\(\int \frac {x (a+b \arccos (c x))}{(d+e x^2)^2} \, dx\) [636]

Optimal result

Integrand size = 19, antiderivative size = 86 \[ \int \frac {x (a+b \arccos (c x))}{\left (d+e x^2\right )^2} \, dx=\frac {-a-b \arccos (c x)}{2 e \left (d+e x^2\right )}+\frac {b c \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{2 \sqrt {d} e \sqrt {c^2 d+e}} \] Output:

1/2*(-a-b*arccos(c*x))/e/(e*x^2+d)+1/2*b*c*arctan((c^2*d+e)^(1/2)*x/d^(1/2 
)/(-c^2*x^2+1)^(1/2))/d^(1/2)/e/(c^2*d+e)^(1/2)
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00 \[ \int \frac {x (a+b \arccos (c x))}{\left (d+e x^2\right )^2} \, dx=-\frac {\frac {a}{d+e x^2}+\frac {b \arccos (c x)}{d+e x^2}+\frac {b c \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{\sqrt {d} \sqrt {c^2 d+e}}}{2 e} \] Input:

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

Output:

-1/2*(a/(d + e*x^2) + (b*ArcCos[c*x])/(d + e*x^2) + (b*c*ArcTan[(Sqrt[c^2* 
d + e]*x)/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(Sqrt[d]*Sqrt[c^2*d + e]))/e
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {5229, 291, 218}

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.

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

\(\Big \downarrow \) 5229

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

-1/2*(a + b*ArcCos[c*x])/(e*(d + e*x^2)) - (b*c*ArcTan[(Sqrt[c^2*d + e]*x) 
/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(2*Sqrt[d]*e*Sqrt[c^2*d + e])
 

Defintions of rubi rules used

rule 218

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

rule 291

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

rule 5229

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_ 
Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])/(2*e*(p + 1))), x] 
 + Simp[b*(c/(2*e*(p + 1)))   Int[(d + e*x^2)^(p + 1)/Sqrt[1 - c^2*x^2], x] 
, x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[c^2*d + e, 0] && NeQ[p, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(403\) vs. \(2(72)=144\).

Time = 7.11 (sec) , antiderivative size = 404, normalized size of antiderivative = 4.70

