\(\int \frac {a+b \arccos (c x)}{x^3 (d-c^2 d x^2)^2} \, dx\) [46]

Optimal result

Integrand size = 25, antiderivative size = 159 \[ \int \frac {a+b \arccos (c x)}{x^3 \left (d-c^2 d x^2\right )^2} \, dx=-\frac {b c}{2 d^2 x \sqrt {1-c^2 x^2}}+\frac {c^2 (a+b \arccos (c x))}{d^2 \left (1-c^2 x^2\right )}-\frac {a+b \arccos (c x)}{2 d^2 x^2 \left (1-c^2 x^2\right )}-\frac {4 c^2 (a+b \arccos (c x)) \text {arctanh}\left (e^{2 i \arccos (c x)}\right )}{d^2}+\frac {i b c^2 \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )}{d^2}-\frac {i b c^2 \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )}{d^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.94 \[ \int \frac {a+b \arccos (c x)}{x^3 \left (d-c^2 d x^2\right )^2} \, dx=\frac {-\frac {2 a}{x^2}+\frac {2 b c \sqrt {1-c^2 x^2}}{x}+\frac {b c^2 \sqrt {1-c^2 x^2}}{1-c x}-\frac {b c^2 \sqrt {1-c^2 x^2}}{1+c x}-\frac {2 a c^2}{-1+c^2 x^2}-\frac {2 b \arccos (c x)}{x^2}+\frac {b c^2 \arccos (c x)}{1-c x}+\frac {b c^2 \arccos (c x)}{1+c x}-8 b c^2 \arccos (c x) \log \left (1-e^{i \arccos (c x)}\right )-8 b c^2 \arccos (c x) \log \left (1+e^{i \arccos (c x)}\right )+8 b c^2 \arccos (c x) \log \left (1+e^{2 i \arccos (c x)}\right )+8 a c^2 \log (x)-4 a c^2 \log \left (1-c^2 x^2\right )+8 i b c^2 \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )+8 i b c^2 \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )-4 i b c^2 \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )}{4 d^2} \] Input:

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

Output:

((-2*a)/x^2 + (2*b*c*Sqrt[1 - c^2*x^2])/x + (b*c^2*Sqrt[1 - c^2*x^2])/(1 - 
 c*x) - (b*c^2*Sqrt[1 - c^2*x^2])/(1 + c*x) - (2*a*c^2)/(-1 + c^2*x^2) - ( 
2*b*ArcCos[c*x])/x^2 + (b*c^2*ArcCos[c*x])/(1 - c*x) + (b*c^2*ArcCos[c*x]) 
/(1 + c*x) - 8*b*c^2*ArcCos[c*x]*Log[1 - E^(I*ArcCos[c*x])] - 8*b*c^2*ArcC 
os[c*x]*Log[1 + E^(I*ArcCos[c*x])] + 8*b*c^2*ArcCos[c*x]*Log[1 + E^((2*I)* 
ArcCos[c*x])] + 8*a*c^2*Log[x] - 4*a*c^2*Log[1 - c^2*x^2] + (8*I)*b*c^2*Po 
lyLog[2, -E^(I*ArcCos[c*x])] + (8*I)*b*c^2*PolyLog[2, E^(I*ArcCos[c*x])] - 
 (4*I)*b*c^2*PolyLog[2, -E^((2*I)*ArcCos[c*x])])/(4*d^2)
 

Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.25, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {5205, 27, 245, 208, 5209, 208, 5185, 4919, 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.

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

\(\Big \downarrow \) 5205

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 245

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

\(\Big \downarrow \) 208

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

\(\Big \downarrow \) 5209

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

\(\Big \downarrow \) 208

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

\(\Big \downarrow \) 5185

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

\(\Big \downarrow \) 4919

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

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 208

Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), 
x] /; FreeQ[{a, b}, x]
 

rule 245

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + 
 b*x^2)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + 2*(p + 1) + 1)/(a*(m + 1))) 
   Int[x^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, m, p}, x] && ILtQ[Si 
mplify[(m + 1)/2 + p + 1], 0] && NeQ[m, -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 4919

Int[Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b 
_.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n   Int[(c + d*x)^m*Csc[2*a + 2*b*x]^n 
, x], x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[n] && RationalQ[m]
 

rule 5185

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

rule 5205

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. 
)*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b 
*ArcCos[c*x])^n/(d*f*(m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1)) 
)   Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] + Simp[b* 
c*(n/(f*(m + 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] && ILtQ[m, -1]
 

rule 5209

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. 
)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(d + e*x^2)^(p + 1)*((a 
+ b*ArcCos[c*x])^n/(2*d*f*(p + 1))), x] + (Simp[(m + 2*p + 3)/(2*d*(p + 1)) 
   Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^n, x], x] - Simp[b*c 
*(n/(2*f*(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, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] &&  !G 
tQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])
 

Maple [A] (verified)

