3.1.8 \(\int \frac {x^4 (a+b \arccos (c x))}{(d-c^2 d x^2)^2} \, dx\) [8]

3.1.8.1 Optimal result

Integrand size = 25, antiderivative size = 180 \[ \int \frac {x^4 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {b}{2 c^5 d^2 \sqrt {1-c^2 x^2}}-\frac {b \sqrt {1-c^2 x^2}}{c^5 d^2}+\frac {3 x (a+b \arccos (c x))}{2 c^4 d^2}+\frac {x^3 (a+b \arccos (c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {3 (a+b \arccos (c x)) \text {arctanh}\left (e^{i \arccos (c x)}\right )}{c^5 d^2}+\frac {3 i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{2 c^5 d^2}-\frac {3 i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{2 c^5 d^2} \]

output

3/2*x*(a+b*arccos(c*x))/c^4/d^2+1/2*x^3*(a+b*arccos(c*x))/c^2/d^2/(-c^2*x^ 
2+1)-3*(a+b*arccos(c*x))*arctanh(c*x+I*(-c^2*x^2+1)^(1/2))/c^5/d^2+3/2*I*b 
*polylog(2,-c*x-I*(-c^2*x^2+1)^(1/2))/c^5/d^2-3/2*I*b*polylog(2,c*x+I*(-c^ 
2*x^2+1)^(1/2))/c^5/d^2+1/2*b/c^5/d^2/(-c^2*x^2+1)^(1/2)-b*(-c^2*x^2+1)^(1 
/2)/c^5/d^2
 

3.1.8.2 Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.63 \[ \int \frac {x^4 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {a x}{c^4 d^2}-\frac {a x}{2 c^4 d^2 \left (-1+c^2 x^2\right )}+\frac {3 a \log (1-c x)}{4 c^5 d^2}-\frac {3 a \log (1+c x)}{4 c^5 d^2}+\frac {b \left (\frac {\sqrt {1-c^2 x^2}-\arccos (c x)}{4 c^4 \left (c+c^2 x\right )}+\frac {\sqrt {1-c^2 x^2}+\arccos (c x)}{4 c^4 \left (c-c^2 x\right )}+\frac {-\sqrt {1-c^2 x^2}+c x \arccos (c x)}{c^5}-\frac {3 \left (-\frac {i \arccos (c x)^2}{2 c}+\frac {2 \arccos (c x) \log \left (1+e^{i \arccos (c x)}\right )}{c}-\frac {2 i \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{c}\right )}{4 c^4}-\frac {3 i \left (\arccos (c x) \left (\arccos (c x)+4 i \log \left (1-e^{i \arccos (c x)}\right )\right )+4 \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )\right )}{8 c^5}\right )}{d^2} \]

input

Integrate[(x^4*(a + b*ArcCos[c*x]))/(d - c^2*d*x^2)^2,x]
 

output

(a*x)/(c^4*d^2) - (a*x)/(2*c^4*d^2*(-1 + c^2*x^2)) + (3*a*Log[1 - c*x])/(4 
*c^5*d^2) - (3*a*Log[1 + c*x])/(4*c^5*d^2) + (b*((Sqrt[1 - c^2*x^2] - ArcC 
os[c*x])/(4*c^4*(c + c^2*x)) + (Sqrt[1 - c^2*x^2] + ArcCos[c*x])/(4*c^4*(c 
 - c^2*x)) + (-Sqrt[1 - c^2*x^2] + c*x*ArcCos[c*x])/c^5 - (3*(((-1/2*I)*Ar 
cCos[c*x]^2)/c + (2*ArcCos[c*x]*Log[1 + E^(I*ArcCos[c*x])])/c - ((2*I)*Pol 
yLog[2, -E^(I*ArcCos[c*x])])/c))/(4*c^4) - (((3*I)/8)*(ArcCos[c*x]*(ArcCos 
[c*x] + (4*I)*Log[1 - E^(I*ArcCos[c*x])]) + 4*PolyLog[2, E^(I*ArcCos[c*x]) 
]))/c^5))/d^2
 

3.1.8.3 Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.06, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {5207, 27, 243, 53, 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.

