\(\int \frac {(d-c^2 d x^2) (a+b \arccos (c x))^2}{x} \, dx\) [163]

Optimal result

Integrand size = 25, antiderivative size = 178 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arccos (c x))^2}{x} \, dx=\frac {1}{4} b^2 c^2 d x^2-\frac {1}{2} b c d x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {1}{4} d (a+b \arccos (c x))^2+\frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2-\frac {i d (a+b \arccos (c x))^3}{3 b}+d (a+b \arccos (c x))^2 \log \left (1-e^{2 i \arccos (c x)}\right )-i b d (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )+\frac {1}{2} b^2 d \operatorname {PolyLog}\left (3,e^{2 i \arccos (c x)}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.38 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arccos (c x))^2}{x} \, dx=-\frac {1}{24} d \left (12 a^2 c^2 x^2-12 a b c x \sqrt {1-c^2 x^2}+24 a b c^2 x^2 \arccos (c x)+24 i a b \arccos (c x)^2+8 i b^2 \arccos (c x)^3+24 a b \arctan \left (\frac {c x}{-1+\sqrt {1-c^2 x^2}}\right )-3 b^2 \cos (2 \arccos (c x))+6 b^2 \arccos (c x)^2 \cos (2 \arccos (c x))-48 a b \arccos (c x) \log \left (1+e^{2 i \arccos (c x)}\right )-24 b^2 \arccos (c x)^2 \log \left (1+e^{2 i \arccos (c x)}\right )-24 a^2 \log (x)+24 i b (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )-12 b^2 \operatorname {PolyLog}\left (3,-e^{2 i \arccos (c x)}\right )-6 b^2 \arccos (c x) \sin (2 \arccos (c x))\right ) \] Input:

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

Output:

-1/24*(d*(12*a^2*c^2*x^2 - 12*a*b*c*x*Sqrt[1 - c^2*x^2] + 24*a*b*c^2*x^2*A 
rcCos[c*x] + (24*I)*a*b*ArcCos[c*x]^2 + (8*I)*b^2*ArcCos[c*x]^3 + 24*a*b*A 
rcTan[(c*x)/(-1 + Sqrt[1 - c^2*x^2])] - 3*b^2*Cos[2*ArcCos[c*x]] + 6*b^2*A 
rcCos[c*x]^2*Cos[2*ArcCos[c*x]] - 48*a*b*ArcCos[c*x]*Log[1 + E^((2*I)*ArcC 
os[c*x])] - 24*b^2*ArcCos[c*x]^2*Log[1 + E^((2*I)*ArcCos[c*x])] - 24*a^2*L 
og[x] + (24*I)*b*(a + b*ArcCos[c*x])*PolyLog[2, -E^((2*I)*ArcCos[c*x])] - 
12*b^2*PolyLog[3, -E^((2*I)*ArcCos[c*x])] - 6*b^2*ArcCos[c*x]*Sin[2*ArcCos 
[c*x]]))
 

Rubi [A] (verified)

Time = 1.13 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.11, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {5203, 5137, 3042, 4202, 2620, 3011, 2720, 5157, 15, 5153, 7143}

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.

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

\(\Big \downarrow \) 5203

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

\(\Big \downarrow \) 5137

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4202

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 5157

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 5153

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

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

rule 4202

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

rule 5137

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> -Subst[Int[ 
(a + b*x)^n*Tan[x], x], x, ArcCos[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0 
]
 

rule 5153

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

rule 5157

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcCos[c*x])^n/2), x] + (Simp[(1/2 
)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]   Int[(a + b*ArcCos[c*x])^n/Sqrt[ 
1 - c^2*x^2], x], x] + Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2 
]]   Int[x*(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]
 

rule 5203

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*((a + b*ArcC 
os[c*x])^n/(f*(m + 2*p + 1))), x] + (Simp[2*d*(p/(m + 2*p + 1))   Int[(f*x) 
^m*(d + e*x^2)^(p - 1)*(a + b*ArcCos[c*x])^n, x], x] + Simp[b*c*(n/(f*(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, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] &&  !LtQ[m, -1]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.84

