\(\int \frac {d-c^2 d x^2}{(a+b \arccos (c x))^2} \, dx\) [34]

Optimal result

Integrand size = 22, antiderivative size = 157 \[ \int \frac {d-c^2 d x^2}{(a+b \arccos (c x))^2} \, dx=\frac {d \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \arccos (c x))}-\frac {3 d \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right )}{4 b^2 c}+\frac {3 d \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arccos (c x))}{b}\right )}{4 b^2 c}-\frac {3 d \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )}{4 b^2 c}+\frac {3 d \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arccos (c x))}{b}\right )}{4 b^2 c} \] Output:

d*(-c^2*x^2+1)^(3/2)/b/c/(a+b*arccos(c*x))-3/4*d*cos(a/b)*Ci((a+b*arccos(c 
*x))/b)/b^2/c+3/4*d*cos(3*a/b)*Ci(3*(a+b*arccos(c*x))/b)/b^2/c-3/4*d*sin(a 
/b)*Si((a+b*arccos(c*x))/b)/b^2/c+3/4*d*sin(3*a/b)*Si(3*(a+b*arccos(c*x))/ 
b)/b^2/c
 

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.78 \[ \int \frac {d-c^2 d x^2}{(a+b \arccos (c x))^2} \, dx=\frac {d \left (\frac {4 b \left (1-c^2 x^2\right )^{3/2}}{a+b \arccos (c x)}+3 \left (-\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arccos (c x)\right )+\cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arccos (c x)\right )\right )-\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arccos (c x)\right )+\sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arccos (c x)\right )\right )\right )\right )}{4 b^2 c} \] Input:

Integrate[(d - c^2*d*x^2)/(a + b*ArcCos[c*x])^2,x]
 

Output:

(d*((4*b*(1 - c^2*x^2)^(3/2))/(a + b*ArcCos[c*x]) + 3*(-(Cos[a/b]*CosInteg 
ral[a/b + ArcCos[c*x]]) + Cos[(3*a)/b]*CosIntegral[3*(a/b + ArcCos[c*x])] 
- Sin[a/b]*SinIntegral[a/b + ArcCos[c*x]] + Sin[(3*a)/b]*SinIntegral[3*(a/ 
b + ArcCos[c*x])])))/(4*b^2*c)
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.89, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {5167, 5225, 4906, 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.

\(\displaystyle \int \frac {d-c^2 d x^2}{(a+b \arccos (c x))^2} \, dx\)

\(\Big \downarrow \) 5167

\(\displaystyle \frac {3 c d \int \frac {x \sqrt {1-c^2 x^2}}{a+b \arccos (c x)}dx}{b}+\frac {d \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \arccos (c x))}\)

\(\Big \downarrow \) 5225

\(\displaystyle \frac {d \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \arccos (c x))}-\frac {3 d \int \frac {\cos \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right ) \sin ^2\left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))}{b^2 c}\)

\(\Big \downarrow \) 4906

\(\displaystyle \frac {d \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \arccos (c x))}-\frac {3 d \int \left (\frac {\cos \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{4 (a+b \arccos (c x))}-\frac {\cos \left (\frac {3 a}{b}-\frac {3 (a+b \arccos (c x))}{b}\right )}{4 (a+b \arccos (c x))}\right )d(a+b \arccos (c x))}{b^2 c}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \arccos (c x))}-\frac {3 d \left (\frac {1}{4} \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right )-\frac {1}{4} \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arccos (c x))}{b}\right )+\frac {1}{4} \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )-\frac {1}{4} \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arccos (c x))}{b}\right )\right )}{b^2 c}\)

Input:

Int[(d - c^2*d*x^2)/(a + b*ArcCos[c*x])^2,x]
 

Output:

(d*(1 - c^2*x^2)^(3/2))/(b*c*(a + b*ArcCos[c*x])) - (3*d*((Cos[a/b]*CosInt 
egral[(a + b*ArcCos[c*x])/b])/4 - (Cos[(3*a)/b]*CosIntegral[(3*(a + b*ArcC 
os[c*x]))/b])/4 + (Sin[a/b]*SinIntegral[(a + b*ArcCos[c*x])/b])/4 - (Sin[( 
3*a)/b]*SinIntegral[(3*(a + b*ArcCos[c*x]))/b])/4))/(b^2*c)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 4906

