\(\int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{x^3} \, dx\) [218]

Optimal result

Integrand size = 29, antiderivative size = 398 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{x^3} \, dx=-\frac {b c \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{x \sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 x^2}+\frac {c^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2 \text {arctanh}\left (e^{i \arccos (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {b^2 c^2 \sqrt {d-c^2 d x^2} \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{\sqrt {1-c^2 x^2}}-\frac {i b c^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {i b c^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {b^2 c^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (3,-e^{i \arccos (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {b^2 c^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (3,e^{i \arccos (c x)}\right )}{\sqrt {1-c^2 x^2}} \] Output:

-b*c*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))/x/(-c^2*x^2+1)^(1/2)-1/2*(-c^2 
*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^2/x^2+c^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arcc 
os(c*x))^2*arctanh(c*x+I*(-c^2*x^2+1)^(1/2))/(-c^2*x^2+1)^(1/2)-b^2*c^2*(- 
c^2*d*x^2+d)^(1/2)*arctanh((-c^2*x^2+1)^(1/2))/(-c^2*x^2+1)^(1/2)-I*b*c^2* 
(-c^2*d*x^2+d)^(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)+I*b*c^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))*polylo 
g(2,c*x+I*(-c^2*x^2+1)^(1/2))/(-c^2*x^2+1)^(1/2)+b^2*c^2*(-c^2*d*x^2+d)^(1 
/2)*polylog(3,-c*x-I*(-c^2*x^2+1)^(1/2))/(-c^2*x^2+1)^(1/2)-b^2*c^2*(-c^2* 
d*x^2+d)^(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 = 1.61 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.14 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{x^3} \, dx=\frac {a^2 d \left (-1+c^2 x^2\right )-a^2 c^2 \sqrt {d} x^2 \sqrt {d-c^2 d x^2} \log (x)+a^2 c^2 \sqrt {d} x^2 \sqrt {d-c^2 d x^2} \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )+2 a b d \sqrt {1-c^2 x^2} \left (c x-\sqrt {1-c^2 x^2} \arccos (c x)+c^2 x^2 \arccos (c x) \log \left (1-i e^{i \arccos (c x)}\right )-c^2 x^2 \arccos (c x) \log \left (1+i e^{i \arccos (c x)}\right )+i c^2 x^2 \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )-i c^2 x^2 \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )\right )+b^2 d \sqrt {1-c^2 x^2} \left (\arccos (c x) \left (2 c x-\sqrt {1-c^2 x^2} \arccos (c x)\right )-2 c^2 x^2 \left (\coth ^{-1}\left (\sqrt {1-c^2 x^2}\right )+i \arccos (c x)^2 \arctan \left (e^{i \arccos (c x)}\right )-i \arccos (c x) \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )+i \arccos (c x) \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )+\operatorname {PolyLog}\left (3,-i e^{i \arccos (c x)}\right )-\operatorname {PolyLog}\left (3,i e^{i \arccos (c x)}\right )\right )\right )}{2 x^2 \sqrt {d-c^2 d x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.34 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.66, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {5197, 5139, 243, 73, 221, 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.

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

\(\Big \downarrow \) 5197

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

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 5219

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

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 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 5139

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

rule 5197

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 + 1))), x] + (Simp[b*c*(n/(f*(m + 1)))*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 
] + Simp[(c^2/(f^2*(m + 1)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]   Int[ 
(f*x)^(m + 2)*((a + b*ArcCos[c*x])^n/Sqrt[1 - c^2*x^2]), x], x]) /; FreeQ[{ 
a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[m, -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 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 = 0.65 (sec) , antiderivative size = 630, normalized size of antiderivative = 1.58