Time = 0.75 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.77

method	result	size
derivativedivides	\(c^{2} \left (\frac {a \left (-\frac {1}{4 \left (c x -1\right )}-\ln \left (c x -1\right )-\frac {1}{2 c^{2} x^{2}}+2 \ln \left (c x \right )+\frac {1}{4 c x +4}-\ln \left (c x +1\right )\right )}{d^{2}}+\frac {b \left (-\frac {2 c^{2} x^{2} \arccos \left (c x \right )+c x \sqrt {-c^{2} x^{2}+1}-\arccos \left (c x \right )}{2 c^{2} x^{2} \left (c^{2} x^{2}-1\right )}-2 \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-2 \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )+2 i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+2 i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}\right )\)	\(281\)
default	\(c^{2} \left (\frac {a \left (-\frac {1}{4 \left (c x -1\right )}-\ln \left (c x -1\right )-\frac {1}{2 c^{2} x^{2}}+2 \ln \left (c x \right )+\frac {1}{4 c x +4}-\ln \left (c x +1\right )\right )}{d^{2}}+\frac {b \left (-\frac {2 c^{2} x^{2} \arccos \left (c x \right )+c x \sqrt {-c^{2} x^{2}+1}-\arccos \left (c x \right )}{2 c^{2} x^{2} \left (c^{2} x^{2}-1\right )}-2 \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-2 \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )+2 i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+2 i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}\right )\)	\(281\)
parts	\(\frac {a \left (-\frac {1}{2 x^{2}}+2 c^{2} \ln \left (x \right )-\frac {c^{2}}{4 \left (c x -1\right )}-c^{2} \ln \left (c x -1\right )+\frac {c^{2}}{4 c x +4}-c^{2} \ln \left (c x +1\right )\right )}{d^{2}}+\frac {b \,c^{2} \left (-\frac {2 c^{2} x^{2} \arccos \left (c x \right )+c x \sqrt {-c^{2} x^{2}+1}-\arccos \left (c x \right )}{2 c^{2} x^{2} \left (c^{2} x^{2}-1\right )}-2 \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-2 \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )+2 i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+2 i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}\)	\(290\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((b*arccos(c*x) + a)/(c^4*d^2*x^7 - 2*c^2*d^2*x^5 + d^2*x^3), x)
 

Sympy [F]

\[ \int \frac {a+b \arccos (c x)}{x^3 \left (d-c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a}{c^{4} x^{7} - 2 c^{2} x^{5} + x^{3}}\, dx + \int \frac {b \operatorname {acos}{\left (c x \right )}}{c^{4} x^{7} - 2 c^{2} x^{5} + x^{3}}\, dx}{d^{2}} \] Input:

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

Output:

(Integral(a/(c**4*x**7 - 2*c**2*x**5 + x**3), x) + Integral(b*acos(c*x)/(c 
**4*x**7 - 2*c**2*x**5 + x**3), x))/d**2
 

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \arccos (c x)}{x^3 \left (d-c^2 d x^2\right )^2} \, dx=\frac {2 \left (\int \frac {\mathit {acos} \left (c x \right )}{c^{4} x^{7}-2 c^{2} x^{5}+x^{3}}d x \right ) b \,c^{2} x^{4}-2 \left (\int \frac {\mathit {acos} \left (c x \right )}{c^{4} x^{7}-2 c^{2} x^{5}+x^{3}}d x \right ) b \,x^{2}-2 \,\mathrm {log}\left (c^{2} x -c \right ) a \,c^{4} x^{4}+2 \,\mathrm {log}\left (c^{2} x -c \right ) a \,c^{2} x^{2}-2 \,\mathrm {log}\left (c^{2} x +c \right ) a \,c^{4} x^{4}+2 \,\mathrm {log}\left (c^{2} x +c \right ) a \,c^{2} x^{2}+4 \,\mathrm {log}\left (x \right ) a \,c^{4} x^{4}-4 \,\mathrm {log}\left (x \right ) a \,c^{2} x^{2}-2 a \,c^{4} x^{4}+a}{2 d^{2} x^{2} \left (c^{2} x^{2}-1\right )} \] Input:

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

Output:

(2*int(acos(c*x)/(c**4*x**7 - 2*c**2*x**5 + x**3),x)*b*c**2*x**4 - 2*int(a 
cos(c*x)/(c**4*x**7 - 2*c**2*x**5 + x**3),x)*b*x**2 - 2*log(c**2*x - c)*a* 
c**4*x**4 + 2*log(c**2*x - c)*a*c**2*x**2 - 2*log(c**2*x + c)*a*c**4*x**4 
+ 2*log(c**2*x + c)*a*c**2*x**2 + 4*log(x)*a*c**4*x**4 - 4*log(x)*a*c**2*x 
**2 - 2*a*c**4*x**4 + a)/(2*d**2*x**2*(c**2*x**2 - 1))