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

\(\Big \downarrow \) 5207

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 53

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 5211

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

\(\Big \downarrow \) 241

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

\(\Big \downarrow \) 5165

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4671

\(\displaystyle -\frac {3 \left (-\frac {-b \int \log \left (1-e^{i \arccos (c x)}\right )d\arccos (c x)+b \int \log \left (1+e^{i \arccos (c x)}\right )d\arccos (c x)-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))}{c^3}-\frac {x (a+b \arccos (c x))}{c^2}+\frac {b \sqrt {1-c^2 x^2}}{c^3}\right )}{2 c^2 d^2}+\frac {x^3 (a+b \arccos (c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {2 \sqrt {1-c^2 x^2}}{c^4}+\frac {2}{c^4 \sqrt {1-c^2 x^2}}\right )}{4 c d^2}\)

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {3 \left (-\frac {i b \int e^{-i \arccos (c x)} \log \left (1-e^{i \arccos (c x)}\right )de^{i \arccos (c x)}-i b \int e^{-i \arccos (c x)} \log \left (1+e^{i \arccos (c x)}\right )de^{i \arccos (c x)}-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))}{c^3}-\frac {x (a+b \arccos (c x))}{c^2}+\frac {b \sqrt {1-c^2 x^2}}{c^3}\right )}{2 c^2 d^2}+\frac {x^3 (a+b \arccos (c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {2 \sqrt {1-c^2 x^2}}{c^4}+\frac {2}{c^4 \sqrt {1-c^2 x^2}}\right )}{4 c d^2}\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {3 \left (-\frac {-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{c^3}-\frac {x (a+b \arccos (c x))}{c^2}+\frac {b \sqrt {1-c^2 x^2}}{c^3}\right )}{2 c^2 d^2}+\frac {x^3 (a+b \arccos (c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {2 \sqrt {1-c^2 x^2}}{c^4}+\frac {2}{c^4 \sqrt {1-c^2 x^2}}\right )}{4 c d^2}\)

input

Int[(x^4*(a + b*ArcCos[c*x]))/(d - c^2*d*x^2)^2,x]
 

output

(b*(2/(c^4*Sqrt[1 - c^2*x^2]) + (2*Sqrt[1 - c^2*x^2])/c^4))/(4*c*d^2) + (x 
^3*(a + b*ArcCos[c*x]))/(2*c^2*d^2*(1 - c^2*x^2)) - (3*((b*Sqrt[1 - c^2*x^ 
2])/c^3 - (x*(a + b*ArcCos[c*x]))/c^2 - (-2*(a + b*ArcCos[c*x])*ArcTanh[E^ 
(I*ArcCos[c*x])] + I*b*PolyLog[2, -E^(I*ArcCos[c*x])] - I*b*PolyLog[2, E^( 
I*ArcCos[c*x])])/c^3))/(2*c^2*d^2)
 

3.1.8.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 53

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, 
x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) 
|| LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
 

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 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

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 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]
 

3.1.8.4 Maple [A] (verified)

Time = 2.41 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.39

method	result	size
derivativedivides	\(\frac {\frac {a \left (c x -\frac {1}{4 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{4}\right )}{d^{2}}-\frac {b \sqrt {-c^{2} x^{2}+1}}{d^{2}}+\frac {b \arccos \left (c x \right ) c x}{d^{2}}-\frac {b \arccos \left (c x \right ) c x}{2 d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-c^{2} x^{2}+1}}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {3 b \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2 d^{2}}-\frac {3 i b \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2 d^{2}}-\frac {3 b \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2 d^{2}}+\frac {3 i b \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2 d^{2}}}{c^{5}}\)	\(250\)
default	\(\frac {\frac {a \left (c x -\frac {1}{4 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{4}\right )}{d^{2}}-\frac {b \sqrt {-c^{2} x^{2}+1}}{d^{2}}+\frac {b \arccos \left (c x \right ) c x}{d^{2}}-\frac {b \arccos \left (c x \right ) c x}{2 d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-c^{2} x^{2}+1}}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {3 b \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2 d^{2}}-\frac {3 i b \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2 d^{2}}-\frac {3 b \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2 d^{2}}+\frac {3 i b \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2 d^{2}}}{c^{5}}\)	\(250\)
parts	\(\frac {a \left (\frac {x}{c^{4}}-\frac {1}{4 c^{5} \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{4 c^{5}}-\frac {1}{4 c^{5} \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{4 c^{5}}\right )}{d^{2}}-\frac {b \sqrt {-c^{2} x^{2}+1}}{c^{5} d^{2}}+\frac {b x \arccos \left (c x \right )}{d^{2} c^{4}}-\frac {b x \arccos \left (c x \right )}{2 d^{2} c^{4} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-c^{2} x^{2}+1}}{2 d^{2} c^{5} \left (c^{2} x^{2}-1\right )}-\frac {3 b \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2 d^{2} c^{5}}+\frac {3 i b \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2 c^{5} d^{2}}+\frac {3 b \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2 d^{2} c^{5}}-\frac {3 i b \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2 c^{5} d^{2}}\)	\(282\)