method	result	size
parts	\(-\frac {a}{2 e \left (e \,x^{2}+d \right )}-\frac {b \,c^{2} \arccos \left (c x \right )}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {b \,c^{2} \ln \left (\frac {\frac {2 c^{2} d +2 e}{e}-\frac {2 \sqrt {-c^{2} d e}\, \left (c x -\frac {\sqrt {-c^{2} d e}}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-{\left (c x -\frac {\sqrt {-c^{2} d e}}{e}\right )}^{2}-\frac {2 \sqrt {-c^{2} d e}\, \left (c x -\frac {\sqrt {-c^{2} d e}}{e}\right )}{e}+\frac {c^{2} d +e}{e}}}{c x -\frac {\sqrt {-c^{2} d e}}{e}}\right )}{4 e \sqrt {-c^{2} d e}\, \sqrt {\frac {c^{2} d +e}{e}}}-\frac {b \,c^{2} \ln \left (\frac {\frac {2 c^{2} d +2 e}{e}+\frac {2 \sqrt {-c^{2} d e}\, \left (c x +\frac {\sqrt {-c^{2} d e}}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-{\left (c x +\frac {\sqrt {-c^{2} d e}}{e}\right )}^{2}+\frac {2 \sqrt {-c^{2} d e}\, \left (c x +\frac {\sqrt {-c^{2} d e}}{e}\right )}{e}+\frac {c^{2} d +e}{e}}}{c x +\frac {\sqrt {-c^{2} d e}}{e}}\right )}{4 e \sqrt {-c^{2} d e}\, \sqrt {\frac {c^{2} d +e}{e}}}\)	\(404\)
derivativedivides	\(\frac {-\frac {a \,c^{4}}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}+b \,c^{4} \left (-\frac {\arccos \left (c x \right )}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {-\frac {\ln \left (\frac {\frac {2 c^{2} d +2 e}{e}-\frac {2 \sqrt {-c^{2} d e}\, \left (c x -\frac {\sqrt {-c^{2} d e}}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-{\left (c x -\frac {\sqrt {-c^{2} d e}}{e}\right )}^{2}-\frac {2 \sqrt {-c^{2} d e}\, \left (c x -\frac {\sqrt {-c^{2} d e}}{e}\right )}{e}+\frac {c^{2} d +e}{e}}}{c x -\frac {\sqrt {-c^{2} d e}}{e}}\right )}{2 \sqrt {-c^{2} d e}\, \sqrt {\frac {c^{2} d +e}{e}}}+\frac {\ln \left (\frac {\frac {2 c^{2} d +2 e}{e}+\frac {2 \sqrt {-c^{2} d e}\, \left (c x +\frac {\sqrt {-c^{2} d e}}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-{\left (c x +\frac {\sqrt {-c^{2} d e}}{e}\right )}^{2}+\frac {2 \sqrt {-c^{2} d e}\, \left (c x +\frac {\sqrt {-c^{2} d e}}{e}\right )}{e}+\frac {c^{2} d +e}{e}}}{c x +\frac {\sqrt {-c^{2} d e}}{e}}\right )}{2 \sqrt {-c^{2} d e}\, \sqrt {\frac {c^{2} d +e}{e}}}}{2 e}\right )}{c^{2}}\)	\(412\)
default	\(\frac {-\frac {a \,c^{4}}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}+b \,c^{4} \left (-\frac {\arccos \left (c x \right )}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {-\frac {\ln \left (\frac {\frac {2 c^{2} d +2 e}{e}-\frac {2 \sqrt {-c^{2} d e}\, \left (c x -\frac {\sqrt {-c^{2} d e}}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-{\left (c x -\frac {\sqrt {-c^{2} d e}}{e}\right )}^{2}-\frac {2 \sqrt {-c^{2} d e}\, \left (c x -\frac {\sqrt {-c^{2} d e}}{e}\right )}{e}+\frac {c^{2} d +e}{e}}}{c x -\frac {\sqrt {-c^{2} d e}}{e}}\right )}{2 \sqrt {-c^{2} d e}\, \sqrt {\frac {c^{2} d +e}{e}}}+\frac {\ln \left (\frac {\frac {2 c^{2} d +2 e}{e}+\frac {2 \sqrt {-c^{2} d e}\, \left (c x +\frac {\sqrt {-c^{2} d e}}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-{\left (c x +\frac {\sqrt {-c^{2} d e}}{e}\right )}^{2}+\frac {2 \sqrt {-c^{2} d e}\, \left (c x +\frac {\sqrt {-c^{2} d e}}{e}\right )}{e}+\frac {c^{2} d +e}{e}}}{c x +\frac {\sqrt {-c^{2} d e}}{e}}\right )}{2 \sqrt {-c^{2} d e}\, \sqrt {\frac {c^{2} d +e}{e}}}}{2 e}\right )}{c^{2}}\)	\(412\)

Input:

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

Output:

-1/2*a/e/(e*x^2+d)-1/2*b*c^2/e/(c^2*e*x^2+c^2*d)*arccos(c*x)+1/4*b*c^2/e/( 
-c^2*d*e)^(1/2)/((c^2*d+e)/e)^(1/2)*ln((2*(c^2*d+e)/e-2*(-c^2*d*e)^(1/2)/e 
*(c*x-(-c^2*d*e)^(1/2)/e)+2*((c^2*d+e)/e)^(1/2)*(-(c*x-(-c^2*d*e)^(1/2)/e) 
^2-2*(-c^2*d*e)^(1/2)/e*(c*x-(-c^2*d*e)^(1/2)/e)+(c^2*d+e)/e)^(1/2))/(c*x- 
(-c^2*d*e)^(1/2)/e))-1/4*b*c^2/e/(-c^2*d*e)^(1/2)/((c^2*d+e)/e)^(1/2)*ln(( 
2*(c^2*d+e)/e+2*(-c^2*d*e)^(1/2)/e*(c*x+(-c^2*d*e)^(1/2)/e)+2*((c^2*d+e)/e 
)^(1/2)*(-(c*x+(-c^2*d*e)^(1/2)/e)^2+2*(-c^2*d*e)^(1/2)/e*(c*x+(-c^2*d*e)^ 
(1/2)/e)+(c^2*d+e)/e)^(1/2))/(c*x+(-c^2*d*e)^(1/2)/e))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 242 vs. \(2 (69) = 138\).

