\(\int \frac {(a+b \arccos (c x))^2}{x^4 \sqrt {d-c^2 d x^2}} \, dx\) [245]

Optimal result

Integrand size = 29, antiderivative size = 319 \[ \int \frac {(a+b \arccos (c x))^2}{x^4 \sqrt {d-c^2 d x^2}} \, dx=-\frac {b^2 c^2 \left (1-c^2 x^2\right )}{3 x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{3 x^2 \sqrt {d-c^2 d x^2}}-\frac {2 i c^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{3 d x^3}-\frac {2 c^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{3 d x}+\frac {4 b c^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x)) \log \left (1-e^{2 i \arccos (c x)}\right )}{3 \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 c^3 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )}{3 \sqrt {d-c^2 d x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.65 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.84 \[ \int \frac {(a+b \arccos (c x))^2}{x^4 \sqrt {d-c^2 d x^2}} \, dx=-\frac {\sqrt {1-c^2 x^2} \left (-a b c x+a^2 \sqrt {1-c^2 x^2}+2 a^2 c^2 x^2 \sqrt {1-c^2 x^2}+b^2 c^2 x^2 \sqrt {1-c^2 x^2}+b^2 \left (-2 i c^3 x^3+\sqrt {1-c^2 x^2}+2 c^2 x^2 \sqrt {1-c^2 x^2}\right ) \arccos (c x)^2+b \arccos (c x) \left (-b c x+2 a \sqrt {1-c^2 x^2} \left (1+2 c^2 x^2\right )+4 b c^3 x^3 \log \left (1+e^{2 i \arccos (c x)}\right )\right )+4 a b c^3 x^3 \log (c x)-2 i b^2 c^3 x^3 \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )\right )}{3 x^3 \sqrt {d-c^2 d x^2}} \] Input:

Integrate[(a + b*ArcCos[c*x])^2/(x^4*Sqrt[d - c^2*d*x^2]),x]
 

Output:

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

Rubi [A] (verified)

Time = 1.31 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.80, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {5205, 5139, 242, 5187, 5137, 3042, 4202, 2620, 2715, 2838}

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.

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

\(\Big \downarrow \) 5205

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

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 242

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

\(\Big \downarrow \) 5187

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

\(\Big \downarrow \) 5137

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4202

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

Int[(a + b*ArcCos[c*x])^2/(x^4*Sqrt[d - c^2*d*x^2]),x]
 

Output:

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

Defintions of rubi rules used

rule 242

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ 
(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x 
] && EqQ[m + 2*p + 3, 0] && 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 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 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 5187

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/(d*f*(m + 1))), x] + Simp[b*c*(n/(f*(m + 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*A 
rcCos[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^ 
2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
 

rule 5205

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/(d*f*(m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1)) 
)   Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] + Simp[b* 
c*(n/(f*(m + 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, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[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. 2185 vs. \(2 (303 ) = 606\).

Time = 0.65 (sec) , antiderivative size = 2186, normalized size of antiderivative = 6.85

method	result	size
default	\(\text {Expression too large to display}\)	\(2186\)
parts	\(\text {Expression too large to display}\)	\(2186\)

Input:

int((a+b*arccos(c*x))^2/x^4/(-c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-2/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x^5*c^8-1/3*b^2* 
(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x^3*c^6+2/3*b^2*(-d*(c^2* 
x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x*c^4+1/3*b^2*(-d*(c^2*x^2-1))^(1/ 
2)/(3*c^4*x^4-2*c^2*x^2-1)/d/x*c^2+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x 
^4-2*c^2*x^2-1)/d/x^3*arccos(c*x)^2-2/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^ 
4*x^4-2*c^2*x^2-1)/d*x*(-c^2*x^2+1)*arccos(c*x)*c^4-4/3*I*b^2*(-d*(c^2*x^2 
-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x^3*(-c^2*x^2+1)*arccos(c*x)*c^6+2*I* 
b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x^2*(-c^2*x^2+1)^(1/2 
)*arccos(c*x)^2*c^5+4/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2- 
1)/d*(-c^2*x^2+1)^(1/2)*arccos(c*x)*c^3-2/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/( 
3*c^4*x^4-2*c^2*x^2-1)/d*x*(-c^2*x^2+1)*c^4-8/3*I*a*b*(-c^2*x^2+1)^(1/2)*( 
-d*(c^2*x^2-1))^(1/2)/d/(c^2*x^2-1)*arccos(c*x)*c^3-4/3*I*a*b*(-d*(c^2*x^2 
-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x^3*(-c^2*x^2+1)*c^6+1/3*I*b^2*(-d*(c 
^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*(-c^2*x^2+1)^(1/2)*c^3-2/3*b^2* 
(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x^3*(-c^2*x^2+1)*c^6-b^2* 
(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*(-c^2*x^2+1)^(1/2)*arccos 
(c*x)*c^3-2*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x^3*arcco 
s(c*x)^2*c^6+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x*ar 
ccos(c*x)^2*c^4+4/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d/x 
*arccos(c*x)^2*c^2+2/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^...
 

Fricas [F]

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

integrate((a+b*arccos(c*x))^2/x^4/(-c^2*d*x^2+d)^(1/2),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)/(c^2*d*x^6 - d*x^4), x)
 

Sympy [F]

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

integrate((a+b*acos(c*x))**2/x**4/(-c**2*d*x**2+d)**(1/2),x)
 

Output:

Integral((a + b*acos(c*x))**2/(x**4*sqrt(-d*(c*x - 1)*(c*x + 1))), x)
 

Maxima [F]

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

integrate((a+b*arccos(c*x))^2/x^4/(-c^2*d*x^2+d)^(1/2),x, algorithm="maxim 
a")
 

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {(a+b \arccos (c x))^2}{x^4 \sqrt {d-c^2 d x^2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((a+b*arccos(c*x))^2/x^4/(-c^2*d*x^2+d)^(1/2),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. \[ \int \frac {(a+b \arccos (c x))^2}{x^4 \sqrt {d-c^2 d x^2}} \, dx=\int \frac {{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2}{x^4\,\sqrt {d-c^2\,d\,x^2}} \,d x \] Input:

int((a + b*acos(c*x))^2/(x^4*(d - c^2*d*x^2)^(1/2)),x)
 

Output:

int((a + b*acos(c*x))^2/(x^4*(d - c^2*d*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {(a+b \arccos (c x))^2}{x^4 \sqrt {d-c^2 d x^2}} \, dx=\frac {-2 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{2} x^{2}-\sqrt {-c^{2} x^{2}+1}\, a^{2}+6 \left (\int \frac {\mathit {acos} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, x^{4}}d x \right ) a b \,x^{3}+3 \left (\int \frac {\mathit {acos} \left (c x \right )^{2}}{\sqrt {-c^{2} x^{2}+1}\, x^{4}}d x \right ) b^{2} x^{3}}{3 \sqrt {d}\, x^{3}} \] Input:

int((a+b*acos(c*x))^2/x^4/(-c^2*d*x^2+d)^(1/2),x)
 

Output:

( - 2*sqrt( - c**2*x**2 + 1)*a**2*c**2*x**2 - sqrt( - c**2*x**2 + 1)*a**2 
+ 6*int(acos(c*x)/(sqrt( - c**2*x**2 + 1)*x**4),x)*a*b*x**3 + 3*int(acos(c 
*x)**2/(sqrt( - c**2*x**2 + 1)*x**4),x)*b**2*x**3)/(3*sqrt(d)*x**3)