\(\int \frac {a+b \arccos (c x)}{x (d-c^2 d x^2)^{5/2}} \, dx\) [138]

Optimal result

Integrand size = 27, antiderivative size = 291 \[ \int \frac {a+b \arccos (c x)}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {b c x}{6 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {a+b \arccos (c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {a+b \arccos (c x)}{d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {1-c^2 x^2} (a+b \arccos (c x)) \text {arctanh}\left (e^{i \arccos (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {7 b \sqrt {1-c^2 x^2} \text {arctanh}(c x)}{6 d^2 \sqrt {d-c^2 d x^2}}+\frac {i b \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {i b \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}} \] Output:

-1/6*b*c*x/d^2/(-c^2*x^2+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)+1/3*(a+b*arccos(c*x 
))/d/(-c^2*d*x^2+d)^(3/2)+(a+b*arccos(c*x))/d^2/(-c^2*d*x^2+d)^(1/2)-2*(-c 
^2*x^2+1)^(1/2)*(a+b*arccos(c*x))*arctanh(c*x+I*(-c^2*x^2+1)^(1/2))/d^2/(- 
c^2*d*x^2+d)^(1/2)-7/6*b*(-c^2*x^2+1)^(1/2)*arctanh(c*x)/d^2/(-c^2*d*x^2+d 
)^(1/2)+I*b*(-c^2*x^2+1)^(1/2)*polylog(2,-c*x-I*(-c^2*x^2+1)^(1/2))/d^2/(- 
c^2*d*x^2+d)^(1/2)-I*b*(-c^2*x^2+1)^(1/2)*polylog(2,c*x+I*(-c^2*x^2+1)^(1/ 
2))/d^2/(-c^2*d*x^2+d)^(1/2)
 

Mathematica [A] (verified)

Time = 3.77 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.21 \[ \int \frac {a+b \arccos (c x)}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {a \left (-4+3 c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{3 d^3 \left (-1+c^2 x^2\right )^2}+\frac {a \log (x)}{d^{5/2}}-\frac {a \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )}{d^{5/2}}-\frac {b \left (1-c^2 x^2\right )^{3/2} \left (-14 \arccos (c x) \cot \left (\frac {1}{2} \arccos (c x)\right )-\csc ^2\left (\frac {1}{2} \arccos (c x)\right )-\frac {1}{2} \sqrt {1-c^2 x^2} \arccos (c x) \csc ^4\left (\frac {1}{2} \arccos (c x)\right )+24 \arccos (c x) \log \left (1-i e^{i \arccos (c x)}\right )-24 \arccos (c x) \log \left (1+i e^{i \arccos (c x)}\right )-28 \log \left (\cos \left (\frac {1}{2} \arccos (c x)\right )\right )+28 \log \left (\sin \left (\frac {1}{2} \arccos (c x)\right )\right )+24 i \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )-24 i \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )+\sec ^2\left (\frac {1}{2} \arccos (c x)\right )-\frac {8 \arccos (c x) \sin ^4\left (\frac {1}{2} \arccos (c x)\right )}{\left (1-c^2 x^2\right )^{3/2}}-14 \arccos (c x) \tan \left (\frac {1}{2} \arccos (c x)\right )\right )}{24 d \left (d-c^2 d x^2\right )^{3/2}} \] Input:

Integrate[(a + b*ArcCos[c*x])/(x*(d - c^2*d*x^2)^(5/2)),x]
 

Output:

-1/3*(a*(-4 + 3*c^2*x^2)*Sqrt[d - c^2*d*x^2])/(d^3*(-1 + c^2*x^2)^2) + (a* 
Log[x])/d^(5/2) - (a*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]])/d^(5/2) - (b*(1 
 - c^2*x^2)^(3/2)*(-14*ArcCos[c*x]*Cot[ArcCos[c*x]/2] - Csc[ArcCos[c*x]/2] 
^2 - (Sqrt[1 - c^2*x^2]*ArcCos[c*x]*Csc[ArcCos[c*x]/2]^4)/2 + 24*ArcCos[c* 
x]*Log[1 - I*E^(I*ArcCos[c*x])] - 24*ArcCos[c*x]*Log[1 + I*E^(I*ArcCos[c*x 
])] - 28*Log[Cos[ArcCos[c*x]/2]] + 28*Log[Sin[ArcCos[c*x]/2]] + (24*I)*Pol 
yLog[2, (-I)*E^(I*ArcCos[c*x])] - (24*I)*PolyLog[2, I*E^(I*ArcCos[c*x])] + 
 Sec[ArcCos[c*x]/2]^2 - (8*ArcCos[c*x]*Sin[ArcCos[c*x]/2]^4)/(1 - c^2*x^2) 
^(3/2) - 14*ArcCos[c*x]*Tan[ArcCos[c*x]/2]))/(24*d*(d - c^2*d*x^2)^(3/2))
 

Rubi [A] (verified)

Time = 1.03 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.92, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {5209, 215, 219, 5209, 219, 5219, 3042, 4669, 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.

