\(\int \frac {(d-c^2 d x^2)^2}{x (a+b \arccos (c x))^{3/2}} \, dx\) [442]

Optimal result

Integrand size = 29, antiderivative size = 29 \[ \int \frac {\left (d-c^2 d x^2\right )^2}{x (a+b \arccos (c x))^{3/2}} \, dx=-\frac {2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c x \sqrt {a+b \arccos (c x)}}-\frac {d^2 \sqrt {\frac {\pi }{2}} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{b^{3/2}}-\frac {3 d^2 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{b^{3/2}}-\frac {3 d^2 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{b^{3/2}}-\frac {d^2 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {4 a}{b}\right )}{b^{3/2}}-\frac {2 d^2 \text {Int}\left (\frac {1}{x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}},x\right )}{b c} \] Output:

-2*d^2*(-c^2*x^2+1)^(5/2)/b/c/x/(a+b*arccos(c*x))^(1/2)-1/2*d^2*2^(1/2)*Pi 
^(1/2)*cos(4*a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)*(a+b*arccos(c*x))^(1/2)/b^(1 
/2))/b^(3/2)-3*d^2*Pi^(1/2)*cos(2*a/b)*FresnelC(2*(a+b*arccos(c*x))^(1/2)/ 
b^(1/2)/Pi^(1/2))/b^(3/2)-3*d^2*Pi^(1/2)*FresnelS(2*(a+b*arccos(c*x))^(1/2 
)/b^(1/2)/Pi^(1/2))*sin(2*a/b)/b^(3/2)-1/2*d^2*2^(1/2)*Pi^(1/2)*FresnelS(2 
*2^(1/2)/Pi^(1/2)*(a+b*arccos(c*x))^(1/2)/b^(1/2))*sin(4*a/b)/b^(3/2)-2*d^ 
2*Defer(Int)(1/x^2/(-c^2*x^2+1)^(1/2)/(a+b*arccos(c*x))^(1/2),x)/b/c
 

Mathematica [N/A]

Not integrable

Time = 1.17 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {\left (d-c^2 d x^2\right )^2}{x (a+b \arccos (c x))^{3/2}} \, dx=\int \frac {\left (d-c^2 d x^2\right )^2}{x (a+b \arccos (c x))^{3/2}} \, dx \] Input:

Integrate[(d - c^2*d*x^2)^2/(x*(a + b*ArcCos[c*x])^(3/2)),x]
 

Output:

Integrate[(d - c^2*d*x^2)^2/(x*(a + b*ArcCos[c*x])^(3/2)), x]
 

Rubi [N/A]

Not integrable

Time = 2.32 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}

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.

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

\(\Big \downarrow \) 5215

\(\displaystyle \frac {8 c d^2 \int \frac {\left (1-c^2 x^2\right )^{3/2}}{\sqrt {a+b \arccos (c x)}}dx}{b}+\frac {2 d^2 \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 \sqrt {a+b \arccos (c x)}}dx}{b c}+\frac {2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c x \sqrt {a+b \arccos (c x)}}\)

\(\Big \downarrow \) 5169

\(\displaystyle -\frac {8 d^2 \int \frac {\sin ^4\left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{b^2}+\frac {2 d^2 \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 \sqrt {a+b \arccos (c x)}}dx}{b c}+\frac {2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c x \sqrt {a+b \arccos (c x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {8 d^2 \int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )^4}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{b^2}+\frac {2 d^2 \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 \sqrt {a+b \arccos (c x)}}dx}{b c}+\frac {2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c x \sqrt {a+b \arccos (c x)}}\)

\(\Big \downarrow \) 3793

\(\displaystyle -\frac {8 d^2 \int \left (\frac {\cos \left (\frac {4 a}{b}-\frac {4 (a+b \arccos (c x))}{b}\right )}{8 \sqrt {a+b \arccos (c x)}}-\frac {\cos \left (\frac {2 a}{b}-\frac {2 (a+b \arccos (c x))}{b}\right )}{2 \sqrt {a+b \arccos (c x)}}+\frac {3}{8 \sqrt {a+b \arccos (c x)}}\right )d(a+b \arccos (c x))}{b^2}+\frac {2 d^2 \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 \sqrt {a+b \arccos (c x)}}dx}{b c}+\frac {2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c x \sqrt {a+b \arccos (c x)}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 d^2 \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 \sqrt {a+b \arccos (c x)}}dx}{b c}-\frac {8 d^2 \left (\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )-\frac {1}{2} \sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {3}{4} \sqrt {a+b \arccos (c x)}\right )}{b^2}+\frac {2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c x \sqrt {a+b \arccos (c x)}}\)

