\(\int \frac {(d-c^2 d x^2)^3}{a+b \arccos (c x)} \, dx\) [27]

Optimal result

Integrand size = 24, antiderivative size = 269 \[ \int \frac {\left (d-c^2 d x^2\right )^3}{a+b \arccos (c x)} \, dx=\frac {35 d^3 \operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{64 b c}-\frac {21 d^3 \operatorname {CosIntegral}\left (\frac {3 (a+b \arccos (c x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{64 b c}+\frac {7 d^3 \operatorname {CosIntegral}\left (\frac {5 (a+b \arccos (c x))}{b}\right ) \sin \left (\frac {5 a}{b}\right )}{64 b c}-\frac {d^3 \operatorname {CosIntegral}\left (\frac {7 (a+b \arccos (c x))}{b}\right ) \sin \left (\frac {7 a}{b}\right )}{64 b c}-\frac {35 d^3 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )}{64 b c}+\frac {21 d^3 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arccos (c x))}{b}\right )}{64 b c}-\frac {7 d^3 \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arccos (c x))}{b}\right )}{64 b c}+\frac {d^3 \cos \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 (a+b \arccos (c x))}{b}\right )}{64 b c} \] Output:

35/64*d^3*Ci((a+b*arccos(c*x))/b)*sin(a/b)/b/c-21/64*d^3*Ci(3*(a+b*arccos( 
c*x))/b)*sin(3*a/b)/b/c+7/64*d^3*Ci(5*(a+b*arccos(c*x))/b)*sin(5*a/b)/b/c- 
1/64*d^3*Ci(7*(a+b*arccos(c*x))/b)*sin(7*a/b)/b/c-35/64*d^3*cos(a/b)*Si((a 
+b*arccos(c*x))/b)/b/c+21/64*d^3*cos(3*a/b)*Si(3*(a+b*arccos(c*x))/b)/b/c- 
7/64*d^3*cos(5*a/b)*Si(5*(a+b*arccos(c*x))/b)/b/c+1/64*d^3*cos(7*a/b)*Si(7 
*(a+b*arccos(c*x))/b)/b/c
 

Mathematica [A] (verified)

Time = 1.29 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.68 \[ \int \frac {\left (d-c^2 d x^2\right )^3}{a+b \arccos (c x)} \, dx=\frac {d^3 \left (35 \operatorname {CosIntegral}\left (\frac {a}{b}+\arccos (c x)\right ) \sin \left (\frac {a}{b}\right )-21 \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arccos (c x)\right )\right ) \sin \left (\frac {3 a}{b}\right )+7 \operatorname {CosIntegral}\left (5 \left (\frac {a}{b}+\arccos (c x)\right )\right ) \sin \left (\frac {5 a}{b}\right )-\operatorname {CosIntegral}\left (7 \left (\frac {a}{b}+\arccos (c x)\right )\right ) \sin \left (\frac {7 a}{b}\right )-35 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arccos (c x)\right )+21 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arccos (c x)\right )\right )-7 \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (5 \left (\frac {a}{b}+\arccos (c x)\right )\right )+\cos \left (\frac {7 a}{b}\right ) \text {Si}\left (7 \left (\frac {a}{b}+\arccos (c x)\right )\right )\right )}{64 b c} \] Input:

Integrate[(d - c^2*d*x^2)^3/(a + b*ArcCos[c*x]),x]
 

Output:

(d^3*(35*CosIntegral[a/b + ArcCos[c*x]]*Sin[a/b] - 21*CosIntegral[3*(a/b + 
 ArcCos[c*x])]*Sin[(3*a)/b] + 7*CosIntegral[5*(a/b + ArcCos[c*x])]*Sin[(5* 
a)/b] - CosIntegral[7*(a/b + ArcCos[c*x])]*Sin[(7*a)/b] - 35*Cos[a/b]*SinI 
ntegral[a/b + ArcCos[c*x]] + 21*Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcCos[c 
*x])] - 7*Cos[(5*a)/b]*SinIntegral[5*(a/b + ArcCos[c*x])] + Cos[(7*a)/b]*S 
inIntegral[7*(a/b + ArcCos[c*x])]))/(64*b*c)
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.77, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {5169, 25, 3042, 3793, 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.