\(\displaystyle \int \frac {a+b \arccos (c x)}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 5209

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

\(\Big \downarrow \) 215

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 5209

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 5219

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

Int[(a + b*ArcCos[c*x])/(x*(d - c^2*d*x^2)^(5/2)),x]
 

Output:

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

Defintions of rubi rules used

rule 215

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 
*p])
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 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 4669

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol 
] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Si 
mp[d*(m/f)   Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))], x], 
 x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x 
))], x], x]) /; FreeQ[{c, d, e, f}, x] && IntegerQ[2*k] && IGtQ[m, 0]
 

rule 5209

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/(2*d*f*(p + 1))), x] + (Simp[(m + 2*p + 3)/(2*d*(p + 1)) 
   Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^n, x], x] - Simp[b*c 
*(n/(2*f*(p + 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, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] &&  !G 
tQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])
 

rule 5219

Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)* 
(x_)^2], x_Symbol] :> Simp[(-(c^(m + 1))^(-1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[ 
d + e*x^2]]   Subst[Int[(a + b*x)^n*Cos[x]^m, x], x, ArcCos[c*x]], x] /; Fr 
eeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[m]
 

Maple [A] (verified)

Time = 0.87 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.20

method	result	size
default	\(\frac {a}{3 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {a}{d^{2} \sqrt {-c^{2} d \,x^{2}+d}}-\frac {a \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {5}{2}}}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (6 c^{2} x^{2} \arccos \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-8 \arccos \left (c x \right )\right )}{6 \left (c^{2} x^{2}-1\right )^{2} d^{3}}+\frac {i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (6 i \arccos \left (c x \right ) \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-6 i \arccos \left (c x \right ) \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-7 i \ln \left (i \sqrt {-c^{2} x^{2}+1}+c x -1\right )+7 i \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+6 \operatorname {dilog}\left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-6 \operatorname {dilog}\left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{6 d^{3} \left (c^{2} x^{2}-1\right )}\right )\)	\(350\)
parts	\(\frac {a}{3 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {a}{d^{2} \sqrt {-c^{2} d \,x^{2}+d}}-\frac {a \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {5}{2}}}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (6 c^{2} x^{2} \arccos \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-8 \arccos \left (c x \right )\right )}{6 \left (c^{2} x^{2}-1\right )^{2} d^{3}}+\frac {i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (6 i \arccos \left (c x \right ) \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-6 i \arccos \left (c x \right ) \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-7 i \ln \left (i \sqrt {-c^{2} x^{2}+1}+c x -1\right )+7 i \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+6 \operatorname {dilog}\left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-6 \operatorname {dilog}\left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{6 d^{3} \left (c^{2} x^{2}-1\right )}\right )\)	\(350\)

Input:

int((a+b*arccos(c*x))/x/(-c^2*d*x^2+d)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/3*a/d/(-c^2*d*x^2+d)^(3/2)+a/d^2/(-c^2*d*x^2+d)^(1/2)-a/d^(5/2)*ln((2*d+ 
2*d^(1/2)*(-c^2*d*x^2+d)^(1/2))/x)+b*(-1/6*(-d*(c^2*x^2-1))^(1/2)*(6*c^2*x 
^2*arccos(c*x)-c*x*(-c^2*x^2+1)^(1/2)-8*arccos(c*x))/(c^2*x^2-1)^2/d^3+1/6 
*I*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*(6*I*arccos(c*x)*ln(1+I*(c*x+ 
I*(-c^2*x^2+1)^(1/2)))-6*I*arccos(c*x)*ln(1-I*(c*x+I*(-c^2*x^2+1)^(1/2)))- 
7*I*ln(I*(-c^2*x^2+1)^(1/2)+c*x-1)+7*I*ln(1+c*x+I*(-c^2*x^2+1)^(1/2))+6*di 
log(1+I*(c*x+I*(-c^2*x^2+1)^(1/2)))-6*dilog(1-I*(c*x+I*(-c^2*x^2+1)^(1/2)) 
))/d^3/(c^2*x^2-1))
 

