\(\int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{x^2} \, dx\) [217]

Optimal result

Integrand size = 29, antiderivative size = 227 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{x^2} \, dx=-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{x}-\frac {i c \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}}-\frac {c \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^3}{3 b \sqrt {1-c^2 x^2}}+\frac {2 b c \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) \log \left (1-e^{2 i \arccos (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {i b^2 c \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )}{\sqrt {1-c^2 x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.14 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{x^2} \, dx=-\frac {a^2 \sqrt {d-c^2 d x^2}}{x}+a^2 c \sqrt {d} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-a b c \sqrt {d-c^2 d x^2} \left (\frac {2 \arccos (c x)}{c x}-\frac {\arccos (c x)^2-2 \log (c x)}{\sqrt {1-c^2 x^2}}\right )+\frac {b^2 \sqrt {d-c^2 d x^2} \left (\arccos (c x) \left (-3 \left (-i c x+\sqrt {1-c^2 x^2}\right ) \arccos (c x)+c x \arccos (c x)^2-6 c x \log \left (1+e^{2 i \arccos (c x)}\right )\right )+3 i c x \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )\right )}{3 x \sqrt {1-c^2 x^2}} \] Input:

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

Output:

-((a^2*Sqrt[d - c^2*d*x^2])/x) + a^2*c*Sqrt[d]*ArcTan[(c*x*Sqrt[d - c^2*d* 
x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - a*b*c*Sqrt[d - c^2*d*x^2]*((2*ArcCos[c*x 
])/(c*x) - (ArcCos[c*x]^2 - 2*Log[c*x])/Sqrt[1 - c^2*x^2]) + (b^2*Sqrt[d - 
 c^2*d*x^2]*(ArcCos[c*x]*(-3*((-I)*c*x + Sqrt[1 - c^2*x^2])*ArcCos[c*x] + 
c*x*ArcCos[c*x]^2 - 6*c*x*Log[1 + E^((2*I)*ArcCos[c*x])]) + (3*I)*c*x*Poly 
Log[2, -E^((2*I)*ArcCos[c*x])]))/(3*x*Sqrt[1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 1.12 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.80, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {5197, 5137, 3042, 4202, 2620, 2715, 2838, 5153}

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.

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

\(\Big \downarrow \) 5197

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

\(\Big \downarrow \) 5137

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4202

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 5153

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

Input:

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

Output:

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

Defintions of rubi rules used

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 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 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 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]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 524 vs. \(2 (227 ) = 454\).

Time = 0.56 (sec) , antiderivative size = 525, normalized size of antiderivative = 2.31

method	result	size
default	\(-\frac {a^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{d x}-a^{2} c^{2} x \sqrt {-c^{2} d \,x^{2}+d}-\frac {a^{2} c^{2} d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{3} c}{3 \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \arccos \left (c x \right )^{2}}{x \left (c^{2} x^{2}-1\right )}-\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (2 i \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+2 \arccos \left (c x \right )^{2}+\operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )\right ) c}{c^{2} x^{2}-1}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2} c}{2 \left (c^{2} x^{2}-1\right )}-\frac {2 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c}{c^{2} x^{2}-1}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \arccos \left (c x \right )}{x \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) c}{c^{2} x^{2}-1}\right )\)	\(525\)
parts	\(-\frac {a^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{d x}-a^{2} c^{2} x \sqrt {-c^{2} d \,x^{2}+d}-\frac {a^{2} c^{2} d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{3} c}{3 \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \arccos \left (c x \right )^{2}}{x \left (c^{2} x^{2}-1\right )}-\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (2 i \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+2 \arccos \left (c x \right )^{2}+\operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )\right ) c}{c^{2} x^{2}-1}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2} c}{2 \left (c^{2} x^{2}-1\right )}-\frac {2 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c}{c^{2} x^{2}-1}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \arccos \left (c x \right )}{x \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) c}{c^{2} x^{2}-1}\right )\)	\(525\)

Input:

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

Output:

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

Fricas [F]

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

integrate((-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^2/x^2,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^2, x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-(c*sqrt(d)*arcsin(c*x) + sqrt(-c^2*d*x^2 + d)/x)*a^2 + sqrt(d)*integrate( 
(b^2*arctan2(sqrt(c*x + 1)*sqrt(-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)/x^2, x)
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{x^2} \, dx=\frac {\sqrt {d}\, \left (\mathit {acos} \left (c x \right )^{3} b^{2} c x +3 \mathit {acos} \left (c x \right )^{2} a b c x -3 \mathit {asin} \left (c x \right ) a^{2} c x -3 \sqrt {-c^{2} x^{2}+1}\, a^{2}+6 \left (\int \frac {\mathit {acos} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, x^{2}}d x \right ) a b x +3 \left (\int \frac {\mathit {acos} \left (c x \right )^{2}}{\sqrt {-c^{2} x^{2}+1}\, x^{2}}d x \right ) b^{2} x \right )}{3 x} \] Input:

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

Output:

(sqrt(d)*(acos(c*x)**3*b**2*c*x + 3*acos(c*x)**2*a*b*c*x - 3*asin(c*x)*a** 
2*c*x - 3*sqrt( - c**2*x**2 + 1)*a**2 + 6*int(acos(c*x)/(sqrt( - c**2*x**2 
 + 1)*x**2),x)*a*b*x + 3*int(acos(c*x)**2/(sqrt( - c**2*x**2 + 1)*x**2),x) 
*b**2*x))/(3*x)