\(\int \frac {a+b \arccos (c x)}{(d-c^2 d x^2)^{7/2}} \, dx\) [57]

Optimal result

Integrand size = 24, antiderivative size = 225 \[ \int \frac {a+b \arccos (c x)}{\left (d-c^2 d x^2\right )^{7/2}} \, dx=\frac {b}{20 c d^3 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {2 b}{15 c d^3 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {x (a+b \arccos (c x))}{5 d \left (d-c^2 d x^2\right )^{5/2}}+\frac {4 x (a+b \arccos (c x))}{15 d^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 x (a+b \arccos (c x))}{15 d^3 \sqrt {d-c^2 d x^2}}-\frac {4 b \sqrt {1-c^2 x^2} \log \left (1-c^2 x^2\right )}{15 c d^3 \sqrt {d-c^2 d x^2}} \] Output:

1/20*b/c/d^3/(-c^2*x^2+1)^(3/2)/(-c^2*d*x^2+d)^(1/2)+2/15*b/c/d^3/(-c^2*x^ 
2+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)+1/5*x*(a+b*arccos(c*x))/d/(-c^2*d*x^2+d)^( 
5/2)+4/15*x*(a+b*arccos(c*x))/d^2/(-c^2*d*x^2+d)^(3/2)+8/15*x*(a+b*arccos( 
c*x))/d^3/(-c^2*d*x^2+d)^(1/2)-4/15*b*(-c^2*x^2+1)^(1/2)*ln(-c^2*x^2+1)/c/ 
d^3/(-c^2*d*x^2+d)^(1/2)
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.68 \[ \int \frac {a+b \arccos (c x)}{\left (d-c^2 d x^2\right )^{7/2}} \, dx=\frac {\sqrt {d-c^2 d x^2} \left (-60 a c x+80 a c^3 x^3-32 a c^5 x^5-11 b \sqrt {1-c^2 x^2}+8 b c^2 x^2 \sqrt {1-c^2 x^2}-4 b c x \left (15-20 c^2 x^2+8 c^4 x^4\right ) \arccos (c x)+16 b \left (1-c^2 x^2\right )^{5/2} \log \left (-1+c^2 x^2\right )\right )}{60 c d^4 \left (-1+c^2 x^2\right )^3} \] Input:

Integrate[(a + b*ArcCos[c*x])/(d - c^2*d*x^2)^(7/2),x]
 

Output:

(Sqrt[d - c^2*d*x^2]*(-60*a*c*x + 80*a*c^3*x^3 - 32*a*c^5*x^5 - 11*b*Sqrt[ 
1 - c^2*x^2] + 8*b*c^2*x^2*Sqrt[1 - c^2*x^2] - 4*b*c*x*(15 - 20*c^2*x^2 + 
8*c^4*x^4)*ArcCos[c*x] + 16*b*(1 - c^2*x^2)^(5/2)*Log[-1 + c^2*x^2]))/(60* 
c*d^4*(-1 + c^2*x^2)^3)
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5163, 241, 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.

\(\displaystyle \int \frac {a+b \arccos (c x)}{\left (d-c^2 d x^2\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 5163

\(\displaystyle \frac {4 \int \frac {a+b \arccos (c x)}{\left (d-c^2 d x^2\right )^{5/2}}dx}{5 d}+\frac {b c \sqrt {1-c^2 x^2} \int \frac {x}{\left (1-c^2 x^2\right )^3}dx}{5 d^3 \sqrt {d-c^2 d x^2}}+\frac {x (a+b \arccos (c x))}{5 d \left (d-c^2 d x^2\right )^{5/2}}\)

\(\Big \downarrow \) 241

\(\displaystyle \frac {4 \int \frac {a+b \arccos (c x)}{\left (d-c^2 d x^2\right )^{5/2}}dx}{5 d}+\frac {x (a+b \arccos (c x))}{5 d \left (d-c^2 d x^2\right )^{5/2}}+\frac {b}{20 c d^3 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 5163

\(\displaystyle \frac {4 \left (\frac {2 \int \frac {a+b \arccos (c x)}{\left (d-c^2 d x^2\right )^{3/2}}dx}{3 d}+\frac {b c \sqrt {1-c^2 x^2} \int \frac {x}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {x (a+b \arccos (c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}\right )}{5 d}+\frac {x (a+b \arccos (c x))}{5 d \left (d-c^2 d x^2\right )^{5/2}}+\frac {b}{20 c d^3 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 241

\(\displaystyle \frac {4 \left (\frac {2 \int \frac {a+b \arccos (c x)}{\left (d-c^2 d x^2\right )^{3/2}}dx}{3 d}+\frac {x (a+b \arccos (c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {b}{6 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}\right )}{5 d}+\frac {x (a+b \arccos (c x))}{5 d \left (d-c^2 d x^2\right )^{5/2}}+\frac {b}{20 c d^3 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 5161

