\(\int \frac {(d-c^2 d x^2) (a+b \arccos (c x))}{x} \, dx\) [8]

Optimal result

Integrand size = 23, antiderivative size = 121 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arccos (c x))}{x} \, dx=-\frac {1}{4} b c d x \sqrt {1-c^2 x^2}-\frac {1}{4} b d \arccos (c x)+\frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \arccos (c x))-\frac {i d (a+b \arccos (c x))^2}{2 b}+d (a+b \arccos (c x)) \log \left (1-e^{2 i \arccos (c x)}\right )-\frac {1}{2} i b d \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.01 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arccos (c x))}{x} \, dx=-\frac {1}{2} a c^2 d x^2+\frac {1}{4} b c d x \sqrt {1-c^2 x^2}-\frac {1}{2} b c^2 d x^2 \arccos (c x)-\frac {1}{2} i b d \arccos (c x)^2-\frac {1}{4} b d \arcsin (c x)+b d \arccos (c x) \log \left (1+e^{2 i \arccos (c x)}\right )+a d \log (x)-\frac {1}{2} i b d \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.12, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {5189, 211, 223, 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.

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

\(\Big \downarrow \) 5189

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 5137

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4202

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt 
[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
 

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 5189

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

Maple [A] (verified)

Time = 0.60 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.08

method	result	size
parts	\(-d a \left (\frac {c^{2} x^{2}}{2}-\ln \left (x \right )\right )-\frac {i b d \arccos \left (c x \right )^{2}}{2}+\frac {b c d x \sqrt {-c^{2} x^{2}+1}}{4}-\frac {\arccos \left (c x \right ) b \,c^{2} d \,x^{2}}{2}+\frac {b d \arccos \left (c x \right )}{4}+d b \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i d b \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\)	\(131\)
derivativedivides	\(-d a \left (\frac {c^{2} x^{2}}{2}-\ln \left (c x \right )\right )-\frac {i b d \arccos \left (c x \right )^{2}}{2}+\frac {b c d x \sqrt {-c^{2} x^{2}+1}}{4}-\frac {\arccos \left (c x \right ) b \,c^{2} d \,x^{2}}{2}+\frac {b d \arccos \left (c x \right )}{4}+d b \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i d b \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\)	\(133\)
default	\(-d a \left (\frac {c^{2} x^{2}}{2}-\ln \left (c x \right )\right )-\frac {i b d \arccos \left (c x \right )^{2}}{2}+\frac {b c d x \sqrt {-c^{2} x^{2}+1}}{4}-\frac {\arccos \left (c x \right ) b \,c^{2} d \,x^{2}}{2}+\frac {b d \arccos \left (c x \right )}{4}+d b \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i d b \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\)	\(133\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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