Time = 0.17 (sec) , antiderivative size = 520, normalized size of antiderivative = 6.05 \[ \int \frac {x (a+b \arccos (c x))}{\left (d+e x^2\right )^2} \, dx=\left [-\frac {4 \, a c^{2} d^{2} - 4 \, {\left (b c^{2} d e + b e^{2}\right )} x^{2} \arccos \left (c x\right ) + 4 \, a d e + {\left (b c e x^{2} + b c d\right )} \sqrt {-c^{2} d^{2} - d e} \log \left (\frac {{\left (8 \, c^{4} d^{2} + 8 \, c^{2} d e + e^{2}\right )} x^{4} - 2 \, {\left (4 \, c^{2} d^{2} + 3 \, d e\right )} x^{2} + 4 \, \sqrt {-c^{2} d^{2} - d e} \sqrt {-c^{2} x^{2} + 1} {\left ({\left (2 \, c^{2} d + e\right )} x^{3} - d x\right )} + d^{2}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}\right ) + 4 \, {\left (b c^{2} d^{2} + b d e + {\left (b c^{2} d e + b e^{2}\right )} x^{2}\right )} \arctan \left (\frac {\sqrt {-c^{2} x^{2} + 1} c x}{c^{2} x^{2} - 1}\right )}{8 \, {\left (c^{2} d^{3} e + d^{2} e^{2} + {\left (c^{2} d^{2} e^{2} + d e^{3}\right )} x^{2}\right )}}, -\frac {2 \, a c^{2} d^{2} - 2 \, {\left (b c^{2} d e + b e^{2}\right )} x^{2} \arccos \left (c x\right ) + 2 \, a d e - {\left (b c e x^{2} + b c d\right )} \sqrt {c^{2} d^{2} + d e} \arctan \left (\frac {\sqrt {c^{2} d^{2} + d e} \sqrt {-c^{2} x^{2} + 1} {\left ({\left (2 \, c^{2} d + e\right )} x^{2} - d\right )}}{2 \, {\left ({\left (c^{4} d^{2} + c^{2} d e\right )} x^{3} - {\left (c^{2} d^{2} + d e\right )} x\right )}}\right ) + 2 \, {\left (b c^{2} d^{2} + b d e + {\left (b c^{2} d e + b e^{2}\right )} x^{2}\right )} \arctan \left (\frac {\sqrt {-c^{2} x^{2} + 1} c x}{c^{2} x^{2} - 1}\right )}{4 \, {\left (c^{2} d^{3} e + d^{2} e^{2} + {\left (c^{2} d^{2} e^{2} + d e^{3}\right )} x^{2}\right )}}\right ] \] Input:

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

Output:

[-1/8*(4*a*c^2*d^2 - 4*(b*c^2*d*e + b*e^2)*x^2*arccos(c*x) + 4*a*d*e + (b* 
c*e*x^2 + b*c*d)*sqrt(-c^2*d^2 - d*e)*log(((8*c^4*d^2 + 8*c^2*d*e + e^2)*x 
^4 - 2*(4*c^2*d^2 + 3*d*e)*x^2 + 4*sqrt(-c^2*d^2 - d*e)*sqrt(-c^2*x^2 + 1) 
*((2*c^2*d + e)*x^3 - d*x) + d^2)/(e^2*x^4 + 2*d*e*x^2 + d^2)) + 4*(b*c^2* 
d^2 + b*d*e + (b*c^2*d*e + b*e^2)*x^2)*arctan(sqrt(-c^2*x^2 + 1)*c*x/(c^2* 
x^2 - 1)))/(c^2*d^3*e + d^2*e^2 + (c^2*d^2*e^2 + d*e^3)*x^2), -1/4*(2*a*c^ 
2*d^2 - 2*(b*c^2*d*e + b*e^2)*x^2*arccos(c*x) + 2*a*d*e - (b*c*e*x^2 + b*c 
*d)*sqrt(c^2*d^2 + d*e)*arctan(1/2*sqrt(c^2*d^2 + d*e)*sqrt(-c^2*x^2 + 1)* 
((2*c^2*d + e)*x^2 - d)/((c^4*d^2 + c^2*d*e)*x^3 - (c^2*d^2 + d*e)*x)) + 2 
*(b*c^2*d^2 + b*d*e + (b*c^2*d*e + b*e^2)*x^2)*arctan(sqrt(-c^2*x^2 + 1)*c 
*x/(c^2*x^2 - 1)))/(c^2*d^3*e + d^2*e^2 + (c^2*d^2*e^2 + d*e^3)*x^2)]
 

