Integrand size = 24, antiderivative size = 237 \[ \int \frac {\left (c-a^2 c x^2\right )^{5/2}}{\arccos (a x)^{3/2}} \, dx=-\frac {2 \sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^{5/2}}{a \sqrt {\arccos (a x)}}-\frac {3 c^2 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{2 a \sqrt {1-a^2 x^2}}-\frac {c^2 \sqrt {3 \pi } \sqrt {c-a^2 c x^2} \operatorname {FresnelS}\left (2 \sqrt {\frac {3}{\pi }} \sqrt {\arccos (a x)}\right )}{8 a \sqrt {1-a^2 x^2}}-\frac {15 c^2 \sqrt {\pi } \sqrt {c-a^2 c x^2} \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{8 a \sqrt {1-a^2 x^2}} \] Output:
-2*(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^(5/2)/a/arccos(a*x)^(1/2)-3/4*c^2*2^( 1/2)*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)*FresnelS(2*2^(1/2)/Pi^(1/2)*arccos(a*x) ^(1/2))/a/(-a^2*x^2+1)^(1/2)-1/8*c^2*3^(1/2)*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2) *FresnelS(2*3^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))/a/(-a^2*x^2+1)^(1/2)-15/8* c^2*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)*FresnelS(2*arccos(a*x)^(1/2)/Pi^(1/2))/a /(-a^2*x^2+1)^(1/2)
Result contains complex when optimal does not.
Time = 0.79 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.70 \[ \int \frac {\left (c-a^2 c x^2\right )^{5/2}}{\arccos (a x)^{3/2}} \, dx=-\frac {c^2 e^{-6 i \arccos (a x)} \sqrt {c-a^2 c x^2} \left (1-6 e^{2 i \arccos (a x)}+15 e^{4 i \arccos (a x)}-20 e^{6 i \arccos (a x)}+15 e^{8 i \arccos (a x)}-6 e^{10 i \arccos (a x)}+e^{12 i \arccos (a x)}+64 e^{6 i \arccos (a x)} \sqrt {\pi } \sqrt {\arccos (a x)} \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )+\sqrt {2} e^{6 i \arccos (a x)} \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-2 i \arccos (a x)\right )+\sqrt {2} e^{6 i \arccos (a x)} \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},2 i \arccos (a x)\right )+12 e^{6 i \arccos (a x)} \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-4 i \arccos (a x)\right )+12 e^{6 i \arccos (a x)} \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},4 i \arccos (a x)\right )-\sqrt {6} e^{6 i \arccos (a x)} \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-6 i \arccos (a x)\right )-\sqrt {6} e^{6 i \arccos (a x)} \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},6 i \arccos (a x)\right )\right )}{32 a \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}} \] Input:
Integrate[(c - a^2*c*x^2)^(5/2)/ArcCos[a*x]^(3/2),x]
Output:
-1/32*(c^2*Sqrt[c - a^2*c*x^2]*(1 - 6*E^((2*I)*ArcCos[a*x]) + 15*E^((4*I)* ArcCos[a*x]) - 20*E^((6*I)*ArcCos[a*x]) + 15*E^((8*I)*ArcCos[a*x]) - 6*E^( (10*I)*ArcCos[a*x]) + E^((12*I)*ArcCos[a*x]) + 64*E^((6*I)*ArcCos[a*x])*Sq rt[Pi]*Sqrt[ArcCos[a*x]]*FresnelS[(2*Sqrt[ArcCos[a*x]])/Sqrt[Pi]] + Sqrt[2 ]*E^((6*I)*ArcCos[a*x])*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-2*I)*ArcCos[a* x]] + Sqrt[2]*E^((6*I)*ArcCos[a*x])*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, (2*I)*A rcCos[a*x]] + 12*E^((6*I)*ArcCos[a*x])*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, ( -4*I)*ArcCos[a*x]] + 12*E^((6*I)*ArcCos[a*x])*Sqrt[I*ArcCos[a*x]]*Gamma[1/ 2, (4*I)*ArcCos[a*x]] - Sqrt[6]*E^((6*I)*ArcCos[a*x])*Sqrt[(-I)*ArcCos[a*x ]]*Gamma[1/2, (-6*I)*ArcCos[a*x]] - Sqrt[6]*E^((6*I)*ArcCos[a*x])*Sqrt[I*A rcCos[a*x]]*Gamma[1/2, (6*I)*ArcCos[a*x]]))/(a*E^((6*I)*ArcCos[a*x])*Sqrt[ 1 - a^2*x^2]*Sqrt[ArcCos[a*x]])
Time = 0.53 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.73, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5167, 5225, 4906, 2009}
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.
