\(\int \frac {a+b \arccos (c x)}{(d-c^2 d x^2)^{5/2}} \, dx\) [56]

Optimal result

Integrand size = 24, antiderivative size = 154 \[ \int \frac {a+b \arccos (c x)}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {b}{6 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {x (a+b \arccos (c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {2 x (a+b \arccos (c x))}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \log \left (1-c^2 x^2\right )}{3 c d^2 \sqrt {d-c^2 d x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.73 \[ \int \frac {a+b \arccos (c x)}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {\sqrt {d-c^2 d x^2} \left (6 a c x-4 a c^3 x^3+b \sqrt {1-c^2 x^2}+b \left (6 c x-4 c^3 x^3\right ) \arccos (c x)-2 b \left (1-c^2 x^2\right )^{3/2} \log \left (-1+c^2 x^2\right )\right )}{6 c d^3 \left (-1+c^2 x^2\right )^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5163, 241, 5161, 240}

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.

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

\(\Big \downarrow \) 5163

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

\(\Big \downarrow \) 241

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

\(\Big \downarrow \) 5161

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

\(\Big \downarrow \) 240

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 240

Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x 
^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
 

rule 241

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

rule 5161

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x 
_Symbol] :> Simp[x*((a + b*ArcCos[c*x])^n/(d*Sqrt[d + e*x^2])), x] + Simp[b 
*c*(n/d)*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]   Int[x*((a + b*ArcCos[c*x 
])^(n - 1)/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d 
 + e, 0] && GtQ[n, 0]
 

rule 5163

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_ 
Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(2*d*(p + 1 
))), x] + (Simp[(2*p + 3)/(2*d*(p + 1))   Int[(d + e*x^2)^(p + 1)*(a + b*Ar 
cCos[c*x])^n, x], x] - Simp[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2 
*x^2)^p]   Int[x*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x 
]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, 
 -1] && NeQ[p, -3/2]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.41 (sec) , antiderivative size = 472, normalized size of antiderivative = 3.06

method	result	size
default	\(a \left (\frac {x}{3 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{2} \sqrt {-c^{2} d \,x^{2}+d}}\right )-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-3 c x +2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-2 i \sqrt {-c^{2} x^{2}+1}\right ) \left (8 i \ln \left (\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) x^{6} c^{6}+8 \ln \left (\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}\, x^{5} c^{5}-24 i \ln \left (\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) x^{4} c^{4}+2 i x^{4} c^{4}-20 \ln \left (\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+2 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+24 i \ln \left (\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) x^{2} c^{2}+6 c^{2} x^{2} \arccos \left (c x \right )-4 i c^{2} x^{2}+12 \ln \left (\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}\, x c -3 c x \sqrt {-c^{2} x^{2}+1}-8 i \ln \left (\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right )-8 \arccos \left (c x \right )+2 i\right )}{6 d^{3} \left (3 c^{6} x^{6}-10 c^{4} x^{4}+11 c^{2} x^{2}-4\right ) c}\)	\(472\)
parts	\(a \left (\frac {x}{3 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{2} \sqrt {-c^{2} d \,x^{2}+d}}\right )-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-3 c x +2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-2 i \sqrt {-c^{2} x^{2}+1}\right ) \left (8 i \ln \left (\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) x^{6} c^{6}+8 \ln \left (\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}\, x^{5} c^{5}-24 i \ln \left (\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) x^{4} c^{4}+2 i x^{4} c^{4}-20 \ln \left (\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+2 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+24 i \ln \left (\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) x^{2} c^{2}+6 c^{2} x^{2} \arccos \left (c x \right )-4 i c^{2} x^{2}+12 \ln \left (\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}\, x c -3 c x \sqrt {-c^{2} x^{2}+1}-8 i \ln \left (\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right )-8 \arccos \left (c x \right )+2 i\right )}{6 d^{3} \left (3 c^{6} x^{6}-10 c^{4} x^{4}+11 c^{2} x^{2}-4\right ) c}\)	\(472\)

Input:

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

Output:

a*(1/3/d*x/(-c^2*d*x^2+d)^(3/2)+2/3/d^2*x/(-c^2*d*x^2+d)^(1/2))-1/6*b*(-d* 
(c^2*x^2-1))^(1/2)*(2*c^3*x^3-3*c*x+2*I*(-c^2*x^2+1)^(1/2)*x^2*c^2-2*I*(-c 
^2*x^2+1)^(1/2))*(8*I*ln((c*x+I*(-c^2*x^2+1)^(1/2))^2-1)*x^6*c^6+8*ln((c*x 
+I*(-c^2*x^2+1)^(1/2))^2-1)*(-c^2*x^2+1)^(1/2)*x^5*c^5-24*I*ln((c*x+I*(-c^ 
2*x^2+1)^(1/2))^2-1)*x^4*c^4+2*I*x^4*c^4-20*ln((c*x+I*(-c^2*x^2+1)^(1/2))^ 
2-1)*(-c^2*x^2+1)^(1/2)*x^3*c^3+2*c^3*x^3*(-c^2*x^2+1)^(1/2)+24*I*ln((c*x+ 
I*(-c^2*x^2+1)^(1/2))^2-1)*x^2*c^2+6*c^2*x^2*arccos(c*x)-4*I*c^2*x^2+12*ln 
((c*x+I*(-c^2*x^2+1)^(1/2))^2-1)*(-c^2*x^2+1)^(1/2)*x*c-3*c*x*(-c^2*x^2+1) 
^(1/2)-8*I*ln((c*x+I*(-c^2*x^2+1)^(1/2))^2-1)-8*arccos(c*x)+2*I)/d^3/(3*c^ 
6*x^6-10*c^4*x^4+11*c^2*x^2-4)/c
 

Fricas [F]

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

integrate((a+b*arccos(c*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^6 - 3*c^4*d^ 
3*x^4 + 3*c^2*d^3*x^2 - d^3), x)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.92 \[ \int \frac {a+b \arccos (c x)}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {1}{6} \, b c {\left (\frac {1}{c^{4} d^{\frac {5}{2}} x^{2} - c^{2} d^{\frac {5}{2}}} + \frac {2 \, \log \left (c x + 1\right )}{c^{2} d^{\frac {5}{2}}} + \frac {2 \, \log \left (c x - 1\right )}{c^{2} d^{\frac {5}{2}}}\right )} + \frac {1}{3} \, b {\left (\frac {2 \, x}{\sqrt {-c^{2} d x^{2} + d} d^{2}} + \frac {x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d}\right )} \arccos \left (c x\right ) + \frac {1}{3} \, a {\left (\frac {2 \, x}{\sqrt {-c^{2} d x^{2} + d} d^{2}} + \frac {x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d}\right )} \] Input:

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

Output:

-1/6*b*c*(1/(c^4*d^(5/2)*x^2 - c^2*d^(5/2)) + 2*log(c*x + 1)/(c^2*d^(5/2)) 
 + 2*log(c*x - 1)/(c^2*d^(5/2))) + 1/3*b*(2*x/(sqrt(-c^2*d*x^2 + d)*d^2) + 
 x/((-c^2*d*x^2 + d)^(3/2)*d))*arccos(c*x) + 1/3*a*(2*x/(sqrt(-c^2*d*x^2 + 
 d)*d^2) + x/((-c^2*d*x^2 + d)^(3/2)*d))
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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