\(\int \frac {a+b \arccos (c x)}{(d-c^2 d x^2)^3} \, dx\) [6]

Optimal result

Integrand size = 22, antiderivative size = 198 \[ \int \frac {a+b \arccos (c x)}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {b}{12 c d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {3 b}{8 c d^3 \sqrt {1-c^2 x^2}}+\frac {x (a+b \arccos (c x))}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {3 x (a+b \arccos (c x))}{8 d^3 \left (1-c^2 x^2\right )}+\frac {3 (2 a+b \pi -b (\pi -2 \arccos (c x))) \text {arctanh}\left (e^{i \arccos (c x)}\right )}{8 c d^3}-\frac {3 i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{8 c d^3}+\frac {3 i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{8 c d^3} \] Output:

1/12*b/c/d^3/(-c^2*x^2+1)^(3/2)+3/8*b/c/d^3/(-c^2*x^2+1)^(1/2)+1/4*x*(a+b* 
arccos(c*x))/d^3/(-c^2*x^2+1)^2+3/8*x*(a+b*arccos(c*x))/d^3/(-c^2*x^2+1)+3 
/8*(2*a+b*Pi-b*(Pi-2*arccos(c*x)))*arctanh(c*x+I*(-c^2*x^2+1)^(1/2))/c/d^3 
-3/8*I*b*polylog(2,-c*x-I*(-c^2*x^2+1)^(1/2))/c/d^3+3/8*I*b*polylog(2,c*x+ 
I*(-c^2*x^2+1)^(1/2))/c/d^3
 

Mathematica [A] (verified)

Time = 0.47 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.62 \[ \int \frac {a+b \arccos (c x)}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {\frac {4 a x}{\left (-1+c^2 x^2\right )^2}-\frac {6 a x}{-1+c^2 x^2}+\frac {b \left ((2+c x) \sqrt {1-c^2 x^2}-3 \arccos (c x)\right )}{3 c (1+c x)^2}+\frac {3 b \left (\sqrt {1-c^2 x^2}-\arccos (c x)\right )}{c+c^2 x}+\frac {3 b \left (\sqrt {1-c^2 x^2}+\arccos (c x)\right )}{c-c^2 x}+\frac {b \left ((2-c x) \sqrt {1-c^2 x^2}+3 \arccos (c x)\right )}{3 c (-1+c x)^2}-\frac {3 a \log (1-c x)}{c}+\frac {3 a \log (1+c x)}{c}-\frac {3 i b \left (\arccos (c x) \left (\arccos (c x)+4 i \log \left (1+e^{i \arccos (c x)}\right )\right )+4 \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )\right )}{2 c}+\frac {3 i b \left (\arccos (c x) \left (\arccos (c x)+4 i \log \left (1-e^{i \arccos (c x)}\right )\right )+4 \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )\right )}{2 c}}{16 d^3} \] Input:

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

Output:

((4*a*x)/(-1 + c^2*x^2)^2 - (6*a*x)/(-1 + c^2*x^2) + (b*((2 + c*x)*Sqrt[1 
- c^2*x^2] - 3*ArcCos[c*x]))/(3*c*(1 + c*x)^2) + (3*b*(Sqrt[1 - c^2*x^2] - 
 ArcCos[c*x]))/(c + c^2*x) + (3*b*(Sqrt[1 - c^2*x^2] + ArcCos[c*x]))/(c - 
c^2*x) + (b*((2 - c*x)*Sqrt[1 - c^2*x^2] + 3*ArcCos[c*x]))/(3*c*(-1 + c*x) 
^2) - (3*a*Log[1 - c*x])/c + (3*a*Log[1 + c*x])/c - (((3*I)/2)*b*(ArcCos[c 
*x]*(ArcCos[c*x] + (4*I)*Log[1 + E^(I*ArcCos[c*x])]) + 4*PolyLog[2, -E^(I* 
ArcCos[c*x])]))/c + (((3*I)/2)*b*(ArcCos[c*x]*(ArcCos[c*x] + (4*I)*Log[1 - 
 E^(I*ArcCos[c*x])]) + 4*PolyLog[2, E^(I*ArcCos[c*x])]))/c)/(16*d^3)
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.88, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {5163, 27, 241, 5163, 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.

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

\(\Big \downarrow \) 5163

\(\displaystyle \frac {3 \int \frac {a+b \arccos (c x)}{d^2 \left (1-c^2 x^2\right )^2}dx}{4 d}+\frac {b c \int \frac {x}{\left (1-c^2 x^2\right )^{5/2}}dx}{4 d^3}+\frac {x (a+b \arccos (c x))}{4 d^3 \left (1-c^2 x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3 \int \frac {a+b \arccos (c x)}{\left (1-c^2 x^2\right )^2}dx}{4 d^3}+\frac {b c \int \frac {x}{\left (1-c^2 x^2\right )^{5/2}}dx}{4 d^3}+\frac {x (a+b \arccos (c x))}{4 d^3 \left (1-c^2 x^2\right )^2}\)

