Integrand size = 12, antiderivative size = 137 \[ \int \frac {1}{(a+b \arccos (c x))^{3/2}} \, dx=\frac {2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arccos (c x)}}-\frac {2 \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{b^{3/2} c}-\frac {2 \sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{b^{3/2} c} \] Output:
2*(-c^2*x^2+1)^(1/2)/b/c/(a+b*arccos(c*x))^(1/2)-2*2^(1/2)*Pi^(1/2)*cos(a/ b)*FresnelC(2^(1/2)/Pi^(1/2)*(a+b*arccos(c*x))^(1/2)/b^(1/2))/b^(3/2)/c-2* 2^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*arccos(c*x))^(1/2)/b^(1/2) )*sin(a/b)/b^(3/2)/c
Result contains complex when optimal does not.
Time = 0.13 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.09 \[ \int \frac {1}{(a+b \arccos (c x))^{3/2}} \, dx=-\frac {i e^{-\frac {i a}{b}} \left (2 i e^{\frac {i a}{b}} \sqrt {1-c^2 x^2}-\sqrt {-\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {i (a+b \arccos (c x))}{b}\right )+e^{\frac {2 i a}{b}} \sqrt {\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {i (a+b \arccos (c x))}{b}\right )\right )}{b c \sqrt {a+b \arccos (c x)}} \] Input:
Integrate[(a + b*ArcCos[c*x])^(-3/2),x]
Output:
((-I)*((2*I)*E^((I*a)/b)*Sqrt[1 - c^2*x^2] - Sqrt[((-I)*(a + b*ArcCos[c*x] ))/b]*Gamma[1/2, ((-I)*(a + b*ArcCos[c*x]))/b] + E^(((2*I)*a)/b)*Sqrt[(I*( a + b*ArcCos[c*x]))/b]*Gamma[1/2, (I*(a + b*ArcCos[c*x]))/b]))/(b*c*E^((I* a)/b)*Sqrt[a + b*ArcCos[c*x]])
Time = 0.77 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.01, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {5133, 5225, 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.
\(\displaystyle \int \frac {1}{(a+b \arccos (c x))^{3/2}} \, dx\) |
\(\Big \downarrow \) 5133 |
\(\displaystyle \frac {2 c \int \frac {x}{\sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}dx}{b}+\frac {2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arccos (c x)}}\) |
\(\Big \downarrow \) 5225 |
\(\displaystyle \frac {2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arccos (c x)}}-\frac {2 \int \frac {\cos \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{b^2 c}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arccos (c x)}}-\frac {2 \int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{b^2 c}\) |
\(\Big \downarrow \) 3787 |
\(\displaystyle \frac {2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arccos (c x)}}-\frac {2 \left (\cos \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))-\sin \left (\frac {a}{b}\right ) \int -\frac {\sin \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))\right )}{b^2 c}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arccos (c x)}}-\frac {2 \left (\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))+\cos \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))\right )}{b^2 c}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arccos (c x)}}-\frac {2 \left (\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))+\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arccos (c x)}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))\right )}{b^2 c}\) |
\(\Big \downarrow \) 3785 |
\(\displaystyle \frac {2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arccos (c x)}}-\frac {2 \left (\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))+2 \cos \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arccos (c x)}{b}\right )d\sqrt {a+b \arccos (c x)}\right )}{b^2 c}\) |
\(\Big \downarrow \) 3786 |
\(\displaystyle \frac {2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arccos (c x)}}-\frac {2 \left (2 \sin \left (\frac {a}{b}\right ) \int \sin \left (\frac {a+b \arccos (c x)}{b}\right )d\sqrt {a+b \arccos (c x)}+2 \cos \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arccos (c x)}{b}\right )d\sqrt {a+b \arccos (c x)}\right )}{b^2 c}\) |
\(\Big \downarrow \) 3832 |
\(\displaystyle \frac {2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arccos (c x)}}-\frac {2 \left (2 \cos \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arccos (c x)}{b}\right )d\sqrt {a+b \arccos (c x)}+\sqrt {2 \pi } \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )\right )}{b^2 c}\) |
