\(\int \frac {(a+b \arccos (c x))^2}{x (d-c^2 d x^2)^{3/2}} \, dx\) [252]

Optimal result

Integrand size = 29, antiderivative size = 467 \[ \int \frac {(a+b \arccos (c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {(a+b \arccos (c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {4 i b \sqrt {1-c^2 x^2} (a+b \arccos (c x)) \arctan \left (e^{i \arccos (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2 \text {arctanh}\left (e^{i \arccos (c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {1-c^2 x^2} (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {1-c^2 x^2} (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arccos (c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arccos (c x)}\right )}{d \sqrt {d-c^2 d x^2}} \] Output:

(a+b*arccos(c*x))^2/d/(-c^2*d*x^2+d)^(1/2)+4*I*b*(-c^2*x^2+1)^(1/2)*(a+b*a 
rccos(c*x))*arctan(c*x+I*(-c^2*x^2+1)^(1/2))/d/(-c^2*d*x^2+d)^(1/2)-2*(-c^ 
2*x^2+1)^(1/2)*(a+b*arccos(c*x))^2*arctanh(c*x+I*(-c^2*x^2+1)^(1/2))/d/(-c 
^2*d*x^2+d)^(1/2)+2*I*b*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))*polylog(2,-c* 
x-I*(-c^2*x^2+1)^(1/2))/d/(-c^2*d*x^2+d)^(1/2)-2*I*b^2*(-c^2*x^2+1)^(1/2)* 
polylog(2,-I*(c*x+I*(-c^2*x^2+1)^(1/2)))/d/(-c^2*d*x^2+d)^(1/2)+2*I*b^2*(- 
c^2*x^2+1)^(1/2)*polylog(2,I*(c*x+I*(-c^2*x^2+1)^(1/2)))/d/(-c^2*d*x^2+d)^ 
(1/2)-2*I*b*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))*polylog(2,c*x+I*(-c^2*x^2 
+1)^(1/2))/d/(-c^2*d*x^2+d)^(1/2)-2*b^2*(-c^2*x^2+1)^(1/2)*polylog(3,-c*x- 
I*(-c^2*x^2+1)^(1/2))/d/(-c^2*d*x^2+d)^(1/2)+2*b^2*(-c^2*x^2+1)^(1/2)*poly 
log(3,c*x+I*(-c^2*x^2+1)^(1/2))/d/(-c^2*d*x^2+d)^(1/2)
 

Mathematica [A] (verified)

Time = 1.48 (sec) , antiderivative size = 686, normalized size of antiderivative = 1.47 \[ \int \frac {(a+b \arccos (c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {a^2 \sqrt {-d \left (-1+c^2 x^2\right )}}{d^2 \left (-1+c^2 x^2\right )}+\frac {a^2 \log (c x)}{d^{3/2}}-\frac {a^2 \log \left (d+\sqrt {d} \sqrt {-d \left (-1+c^2 x^2\right )}\right )}{d^{3/2}}-\frac {2 a b \left (-\arccos (c x)+\sqrt {1-c^2 x^2} \arccos (c x) \log \left (1-i e^{i \arccos (c x)}\right )-\sqrt {1-c^2 x^2} \arccos (c x) \log \left (1+i e^{i \arccos (c x)}\right )-\sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \arccos (c x)\right )\right )+\sqrt {1-c^2 x^2} \log \left (\sin \left (\frac {1}{2} \arccos (c x)\right )\right )+i \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )-i \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )\right )}{d \sqrt {d \left (1-c^2 x^2\right )}}-\frac {b^2 \left (-\arccos (c x)^2+2 \sqrt {1-c^2 x^2} \arccos (c x) \log \left (1-e^{i \arccos (c x)}\right )+\sqrt {1-c^2 x^2} \arccos (c x)^2 \log \left (1-i e^{i \arccos (c x)}\right )-\sqrt {1-c^2 x^2} \arccos (c x)^2 \log \left (1+i e^{i \arccos (c x)}\right )-2 \sqrt {1-c^2 x^2} \arccos (c x) \log \left (1+e^{i \arccos (c x)}\right )+2 i \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )+2 i \sqrt {1-c^2 x^2} \arccos (c x) \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )-2 i \sqrt {1-c^2 x^2} \arccos (c x) \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )-2 i \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )-2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arccos (c x)}\right )+2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arccos (c x)}\right )\right )}{d \sqrt {d \left (1-c^2 x^2\right )}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 2.12 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.61, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {5209, 5165, 3042, 4671, 2715, 2838, 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.

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

\(\Big \downarrow \) 5209

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

\(\Big \downarrow \) 5165

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 5219

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

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

Defintions of rubi rules used

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 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 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 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 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 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]
 

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])
 

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 = 0.76 (sec) , antiderivative size = 640, normalized size of antiderivative = 1.37

method	result	size
default	\(\frac {a^{2}}{d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {a^{2} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {3}{2}}}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right )^{2}}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \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 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, 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 \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 i \operatorname {dilog}\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 i \operatorname {dilog}\left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, 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 )\right )}{d^{2} \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right )}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \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 )-i \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+i \ln \left (i \sqrt {-c^{2} x^{2}+1}+c x -1\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 )}{d^{2} \left (c^{2} x^{2}-1\right )}\right )\)	\(640\)
parts	\(\frac {a^{2}}{d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {a^{2} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {3}{2}}}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right )^{2}}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \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 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, 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 \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 i \operatorname {dilog}\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 i \operatorname {dilog}\left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, 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 )\right )}{d^{2} \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right )}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \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 )-i \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+i \ln \left (i \sqrt {-c^{2} x^{2}+1}+c x -1\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 )}{d^{2} \left (c^{2} x^{2}-1\right )}\right )\)	\(640\)

Input:

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

Output:

a^2/d/(-c^2*d*x^2+d)^(1/2)-a^2/d^(3/2)*ln((2*d+2*d^(1/2)*(-c^2*d*x^2+d)^(1 
/2))/x)+b^2*(-(-d*(c^2*x^2-1))^(1/2)/d^2/(c^2*x^2-1)*arccos(c*x)^2+(-c^2*x 
^2+1)^(1/2)*(-d*(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*I*arccos(c*x 
)*polylog(2,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*arccos(c*x)*ln(1+c*x+I*(-c^2*x^2+1)^(1/2))+2*I* 
dilog(c*x+I*(-c^2*x^2+1)^(1/2))+2*I*dilog(1+c*x+I*(-c^2*x^2+1)^(1/2))+2*po 
lylog(3,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))))/d^2/(c^2*x^2-1))+2*a*b*(-(-d*(c^2*x^2-1))^(1/2)/d^2/(c^2*x^2-1)*ar 
ccos(c*x)-I*(-c^2*x^2+1)^(1/2)*(-d*(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)))-I*ln(1+c*x+I*(-c^2*x^2+1)^(1/2))+I*ln(I*(-c^2*x^2+1)^(1/2)+c*x-1)-dil 
og(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))))/ 
d^2/(c^2*x^2-1))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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