a/b)^3*cos_integral(3*a/b + 3*arcco s(c*x))/(b^3*c^3*arccos(c*x) + a*b^2*c^3) - a*c^2*d*cos(a/b)*cos_integral( a/b + arccos(c*x))/(b^3*c^3*arccos(c*x) + a*b^2*c^3) - 3*a*e*cos(a/b)^2*si n(a/b)*sin_integral(3*a/b + 3*arccos(c*x))/(b^3*c^3*arccos(c*x) + a*b^2*c^ 3) - a*c^2*d*sin(a/b)*sin_integral(a/b + arccos(c*x))/(b^3*c^3*arccos(c*x) + a*b^2*c^3) + 9/4*b*e*arccos(c*x)*cos(a/b)*cos_integral(3*a/b + 3*arccos (c*x))/(b^3*c^3*arccos(c*x) + a*b^2*c^3) - 1/4*b*e*arccos(c*x)*cos(a/b)*co s_integral(a/b + arccos(c*x))/(b^3*c^3*arccos(c*x) + a*b^2*c^3) + 3/4*b*e* arccos(c*x)*sin(a/b)*sin_integral(3*a/b + 3*arccos(c*x))/(b^3*c^3*arccos(c *x) + a*b^2*c^3) - 1/4*b*e*arccos(c*x)*sin(a/b)*sin_integral(a/b + arccos( c*x))/(b^3*c^3*arccos(c*x) + a*b^2*c^3) + sqrt(-c^2*x^2 + 1)*b*c^2*d/(b^3* c^3*arccos(c*x) + a*b^2*c^3) + 9/4*a*e*cos(a/b)*cos_integral(3*a/b + 3*arc cos(c*x))/(b^3*c^3*arccos(c*x) + a*b^2*c^3) - 1/4*a*e*cos(a/b)*cos_integra l(a/b + arccos(c*x))/(b^3*c^3*arccos(c*x) + a*b^2*c^3) + 3/4*a*e*sin(a/...
Timed out. \[ \int \frac {d+e x^2}{(a+b \arccos (c x))^2} \, dx=\int \frac {e\,x^2+d}{{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2} \,d x \] Input:
int((d + e*x^2)/(a + b*acos(c*x))^2,x)
Output:
int((d + e*x^2)/(a + b*acos(c*x))^2, x)
\[ \int \frac {d+e x^2}{(a+b \arccos (c x))^2} \, dx=\left (\int \frac {x^{2}}{\mathit {acos} \left (c x \right )^{2} b^{2}+2 \mathit {acos} \left (c x \right ) a b +a^{2}}d x \right ) e +\left (\int \frac {1}{\mathit {acos} \left (c x \right )^{2} b^{2}+2 \mathit {acos} \left (c x \right ) a b +a^{2}}d x \right ) d \] Input:
int((e*x^2+d)/(a+b*acos(c*x))^2,x)
Output:
int(x**2/(acos(c*x)**2*b**2 + 2*acos(c*x)*a*b + a**2),x)*e + int(1/(acos(c *x)**2*b**2 + 2*acos(c*x)*a*b + a**2),x)*d