\(\int \frac {a+b \arccos (c x)}{x^4 \sqrt {d-c^2 d x^2}} \, dx\) [120]

Optimal result

Integrand size = 27, antiderivative size = 147 \[ \int \frac {a+b \arccos (c x)}{x^4 \sqrt {d-c^2 d x^2}} \, dx=-\frac {b c \sqrt {1-c^2 x^2}}{6 x^2 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{3 d x^3}-\frac {2 c^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{3 d x}+\frac {2 b c^3 \sqrt {1-c^2 x^2} \log (x)}{3 \sqrt {d-c^2 d x^2}} \] Output:

-1/6*b*c*(-c^2*x^2+1)^(1/2)/x^2/(-c^2*d*x^2+d)^(1/2)-1/3*(-c^2*d*x^2+d)^(1 
/2)*(a+b*arccos(c*x))/d/x^3-2/3*c^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x)) 
/d/x+2/3*b*c^3*(-c^2*x^2+1)^(1/2)*ln(x)/(-c^2*d*x^2+d)^(1/2)
 

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.92 \[ \int \frac {a+b \arccos (c x)}{x^4 \sqrt {d-c^2 d x^2}} \, dx=\frac {\sqrt {d-c^2 d x^2} \left (b c x+6 b c^3 x^3-2 a \sqrt {1-c^2 x^2}-4 a c^2 x^2 \sqrt {1-c^2 x^2}-2 b \sqrt {1-c^2 x^2} \left (1+2 c^2 x^2\right ) \arccos (c x)-4 b c^3 x^3 \log (x)\right )}{6 d x^3 \sqrt {1-c^2 x^2}} \] Input:

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

Output:

(Sqrt[d - c^2*d*x^2]*(b*c*x + 6*b*c^3*x^3 - 2*a*Sqrt[1 - c^2*x^2] - 4*a*c^ 
2*x^2*Sqrt[1 - c^2*x^2] - 2*b*Sqrt[1 - c^2*x^2]*(1 + 2*c^2*x^2)*ArcCos[c*x 
] - 4*b*c^3*x^3*Log[x]))/(6*d*x^3*Sqrt[1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {5205, 15, 5187, 14}

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.

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

\(\Big \downarrow \) 5205

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

\(\Big \downarrow \) 15

\(\displaystyle \frac {2}{3} c^2 \int \frac {a+b \arccos (c x)}{x^2 \sqrt {d-c^2 d x^2}}dx-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{3 d x^3}+\frac {b c \sqrt {1-c^2 x^2}}{6 x^2 \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 5187

\(\displaystyle \frac {2}{3} c^2 \left (-\frac {b c \sqrt {1-c^2 x^2} \int \frac {1}{x}dx}{\sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{d x}\right )-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{3 d x^3}+\frac {b c \sqrt {1-c^2 x^2}}{6 x^2 \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 14

\(\displaystyle \frac {2}{3} c^2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{d x}-\frac {b c \sqrt {1-c^2 x^2} \log (x)}{\sqrt {d-c^2 d x^2}}\right )-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{3 d x^3}+\frac {b c \sqrt {1-c^2 x^2}}{6 x^2 \sqrt {d-c^2 d x^2}}\)

Input:

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

Output:

(b*c*Sqrt[1 - c^2*x^2])/(6*x^2*Sqrt[d - c^2*d*x^2]) - (Sqrt[d - c^2*d*x^2] 
*(a + b*ArcCos[c*x]))/(3*d*x^3) + (2*c^2*(-((Sqrt[d - c^2*d*x^2]*(a + b*Ar 
cCos[c*x]))/(d*x)) - (b*c*Sqrt[1 - c^2*x^2]*Log[x])/Sqrt[d - c^2*d*x^2]))/ 
3
 

Defintions of rubi rules used

rule 14

Int[(a_.)/(x_), x_Symbol] :> Simp[a*Log[x], x] /; FreeQ[a, x]
 

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && 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 [C] (verified)

Result contains complex when optimal does not.

