\(\int \frac {x^3 (a+b \arccos (c x))^2}{(d-c^2 d x^2)^{3/2}} \, dx\) [248]

Optimal result

Integrand size = 29, antiderivative size = 412 \[ \int \frac {x^3 (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {4 a b x \sqrt {1-c^2 x^2}}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 \left (1-c^2 x^2\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {4 b^2 x \sqrt {1-c^2 x^2} \arccos (c x)}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {2 b x \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \arccos (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{c^4 d^2}+\frac {4 i b \sqrt {1-c^2 x^2} (a+b \arccos (c x)) \arctan \left (e^{i \arccos (c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}} \] Output:

-4*a*b*x*(-c^2*x^2+1)^(1/2)/c^3/d/(-c^2*d*x^2+d)^(1/2)-2*b^2*(-c^2*x^2+1)/ 
c^4/d/(-c^2*d*x^2+d)^(1/2)-4*b^2*x*(-c^2*x^2+1)^(1/2)*arccos(c*x)/c^3/d/(- 
c^2*d*x^2+d)^(1/2)+2*b*x*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))/c^3/d/(-c^2* 
d*x^2+d)^(1/2)+x^2*(a+b*arccos(c*x))^2/c^2/d/(-c^2*d*x^2+d)^(1/2)+2*(-c^2* 
d*x^2+d)^(1/2)*(a+b*arccos(c*x))^2/c^4/d^2+4*I*b*(-c^2*x^2+1)^(1/2)*(a+b*a 
rccos(c*x))*arctan(c*x+I*(-c^2*x^2+1)^(1/2))/c^4/d/(-c^2*d*x^2+d)^(1/2)-2* 
I*b^2*(-c^2*x^2+1)^(1/2)*polylog(2,-I*(c*x+I*(-c^2*x^2+1)^(1/2)))/c^4/d/(- 
c^2*d*x^2+d)^(1/2)+2*I*b^2*(-c^2*x^2+1)^(1/2)*polylog(2,I*(c*x+I*(-c^2*x^2 
+1)^(1/2)))/c^4/d/(-c^2*d*x^2+d)^(1/2)
 

Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 336, normalized size of antiderivative = 0.82 \[ \int \frac {x^3 (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {4 a^2-2 b^2-2 a^2 c^2 x^2+6 a b \arccos (c x)+3 b^2 \arccos (c x)^2+2 b^2 \cos (2 \arccos (c x))-2 a b \arccos (c x) \cos (2 \arccos (c x))-b^2 \arccos (c x)^2 \cos (2 \arccos (c x))-4 b^2 \sqrt {1-c^2 x^2} \arccos (c x) \log \left (1-e^{i \arccos (c x)}\right )+4 b^2 \sqrt {1-c^2 x^2} \arccos (c x) \log \left (1+e^{i \arccos (c x)}\right )+4 a b \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \arccos (c x)\right )\right )-4 a b \sqrt {1-c^2 x^2} \log \left (\sin \left (\frac {1}{2} \arccos (c x)\right )\right )-4 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )+4 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )+2 a b \sin (2 \arccos (c x))+2 b^2 \arccos (c x) \sin (2 \arccos (c x))}{2 c^4 d \sqrt {d-c^2 d x^2}} \] Input:

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

Output:

(4*a^2 - 2*b^2 - 2*a^2*c^2*x^2 + 6*a*b*ArcCos[c*x] + 3*b^2*ArcCos[c*x]^2 + 
 2*b^2*Cos[2*ArcCos[c*x]] - 2*a*b*ArcCos[c*x]*Cos[2*ArcCos[c*x]] - b^2*Arc 
Cos[c*x]^2*Cos[2*ArcCos[c*x]] - 4*b^2*Sqrt[1 - c^2*x^2]*ArcCos[c*x]*Log[1 
- E^(I*ArcCos[c*x])] + 4*b^2*Sqrt[1 - c^2*x^2]*ArcCos[c*x]*Log[1 + E^(I*Ar 
cCos[c*x])] + 4*a*b*Sqrt[1 - c^2*x^2]*Log[Cos[ArcCos[c*x]/2]] - 4*a*b*Sqrt 
[1 - c^2*x^2]*Log[Sin[ArcCos[c*x]/2]] - (4*I)*b^2*Sqrt[1 - c^2*x^2]*PolyLo 
g[2, -E^(I*ArcCos[c*x])] + (4*I)*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, E^(I*Arc 
Cos[c*x])] + 2*a*b*Sin[2*ArcCos[c*x]] + 2*b^2*ArcCos[c*x]*Sin[2*ArcCos[c*x 
]])/(2*c^4*d*Sqrt[d - c^2*d*x^2])
 

