3.2.100 \(\int \frac {1}{(a+b \arccos (c x))^{5/2}} \, dx\) [200]

3.2.100.1 Optimal result

Integrand size = 12, antiderivative size = 163 \[ \int \frac {1}{(a+b \arccos (c x))^{5/2}} \, dx=\frac {2 \sqrt {1-c^2 x^2}}{3 b c (a+b \arccos (c x))^{3/2}}+\frac {4 x}{3 b^2 \sqrt {a+b \arccos (c x)}}+\frac {4 \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c}-\frac {4 \sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{3 b^{5/2} c} \]

output

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

3.2.100.2 Mathematica [F]

\[ \int \frac {1}{(a+b \arccos (c x))^{5/2}} \, dx=\int \frac {1}{(a+b \arccos (c x))^{5/2}} \, dx \]

input

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

output

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

3.2.100.3 Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.05, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5133, 5223, 5135, 25, 3042, 3787, 25, 3042, 3785, 3786, 3832, 3833}

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.

\(\displaystyle \int \frac {1}{(a+b \arccos (c x))^{5/2}} \, dx\)

\(\Big \downarrow \) 5133

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

\(\Big \downarrow \) 5223

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

\(\Big \downarrow \) 5135

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3787

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3785

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

\(\Big \downarrow \) 3786

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

\(\Big \downarrow \) 3832

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

\(\Big \downarrow \) 3833

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

input

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

output

(2*Sqrt[1 - c^2*x^2])/(3*b*c*(a + b*ArcCos[c*x])^(3/2)) + (2*c*((2*x)/(b*c 
*Sqrt[a + b*ArcCos[c*x]]) + (2*(Sqrt[b]*Sqrt[2*Pi]*Cos[a/b]*FresnelS[(Sqrt 
[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]] - Sqrt[b]*Sqrt[2*Pi]*FresnelC[(Sq 
rt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[a/b]))/(b^2*c^2)))/(3*b)
 

3.2.100.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3785

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> S 
imp[2/d   Subst[Int[Cos[f*(x^2/d)], x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, 
d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]
 

rule 3786

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[2/d 
   Subst[Int[Sin[f*(x^2/d)], x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f 
}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]
 

rule 3787

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Cos 
[(d*e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] + Simp[Sin[( 
d*e - c*f)/d]   Int[Cos[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] /; FreeQ[{c, d 
, e, f}, x] && ComplexFreeQ[f] && NeQ[d*e - c*f, 0]
 

rule 3832

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ 
d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
 

rule 3833

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ 
d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
 

rule 5133

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-Sqrt[1 - c 
^2*x^2])*((a + b*ArcCos[c*x])^(n + 1)/(b*c*(n + 1))), x] - Simp[c/(b*(n + 1 
))   Int[x*((a + b*ArcCos[c*x])^(n + 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ 
[{a, b, c}, x] && LtQ[n, -1]
 

rule 5135

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[-(b*c)^(-1) 
  Subst[Int[x^n*Sin[-a/b + x/b], x], x, a + b*ArcCos[c*x]], x] /; FreeQ[{a, 
 b, c, n}, x]
 

rule 5223

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

3.2.100.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(340\) vs. \(2(129)=258\).

Time = 2.09 (sec) , antiderivative size = 341, normalized size of antiderivative = 2.09

method	result	size
default	\(\frac {-\frac {4 \arccos \left (c x \right ) \sqrt {a +b \arccos \left (c x \right )}\, \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}{3}-\frac {4 \arccos \left (c x \right ) \sqrt {a +b \arccos \left (c x \right )}\, \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}{3}-\frac {4 \sqrt {a +b \arccos \left (c x \right )}\, \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, a}{3}-\frac {4 \sqrt {a +b \arccos \left (c x \right )}\, \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, a}{3}+\frac {4 \arccos \left (c x \right ) \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) b}{3}-\frac {2 \sin \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) b}{3}+\frac {4 \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a}{3}}{c \,b^{2} \left (a +b \arccos \left (c x \right )\right )^{\frac {3}{2}}}\)	\(341\)

input

int(1/(a+b*arccos(c*x))^(5/2),x,method=_RETURNVERBOSE)
 

output

2/3/c/b^2*(-2*arccos(c*x)*(a+b*arccos(c*x))^(1/2)*cos(a/b)*FresnelS(2^(1/2 
)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)*2^(1/2)*Pi^(1/2)*(-1/b) 
^(1/2)*b-2*arccos(c*x)*(a+b*arccos(c*x))^(1/2)*sin(a/b)*FresnelC(2^(1/2)/P 
i^(1/2)/(-1/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)*2^(1/2)*Pi^(1/2)*(-1/b)^(1 
/2)*b-2*(a+b*arccos(c*x))^(1/2)*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(-1/b)^ 
(1/2)*(a+b*arccos(c*x))^(1/2)/b)*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*a-2*(a+b*ar 
ccos(c*x))^(1/2)*sin(a/b)*FresnelC(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcc 
os(c*x))^(1/2)/b)*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*a+2*arccos(c*x)*cos(-(a+b* 
arccos(c*x))/b+a/b)*b-sin(-(a+b*arccos(c*x))/b+a/b)*b+2*cos(-(a+b*arccos(c 
*x))/b+a/b)*a)/(a+b*arccos(c*x))^(3/2)
 

3.2.100.5 Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{(a+b \arccos (c x))^{5/2}} \, dx=\text {Exception raised: TypeError} \]

input

integrate(1/(a+b*arccos(c*x))^(5/2),x, algorithm="fricas")
 

output

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

3.2.100.6 Sympy [F]

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

input

integrate(1/(a+b*acos(c*x))**(5/2),x)
 

output

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

3.2.100.7 Maxima [F]

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

input

integrate(1/(a+b*arccos(c*x))^(5/2),x, algorithm="maxima")
 

output

integrate((b*arccos(c*x) + a)^(-5/2), x)
 

3.2.100.8 Giac [F]

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

input

integrate(1/(a+b*arccos(c*x))^(5/2),x, algorithm="giac")
 

output

integrate((b*arccos(c*x) + a)^(-5/2), x)
 

3.2.100.9 Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \arccos (c x))^{5/2}} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^{5/2}} \,d x \]

input

int(1/(a + b*acos(c*x))^(5/2),x)
 

output

int(1/(a + b*acos(c*x))^(5/2), x)