\(\int \frac {(d+e x^2)^2}{a+b \arccos (c x)} \, dx\) [91]

Optimal result

Integrand size = 20, antiderivative size = 388 \[ \int \frac {\left (d+e x^2\right )^2}{a+b \arccos (c x)} \, dx=\frac {d^2 \operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{b c}+\frac {d e \operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{2 b c^3}+\frac {e^2 \operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{8 b c^5}+\frac {d e \operatorname {CosIntegral}\left (\frac {3 (a+b \arccos (c x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{2 b c^3}+\frac {3 e^2 \operatorname {CosIntegral}\left (\frac {3 (a+b \arccos (c x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{16 b c^5}+\frac {e^2 \operatorname {CosIntegral}\left (\frac {5 (a+b \arccos (c x))}{b}\right ) \sin \left (\frac {5 a}{b}\right )}{16 b c^5}-\frac {d^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )}{b c}-\frac {d e \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )}{2 b c^3}-\frac {e^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )}{8 b c^5}-\frac {d e \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arccos (c x))}{b}\right )}{2 b c^3}-\frac {3 e^2 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arccos (c x))}{b}\right )}{16 b c^5}-\frac {e^2 \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arccos (c x))}{b}\right )}{16 b c^5} \] Output:

d^2*Ci((a+b*arccos(c*x))/b)*sin(a/b)/b/c+1/2*d*e*Ci((a+b*arccos(c*x))/b)*s 
in(a/b)/b/c^3+1/8*e^2*Ci((a+b*arccos(c*x))/b)*sin(a/b)/b/c^5+1/2*d*e*Ci(3* 
(a+b*arccos(c*x))/b)*sin(3*a/b)/b/c^3+3/16*e^2*Ci(3*(a+b*arccos(c*x))/b)*s 
in(3*a/b)/b/c^5+1/16*e^2*Ci(5*(a+b*arccos(c*x))/b)*sin(5*a/b)/b/c^5-d^2*co 
s(a/b)*Si((a+b*arccos(c*x))/b)/b/c-1/2*d*e*cos(a/b)*Si((a+b*arccos(c*x))/b 
)/b/c^3-1/8*e^2*cos(a/b)*Si((a+b*arccos(c*x))/b)/b/c^5-1/2*d*e*cos(3*a/b)* 
Si(3*(a+b*arccos(c*x))/b)/b/c^3-3/16*e^2*cos(3*a/b)*Si(3*(a+b*arccos(c*x)) 
/b)/b/c^5-1/16*e^2*cos(5*a/b)*Si(5*(a+b*arccos(c*x))/b)/b/c^5
 

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.65 \[ \int \frac {\left (d+e x^2\right )^2}{a+b \arccos (c x)} \, dx=-\frac {-2 \left (8 c^4 d^2+4 c^2 d e+e^2\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arccos (c x)\right ) \sin \left (\frac {a}{b}\right )-e \left (8 c^2 d+3 e\right ) \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arccos (c x)\right )\right ) \sin \left (\frac {3 a}{b}\right )-e^2 \operatorname {CosIntegral}\left (5 \left (\frac {a}{b}+\arccos (c x)\right )\right ) \sin \left (\frac {5 a}{b}\right )+16 c^4 d^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arccos (c x)\right )+8 c^2 d e \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arccos (c x)\right )+2 e^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arccos (c x)\right )+8 c^2 d e \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arccos (c x)\right )\right )+3 e^2 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arccos (c x)\right )\right )+e^2 \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (5 \left (\frac {a}{b}+\arccos (c x)\right )\right )}{16 b c^5} \] Input:

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

Output:

-1/16*(-2*(8*c^4*d^2 + 4*c^2*d*e + e^2)*CosIntegral[a/b + ArcCos[c*x]]*Sin 
[a/b] - e*(8*c^2*d + 3*e)*CosIntegral[3*(a/b + ArcCos[c*x])]*Sin[(3*a)/b] 
- e^2*CosIntegral[5*(a/b + ArcCos[c*x])]*Sin[(5*a)/b] + 16*c^4*d^2*Cos[a/b 
]*SinIntegral[a/b + ArcCos[c*x]] + 8*c^2*d*e*Cos[a/b]*SinIntegral[a/b + Ar 
cCos[c*x]] + 2*e^2*Cos[a/b]*SinIntegral[a/b + ArcCos[c*x]] + 8*c^2*d*e*Cos 
[(3*a)/b]*SinIntegral[3*(a/b + ArcCos[c*x])] + 3*e^2*Cos[(3*a)/b]*SinInteg 
ral[3*(a/b + ArcCos[c*x])] + e^2*Cos[(5*a)/b]*SinIntegral[5*(a/b + ArcCos[ 
c*x])])/(b*c^5)
 

Rubi [A] (verified)

Time = 1.01 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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.