Rubi [A] (verified)

Time = 1.36 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.68, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {5207, 5183, 2009, 5211, 241, 5165, 3042, 4671, 2715, 2838}

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.

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

\(\Big \downarrow \) 5207

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

\(\Big \downarrow \) 5183

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 5211

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

\(\Big \downarrow \) 241

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

\(\Big \downarrow \) 5165

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4671

\(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {-b \int \log \left (1-e^{i \arccos (c x)}\right )d\arccos (c x)+b \int \log \left (1+e^{i \arccos (c x)}\right )d\arccos (c x)-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))}{c^3}-\frac {x (a+b \arccos (c x))}{c^2}+\frac {b \sqrt {1-c^2 x^2}}{c^3}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \arccos (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {i b \int e^{-i \arccos (c x)} \log \left (1-e^{i \arccos (c x)}\right )de^{i \arccos (c x)}-i b \int e^{-i \arccos (c x)} \log \left (1+e^{i \arccos (c x)}\right )de^{i \arccos (c x)}-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))}{c^3}-\frac {x (a+b \arccos (c x))}{c^2}+\frac {b \sqrt {1-c^2 x^2}}{c^3}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \arccos (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{c^3}-\frac {x (a+b \arccos (c x))}{c^2}+\frac {b \sqrt {1-c^2 x^2}}{c^3}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \arccos (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\)

Input:

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

Output:

(x^2*(a + b*ArcCos[c*x])^2)/(c^2*d*Sqrt[d - c^2*d*x^2]) - (2*(-((Sqrt[d - 
c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(c^2*d)) - (2*b*Sqrt[1 - c^2*x^2]*(a*x - 
 (b*Sqrt[1 - c^2*x^2])/c + b*x*ArcCos[c*x]))/(c*Sqrt[d - c^2*d*x^2])))/(c^ 
2*d) + (2*b*Sqrt[1 - c^2*x^2]*((b*Sqrt[1 - c^2*x^2])/c^3 - (x*(a + b*ArcCo 
s[c*x]))/c^2 - (-2*(a + b*ArcCos[c*x])*ArcTanh[E^(I*ArcCos[c*x])] + I*b*Po 
lyLog[2, -E^(I*ArcCos[c*x])] - I*b*PolyLog[2, E^(I*ArcCos[c*x])])/c^3))/(c 
*d*Sqrt[d - c^2*d*x^2])
 

Defintions of rubi rules used

rule 241

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ 
(2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
 

rule 2009

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

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

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

rule 4671

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 
2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f)   Int[(c + 
d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f)   Int[(c + d*x 
)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG 
tQ[m, 0]
 

rule 5165

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

rule 5183

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]
 

rule 5207

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

rule 5211

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

Maple [A] (verified)

Time = 0.61 (sec) , antiderivative size = 658, normalized size of antiderivative = 1.60