Fricas [F]

\[ \int \frac {a+b \arccos (c x)}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {b \arccos \left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x} \,d x } \] Input:

integrate((a+b*arccos(c*x))/x/(-c^2*d*x^2+d)^(5/2),x, algorithm="fricas")
 

Output:

integral(-sqrt(-c^2*d*x^2 + d)*(b*arccos(c*x) + a)/(c^6*d^3*x^7 - 3*c^4*d^ 
3*x^5 + 3*c^2*d^3*x^3 - d^3*x), x)
 

Sympy [F]

\[ \int \frac {a+b \arccos (c x)}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {a + b \operatorname {acos}{\left (c x \right )}}{x \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:

integrate((a+b*acos(c*x))/x/(-c**2*d*x**2+d)**(5/2),x)
 

Output:

Integral((a + b*acos(c*x))/(x*(-d*(c*x - 1)*(c*x + 1))**(5/2)), x)
 

Maxima [F]

\[ \int \frac {a+b \arccos (c x)}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {b \arccos \left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x} \,d x } \] Input:

integrate((a+b*arccos(c*x))/x/(-c^2*d*x^2+d)^(5/2),x, algorithm="maxima")
 

Output:

-1/3*a*(3*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x))/d^(5/2) 
- 3/(sqrt(-c^2*d*x^2 + d)*d^2) - 1/((-c^2*d*x^2 + d)^(3/2)*d)) + b*integra 
te(arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)/((c^4*d^2*x^5 - 2*c^2*d^2*x^ 
3 + d^2*x)*sqrt(c*x + 1)*sqrt(-c*x + 1)), x)/sqrt(d)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {a+b \arccos (c x)}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((a+b*arccos(c*x))/x/(-c^2*d*x^2+d)^(5/2),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. \[ \int \frac {a+b \arccos (c x)}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {a+b\,\mathrm {acos}\left (c\,x\right )}{x\,{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \] Input:

int((a + b*acos(c*x))/(x*(d - c^2*d*x^2)^(5/2)),x)
                                                                                    
                                                                                    
 

Output:

int((a + b*acos(c*x))/(x*(d - c^2*d*x^2)^(5/2)), x)
 

Reduce [F]

\[ \int \frac {a+b \arccos (c x)}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acos} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{5}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{3}+\sqrt {-c^{2} x^{2}+1}\, x}d x \right ) b \,c^{2} x^{2}-3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acos} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{5}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{3}+\sqrt {-c^{2} x^{2}+1}\, x}d x \right ) b +3 \sqrt {-c^{2} x^{2}+1}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right )\right ) a \,c^{2} x^{2}-3 \sqrt {-c^{2} x^{2}+1}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right )\right ) a -4 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} x^{2}+4 \sqrt {-c^{2} x^{2}+1}\, a +3 a \,c^{2} x^{2}-4 a}{3 \sqrt {d}\, \sqrt {-c^{2} x^{2}+1}\, d^{2} \left (c^{2} x^{2}-1\right )} \] Input:

int((a+b*acos(c*x))/x/(-c^2*d*x^2+d)^(5/2),x)
 

Output:

(3*sqrt( - c**2*x**2 + 1)*int(acos(c*x)/(sqrt( - c**2*x**2 + 1)*c**4*x**5 
- 2*sqrt( - c**2*x**2 + 1)*c**2*x**3 + sqrt( - c**2*x**2 + 1)*x),x)*b*c**2 
*x**2 - 3*sqrt( - c**2*x**2 + 1)*int(acos(c*x)/(sqrt( - c**2*x**2 + 1)*c** 
4*x**5 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**3 + sqrt( - c**2*x**2 + 1)*x),x) 
*b + 3*sqrt( - c**2*x**2 + 1)*log(tan(asin(c*x)/2))*a*c**2*x**2 - 3*sqrt( 
- c**2*x**2 + 1)*log(tan(asin(c*x)/2))*a - 4*sqrt( - c**2*x**2 + 1)*a*c**2 
*x**2 + 4*sqrt( - c**2*x**2 + 1)*a + 3*a*c**2*x**2 - 4*a)/(3*sqrt(d)*sqrt( 
 - c**2*x**2 + 1)*d**2*(c**2*x**2 - 1))