method	result	size
default	\(a^{2} \left (-\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{2 d \,x^{2}}-\frac {c^{2} \left (\sqrt {-c^{2} d \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )\right )}{2}\right )+b^{2} \left (-\frac {\left (c^{2} x^{2} \arccos \left (c x \right )+2 c x \sqrt {-c^{2} x^{2}+1}-\arccos \left (c x \right )\right ) \arccos \left (c x \right ) \sqrt {-d \left (c^{2} x^{2}-1\right )}}{2 \left (c^{2} x^{2}-1\right ) x^{2}}+\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (i \arccos \left (c x \right )^{2} \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-i \arccos \left (c x \right )^{2} \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )+2 \arccos \left (c x \right ) \operatorname {polylog}\left (2, i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-2 \arccos \left (c x \right ) \operatorname {polylog}\left (2, -i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )+2 i \operatorname {polylog}\left (3, i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-2 i \operatorname {polylog}\left (3, -i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-4 \arctan \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right ) c^{2}}{2 c^{2} x^{2}-2}\right )+2 a b \left (-\frac {\left (c^{2} x^{2} \arccos \left (c x \right )+c x \sqrt {-c^{2} x^{2}+1}-\arccos \left (c x \right )\right ) \sqrt {-d \left (c^{2} x^{2}-1\right )}}{2 \left (c^{2} x^{2}-1\right ) x^{2}}+\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (i \arccos \left (c x \right ) \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-i \arccos \left (c x \right ) \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-\operatorname {dilog}\left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )+\operatorname {dilog}\left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )\right ) c^{2}}{2 c^{2} x^{2}-2}\right )\)	\(630\)
parts	\(a^{2} \left (-\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{2 d \,x^{2}}-\frac {c^{2} \left (\sqrt {-c^{2} d \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )\right )}{2}\right )+b^{2} \left (-\frac {\left (c^{2} x^{2} \arccos \left (c x \right )+2 c x \sqrt {-c^{2} x^{2}+1}-\arccos \left (c x \right )\right ) \arccos \left (c x \right ) \sqrt {-d \left (c^{2} x^{2}-1\right )}}{2 \left (c^{2} x^{2}-1\right ) x^{2}}+\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (i \arccos \left (c x \right )^{2} \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-i \arccos \left (c x \right )^{2} \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )+2 \arccos \left (c x \right ) \operatorname {polylog}\left (2, i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-2 \arccos \left (c x \right ) \operatorname {polylog}\left (2, -i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )+2 i \operatorname {polylog}\left (3, i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-2 i \operatorname {polylog}\left (3, -i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-4 \arctan \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right ) c^{2}}{2 c^{2} x^{2}-2}\right )+2 a b \left (-\frac {\left (c^{2} x^{2} \arccos \left (c x \right )+c x \sqrt {-c^{2} x^{2}+1}-\arccos \left (c x \right )\right ) \sqrt {-d \left (c^{2} x^{2}-1\right )}}{2 \left (c^{2} x^{2}-1\right ) x^{2}}+\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (i \arccos \left (c x \right ) \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-i \arccos \left (c x \right ) \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-\operatorname {dilog}\left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )+\operatorname {dilog}\left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )\right ) c^{2}}{2 c^{2} x^{2}-2}\right )\)	\(630\)

Input:

int((-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^2/x^3,x,method=_RETURNVERBOSE)
 

Output:

a^2*(-1/2/d/x^2*(-c^2*d*x^2+d)^(3/2)-1/2*c^2*((-c^2*d*x^2+d)^(1/2)-d^(1/2) 
*ln((2*d+2*d^(1/2)*(-c^2*d*x^2+d)^(1/2))/x)))+b^2*(-1/2*(c^2*x^2*arccos(c* 
x)+2*c*x*(-c^2*x^2+1)^(1/2)-arccos(c*x))*arccos(c*x)*(-d*(c^2*x^2-1))^(1/2 
)/(c^2*x^2-1)/x^2+I*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*(I*arccos(c* 
x)^2*ln(1-I*(c*x+I*(-c^2*x^2+1)^(1/2)))-I*arccos(c*x)^2*ln(1+I*(c*x+I*(-c^ 
2*x^2+1)^(1/2)))+2*arccos(c*x)*polylog(2,I*(c*x+I*(-c^2*x^2+1)^(1/2)))-2*a 
rccos(c*x)*polylog(2,-I*(c*x+I*(-c^2*x^2+1)^(1/2)))+2*I*polylog(3,I*(c*x+I 
*(-c^2*x^2+1)^(1/2)))-2*I*polylog(3,-I*(c*x+I*(-c^2*x^2+1)^(1/2)))-4*arcta 
n(c*x+I*(-c^2*x^2+1)^(1/2)))*c^2/(2*c^2*x^2-2))+2*a*b*(-1/2*(c^2*x^2*arcco 
s(c*x)+c*x*(-c^2*x^2+1)^(1/2)-arccos(c*x))*(-d*(c^2*x^2-1))^(1/2)/(c^2*x^2 
-1)/x^2+I*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*(I*arccos(c*x)*ln(1-I* 
(c*x+I*(-c^2*x^2+1)^(1/2)))-I*arccos(c*x)*ln(1+I*(c*x+I*(-c^2*x^2+1)^(1/2) 
))-dilog(1+I*(c*x+I*(-c^2*x^2+1)^(1/2)))+dilog(1-I*(c*x+I*(-c^2*x^2+1)^(1/ 
2))))*c^2/(2*c^2*x^2-2))
 

Fricas [F]

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

integrate((-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^2/x^3,x, algorithm="frica 
s")
 

Output:

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

Sympy [F]

\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{x^3} \, dx=\int \frac {\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acos}{\left (c x \right )}\right )^{2}}{x^{3}}\, dx \] Input:

integrate((-c**2*d*x**2+d)**(1/2)*(a+b*acos(c*x))**2/x**3,x)
                                                                                    
                                                                                    
 

Output:

Integral(sqrt(-d*(c*x - 1)*(c*x + 1))*(a + b*acos(c*x))**2/x**3, x)
 

Maxima [F]

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

integrate((-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^2/x^3,x, algorithm="maxim 
a")
 

Output:

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

Giac [F(-2)]

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

integrate((-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^2/x^3,x, algorithm="giac" 
)
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

int(((a + b*acos(c*x))^2*(d - c^2*d*x^2)^(1/2))/x^3,x)
 

Output:

int(((a + b*acos(c*x))^2*(d - c^2*d*x^2)^(1/2))/x^3, x)
 

Reduce [F]

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

int((-c^2*d*x^2+d)^(1/2)*(a+b*acos(c*x))^2/x^3,x)
 

Output:

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