\(\displaystyle \int \frac {\left (c-a^2 c x^2\right )^{5/2}}{\arccos (a x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 5167 |
\(\displaystyle \frac {12 a c^2 \sqrt {c-a^2 c x^2} \int \frac {x \left (1-a^2 x^2\right )^2}{\sqrt {\arccos (a x)}}dx}{\sqrt {1-a^2 x^2}}+\frac {2 \sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^{5/2}}{a \sqrt {\arccos (a x)}}\) |
\(\Big \downarrow \) 5225 |
\(\displaystyle \frac {2 \sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^{5/2}}{a \sqrt {\arccos (a x)}}-\frac {12 c^2 \sqrt {c-a^2 c x^2} \int \frac {a x \left (1-a^2 x^2\right )^{5/2}}{\sqrt {\arccos (a x)}}d\arccos (a x)}{a \sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 4906 |
\(\displaystyle \frac {2 \sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^{5/2}}{a \sqrt {\arccos (a x)}}-\frac {12 c^2 \sqrt {c-a^2 c x^2} \int \left (\frac {5 \sin (2 \arccos (a x))}{32 \sqrt {\arccos (a x)}}-\frac {\sin (4 \arccos (a x))}{8 \sqrt {\arccos (a x)}}+\frac {\sin (6 \arccos (a x))}{32 \sqrt {\arccos (a x)}}\right )d\arccos (a x)}{a \sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 \sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^{5/2}}{a \sqrt {\arccos (a x)}}-\frac {12 c^2 \sqrt {c-a^2 c x^2} \left (-\frac {1}{8} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{32} \sqrt {\frac {\pi }{3}} \operatorname {FresnelS}\left (2 \sqrt {\frac {3}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {5}{32} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )\right )}{a \sqrt {1-a^2 x^2}}\) |
Input:
Int[(c - a^2*c*x^2)^(5/2)/ArcCos[a*x]^(3/2),x]
Output:
(2*Sqrt[1 - a^2*x^2]*(c - a^2*c*x^2)^(5/2))/(a*Sqrt[ArcCos[a*x]]) - (12*c^ 2*Sqrt[c - a^2*c*x^2]*(-1/8*(Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcCos[ a*x]]]) + (Sqrt[Pi/3]*FresnelS[2*Sqrt[3/Pi]*Sqrt[ArcCos[a*x]]])/32 + (5*Sq rt[Pi]*FresnelS[(2*Sqrt[ArcCos[a*x]])/Sqrt[Pi]])/32))/(a*Sqrt[1 - a^2*x^2] )
Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x ]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG tQ[p, 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_)^2)^(p_.), x_ Symbol] :> Simp[(-Sqrt[1 - c^2*x^2])*(d + e*x^2)^p*((a + b*ArcCos[c*x])^(n + 1)/(b*c*(n + 1))), x] - Simp[c*((2*p + 1)/(b*(n + 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, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, - 1]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 2)^(p_.), x_Symbol] :> Simp[(-(b*c^(m + 1))^(-1))*Simp[(d + e*x^2)^p/(1 - c ^2*x^2)^p] Subst[Int[x^n*Cos[-a/b + x/b]^m*Sin[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e , 0] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
\[\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{\arccos \left (a x \right )^{\frac {3}{2}}}d x\]
Input:
int((-a^2*c*x^2+c)^(5/2)/arccos(a*x)^(3/2),x)
Output:
int((-a^2*c*x^2+c)^(5/2)/arccos(a*x)^(3/2),x)
Exception generated. \[ \int \frac {\left (c-a^2 c x^2\right )^{5/2}}{\arccos (a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-a^2*c*x^2+c)^(5/2)/arccos(a*x)^(3/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
Timed out. \[ \int \frac {\left (c-a^2 c x^2\right )^{5/2}}{\arccos (a x)^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((-a**2*c*x**2+c)**(5/2)/acos(a*x)**(3/2),x)
Output:
Timed out
Exception generated. \[ \int \frac {\left (c-a^2 c x^2\right )^{5/2}}{\arccos (a x)^{3/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((-a^2*c*x^2+c)^(5/2)/arccos(a*x)^(3/2),x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
\[ \int \frac {\left (c-a^2 c x^2\right )^{5/2}}{\arccos (a x)^{3/2}} \, dx=\int { \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}{\arccos \left (a x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((-a^2*c*x^2+c)^(5/2)/arccos(a*x)^(3/2),x, algorithm="giac")
Output:
integrate((-a^2*c*x^2 + c)^(5/2)/arccos(a*x)^(3/2), x)
Timed out. \[ \int \frac {\left (c-a^2 c x^2\right )^{5/2}}{\arccos (a x)^{3/2}} \, dx=\int \frac {{\left (c-a^2\,c\,x^2\right )}^{5/2}}{{\mathrm {acos}\left (a\,x\right )}^{3/2}} \,d x \] Input:
int((c - a^2*c*x^2)^(5/2)/acos(a*x)^(3/2),x)
Output:
int((c - a^2*c*x^2)^(5/2)/acos(a*x)^(3/2), x)
\[ \int \frac {\left (c-a^2 c x^2\right )^{5/2}}{\arccos (a x)^{3/2}} \, dx=\sqrt {c}\, c^{2} \left (\left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acos} \left (a x \right )}\, x^{4}}{\mathit {acos} \left (a x \right )^{2}}d x \right ) a^{4}-2 \left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acos} \left (a x \right )}\, x^{2}}{\mathit {acos} \left (a x \right )^{2}}d x \right ) a^{2}+\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acos} \left (a x \right )}}{\mathit {acos} \left (a x \right )^{2}}d x \right ) \] Input:
int((-a^2*c*x^2+c)^(5/2)/acos(a*x)^(3/2),x)
Output:
sqrt(c)*c**2*(int((sqrt( - a**2*x**2 + 1)*sqrt(acos(a*x))*x**4)/acos(a*x)* *2,x)*a**4 - 2*int((sqrt( - a**2*x**2 + 1)*sqrt(acos(a*x))*x**2)/acos(a*x) **2,x)*a**2 + int((sqrt( - a**2*x**2 + 1)*sqrt(acos(a*x)))/acos(a*x)**2,x) )