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

\(\Big \downarrow \) 5169

\(\displaystyle -\frac {d^3 \int -\frac {\sin ^7\left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))}{b c}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {d^3 \int \frac {\sin ^7\left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))}{b c}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {d^3 \int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )^7}{a+b \arccos (c x)}d(a+b \arccos (c x))}{b c}\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {d^3 \int \left (-\frac {\sin \left (\frac {7 a}{b}-\frac {7 (a+b \arccos (c x))}{b}\right )}{64 (a+b \arccos (c x))}+\frac {7 \sin \left (\frac {5 a}{b}-\frac {5 (a+b \arccos (c x))}{b}\right )}{64 (a+b \arccos (c x))}-\frac {21 \sin \left (\frac {3 a}{b}-\frac {3 (a+b \arccos (c x))}{b}\right )}{64 (a+b \arccos (c x))}+\frac {35 \sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{64 (a+b \arccos (c x))}\right )d(a+b \arccos (c x))}{b c}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {d^3 \left (-\frac {35}{64} \sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right )+\frac {21}{64} \sin \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arccos (c x))}{b}\right )-\frac {7}{64} \sin \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {5 (a+b \arccos (c x))}{b}\right )+\frac {1}{64} \sin \left (\frac {7 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {7 (a+b \arccos (c x))}{b}\right )+\frac {35}{64} \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )-\frac {21}{64} \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arccos (c x))}{b}\right )+\frac {7}{64} \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arccos (c x))}{b}\right )-\frac {1}{64} \cos \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 (a+b \arccos (c x))}{b}\right )\right )}{b c}\)

Input:

Int[(d - c^2*d*x^2)^3/(a + b*ArcCos[c*x]),x]
 

Output:

-((d^3*((-35*CosIntegral[(a + b*ArcCos[c*x])/b]*Sin[a/b])/64 + (21*CosInte 
gral[(3*(a + b*ArcCos[c*x]))/b]*Sin[(3*a)/b])/64 - (7*CosIntegral[(5*(a + 
b*ArcCos[c*x]))/b]*Sin[(5*a)/b])/64 + (CosIntegral[(7*(a + b*ArcCos[c*x])) 
/b]*Sin[(7*a)/b])/64 + (35*Cos[a/b]*SinIntegral[(a + b*ArcCos[c*x])/b])/64 
 - (21*Cos[(3*a)/b]*SinIntegral[(3*(a + b*ArcCos[c*x]))/b])/64 + (7*Cos[(5 
*a)/b]*SinIntegral[(5*(a + b*ArcCos[c*x]))/b])/64 - (Cos[(7*a)/b]*SinInteg 
ral[(7*(a + b*ArcCos[c*x]))/b])/64))/(b*c))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2009

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

rule 5169

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x 
_Symbol] :> Simp[(-(b*c)^(-1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   Subst[ 
Int[x^n*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, 0]
 