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b 
_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x 
]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG 
tQ[p, 0]
 

rule 5167

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_)^2)^(p_.), x_ 
Symbol] :> Simp[(-Sqrt[1 - c^2*x^2])*(d + e*x^2)^p*((a + b*ArcCos[c*x])^(n 
+ 1)/(b*c*(n + 1))), x] - Simp[c*((2*p + 1)/(b*(n + 1)))*Simp[(d + e*x^2)^p 
/(1 - c^2*x^2)^p]   Int[x*(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] && LtQ[n, - 
1]
 

rule 5225

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 
2)^(p_.), x_Symbol] :> Simp[(-(b*c^(m + 1))^(-1))*Simp[(d + e*x^2)^p/(1 - c 
^2*x^2)^p]   Subst[Int[x^n*Cos[-a/b + x/b]^m*Sin[-a/b + x/b]^(2*p + 1), x], 
 x, a + b*ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e 
, 0] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
 

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.52

method	result	size
derivativedivides	\(\frac {d \left (3 \arccos \left (c x \right ) \operatorname {Si}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b +3 \arccos \left (c x \right ) \operatorname {Ci}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) b -3 \arccos \left (c x \right ) \operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b -3 \arccos \left (c x \right ) \operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b +3 \,\operatorname {Si}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a +3 \,\operatorname {Ci}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a -3 \,\operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a -3 \,\operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a +3 \sqrt {-c^{2} x^{2}+1}\, b -\sin \left (3 \arccos \left (c x \right )\right ) b \right )}{4 c \left (a +b \arccos \left (c x \right )\right ) b^{2}}\)	\(238\)
default	\(\frac {d \left (3 \arccos \left (c x \right ) \operatorname {Si}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b +3 \arccos \left (c x \right ) \operatorname {Ci}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) b -3 \arccos \left (c x \right ) \operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b -3 \arccos \left (c x \right ) \operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b +3 \,\operatorname {Si}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a +3 \,\operatorname {Ci}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a -3 \,\operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a -3 \,\operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a +3 \sqrt {-c^{2} x^{2}+1}\, b -\sin \left (3 \arccos \left (c x \right )\right ) b \right )}{4 c \left (a +b \arccos \left (c x \right )\right ) b^{2}}\)	\(238\)

Input:

int((-c^2*d*x^2+d)/(a+b*arccos(c*x))^2,x,method=_RETURNVERBOSE)
 

Output:

1/4/c*d*(3*arccos(c*x)*Si(3*arccos(c*x)+3*a/b)*sin(3*a/b)*b+3*arccos(c*x)* 
Ci(3*arccos(c*x)+3*a/b)*cos(3*a/b)*b-3*arccos(c*x)*Si(arccos(c*x)+a/b)*sin 
(a/b)*b-3*arccos(c*x)*Ci(arccos(c*x)+a/b)*cos(a/b)*b+3*Si(3*arccos(c*x)+3* 
a/b)*sin(3*a/b)*a+3*Ci(3*arccos(c*x)+3*a/b)*cos(3*a/b)*a-3*Si(arccos(c*x)+ 
a/b)*sin(a/b)*a-3*Ci(arccos(c*x)+a/b)*cos(a/b)*a+3*(-c^2*x^2+1)^(1/2)*b-si 
n(3*arccos(c*x))*b)/(a+b*arccos(c*x))/b^2
 

Fricas [F]

\[ \int \frac {d-c^2 d x^2}{(a+b \arccos (c x))^2} \, dx=\int { -\frac {c^{2} d x^{2} - d}{{\left (b \arccos \left (c x\right ) + a\right )}^{2}} \,d x } \] Input:

integrate((-c^2*d*x^2+d)/(a+b*arccos(c*x))^2,x, algorithm="fricas")
 

Output:

integral(-(c^2*d*x^2 - d)/(b^2*arccos(c*x)^2 + 2*a*b*arccos(c*x) + a^2), x 
)
 

Sympy [F]

\[ \int \frac {d-c^2 d x^2}{(a+b \arccos (c x))^2} \, dx=- d \left (\int \frac {c^{2} x^{2}}{a^{2} + 2 a b \operatorname {acos}{\left (c x \right )} + b^{2} \operatorname {acos}^{2}{\left (c x \right )}}\, dx + \int \left (- \frac {1}{a^{2} + 2 a b \operatorname {acos}{\left (c x \right )} + b^{2} \operatorname {acos}^{2}{\left (c x \right )}}\right )\, dx\right ) \] Input:

integrate((-c**2*d*x**2+d)/(a+b*acos(c*x))**2,x)
 

Output:

-d*(Integral(c**2*x**2/(a**2 + 2*a*b*acos(c*x) + b**2*acos(c*x)**2), x) + 
Integral(-1/(a**2 + 2*a*b*acos(c*x) + b**2*acos(c*x)**2), x))
 

Maxima [F]

\[ \int \frac {d-c^2 d x^2}{(a+b \arccos (c x))^2} \, dx=\int { -\frac {c^{2} d x^{2} - d}{{\left (b \arccos \left (c x\right ) + a\right )}^{2}} \,d x } \] Input:

integrate((-c^2*d*x^2+d)/(a+b*arccos(c*x))^2,x, algorithm="maxima")
 

Output:

-((c^2*d*x^2 - d)*sqrt(c*x + 1)*sqrt(-c*x + 1) - 3*(b^2*c^2*d*arctan2(sqrt 
(c*x + 1)*sqrt(-c*x + 1), c*x) + a*b*c^2*d)*integrate(sqrt(c*x + 1)*sqrt(- 
c*x + 1)*x/(b^2*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) + a*b), x))/(b^ 
2*c*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) + a*b*c)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 610 vs. \(2 (147) = 294\).

