\(\int \frac {\sqrt {d+c d x} \sqrt {e-c e x} (a+b \arccos (c x))^2}{x} \, dx\) [581]

Optimal result

Integrand size = 35, antiderivative size = 432 \[ \int \frac {\sqrt {d+c d x} \sqrt {e-c e x} (a+b \arccos (c x))^2}{x} \, dx=-2 b^2 \sqrt {d+c d x} \sqrt {e-c e x}-\frac {2 a b c x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {1-c^2 x^2}}-\frac {2 b^2 c x \sqrt {d+c d x} \sqrt {e-c e x} \arccos (c x)}{\sqrt {1-c^2 x^2}}+\sqrt {d+c d x} \sqrt {e-c e x} (a+b \arccos (c x))^2-\frac {2 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arccos (c x))^2 \text {arctanh}\left (e^{i \arccos (c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 i b \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 i b \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 b^2 \sqrt {d+c d x} \sqrt {e-c e x} \operatorname {PolyLog}\left (3,-e^{i \arccos (c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 b^2 \sqrt {d+c d x} \sqrt {e-c e x} \operatorname {PolyLog}\left (3,e^{i \arccos (c x)}\right )}{\sqrt {1-c^2 x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.91 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {d+c d x} \sqrt {e-c e x} (a+b \arccos (c x))^2}{x} \, dx=a^2 \sqrt {d+c d x} \sqrt {e-c e x}+a^2 \sqrt {d} \sqrt {e} \log (c x)-a^2 \sqrt {d} \sqrt {e} \log \left (d e+\sqrt {d} \sqrt {e} \sqrt {d+c d x} \sqrt {e-c e x}\right )+\frac {2 a b \sqrt {d+c d x} \sqrt {e-c e x} \left (c x+\sqrt {1-c^2 x^2} \arccos (c x)-\arccos (c x) \log \left (1-i e^{i \arccos (c x)}\right )+\arccos (c x) \log \left (1+i e^{i \arccos (c x)}\right )-i \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )+i \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )\right )}{\sqrt {1-c^2 x^2}}+\frac {b^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (-2 \sqrt {1-c^2 x^2}+2 c x \arccos (c x)+\sqrt {1-c^2 x^2} \arccos (c x)^2-\arccos (c x)^2 \log \left (1-i e^{i \arccos (c x)}\right )+\arccos (c x)^2 \log \left (1+i e^{i \arccos (c x)}\right )-2 i \arccos (c x) \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )+2 i \arccos (c x) \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )+2 \operatorname {PolyLog}\left (3,-i e^{i \arccos (c x)}\right )-2 \operatorname {PolyLog}\left (3,i e^{i \arccos (c x)}\right )\right )}{\sqrt {1-c^2 x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.51 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.52, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {5239, 5199, 2009, 5219, 3042, 4669, 3011, 2720, 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.

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

\(\Big \downarrow \) 5239

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

\(\Big \downarrow \) 5199

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 5219

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

(Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^2 
+ 2*b*c*(a*x - (b*Sqrt[1 - c^2*x^2])/c + b*x*ArcCos[c*x]) + (2*I)*(a + b*A 
rcCos[c*x])^2*ArcTan[E^(I*ArcCos[c*x])] - 2*b*(I*(a + b*ArcCos[c*x])*PolyL 
og[2, (-I)*E^(I*ArcCos[c*x])] - b*PolyLog[3, (-I)*E^(I*ArcCos[c*x])]) + 2* 
b*(I*(a + b*ArcCos[c*x])*PolyLog[2, I*E^(I*ArcCos[c*x])] - b*PolyLog[3, I* 
E^(I*ArcCos[c*x])])))/Sqrt[1 - c^2*x^2]
 

Defintions of rubi rules used

rule 2009

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

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 4669

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

rule 5199

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + 
(e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*ArcC 
os[c*x])^n/(f*(m + 2))), x] + (Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/Sqrt[1 
 - c^2*x^2]]   Int[(f*x)^m*((a + b*ArcCos[c*x])^n/Sqrt[1 - c^2*x^2]), x], x 
] + Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]   Int[ 
(f*x)^(m + 1)*(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] && (IGtQ[m, -2] || EqQ[n, 1])
 

rule 5219

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

rule 5239

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((h_.)*(x_))^(m_.)*((d_) + (e_ 
.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(q_), x_Symbol] :> Simp[((-d^2)*(g/e))^In 
tPart[q]*(d + e*x)^FracPart[q]*((f + g*x)^FracPart[q]/(1 - c^2*x^2)^FracPar 
t[q])   Int[(h*x)^m*(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcCos[c*x])^n 
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e*f + d*g, 0] & 
& EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]
 