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.70

method	result	size
derivativedivides	\(\frac {d^{3} \left (\operatorname {Si}\left (7 \arccos \left (c x \right )+\frac {7 a}{b}\right ) \cos \left (\frac {7 a}{b}\right )-\operatorname {Ci}\left (7 \arccos \left (c x \right )+\frac {7 a}{b}\right ) \sin \left (\frac {7 a}{b}\right )-7 \,\operatorname {Si}\left (5 \arccos \left (c x \right )+\frac {5 a}{b}\right ) \cos \left (\frac {5 a}{b}\right )+7 \,\operatorname {Ci}\left (5 \arccos \left (c x \right )+\frac {5 a}{b}\right ) \sin \left (\frac {5 a}{b}\right )+21 \,\operatorname {Si}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right )-21 \,\operatorname {Ci}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right )-35 \,\operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right )+35 \,\operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right )\right )}{64 c b}\)	\(188\)
default	\(\frac {d^{3} \left (\operatorname {Si}\left (7 \arccos \left (c x \right )+\frac {7 a}{b}\right ) \cos \left (\frac {7 a}{b}\right )-\operatorname {Ci}\left (7 \arccos \left (c x \right )+\frac {7 a}{b}\right ) \sin \left (\frac {7 a}{b}\right )-7 \,\operatorname {Si}\left (5 \arccos \left (c x \right )+\frac {5 a}{b}\right ) \cos \left (\frac {5 a}{b}\right )+7 \,\operatorname {Ci}\left (5 \arccos \left (c x \right )+\frac {5 a}{b}\right ) \sin \left (\frac {5 a}{b}\right )+21 \,\operatorname {Si}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right )-21 \,\operatorname {Ci}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right )-35 \,\operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right )+35 \,\operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right )\right )}{64 c b}\)	\(188\)

Input:

int((-c^2*d*x^2+d)^3/(a+b*arccos(c*x)),x,method=_RETURNVERBOSE)
 

Output:

1/64/c*d^3*(Si(7*arccos(c*x)+7*a/b)*cos(7*a/b)-Ci(7*arccos(c*x)+7*a/b)*sin 
(7*a/b)-7*Si(5*arccos(c*x)+5*a/b)*cos(5*a/b)+7*Ci(5*arccos(c*x)+5*a/b)*sin 
(5*a/b)+21*Si(3*arccos(c*x)+3*a/b)*cos(3*a/b)-21*Ci(3*arccos(c*x)+3*a/b)*s 
in(3*a/b)-35*Si(arccos(c*x)+a/b)*cos(a/b)+35*Ci(arccos(c*x)+a/b)*sin(a/b)) 
/b
 

Fricas [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^3}{a+b \arccos (c x)} \, dx=\int { -\frac {{\left (c^{2} d x^{2} - d\right )}^{3}}{b \arccos \left (c x\right ) + a} \,d x } \] Input:

integrate((-c^2*d*x^2+d)^3/(a+b*arccos(c*x)),x, algorithm="fricas")
 

Output:

integral(-(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3)/(b*arccos(c* 
x) + a), x)
 

Sympy [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^3}{a+b \arccos (c x)} \, dx=- d^{3} \left (\int \frac {3 c^{2} x^{2}}{a + b \operatorname {acos}{\left (c x \right )}}\, dx + \int \left (- \frac {3 c^{4} x^{4}}{a + b \operatorname {acos}{\left (c x \right )}}\right )\, dx + \int \frac {c^{6} x^{6}}{a + b \operatorname {acos}{\left (c x \right )}}\, dx + \int \left (- \frac {1}{a + b \operatorname {acos}{\left (c x \right )}}\right )\, dx\right ) \] Input:

integrate((-c**2*d*x**2+d)**3/(a+b*acos(c*x)),x)
 

Output:

-d**3*(Integral(3*c**2*x**2/(a + b*acos(c*x)), x) + Integral(-3*c**4*x**4/ 
(a + b*acos(c*x)), x) + Integral(c**6*x**6/(a + b*acos(c*x)), x) + Integra 
l(-1/(a + b*acos(c*x)), x))
 

Maxima [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^3}{a+b \arccos (c x)} \, dx=\int { -\frac {{\left (c^{2} d x^{2} - d\right )}^{3}}{b \arccos \left (c x\right ) + a} \,d x } \] Input:

integrate((-c^2*d*x^2+d)^3/(a+b*arccos(c*x)),x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 674 vs. \(2 (253) = 506\).