input

int(x^4*(a+b*arccos(c*x))/(-c^2*d*x^2+d)^2,x,method=_RETURNVERBOSE)
 

output

1/c^5*(a/d^2*(c*x-1/4/(c*x-1)+3/4*ln(c*x-1)-1/4/(c*x+1)-3/4*ln(c*x+1))-b/d 
^2*(-c^2*x^2+1)^(1/2)+b/d^2*arccos(c*x)*c*x-1/2*b/d^2/(c^2*x^2-1)*arccos(c 
*x)*c*x-1/2*b/d^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)+3/2*b/d^2*arccos(c*x)*ln( 
1-c*x-I*(-c^2*x^2+1)^(1/2))-3/2*I*b/d^2*polylog(2,c*x+I*(-c^2*x^2+1)^(1/2) 
)-3/2*b/d^2*arccos(c*x)*ln(1+c*x+I*(-c^2*x^2+1)^(1/2))+3/2*I*b/d^2*polylog 
(2,-c*x-I*(-c^2*x^2+1)^(1/2)))
 

3.1.8.5 Fricas [F]

\[ \int \frac {x^4 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )} x^{4}}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]

input

integrate(x^4*(a+b*arccos(c*x))/(-c^2*d*x^2+d)^2,x, algorithm="fricas")
 

output

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

3.1.8.6 Sympy [F]

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

input

integrate(x**4*(a+b*acos(c*x))/(-c**2*d*x**2+d)**2,x)
 

output

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

3.1.8.7 Maxima [F]

\[ \int \frac {x^4 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )} x^{4}}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]

input

integrate(x^4*(a+b*arccos(c*x))/(-c^2*d*x^2+d)^2,x, algorithm="maxima")
 

output

-1/4*a*(2*x/(c^6*d^2*x^2 - c^4*d^2) - 4*x/(c^4*d^2) + 3*log(c*x + 1)/(c^5* 
d^2) - 3*log(c*x - 1)/(c^5*d^2)) + 1/4*((4*c^3*x^3 - 6*c*x - 3*(c^2*x^2 - 
1)*log(c*x + 1) + 3*(c^2*x^2 - 1)*log(-c*x + 1))*arctan2(sqrt(c*x + 1)*sqr 
t(-c*x + 1), c*x) + 4*(c^7*d^2*x^2 - c^5*d^2)*integrate(-1/4*(4*c^3*x^3 - 
6*c*x - 3*(c^2*x^2 - 1)*log(c*x + 1) + 3*(c^2*x^2 - 1)*log(-c*x + 1))*sqrt 
(c*x + 1)*sqrt(-c*x + 1)/(c^8*d^2*x^4 - 2*c^6*d^2*x^2 + c^4*d^2), x))*b/(c 
^7*d^2*x^2 - c^5*d^2)
 

3.1.8.8 Giac [F]

\[ \int \frac {x^4 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )} x^{4}}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]

input

integrate(x^4*(a+b*arccos(c*x))/(-c^2*d*x^2+d)^2,x, algorithm="giac")
 

output

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

3.1.8.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^2} \,d x \]

input

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

output

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