method	result	size
default	\(a^{2} \left (-\frac {x^{2}}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {2}{d \,c^{4} \sqrt {-c^{2} d \,x^{2}+d}}\right )+b^{2} \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )^{2}-2+2 i \arccos \left (c x \right )\right )}{2 d^{2} c^{4} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )^{2}-2-2 i \arccos \left (c x \right )\right )}{2 d^{2} c^{4} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right )^{2}}{d^{2} c^{4} \left (c^{2} x^{2}-1\right )}-\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-i \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} c^{4} \left (c^{2} x^{2}-1\right )}\right )-\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, x}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}+\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right ) x^{2}}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}-\frac {4 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right )}{d^{2} c^{4} \left (c^{2} x^{2}-1\right )}-\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2} c^{4} \left (c^{2} x^{2}-1\right )}+\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i \sqrt {-c^{2} x^{2}+1}+c x -1\right )}{d^{2} c^{4} \left (c^{2} x^{2}-1\right )}\)	\(658\)
parts	\(a^{2} \left (-\frac {x^{2}}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {2}{d \,c^{4} \sqrt {-c^{2} d \,x^{2}+d}}\right )+b^{2} \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )^{2}-2+2 i \arccos \left (c x \right )\right )}{2 d^{2} c^{4} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )^{2}-2-2 i \arccos \left (c x \right )\right )}{2 d^{2} c^{4} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right )^{2}}{d^{2} c^{4} \left (c^{2} x^{2}-1\right )}-\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-i \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} c^{4} \left (c^{2} x^{2}-1\right )}\right )-\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, x}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}+\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right ) x^{2}}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}-\frac {4 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right )}{d^{2} c^{4} \left (c^{2} x^{2}-1\right )}-\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2} c^{4} \left (c^{2} x^{2}-1\right )}+\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i \sqrt {-c^{2} x^{2}+1}+c x -1\right )}{d^{2} c^{4} \left (c^{2} x^{2}-1\right )}\)	\(658\)

Input:

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

Output:

a^2*(-x^2/c^2/d/(-c^2*d*x^2+d)^(1/2)+2/d/c^4/(-c^2*d*x^2+d)^(1/2))+b^2*(1/ 
2*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*c*x+c^2*x^2-1)*(arccos(c*x) 
^2-2+2*I*arccos(c*x))/d^2/c^4/(c^2*x^2-1)+1/2*(-d*(c^2*x^2-1))^(1/2)*(-I*( 
-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(arccos(c*x)^2-2-2*I*arccos(c*x))/d^2/c^4 
/(c^2*x^2-1)-(-d*(c^2*x^2-1))^(1/2)/d^2/c^4/(c^2*x^2-1)*arccos(c*x)^2-2*I* 
(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*(I*arccos(c*x)*ln(1-c*x-I*(-c^2* 
x^2+1)^(1/2))-I*arccos(c*x)*ln(1+c*x+I*(-c^2*x^2+1)^(1/2))-polylog(2,-c*x- 
I*(-c^2*x^2+1)^(1/2))+polylog(2,c*x+I*(-c^2*x^2+1)^(1/2)))/d^2/c^4/(c^2*x^ 
2-1))-2*a*b*(-d*(c^2*x^2-1))^(1/2)/d^2/c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)* 
x+2*a*b*(-d*(c^2*x^2-1))^(1/2)/d^2/c^2/(c^2*x^2-1)*arccos(c*x)*x^2-4*a*b*( 
-d*(c^2*x^2-1))^(1/2)/d^2/c^4/(c^2*x^2-1)*arccos(c*x)-2*a*b*(-d*(c^2*x^2-1 
))^(1/2)*(-c^2*x^2+1)^(1/2)/d^2/c^4/(c^2*x^2-1)*ln(1+c*x+I*(-c^2*x^2+1)^(1 
/2))+2*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/d^2/c^4/(c^2*x^2-1)*l 
n(I*(-c^2*x^2+1)^(1/2)+c*x-1)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

a*b*c*(2*x/(c^4*d^(3/2)) + log(c*x + 1)/(c^5*d^(3/2)) - log(c*x - 1)/(c^5* 
d^(3/2))) - 2*a*b*(x^2/(sqrt(-c^2*d*x^2 + d)*c^2*d) - 2/(sqrt(-c^2*d*x^2 + 
 d)*c^4*d))*arccos(c*x) - a^2*(x^2/(sqrt(-c^2*d*x^2 + d)*c^2*d) - 2/(sqrt( 
-c^2*d*x^2 + d)*c^4*d)) + ((c^2*x^2 - 2)*sqrt(c*x + 1)*sqrt(-c*x + 1)*sqrt 
(d)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^2 + (c^6*d^2*x^2 - c^4*d^2) 
*sqrt(d)*integrate(2*(c^2*x^4 - 2*x^2)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1 
), c*x)/(c^3*d^2*x^2 - c*d^2), x))*b^2/(c^6*d^2*x^2 - c^4*d^2)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x^3 (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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