\(\displaystyle \frac {4 \left (\frac {2 \left (\frac {b c \sqrt {1-c^2 x^2} \int \frac {x}{1-c^2 x^2}dx}{d \sqrt {d-c^2 d x^2}}+\frac {x (a+b \arccos (c x))}{d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \arccos (c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {b}{6 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}\right )}{5 d}+\frac {x (a+b \arccos (c x))}{5 d \left (d-c^2 d x^2\right )^{5/2}}+\frac {b}{20 c d^3 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 240

\(\displaystyle \frac {4 \left (\frac {x (a+b \arccos (c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {2 \left (\frac {x (a+b \arccos (c x))}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \log \left (1-c^2 x^2\right )}{2 c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {b}{6 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}\right )}{5 d}+\frac {x (a+b \arccos (c x))}{5 d \left (d-c^2 d x^2\right )^{5/2}}+\frac {b}{20 c d^3 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2}}\)

Input:

Int[(a + b*ArcCos[c*x])/(d - c^2*d*x^2)^(7/2),x]
 

Output:

b/(20*c*d^3*(1 - c^2*x^2)^(3/2)*Sqrt[d - c^2*d*x^2]) + (x*(a + b*ArcCos[c* 
x]))/(5*d*(d - c^2*d*x^2)^(5/2)) + (4*(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]*Lo 
g[1 - c^2*x^2])/(2*c*d*Sqrt[d - c^2*d*x^2])))/(3*d)))/(5*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 = 0.50 (sec) , antiderivative size = 2280, normalized size of antiderivative = 10.13

method	result	size
default	\(\text {Expression too large to display}\)	\(2280\)
parts	\(\text {Expression too large to display}\)	\(2280\)

Input:

int((a+b*arccos(c*x))/(-c^2*d*x^2+d)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

-176/15*b*(-d*(c^2*x^2-1))^(1/2)/d^4/(40*c^10*x^10-215*c^8*x^8+469*c^6*x^6 
-517*c^4*x^4+287*c^2*x^2-64)/c*(-c^2*x^2+1)^(1/2)-22*I*b*(-d*(c^2*x^2-1))^ 
(1/2)/d^4/(40*c^10*x^10-215*c^8*x^8+469*c^6*x^6-517*c^4*x^4+287*c^2*x^2-64 
)*x-64*b*(-d*(c^2*x^2-1))^(1/2)/d^4/(40*c^10*x^10-215*c^8*x^8+469*c^6*x^6- 
517*c^4*x^4+287*c^2*x^2-64)*arccos(c*x)*x+64/3*I*b*(-d*(c^2*x^2-1))^(1/2)/ 
d^4/(40*c^10*x^10-215*c^8*x^8+469*c^6*x^6-517*c^4*x^4+287*c^2*x^2-64)*c^7* 
(-c^2*x^2+1)^(1/2)*arccos(c*x)*x^8-280/3*I*b*(-d*(c^2*x^2-1))^(1/2)/d^4/(4 
0*c^10*x^10-215*c^8*x^8+469*c^6*x^6-517*c^4*x^4+287*c^2*x^2-64)*c^5*(-c^2* 
x^2+1)^(1/2)*arccos(c*x)*x^6+784/5*I*b*(-d*(c^2*x^2-1))^(1/2)/d^4/(40*c^10 
*x^10-215*c^8*x^8+469*c^6*x^6-517*c^4*x^4+287*c^2*x^2-64)*c^3*(-c^2*x^2+1) 
^(1/2)*arccos(c*x)*x^4-1784/15*I*b*(-d*(c^2*x^2-1))^(1/2)/d^4/(40*c^10*x^1 
0-215*c^8*x^8+469*c^6*x^6-517*c^4*x^4+287*c^2*x^2-64)*c*(-c^2*x^2+1)^(1/2) 
*arccos(c*x)*x^2+541/3*b*(-d*(c^2*x^2-1))^(1/2)/d^4/(40*c^10*x^10-215*c^8* 
x^8+469*c^6*x^6-517*c^4*x^4+287*c^2*x^2-64)*c^2*arccos(c*x)*x^3+22*I*b*(-d 
*(c^2*x^2-1))^(1/2)/d^4/(40*c^10*x^10-215*c^8*x^8+469*c^6*x^6-517*c^4*x^4+ 
287*c^2*x^2-64)*(-c^2*x^2+1)*x-128/15*I*b*(-d*(c^2*x^2-1))^(1/2)/d^4/(40*c 
^10*x^10-215*c^8*x^8+469*c^6*x^6-517*c^4*x^4+287*c^2*x^2-64)*c^12*x^13+176 
/3*I*b*(-d*(c^2*x^2-1))^(1/2)/d^4/(40*c^10*x^10-215*c^8*x^8+469*c^6*x^6-51 
7*c^4*x^4+287*c^2*x^2-64)*c^10*x^11-2552/15*I*b*(-d*(c^2*x^2-1))^(1/2)/d^4 
/(40*c^10*x^10-215*c^8*x^8+469*c^6*x^6-517*c^4*x^4+287*c^2*x^2-64)*c^8*...
 