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

\(\Big \downarrow \) 5173

\(\displaystyle \int \left (\frac {d^2}{a+b \arccos (c x)}+\frac {2 d e x^2}{a+b \arccos (c x)}+\frac {e^2 x^4}{a+b \arccos (c x)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {e^2 \sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right )}{8 b c^5}+\frac {3 e^2 \sin \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arccos (c x))}{b}\right )}{16 b c^5}+\frac {e^2 \sin \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {5 (a+b \arccos (c x))}{b}\right )}{16 b c^5}-\frac {e^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )}{8 b c^5}-\frac {3 e^2 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arccos (c x))}{b}\right )}{16 b c^5}-\frac {e^2 \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arccos (c x))}{b}\right )}{16 b c^5}+\frac {d e \sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right )}{2 b c^3}+\frac {d e \sin \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arccos (c x))}{b}\right )}{2 b c^3}-\frac {d e \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )}{2 b c^3}-\frac {d e \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arccos (c x))}{b}\right )}{2 b c^3}+\frac {d^2 \sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right )}{b c}-\frac {d^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )}{b c}\)

Input:

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

Output:

(d^2*CosIntegral[(a + b*ArcCos[c*x])/b]*Sin[a/b])/(b*c) + (d*e*CosIntegral 
[(a + b*ArcCos[c*x])/b]*Sin[a/b])/(2*b*c^3) + (e^2*CosIntegral[(a + b*ArcC 
os[c*x])/b]*Sin[a/b])/(8*b*c^5) + (d*e*CosIntegral[(3*(a + b*ArcCos[c*x])) 
/b]*Sin[(3*a)/b])/(2*b*c^3) + (3*e^2*CosIntegral[(3*(a + b*ArcCos[c*x]))/b 
]*Sin[(3*a)/b])/(16*b*c^5) + (e^2*CosIntegral[(5*(a + b*ArcCos[c*x]))/b]*S 
in[(5*a)/b])/(16*b*c^5) - (d^2*Cos[a/b]*SinIntegral[(a + b*ArcCos[c*x])/b] 
)/(b*c) - (d*e*Cos[a/b]*SinIntegral[(a + b*ArcCos[c*x])/b])/(2*b*c^3) - (e 
^2*Cos[a/b]*SinIntegral[(a + b*ArcCos[c*x])/b])/(8*b*c^5) - (d*e*Cos[(3*a) 
/b]*SinIntegral[(3*(a + b*ArcCos[c*x]))/b])/(2*b*c^3) - (3*e^2*Cos[(3*a)/b 
]*SinIntegral[(3*(a + b*ArcCos[c*x]))/b])/(16*b*c^5) - (e^2*Cos[(5*a)/b]*S 
inIntegral[(5*(a + b*ArcCos[c*x]))/b])/(16*b*c^5)
 

Defintions of rubi rules used

rule 2009

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

rule 5173

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])
 

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.80

method	result	size
derivativedivides	\(-\frac {16 \,\operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) c^{4} d^{2}-16 \,\operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) c^{4} d^{2}+8 \,\operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) c^{2} d e -8 \,\operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) c^{2} d e +8 \,\operatorname {Si}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) c^{2} d e -8 \,\operatorname {Ci}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) c^{2} d e +2 \,\operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) e^{2}-2 \,\operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) e^{2}+3 \,\operatorname {Si}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) e^{2}-3 \,\operatorname {Ci}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) e^{2}+\operatorname {Si}\left (5 \arccos \left (c x \right )+\frac {5 a}{b}\right ) \cos \left (\frac {5 a}{b}\right ) e^{2}-\operatorname {Ci}\left (5 \arccos \left (c x \right )+\frac {5 a}{b}\right ) \sin \left (\frac {5 a}{b}\right ) e^{2}}{16 c^{5} b}\)	\(311\)
default	\(-\frac {16 \,\operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) c^{4} d^{2}-16 \,\operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) c^{4} d^{2}+8 \,\operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) c^{2} d e -8 \,\operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) c^{2} d e +8 \,\operatorname {Si}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) c^{2} d e -8 \,\operatorname {Ci}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) c^{2} d e +2 \,\operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) e^{2}-2 \,\operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) e^{2}+3 \,\operatorname {Si}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) e^{2}-3 \,\operatorname {Ci}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) e^{2}+\operatorname {Si}\left (5 \arccos \left (c x \right )+\frac {5 a}{b}\right ) \cos \left (\frac {5 a}{b}\right ) e^{2}-\operatorname {Ci}\left (5 \arccos \left (c x \right )+\frac {5 a}{b}\right ) \sin \left (\frac {5 a}{b}\right ) e^{2}}{16 c^{5} b}\)	\(311\)

Input:

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

Output:

-1/16/c^5*(16*Si(arccos(c*x)+a/b)*cos(a/b)*c^4*d^2-16*Ci(arccos(c*x)+a/b)* 
sin(a/b)*c^4*d^2+8*Si(arccos(c*x)+a/b)*cos(a/b)*c^2*d*e-8*Ci(arccos(c*x)+a 
/b)*sin(a/b)*c^2*d*e+8*Si(3*arccos(c*x)+3*a/b)*cos(3*a/b)*c^2*d*e-8*Ci(3*a 
rccos(c*x)+3*a/b)*sin(3*a/b)*c^2*d*e+2*Si(arccos(c*x)+a/b)*cos(a/b)*e^2-2* 
Ci(arccos(c*x)+a/b)*sin(a/b)*e^2+3*Si(3*arccos(c*x)+3*a/b)*cos(3*a/b)*e^2- 
3*Ci(3*arccos(c*x)+3*a/b)*sin(3*a/b)*e^2+Si(5*arccos(c*x)+5*a/b)*cos(5*a/b 
)*e^2-Ci(5*arccos(c*x)+5*a/b)*sin(5*a/b)*e^2)/b
 

Fricas [F]

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

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

Output:

integral((e^2*x^4 + 2*d*e*x^2 + d^2)/(b*arccos(c*x) + a), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

integrate((e*x^2 + d)^2/(b*arccos(c*x) + a), x)
 

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 635, normalized size of antiderivative = 1.64 \[ \int \frac {\left (d+e x^2\right )^2}{a+b \arccos (c x)} \, dx =\text {Too large to display} \] Input:

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

Output:

e^2*cos(a/b)^4*cos_integral(5*a/b + 5*arccos(c*x))*sin(a/b)/(b*c^5) + 2*d* 
e*cos(a/b)^2*cos_integral(3*a/b + 3*arccos(c*x))*sin(a/b)/(b*c^3) + d^2*co 
s_integral(a/b + arccos(c*x))*sin(a/b)/(b*c) - e^2*cos(a/b)^5*sin_integral 
(5*a/b + 5*arccos(c*x))/(b*c^5) - 2*d*e*cos(a/b)^3*sin_integral(3*a/b + 3* 
arccos(c*x))/(b*c^3) - d^2*cos(a/b)*sin_integral(a/b + arccos(c*x))/(b*c) 
- 3/4*e^2*cos(a/b)^2*cos_integral(5*a/b + 5*arccos(c*x))*sin(a/b)/(b*c^5) 
- 1/2*d*e*cos_integral(3*a/b + 3*arccos(c*x))*sin(a/b)/(b*c^3) + 3/4*e^2*c 
os(a/b)^2*cos_integral(3*a/b + 3*arccos(c*x))*sin(a/b)/(b*c^5) + 1/2*d*e*c 
os_integral(a/b + arccos(c*x))*sin(a/b)/(b*c^3) + 5/4*e^2*cos(a/b)^3*sin_i 
ntegral(5*a/b + 5*arccos(c*x))/(b*c^5) + 3/2*d*e*cos(a/b)*sin_integral(3*a 
/b + 3*arccos(c*x))/(b*c^3) - 3/4*e^2*cos(a/b)^3*sin_integral(3*a/b + 3*ar 
ccos(c*x))/(b*c^5) - 1/2*d*e*cos(a/b)*sin_integral(a/b + arccos(c*x))/(b*c 
^3) + 1/16*e^2*cos_integral(5*a/b + 5*arccos(c*x))*sin(a/b)/(b*c^5) - 3/16 
*e^2*cos_integral(3*a/b + 3*arccos(c*x))*sin(a/b)/(b*c^5) + 1/8*e^2*cos_in 
tegral(a/b + arccos(c*x))*sin(a/b)/(b*c^5) - 5/16*e^2*cos(a/b)*sin_integra 
l(5*a/b + 5*arccos(c*x))/(b*c^5) + 9/16*e^2*cos(a/b)*sin_integral(3*a/b + 
3*arccos(c*x))/(b*c^5) - 1/8*e^2*cos(a/b)*sin_integral(a/b + arccos(c*x))/ 
(b*c^5)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (d+e x^2\right )^2}{a+b \arccos (c x)} \, dx=\left (\int \frac {x^{4}}{\mathit {acos} \left (c x \right ) b +a}d x \right ) e^{2}+2 \left (\int \frac {x^{2}}{\mathit {acos} \left (c x \right ) b +a}d x \right ) d e +\left (\int \frac {1}{\mathit {acos} \left (c x \right ) b +a}d x \right ) d^{2} \] Input:

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

Output:

int(x**4/(acos(c*x)*b + a),x)*e**2 + 2*int(x**2/(acos(c*x)*b + a),x)*d*e + 
 int(1/(acos(c*x)*b + a),x)*d**2