\(\Big \downarrow \) 5227

\(\displaystyle \frac {2 d^2 \int \left (\frac {x^2 c^4}{\sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}-\frac {2 c^2}{\sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}+\frac {1}{x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}\right )dx}{b c}-\frac {8 d^2 \left (\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )-\frac {1}{2} \sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {3}{4} \sqrt {a+b \arccos (c x)}\right )}{b^2}+\frac {2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c x \sqrt {a+b \arccos (c x)}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 d^2 \left (\int \frac {1}{x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}dx-\frac {\sqrt {\pi } c \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{2 \sqrt {b}}-\frac {\sqrt {\pi } c \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{2 \sqrt {b}}+\frac {3 c \sqrt {a+b \arccos (c x)}}{b}\right )}{b c}-\frac {8 d^2 \left (\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )-\frac {1}{2} \sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {3}{4} \sqrt {a+b \arccos (c x)}\right )}{b^2}+\frac {2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c x \sqrt {a+b \arccos (c x)}}\)

Input:

Int[(d - c^2*d*x^2)^2/(x*(a + b*ArcCos[c*x])^(3/2)),x]
 

Output:

$Aborted
 

Maple [N/A]

Not integrable

Time = 0.88 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93

\[\int \frac {\left (-c^{2} d \,x^{2}+d \right )^{2}}{x \left (a +b \arccos \left (c x \right )\right )^{\frac {3}{2}}}d x\]

Input:

int((-c^2*d*x^2+d)^2/x/(a+b*arccos(c*x))^(3/2),x)
 

Output:

int((-c^2*d*x^2+d)^2/x/(a+b*arccos(c*x))^(3/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^2}{x (a+b \arccos (c x))^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-c^2*d*x^2+d)^2/x/(a+b*arccos(c*x))^(3/2),x, algorithm="fricas" 
)
 

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [N/A]

Not integrable

Time = 5.27 (sec) , antiderivative size = 133, normalized size of antiderivative = 4.59 \[ \int \frac {\left (d-c^2 d x^2\right )^2}{x (a+b \arccos (c x))^{3/2}} \, dx=d^{2} \left (\int \left (- \frac {2 c^{2} x^{2}}{a x \sqrt {a + b \operatorname {acos}{\left (c x \right )}} + b x \sqrt {a + b \operatorname {acos}{\left (c x \right )}} \operatorname {acos}{\left (c x \right )}}\right )\, dx + \int \frac {c^{4} x^{4}}{a x \sqrt {a + b \operatorname {acos}{\left (c x \right )}} + b x \sqrt {a + b \operatorname {acos}{\left (c x \right )}} \operatorname {acos}{\left (c x \right )}}\, dx + \int \frac {1}{a x \sqrt {a + b \operatorname {acos}{\left (c x \right )}} + b x \sqrt {a + b \operatorname {acos}{\left (c x \right )}} \operatorname {acos}{\left (c x \right )}}\, dx\right ) \] Input:

integrate((-c**2*d*x**2+d)**2/x/(a+b*acos(c*x))**(3/2),x)
 

Output:

d**2*(Integral(-2*c**2*x**2/(a*x*sqrt(a + b*acos(c*x)) + b*x*sqrt(a + b*ac 
os(c*x))*acos(c*x)), x) + Integral(c**4*x**4/(a*x*sqrt(a + b*acos(c*x)) + 
b*x*sqrt(a + b*acos(c*x))*acos(c*x)), x) + Integral(1/(a*x*sqrt(a + b*acos 
(c*x)) + b*x*sqrt(a + b*acos(c*x))*acos(c*x)), x))
 

Maxima [N/A]

Not integrable

Time = 1.11 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03 \[ \int \frac {\left (d-c^2 d x^2\right )^2}{x (a+b \arccos (c x))^{3/2}} \, dx=\int { \frac {{\left (c^{2} d x^{2} - d\right )}^{2}}{{\left (b \arccos \left (c x\right ) + a\right )}^{\frac {3}{2}} x} \,d x } \] Input:

integrate((-c^2*d*x^2+d)^2/x/(a+b*arccos(c*x))^(3/2),x, algorithm="maxima" 
)
 

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^2}{x (a+b \arccos (c x))^{3/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((-c^2*d*x^2+d)^2/x/(a+b*arccos(c*x))^(3/2),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 [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d-c^2 d x^2\right )^2}{x (a+b \arccos (c x))^{3/2}} \, dx=\int \frac {{\left (d-c^2\,d\,x^2\right )}^2}{x\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 130, normalized size of antiderivative = 4.48 \[ \int \frac {\left (d-c^2 d x^2\right )^2}{x (a+b \arccos (c x))^{3/2}} \, dx=d^{2} \left (\int \frac {\sqrt {\mathit {acos} \left (c x \right ) b +a}}{\mathit {acos} \left (c x \right )^{2} b^{2} x +2 \mathit {acos} \left (c x \right ) a b x +a^{2} x}d x +\left (\int \frac {\sqrt {\mathit {acos} \left (c x \right ) b +a}\, x^{3}}{\mathit {acos} \left (c x \right )^{2} b^{2}+2 \mathit {acos} \left (c x \right ) a b +a^{2}}d x \right ) c^{4}-2 \left (\int \frac {\sqrt {\mathit {acos} \left (c x \right ) b +a}\, x}{\mathit {acos} \left (c x \right )^{2} b^{2}+2 \mathit {acos} \left (c x \right ) a b +a^{2}}d x \right ) c^{2}\right ) \] Input:

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

Output:

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