Time = 0.23 (sec) , antiderivative size = 610, normalized size of antiderivative = 3.89 \[ \int \frac {d-c^2 d x^2}{(a+b \arccos (c x))^2} \, dx =\text {Too large to display} \] Input:

integrate((-c^2*d*x^2+d)/(a+b*arccos(c*x))^2,x, algorithm="giac")
 

Output:

3*b*d*arccos(c*x)*cos(a/b)^3*cos_integral(3*a/b + 3*arccos(c*x))/(b^3*c*ar 
ccos(c*x) + a*b^2*c) + 3*b*d*arccos(c*x)*cos(a/b)^2*sin(a/b)*sin_integral( 
3*a/b + 3*arccos(c*x))/(b^3*c*arccos(c*x) + a*b^2*c) - sqrt(-c^2*x^2 + 1)* 
b*c^2*d*x^2/(b^3*c*arccos(c*x) + a*b^2*c) + 3*a*d*cos(a/b)^3*cos_integral( 
3*a/b + 3*arccos(c*x))/(b^3*c*arccos(c*x) + a*b^2*c) + 3*a*d*cos(a/b)^2*si 
n(a/b)*sin_integral(3*a/b + 3*arccos(c*x))/(b^3*c*arccos(c*x) + a*b^2*c) - 
 9/4*b*d*arccos(c*x)*cos(a/b)*cos_integral(3*a/b + 3*arccos(c*x))/(b^3*c*a 
rccos(c*x) + a*b^2*c) - 3/4*b*d*arccos(c*x)*cos(a/b)*cos_integral(a/b + ar 
ccos(c*x))/(b^3*c*arccos(c*x) + a*b^2*c) - 3/4*b*d*arccos(c*x)*sin(a/b)*si 
n_integral(3*a/b + 3*arccos(c*x))/(b^3*c*arccos(c*x) + a*b^2*c) - 3/4*b*d* 
arccos(c*x)*sin(a/b)*sin_integral(a/b + arccos(c*x))/(b^3*c*arccos(c*x) + 
a*b^2*c) - 9/4*a*d*cos(a/b)*cos_integral(3*a/b + 3*arccos(c*x))/(b^3*c*arc 
cos(c*x) + a*b^2*c) - 3/4*a*d*cos(a/b)*cos_integral(a/b + arccos(c*x))/(b^ 
3*c*arccos(c*x) + a*b^2*c) - 3/4*a*d*sin(a/b)*sin_integral(3*a/b + 3*arcco 
s(c*x))/(b^3*c*arccos(c*x) + a*b^2*c) - 3/4*a*d*sin(a/b)*sin_integral(a/b 
+ arccos(c*x))/(b^3*c*arccos(c*x) + a*b^2*c) + sqrt(-c^2*x^2 + 1)*b*d/(b^3 
*c*arccos(c*x) + a*b^2*c)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {d-c^2 d x^2}{(a+b \arccos (c x))^2} \, dx=\int \frac {d-c^2\,d\,x^2}{{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2} \,d x \] Input:

int((d - c^2*d*x^2)/(a + b*acos(c*x))^2,x)
 

Output:

int((d - c^2*d*x^2)/(a + b*acos(c*x))^2, x)
 

Reduce [F]

\[ \int \frac {d-c^2 d x^2}{(a+b \arccos (c x))^2} \, dx=d \left (-\left (\int \frac {x^{2}}{\mathit {acos} \left (c x \right )^{2} b^{2}+2 \mathit {acos} \left (c x \right ) a b +a^{2}}d x \right ) c^{2}+\int \frac {1}{\mathit {acos} \left (c x \right )^{2} b^{2}+2 \mathit {acos} \left (c x \right ) a b +a^{2}}d x \right ) \] Input:

int((-c^2*d*x^2+d)/(a+b*acos(c*x))^2,x)
 

Output:

d*( - int(x**2/(acos(c*x)**2*b**2 + 2*acos(c*x)*a*b + a**2),x)*c**2 + int( 
1/(acos(c*x)**2*b**2 + 2*acos(c*x)*a*b + a**2),x))