Fricas [F]

\[ \int \frac {a+b \arccos (c x)}{\left (d-c^2 d x^2\right )^{7/2}} \, dx=\int { \frac {b \arccos \left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((a+b*arccos(c*x))/(-c^2*d*x^2+d)^(7/2),x, algorithm="fricas")
 

Output:

integral(sqrt(-c^2*d*x^2 + d)*(b*arccos(c*x) + a)/(c^8*d^4*x^8 - 4*c^6*d^4 
*x^6 + 6*c^4*d^4*x^4 - 4*c^2*d^4*x^2 + d^4), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b \arccos (c x)}{\left (d-c^2 d x^2\right )^{7/2}} \, dx=\text {Timed out} \] Input:

integrate((a+b*acos(c*x))/(-c**2*d*x**2+d)**(7/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {a+b \arccos (c x)}{\left (d-c^2 d x^2\right )^{7/2}} \, dx=\int { \frac {b \arccos \left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((a+b*arccos(c*x))/(-c^2*d*x^2+d)^(7/2),x, algorithm="maxima")
 

Output:

1/15*a*(8*x/(sqrt(-c^2*d*x^2 + d)*d^3) + 4*x/((-c^2*d*x^2 + d)^(3/2)*d^2) 
+ 3*x/((-c^2*d*x^2 + d)^(5/2)*d)) - b*integrate(arctan2(sqrt(c*x + 1)*sqrt 
(-c*x + 1), c*x)/((c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3)*sqrt 
(c*x + 1)*sqrt(-c*x + 1)), x)/sqrt(d)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {a+b \arccos (c x)}{\left (d-c^2 d x^2\right )^{7/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((a+b*arccos(c*x))/(-c^2*d*x^2+d)^(7/2),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. \[ \int \frac {a+b \arccos (c x)}{\left (d-c^2 d x^2\right )^{7/2}} \, dx=\int \frac {a+b\,\mathrm {acos}\left (c\,x\right )}{{\left (d-c^2\,d\,x^2\right )}^{7/2}} \,d x \] Input:

int((a + b*acos(c*x))/(d - c^2*d*x^2)^(7/2),x)
 

Output:

int((a + b*acos(c*x))/(d - c^2*d*x^2)^(7/2), x)
 

Reduce [F]

\[ \int \frac {a+b \arccos (c x)}{\left (d-c^2 d x^2\right )^{7/2}} \, dx=\frac {-15 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acos} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{6} x^{6}-3 \sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}+3 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-\sqrt {-c^{2} x^{2}+1}}d x \right ) b \,c^{4} x^{4}+30 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acos} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{6} x^{6}-3 \sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}+3 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-\sqrt {-c^{2} x^{2}+1}}d x \right ) b \,c^{2} x^{2}-15 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acos} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{6} x^{6}-3 \sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}+3 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-\sqrt {-c^{2} x^{2}+1}}d x \right ) b +8 a \,c^{4} x^{5}-20 a \,c^{2} x^{3}+15 a x}{15 \sqrt {d}\, \sqrt {-c^{2} x^{2}+1}\, d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )} \] Input:

int((a+b*acos(c*x))/(-c^2*d*x^2+d)^(7/2),x)
 

Output:

( - 15*sqrt( - c**2*x**2 + 1)*int(acos(c*x)/(sqrt( - c**2*x**2 + 1)*c**6*x 
**6 - 3*sqrt( - c**2*x**2 + 1)*c**4*x**4 + 3*sqrt( - c**2*x**2 + 1)*c**2*x 
**2 - sqrt( - c**2*x**2 + 1)),x)*b*c**4*x**4 + 30*sqrt( - c**2*x**2 + 1)*i 
nt(acos(c*x)/(sqrt( - c**2*x**2 + 1)*c**6*x**6 - 3*sqrt( - c**2*x**2 + 1)* 
c**4*x**4 + 3*sqrt( - c**2*x**2 + 1)*c**2*x**2 - sqrt( - c**2*x**2 + 1)),x 
)*b*c**2*x**2 - 15*sqrt( - c**2*x**2 + 1)*int(acos(c*x)/(sqrt( - c**2*x**2 
 + 1)*c**6*x**6 - 3*sqrt( - c**2*x**2 + 1)*c**4*x**4 + 3*sqrt( - c**2*x**2 
 + 1)*c**2*x**2 - sqrt( - c**2*x**2 + 1)),x)*b + 8*a*c**4*x**5 - 20*a*c**2 
*x**3 + 15*a*x)/(15*sqrt(d)*sqrt( - c**2*x**2 + 1)*d**3*(c**4*x**4 - 2*c** 
2*x**2 + 1))