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 = 5.21 (sec) , antiderivative size = 771, normalized size of antiderivative = 1.78

method	result	size
default	\(\frac {a^{2} \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \left (-d e \ln \left (\frac {2 \sqrt {d e}\, \sqrt {-d e \left (c^{2} x^{2}-1\right )}+2 d e}{x}\right )+\sqrt {d e}\, \sqrt {-d e \left (c^{2} x^{2}-1\right )}\right )}{\sqrt {d e}\, \sqrt {-d e \left (c^{2} x^{2}-1\right )}}+b^{2} \left (\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -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 c^{2} x^{2}-2}+\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -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 c^{2} x^{2}-2}+\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (-\arccos \left (c x \right )^{2} \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )+\arccos \left (c x \right )^{2} \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )+2 \operatorname {polylog}\left (3, i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )+2 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-2 \operatorname {polylog}\left (3, -i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-2 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{c^{2} x^{2}-1}\right )+2 a b \left (\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )+i\right )}{2 c^{2} x^{2}-2}+\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )-i\right )}{2 c^{2} x^{2}-2}-\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (\arccos \left (c x \right ) \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-\arccos \left (c x \right ) \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-i \operatorname {polylog}\left (2, -i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )+i \operatorname {polylog}\left (2, i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{c^{2} x^{2}-1}\right )\)	\(771\)
parts	\(\frac {a^{2} \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \left (-d e \ln \left (\frac {2 \sqrt {d e}\, \sqrt {-d e \left (c^{2} x^{2}-1\right )}+2 d e}{x}\right )+\sqrt {d e}\, \sqrt {-d e \left (c^{2} x^{2}-1\right )}\right )}{\sqrt {d e}\, \sqrt {-d e \left (c^{2} x^{2}-1\right )}}+b^{2} \left (\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -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 c^{2} x^{2}-2}+\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -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 c^{2} x^{2}-2}+\frac {i \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (i \arccos \left (c x \right )^{2} \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-i \arccos \left (c x \right )^{2} \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )+2 i \operatorname {polylog}\left (3, -i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-2 i \operatorname {polylog}\left (3, i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )+2 \arccos \left (c x \right ) \operatorname {polylog}\left (2, -i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-2 \arccos \left (c x \right ) \operatorname {polylog}\left (2, i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{c^{2} x^{2}-1}\right )+2 a b \left (\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )+i\right )}{2 c^{2} x^{2}-2}+\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )-i\right )}{2 c^{2} x^{2}-2}-\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (\arccos \left (c x \right ) \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-\arccos \left (c x \right ) \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-i \operatorname {polylog}\left (2, -i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )+i \operatorname {polylog}\left (2, i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{c^{2} x^{2}-1}\right )\)	\(776\)

Input:

int((c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(a+b*arccos(c*x))^2/x,x,method=_RETUR 
NVERBOSE)
 

Output:

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

Fricas [F]

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

integrate((c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(a+b*arccos(c*x))^2/x,x, algori 
thm="fricas")
 

Output:

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

Sympy [F]

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

integrate((c*d*x+d)**(1/2)*(-c*e*x+e)**(1/2)*(a+b*acos(c*x))**2/x,x)
 

Output:

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

Maxima [F(-2)]

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

integrate((c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(a+b*arccos(c*x))^2/x,x, algori 
thm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F]

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

integrate((c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(a+b*arccos(c*x))^2/x,x, algori 
thm="giac")
 

Output:

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

Mupad [F(-1)]

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

int(((a + b*acos(c*x))^2*(d + c*d*x)^(1/2)*(e - c*e*x)^(1/2))/x,x)
 

Output:

int(((a + b*acos(c*x))^2*(d + c*d*x)^(1/2)*(e - c*e*x)^(1/2))/x, x)
 

Reduce [F]

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

int((c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(a+b*acos(c*x))^2/x,x)
 

Output:

sqrt(e)*sqrt(d)*(sqrt(c*x + 1)*sqrt( - c*x + 1)*a**2 + 2*int((sqrt(c*x + 1 
)*sqrt( - c*x + 1)*acos(c*x))/x,x)*a*b + int((sqrt(c*x + 1)*sqrt( - c*x + 
1)*acos(c*x)**2)/x,x)*b**2 - log( - sqrt(2) + tan(asin(sqrt( - c*x + 1)/sq 
rt(2))/2) - 1)*a**2 + log( - sqrt(2) + tan(asin(sqrt( - c*x + 1)/sqrt(2))/ 
2) + 1)*a**2 - log(sqrt(2) + tan(asin(sqrt( - c*x + 1)/sqrt(2))/2) - 1)*a* 
*2 + log(sqrt(2) + tan(asin(sqrt( - c*x + 1)/sqrt(2))/2) + 1)*a**2)