\(\Big \downarrow \) 3833 |
\(\displaystyle \frac {2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arccos (c x)}}-\frac {2 \left (\sqrt {2 \pi } \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\sqrt {2 \pi } \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )\right )}{b^2 c}\) |
Input:
Int[(a + b*ArcCos[c*x])^(-3/2),x]
Output:
(2*Sqrt[1 - c^2*x^2])/(b*c*Sqrt[a + b*ArcCos[c*x]]) - (2*(Sqrt[b]*Sqrt[2*P i]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]] + Sqrt[ b]*Sqrt[2*Pi]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[a /b]))/(b^2*c)
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]
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]
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]
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]
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]
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]
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]
Time = 0.10 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.15
method | result | size |
default | \(-\frac {2 \left (\sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}\, \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right )-\sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}\, \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right )+\sin \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right )\right )}{c b \sqrt {a +b \arccos \left (c x \right )}}\) | \(157\) |
Input:
int(1/(a+b*arccos(c*x))^(3/2),x,method=_RETURNVERBOSE)
Output:
-2/c/b*((-1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arccos(c*x))^(1/2)*cos(a/b)*Fre snelC(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)-(-1/b)^(1/2 )*Pi^(1/2)*2^(1/2)*(a+b*arccos(c*x))^(1/2)*sin(a/b)*FresnelS(2^(1/2)/Pi^(1 /2)/(-1/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)+sin(-(a+b*arccos(c*x))/b+a/b)) /(a+b*arccos(c*x))^(1/2)
Exception generated. \[ \int \frac {1}{(a+b \arccos (c x))^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/(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)
\[ \int \frac {1}{(a+b \arccos (c x))^{3/2}} \, dx=\int \frac {1}{\left (a + b \operatorname {acos}{\left (c x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/(a+b*acos(c*x))**(3/2),x)
Output:
Integral((a + b*acos(c*x))**(-3/2), x)
\[ \int \frac {1}{(a+b \arccos (c x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \arccos \left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(a+b*arccos(c*x))^(3/2),x, algorithm="maxima")
Output:
integrate((b*arccos(c*x) + a)^(-3/2), x)
\[ \int \frac {1}{(a+b \arccos (c x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \arccos \left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(a+b*arccos(c*x))^(3/2),x, algorithm="giac")
Output:
integrate((b*arccos(c*x) + a)^(-3/2), x)
Timed out. \[ \int \frac {1}{(a+b \arccos (c x))^{3/2}} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^{3/2}} \,d x \] Input:
int(1/(a + b*acos(c*x))^(3/2),x)
Output:
int(1/(a + b*acos(c*x))^(3/2), x)
\[ \int \frac {1}{(a+b \arccos (c x))^{3/2}} \, dx=\frac {-2 \mathit {acos} \left (c x \right ) \left (\int \frac {\sqrt {\mathit {acos} \left (c x \right ) b +a}\, \sqrt {-c^{2} x^{2}+1}\, x}{\mathit {acos} \left (c x \right ) b \,c^{2} x^{2}-\mathit {acos} \left (c x \right ) b +a \,c^{2} x^{2}-a}d x \right ) b \,c^{2}+2 \sqrt {\mathit {acos} \left (c x \right ) b +a}\, \sqrt {-c^{2} x^{2}+1}-2 \left (\int \frac {\sqrt {\mathit {acos} \left (c x \right ) b +a}\, \sqrt {-c^{2} x^{2}+1}\, x}{\mathit {acos} \left (c x \right ) b \,c^{2} x^{2}-\mathit {acos} \left (c x \right ) b +a \,c^{2} x^{2}-a}d x \right ) a \,c^{2}}{b c \left (\mathit {acos} \left (c x \right ) b +a \right )} \] Input:
int(1/(a+b*acos(c*x))^(3/2),x)
Output:
(2*( - acos(c*x)*int((sqrt(acos(c*x)*b + a)*sqrt( - c**2*x**2 + 1)*x)/(aco s(c*x)*b*c**2*x**2 - acos(c*x)*b + a*c**2*x**2 - a),x)*b*c**2 + sqrt(acos( c*x)*b + a)*sqrt( - c**2*x**2 + 1) - int((sqrt(acos(c*x)*b + a)*sqrt( - c* *2*x**2 + 1)*x)/(acos(c*x)*b*c**2*x**2 - acos(c*x)*b + a*c**2*x**2 - a),x) *a*c**2))/(b*c*(acos(c*x)*b + a))