\(\Big \downarrow \) 241

\(\displaystyle \frac {3 \int \frac {a+b \arccos (c x)}{\left (1-c^2 x^2\right )^2}dx}{4 d^3}+\frac {x (a+b \arccos (c x))}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c d^3 \left (1-c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 5163

\(\displaystyle \frac {3 \left (\frac {1}{2} \int \frac {a+b \arccos (c x)}{1-c^2 x^2}dx+\frac {1}{2} b c \int \frac {x}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {x (a+b \arccos (c x))}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}+\frac {x (a+b \arccos (c x))}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c d^3 \left (1-c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 241

\(\displaystyle \frac {3 \left (\frac {1}{2} \int \frac {a+b \arccos (c x)}{1-c^2 x^2}dx+\frac {x (a+b \arccos (c x))}{2 \left (1-c^2 x^2\right )}+\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{4 d^3}+\frac {x (a+b \arccos (c x))}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c d^3 \left (1-c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 5165

\(\displaystyle \frac {3 \left (-\frac {\int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}d\arccos (c x)}{2 c}+\frac {x (a+b \arccos (c x))}{2 \left (1-c^2 x^2\right )}+\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{4 d^3}+\frac {x (a+b \arccos (c x))}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c d^3 \left (1-c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3 \left (-\frac {\int (a+b \arccos (c x)) \csc (\arccos (c x))d\arccos (c x)}{2 c}+\frac {x (a+b \arccos (c x))}{2 \left (1-c^2 x^2\right )}+\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{4 d^3}+\frac {x (a+b \arccos (c x))}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c d^3 \left (1-c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 4671

\(\displaystyle \frac {3 \left (-\frac {-b \int \log \left (1-e^{i \arccos (c x)}\right )d\arccos (c x)+b \int \log \left (1+e^{i \arccos (c x)}\right )d\arccos (c x)-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))}{2 c}+\frac {x (a+b \arccos (c x))}{2 \left (1-c^2 x^2\right )}+\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{4 d^3}+\frac {x (a+b \arccos (c x))}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c d^3 \left (1-c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {3 \left (-\frac {i b \int e^{-i \arccos (c x)} \log \left (1-e^{i \arccos (c x)}\right )de^{i \arccos (c x)}-i b \int e^{-i \arccos (c x)} \log \left (1+e^{i \arccos (c x)}\right )de^{i \arccos (c x)}-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))}{2 c}+\frac {x (a+b \arccos (c x))}{2 \left (1-c^2 x^2\right )}+\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{4 d^3}+\frac {x (a+b \arccos (c x))}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c d^3 \left (1-c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {3 \left (-\frac {-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{2 c}+\frac {x (a+b \arccos (c x))}{2 \left (1-c^2 x^2\right )}+\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{4 d^3}+\frac {x (a+b \arccos (c x))}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c d^3 \left (1-c^2 x^2\right )^{3/2}}\)

Input:

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

Output:

b/(12*c*d^3*(1 - c^2*x^2)^(3/2)) + (x*(a + b*ArcCos[c*x]))/(4*d^3*(1 - c^2 
*x^2)^2) + (3*(b/(2*c*Sqrt[1 - c^2*x^2]) + (x*(a + b*ArcCos[c*x]))/(2*(1 - 
 c^2*x^2)) - (-2*(a + b*ArcCos[c*x])*ArcTanh[E^(I*ArcCos[c*x])] + I*b*Poly 
Log[2, -E^(I*ArcCos[c*x])] - I*b*PolyLog[2, E^(I*ArcCos[c*x])])/(2*c)))/(4 
*d^3)
 

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 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 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 5163

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_ 
Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(2*d*(p + 1 
))), x] + (Simp[(2*p + 3)/(2*d*(p + 1))   Int[(d + e*x^2)^(p + 1)*(a + b*Ar 
cCos[c*x])^n, x], x] - Simp[b*c*(n/(2*(p + 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}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, 
 -1] && NeQ[p, -3/2]
 

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]
 