Time = 0.73 (sec) , antiderivative size = 851, normalized size of antiderivative = 5.79

method	result	size
default	\(a \left (-\frac {\sqrt {-c^{2} d \,x^{2}+d}}{3 d \,x^{3}}-\frac {2 c^{2} \sqrt {-c^{2} d \,x^{2}+d}}{3 d x}\right )+\frac {2 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{2} \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{5}}{\left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}-\frac {2 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{5} c^{8}}{3 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}-\frac {4 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{3}}{3 d \left (c^{2} x^{2}-1\right )}-\frac {2 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{3} \left (-c^{2} x^{2}+1\right ) c^{6}}{3 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}-\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{3} \arccos \left (c x \right ) c^{6}}{\left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}+\frac {2 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{3}}{3 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x \left (-c^{2} x^{2}+1\right ) c^{4}}{3 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{3} c^{6}}{3 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x \arccos \left (c x \right ) c^{4}}{3 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x \,c^{4}}{3 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{3} \sqrt {-c^{2} x^{2}+1}}{2 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}+\frac {4 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right ) c^{2}}{3 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d x}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, c}{6 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d \,x^{2}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right )}{3 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d \,x^{3}}+\frac {2 b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) c^{3}}{3 d \left (c^{2} x^{2}-1\right )}\)	\(851\)
parts	\(a \left (-\frac {\sqrt {-c^{2} d \,x^{2}+d}}{3 d \,x^{3}}-\frac {2 c^{2} \sqrt {-c^{2} d \,x^{2}+d}}{3 d x}\right )+\frac {2 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{2} \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{5}}{\left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}-\frac {2 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{5} c^{8}}{3 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}-\frac {4 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{3}}{3 d \left (c^{2} x^{2}-1\right )}-\frac {2 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{3} \left (-c^{2} x^{2}+1\right ) c^{6}}{3 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}-\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{3} \arccos \left (c x \right ) c^{6}}{\left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}+\frac {2 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{3}}{3 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x \left (-c^{2} x^{2}+1\right ) c^{4}}{3 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{3} c^{6}}{3 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x \arccos \left (c x \right ) c^{4}}{3 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x \,c^{4}}{3 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{3} \sqrt {-c^{2} x^{2}+1}}{2 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}+\frac {4 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right ) c^{2}}{3 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d x}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, c}{6 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d \,x^{2}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right )}{3 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d \,x^{3}}+\frac {2 b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) c^{3}}{3 d \left (c^{2} x^{2}-1\right )}\)	\(851\)

Input:

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

Output:

a*(-1/3/d/x^3*(-c^2*d*x^2+d)^(1/2)-2/3*c^2/d/x*(-c^2*d*x^2+d)^(1/2))+2*I*b 
*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x^2*arccos(c*x)*(-c^2*x^ 
2+1)^(1/2)*c^5-2/3*I*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x^ 
5*c^8-4/3*I*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/d/(c^2*x^2-1)*arcc 
os(c*x)*c^3-2/3*I*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^6-2*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x^3*a 
rccos(c*x)*c^6+2/3*I*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*ar 
ccos(c*x)*(-c^2*x^2+1)^(1/2)*c^3-1/3*I*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+1/3*I*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x 
^4-2*c^2*x^2-1)/d*x^3*c^6+1/3*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^ 
2-1)/d*x*arccos(c*x)*c^4+1/3*I*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x 
^2-1)/d*x*c^4-1/2*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*c^3*( 
-c^2*x^2+1)^(1/2)+4/3*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d/x 
*arccos(c*x)*c^2-1/6*b*(-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)*c+1/3*b*(-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)+2/3*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d/(c^2 
*x^2-1)*ln(1+(c*x+I*(-c^2*x^2+1)^(1/2))^2)*c^3
 