Time = 0.16 (sec) , antiderivative size = 674, normalized size of antiderivative = 2.51 \[ \int \frac {\left (d-c^2 d x^2\right )^3}{a+b \arccos (c x)} \, dx =\text {Too large to display} \] Input:

integrate((-c^2*d*x^2+d)^3/(a+b*arccos(c*x)),x, algorithm="giac")
 

Output:

-d^3*cos(a/b)^6*cos_integral(7*a/b + 7*arccos(c*x))*sin(a/b)/(b*c) + d^3*c 
os(a/b)^7*sin_integral(7*a/b + 7*arccos(c*x))/(b*c) + 5/4*d^3*cos(a/b)^4*c 
os_integral(7*a/b + 7*arccos(c*x))*sin(a/b)/(b*c) + 7/4*d^3*cos(a/b)^4*cos 
_integral(5*a/b + 5*arccos(c*x))*sin(a/b)/(b*c) - 7/4*d^3*cos(a/b)^5*sin_i 
ntegral(7*a/b + 7*arccos(c*x))/(b*c) - 7/4*d^3*cos(a/b)^5*sin_integral(5*a 
/b + 5*arccos(c*x))/(b*c) - 3/8*d^3*cos(a/b)^2*cos_integral(7*a/b + 7*arcc 
os(c*x))*sin(a/b)/(b*c) - 21/16*d^3*cos(a/b)^2*cos_integral(5*a/b + 5*arcc 
os(c*x))*sin(a/b)/(b*c) - 21/16*d^3*cos(a/b)^2*cos_integral(3*a/b + 3*arcc 
os(c*x))*sin(a/b)/(b*c) + 7/8*d^3*cos(a/b)^3*sin_integral(7*a/b + 7*arccos 
(c*x))/(b*c) + 35/16*d^3*cos(a/b)^3*sin_integral(5*a/b + 5*arccos(c*x))/(b 
*c) + 21/16*d^3*cos(a/b)^3*sin_integral(3*a/b + 3*arccos(c*x))/(b*c) + 1/6 
4*d^3*cos_integral(7*a/b + 7*arccos(c*x))*sin(a/b)/(b*c) + 7/64*d^3*cos_in 
tegral(5*a/b + 5*arccos(c*x))*sin(a/b)/(b*c) + 21/64*d^3*cos_integral(3*a/ 
b + 3*arccos(c*x))*sin(a/b)/(b*c) + 35/64*d^3*cos_integral(a/b + arccos(c* 
x))*sin(a/b)/(b*c) - 7/64*d^3*cos(a/b)*sin_integral(7*a/b + 7*arccos(c*x)) 
/(b*c) - 35/64*d^3*cos(a/b)*sin_integral(5*a/b + 5*arccos(c*x))/(b*c) - 63 
/64*d^3*cos(a/b)*sin_integral(3*a/b + 3*arccos(c*x))/(b*c) - 35/64*d^3*cos 
(a/b)*sin_integral(a/b + arccos(c*x))/(b*c)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^3}{a+b \arccos (c x)} \, dx=\int \frac {{\left (d-c^2\,d\,x^2\right )}^3}{a+b\,\mathrm {acos}\left (c\,x\right )} \,d x \] Input:

int((d - c^2*d*x^2)^3/(a + b*acos(c*x)),x)
 

Output:

int((d - c^2*d*x^2)^3/(a + b*acos(c*x)), x)
 

Reduce [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^3}{a+b \arccos (c x)} \, dx=d^{3} \left (-\left (\int \frac {x^{6}}{\mathit {acos} \left (c x \right ) b +a}d x \right ) c^{6}+3 \left (\int \frac {x^{4}}{\mathit {acos} \left (c x \right ) b +a}d x \right ) c^{4}-3 \left (\int \frac {x^{2}}{\mathit {acos} \left (c x \right ) b +a}d x \right ) c^{2}+\int \frac {1}{\mathit {acos} \left (c x \right ) b +a}d x \right ) \] Input:

int((-c^2*d*x^2+d)^3/(a+b*acos(c*x)),x)
 

Output:

d**3*( - int(x**6/(acos(c*x)*b + a),x)*c**6 + 3*int(x**4/(acos(c*x)*b + a) 
,x)*c**4 - 3*int(x**2/(acos(c*x)*b + a),x)*c**2 + int(1/(acos(c*x)*b + a), 
x))