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.27

method	result	size
derivativedivides	\(\frac {-\frac {a \left (-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {3}{16 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{16}+\frac {1}{16 \left (c x +1\right )^{2}}+\frac {3}{16 \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{16}\right )}{d^{3}}-\frac {b \left (\frac {9 c^{3} x^{3} \arccos \left (c x \right )+9 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}-15 c x \arccos \left (c x \right )-11 \sqrt {-c^{2} x^{2}+1}}{24 c^{4} x^{4}-48 c^{2} x^{2}+24}+\frac {3 \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{8}-\frac {3 \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{8}-\frac {3 i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{8}+\frac {3 i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{8}\right )}{d^{3}}}{c}\)	\(252\)
default	\(\frac {-\frac {a \left (-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {3}{16 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{16}+\frac {1}{16 \left (c x +1\right )^{2}}+\frac {3}{16 \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{16}\right )}{d^{3}}-\frac {b \left (\frac {9 c^{3} x^{3} \arccos \left (c x \right )+9 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}-15 c x \arccos \left (c x \right )-11 \sqrt {-c^{2} x^{2}+1}}{24 c^{4} x^{4}-48 c^{2} x^{2}+24}+\frac {3 \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{8}-\frac {3 \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{8}-\frac {3 i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{8}+\frac {3 i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{8}\right )}{d^{3}}}{c}\)	\(252\)
parts	\(-\frac {a \left (-\frac {1}{16 c \left (c x -1\right )^{2}}+\frac {3}{16 c \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{16 c}+\frac {1}{16 c \left (c x +1\right )^{2}}+\frac {3}{16 c \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{16 c}\right )}{d^{3}}-\frac {b \left (\frac {9 c^{3} x^{3} \arccos \left (c x \right )+9 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}-15 c x \arccos \left (c x \right )-11 \sqrt {-c^{2} x^{2}+1}}{24 c^{4} x^{4}-48 c^{2} x^{2}+24}+\frac {3 \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{8}-\frac {3 \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{8}-\frac {3 i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{8}+\frac {3 i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{8}\right )}{d^{3} c}\)	\(269\)

Input:

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

Output:

1/c*(-a/d^3*(-1/16/(c*x-1)^2+3/16/(c*x-1)+3/16*ln(c*x-1)+1/16/(c*x+1)^2+3/ 
16/(c*x+1)-3/16*ln(c*x+1))-b/d^3*(1/24*(9*c^3*x^3*arccos(c*x)+9*c^2*x^2*(- 
c^2*x^2+1)^(1/2)-15*c*x*arccos(c*x)-11*(-c^2*x^2+1)^(1/2))/(c^4*x^4-2*c^2* 
x^2+1)+3/8*arccos(c*x)*ln(1-c*x-I*(-c^2*x^2+1)^(1/2))-3/8*arccos(c*x)*ln(1 
+c*x+I*(-c^2*x^2+1)^(1/2))-3/8*I*polylog(2,c*x+I*(-c^2*x^2+1)^(1/2))+3/8*I 
*polylog(2,-c*x-I*(-c^2*x^2+1)^(1/2))))
 

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-1/16*a*(2*(3*c^2*x^3 - 5*x)/(c^4*d^3*x^4 - 2*c^2*d^3*x^2 + d^3) - 3*log(c 
*x + 1)/(c*d^3) + 3*log(c*x - 1)/(c*d^3)) - 1/16*((6*c^3*x^3 - 10*c*x - 3* 
(c^4*x^4 - 2*c^2*x^2 + 1)*log(c*x + 1) + 3*(c^4*x^4 - 2*c^2*x^2 + 1)*log(- 
c*x + 1))*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) + 16*(c^5*d^3*x^4 - 2 
*c^3*d^3*x^2 + c*d^3)*integrate(-1/16*(6*c^3*x^3 - 10*c*x - 3*(c^4*x^4 - 2 
*c^2*x^2 + 1)*log(c*x + 1) + 3*(c^4*x^4 - 2*c^2*x^2 + 1)*log(-c*x + 1))*sq 
rt(c*x + 1)*sqrt(-c*x + 1)/(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - 
d^3), x))*b/(c^5*d^3*x^4 - 2*c^3*d^3*x^2 + c*d^3)
 

Giac [F(-2)]

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

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

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \arccos (c x)}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {-16 \left (\int \frac {\mathit {acos} \left (c x \right )}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) b \,c^{5} x^{4}+32 \left (\int \frac {\mathit {acos} \left (c x \right )}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) b \,c^{3} x^{2}-16 \left (\int \frac {\mathit {acos} \left (c x \right )}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) b c -3 \,\mathrm {log}\left (c^{2} x -c \right ) a \,c^{4} x^{4}+6 \,\mathrm {log}\left (c^{2} x -c \right ) a \,c^{2} x^{2}-3 \,\mathrm {log}\left (c^{2} x -c \right ) a +3 \,\mathrm {log}\left (c^{2} x +c \right ) a \,c^{4} x^{4}-6 \,\mathrm {log}\left (c^{2} x +c \right ) a \,c^{2} x^{2}+3 \,\mathrm {log}\left (c^{2} x +c \right ) a -6 a \,c^{3} x^{3}+10 a c x}{16 c \,d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )} \] Input:

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

Output:

( - 16*int(acos(c*x)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1),x)*b*c**5 
*x**4 + 32*int(acos(c*x)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1),x)*b* 
c**3*x**2 - 16*int(acos(c*x)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1),x 
)*b*c - 3*log(c**2*x - c)*a*c**4*x**4 + 6*log(c**2*x - c)*a*c**2*x**2 - 3* 
log(c**2*x - c)*a + 3*log(c**2*x + c)*a*c**4*x**4 - 6*log(c**2*x + c)*a*c* 
*2*x**2 + 3*log(c**2*x + c)*a - 6*a*c**3*x**3 + 10*a*c*x)/(16*c*d**3*(c**4 
*x**4 - 2*c**2*x**2 + 1))