method	result	size
parts	\(-d \,a^{2} \left (\frac {c^{2} x^{2}}{2}-\ln \left (x \right )\right )-i d a b \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {c d x \,b^{2} \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{2}-\frac {d \,b^{2} \arccos \left (c x \right )^{2} c^{2} x^{2}}{2}+\frac {b^{2} c^{2} d \,x^{2}}{4}+\frac {d \,b^{2} \arccos \left (c x \right )^{2}}{4}-\frac {d \,b^{2}}{8}+d \,b^{2} \arccos \left (c x \right )^{2} \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i d \,b^{2} \arccos \left (c x \right )^{3}}{3}+\frac {d \,b^{2} \operatorname {polylog}\left (3, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-i d a b \arccos \left (c x \right )^{2}+\frac {a b c d x \sqrt {-c^{2} x^{2}+1}}{2}-\arccos \left (c x \right ) a b \,c^{2} d \,x^{2}+\frac {\arccos \left (c x \right ) a b d}{2}+2 d a b \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i d \,b^{2} \arccos \left (c x \right ) \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )\)	\(327\)
derivativedivides	\(-d \,a^{2} \left (\frac {c^{2} x^{2}}{2}-\ln \left (c x \right )\right )-i d a b \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {c d x \,b^{2} \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{2}-\frac {d \,b^{2} \arccos \left (c x \right )^{2} c^{2} x^{2}}{2}+\frac {d \,b^{2} \arccos \left (c x \right )^{2}}{4}+\frac {b^{2} c^{2} d \,x^{2}}{4}-\frac {d \,b^{2}}{8}+d \,b^{2} \arccos \left (c x \right )^{2} \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i d \,b^{2} \arccos \left (c x \right )^{3}}{3}+\frac {d \,b^{2} \operatorname {polylog}\left (3, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-i d a b \arccos \left (c x \right )^{2}+\frac {a b c d x \sqrt {-c^{2} x^{2}+1}}{2}-\arccos \left (c x \right ) a b \,c^{2} d \,x^{2}+\frac {\arccos \left (c x \right ) a b d}{2}+2 d a b \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i d \,b^{2} \arccos \left (c x \right ) \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )\)	\(329\)
default	\(-d \,a^{2} \left (\frac {c^{2} x^{2}}{2}-\ln \left (c x \right )\right )-i d a b \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {c d x \,b^{2} \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{2}-\frac {d \,b^{2} \arccos \left (c x \right )^{2} c^{2} x^{2}}{2}+\frac {d \,b^{2} \arccos \left (c x \right )^{2}}{4}+\frac {b^{2} c^{2} d \,x^{2}}{4}-\frac {d \,b^{2}}{8}+d \,b^{2} \arccos \left (c x \right )^{2} \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i d \,b^{2} \arccos \left (c x \right )^{3}}{3}+\frac {d \,b^{2} \operatorname {polylog}\left (3, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-i d a b \arccos \left (c x \right )^{2}+\frac {a b c d x \sqrt {-c^{2} x^{2}+1}}{2}-\arccos \left (c x \right ) a b \,c^{2} d \,x^{2}+\frac {\arccos \left (c x \right ) a b d}{2}+2 d a b \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i d \,b^{2} \arccos \left (c x \right ) \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )\)	\(329\)

Input:

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

Output:

-d*a^2*(1/2*c^2*x^2-ln(x))-I*d*a*b*polylog(2,-(c*x+I*(-c^2*x^2+1)^(1/2))^2 
)+1/2*c*d*x*b^2*arccos(c*x)*(-c^2*x^2+1)^(1/2)-1/2*d*b^2*arccos(c*x)^2*c^2 
*x^2+1/4*b^2*c^2*d*x^2+1/4*d*b^2*arccos(c*x)^2-1/8*d*b^2+d*b^2*arccos(c*x) 
^2*ln(1+(c*x+I*(-c^2*x^2+1)^(1/2))^2)-1/3*I*d*b^2*arccos(c*x)^3+1/2*d*b^2* 
polylog(3,-(c*x+I*(-c^2*x^2+1)^(1/2))^2)-I*d*a*b*arccos(c*x)^2+1/2*a*b*c*d 
*x*(-c^2*x^2+1)^(1/2)-arccos(c*x)*a*b*c^2*d*x^2+1/2*arccos(c*x)*a*b*d+2*d* 
a*b*arccos(c*x)*ln(1+(c*x+I*(-c^2*x^2+1)^(1/2))^2)-I*d*b^2*arccos(c*x)*pol 
ylog(2,-(c*x+I*(-c^2*x^2+1)^(1/2))^2)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

-d*(Integral(-a**2/x, x) + Integral(a**2*c**2*x, x) + Integral(-b**2*acos( 
c*x)**2/x, x) + Integral(-2*a*b*acos(c*x)/x, x) + Integral(b**2*c**2*x*aco 
s(c*x)**2, x) + Integral(2*a*b*c**2*x*acos(c*x), x))
 

Maxima [F]

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

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

Output:

-1/2*a^2*c^2*d*x^2 + a^2*d*log(x) - integrate(((b^2*c^2*d*x^2 - b^2*d)*arc 
tan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^2 + 2*(a*b*c^2*d*x^2 - a*b*d)*arct 
an2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x))/x, x)
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arccos (c x))^2}{x} \, dx=\frac {d \left (-2 \mathit {acos} \left (c x \right )^{2} b^{2} c^{2} x^{2}+\mathit {acos} \left (c x \right )^{2} b^{2}+2 \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) b^{2} c x -4 \mathit {acos} \left (c x \right ) a b \,c^{2} x^{2}-2 \mathit {asin} \left (c x \right ) a b +2 \sqrt {-c^{2} x^{2}+1}\, a b c x +8 \left (\int \frac {\mathit {acos} \left (c x \right )}{x}d x \right ) a b +4 \left (\int \frac {\mathit {acos} \left (c x \right )^{2}}{x}d x \right ) b^{2}+4 \,\mathrm {log}\left (x \right ) a^{2}-2 a^{2} c^{2} x^{2}+b^{2} c^{2} x^{2}\right )}{4} \] Input:

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

Output:

(d*( - 2*acos(c*x)**2*b**2*c**2*x**2 + acos(c*x)**2*b**2 + 2*sqrt( - c**2* 
x**2 + 1)*acos(c*x)*b**2*c*x - 4*acos(c*x)*a*b*c**2*x**2 - 2*asin(c*x)*a*b 
 + 2*sqrt( - c**2*x**2 + 1)*a*b*c*x + 8*int(acos(c*x)/x,x)*a*b + 4*int(aco 
s(c*x)**2/x,x)*b**2 + 4*log(x)*a**2 - 2*a**2*c**2*x**2 + b**2*c**2*x**2))/ 
4