Fricas [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 432, normalized size of antiderivative = 2.94 \[ \int \frac {a+b \arccos (c x)}{x^4 \sqrt {d-c^2 d x^2}} \, dx=\left [\frac {2 \, {\left (b c^{5} x^{5} - b c^{3} x^{3}\right )} \sqrt {d} \log \left (\frac {c^{2} d x^{6} + c^{2} d x^{2} - d x^{4} + \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} {\left (x^{4} - 1\right )} \sqrt {d} - d}{c^{2} x^{4} - x^{2}}\right ) + \sqrt {-c^{2} d x^{2} + d} {\left (b c x^{3} - b c x\right )} \sqrt {-c^{2} x^{2} + 1} - 2 \, {\left (2 \, a c^{4} x^{4} - a c^{2} x^{2} + {\left (2 \, b c^{4} x^{4} - b c^{2} x^{2} - b\right )} \arccos \left (c x\right ) - a\right )} \sqrt {-c^{2} d x^{2} + d}}{6 \, {\left (c^{2} d x^{5} - d x^{3}\right )}}, -\frac {4 \, {\left (b c^{5} x^{5} - b c^{3} x^{3}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} {\left (x^{2} - 1\right )} \sqrt {-d}}{c^{2} d x^{4} + {\left (c^{2} - 1\right )} d x^{2} - d}\right ) - \sqrt {-c^{2} d x^{2} + d} {\left (b c x^{3} - b c x\right )} \sqrt {-c^{2} x^{2} + 1} + 2 \, {\left (2 \, a c^{4} x^{4} - a c^{2} x^{2} + {\left (2 \, b c^{4} x^{4} - b c^{2} x^{2} - b\right )} \arccos \left (c x\right ) - a\right )} \sqrt {-c^{2} d x^{2} + d}}{6 \, {\left (c^{2} d x^{5} - d x^{3}\right )}}\right ] \] Input:

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

Output:

[1/6*(2*(b*c^5*x^5 - b*c^3*x^3)*sqrt(d)*log((c^2*d*x^6 + c^2*d*x^2 - d*x^4 
 + sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1)*(x^4 - 1)*sqrt(d) - d)/(c^2*x^4 
 - x^2)) + sqrt(-c^2*d*x^2 + d)*(b*c*x^3 - b*c*x)*sqrt(-c^2*x^2 + 1) - 2*( 
2*a*c^4*x^4 - a*c^2*x^2 + (2*b*c^4*x^4 - b*c^2*x^2 - b)*arccos(c*x) - a)*s 
qrt(-c^2*d*x^2 + d))/(c^2*d*x^5 - d*x^3), -1/6*(4*(b*c^5*x^5 - b*c^3*x^3)* 
sqrt(-d)*arctan(sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1)*(x^2 - 1)*sqrt(-d) 
/(c^2*d*x^4 + (c^2 - 1)*d*x^2 - d)) - sqrt(-c^2*d*x^2 + d)*(b*c*x^3 - b*c* 
x)*sqrt(-c^2*x^2 + 1) + 2*(2*a*c^4*x^4 - a*c^2*x^2 + (2*b*c^4*x^4 - b*c^2* 
x^2 - b)*arccos(c*x) - a)*sqrt(-c^2*d*x^2 + d))/(c^2*d*x^5 - d*x^3)]
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.84 \[ \int \frac {a+b \arccos (c x)}{x^4 \sqrt {d-c^2 d x^2}} \, dx=-\frac {1}{6} \, {\left (\frac {4 \, c^{2} \log \left (x\right )}{\sqrt {d}} - \frac {1}{\sqrt {d} x^{2}}\right )} b c - \frac {1}{3} \, b {\left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} c^{2}}{d x} + \frac {\sqrt {-c^{2} d x^{2} + d}}{d x^{3}}\right )} \arccos \left (c x\right ) - \frac {1}{3} \, a {\left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} c^{2}}{d x} + \frac {\sqrt {-c^{2} d x^{2} + d}}{d x^{3}}\right )} \] Input:

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

Output:

-1/6*(4*c^2*log(x)/sqrt(d) - 1/(sqrt(d)*x^2))*b*c - 1/3*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*s 
qrt(-c^2*d*x^2 + d)*c^2/(d*x) + sqrt(-c^2*d*x^2 + d)/(d*x^3))
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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