Sympy [F]

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

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

Output:

Integral(x*(a + b*acos(c*x))/(d + e*x**2)**2, x)
 

Maxima [F(-2)]

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

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 1401, normalized size of antiderivative = 16.29 \[ \int \frac {x (a+b \arccos (c x))}{\left (d+e x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 2*acos(c*x)*b*c**3*d**3 - 2*acos(c*x)*b*c*d**2*e - 2*sqrt(e)*sqrt(d)*s 
qrt(c**2*d + e)*sqrt( - 2*sqrt(e)*sqrt(c**2*d + e) + c**2*d + 2*e)*atan((t 
an(asin(c*x)/2)*c*d)/(sqrt(d)*sqrt( - 2*sqrt(e)*sqrt(c**2*d + e) + c**2*d 
+ 2*e)))*b*d - 2*sqrt(e)*sqrt(d)*sqrt(c**2*d + e)*sqrt( - 2*sqrt(e)*sqrt(c 
**2*d + e) + c**2*d + 2*e)*atan((tan(asin(c*x)/2)*c*d)/(sqrt(d)*sqrt( - 2* 
sqrt(e)*sqrt(c**2*d + e) + c**2*d + 2*e)))*b*e*x**2 - 2*sqrt(d)*sqrt( - 2* 
sqrt(e)*sqrt(c**2*d + e) + c**2*d + 2*e)*atan((tan(asin(c*x)/2)*c*d)/(sqrt 
(d)*sqrt( - 2*sqrt(e)*sqrt(c**2*d + e) + c**2*d + 2*e)))*b*c**2*d**2 - 2*s 
qrt(d)*sqrt( - 2*sqrt(e)*sqrt(c**2*d + e) + c**2*d + 2*e)*atan((tan(asin(c 
*x)/2)*c*d)/(sqrt(d)*sqrt( - 2*sqrt(e)*sqrt(c**2*d + e) + c**2*d + 2*e)))* 
b*c**2*d*e*x**2 - 2*sqrt(d)*sqrt( - 2*sqrt(e)*sqrt(c**2*d + e) + c**2*d + 
2*e)*atan((tan(asin(c*x)/2)*c*d)/(sqrt(d)*sqrt( - 2*sqrt(e)*sqrt(c**2*d + 
e) + c**2*d + 2*e)))*b*d*e - 2*sqrt(d)*sqrt( - 2*sqrt(e)*sqrt(c**2*d + e) 
+ c**2*d + 2*e)*atan((tan(asin(c*x)/2)*c*d)/(sqrt(d)*sqrt( - 2*sqrt(e)*sqr 
t(c**2*d + e) + c**2*d + 2*e)))*b*e**2*x**2 - sqrt(e)*sqrt(d)*sqrt(c**2*d 
+ e)*sqrt( - 2*sqrt(e)*sqrt(c**2*d + e) - c**2*d - 2*e)*log( - sqrt( - 2*s 
qrt(e)*sqrt(c**2*d + e) - c**2*d - 2*e) + sqrt(d)*tan(asin(c*x)/2)*c)*b*d 
- sqrt(e)*sqrt(d)*sqrt(c**2*d + e)*sqrt( - 2*sqrt(e)*sqrt(c**2*d + e) - c* 
*2*d - 2*e)*log( - sqrt( - 2*sqrt(e)*sqrt(c**2*d + e) - c**2*d - 2*e) + sq 
rt(d)*tan(asin(c*x)/2)*c)*b*e*x**2 + sqrt(e)*sqrt(d)*sqrt(c**2*d + e)*s...