Integrand size = 10, antiderivative size = 111 \[ \int \arccos (a+b x)^{5/2} \, dx=-\frac {15 (a+b x) \sqrt {\arccos (a+b x)}}{4 b}-\frac {5 \sqrt {1-(a+b x)^2} \arccos (a+b x)^{3/2}}{2 b}+\frac {(a+b x) \arccos (a+b x)^{5/2}}{b}+\frac {15 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a+b x)}\right )}{4 b} \]
(b*x+a)*arccos(b*x+a)^(5/2)/b+15/8*FresnelC(2^(1/2)/Pi^(1/2)*arccos(b*x+a) ^(1/2))*2^(1/2)*Pi^(1/2)/b-5/2*arccos(b*x+a)^(3/2)*(1-(b*x+a)^2)^(1/2)/b-1 5/4*(b*x+a)*arccos(b*x+a)^(1/2)/b
Result contains complex when optimal does not.
Time = 0.04 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.71 \[ \int \arccos (a+b x)^{5/2} \, dx=-\frac {i \left (\sqrt {-i \arccos (a+b x)} \Gamma \left (\frac {7}{2},-i \arccos (a+b x)\right )-\sqrt {i \arccos (a+b x)} \Gamma \left (\frac {7}{2},i \arccos (a+b x)\right )\right )}{2 b \sqrt {\arccos (a+b x)}} \]
((-1/2*I)*(Sqrt[(-I)*ArcCos[a + b*x]]*Gamma[7/2, (-I)*ArcCos[a + b*x]] - S qrt[I*ArcCos[a + b*x]]*Gamma[7/2, I*ArcCos[a + b*x]]))/(b*Sqrt[ArcCos[a + b*x]])
Time = 0.53 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.95, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5303, 5131, 5183, 5131, 5225, 3042, 3785, 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.
\(\displaystyle \int \arccos (a+b x)^{5/2} \, dx\) |
\(\Big \downarrow \) 5303 |
\(\displaystyle \frac {\int \arccos (a+b x)^{5/2}d(a+b x)}{b}\) |
\(\Big \downarrow \) 5131 |
\(\displaystyle \frac {\frac {5}{2} \int \frac {(a+b x) \arccos (a+b x)^{3/2}}{\sqrt {1-(a+b x)^2}}d(a+b x)+(a+b x) \arccos (a+b x)^{5/2}}{b}\) |
\(\Big \downarrow \) 5183 |
\(\displaystyle \frac {\frac {5}{2} \left (-\frac {3}{2} \int \sqrt {\arccos (a+b x)}d(a+b x)-\sqrt {1-(a+b x)^2} \arccos (a+b x)^{3/2}\right )+(a+b x) \arccos (a+b x)^{5/2}}{b}\) |
\(\Big \downarrow \) 5131 |
\(\displaystyle \frac {\frac {5}{2} \left (-\frac {3}{2} \left (\frac {1}{2} \int \frac {a+b x}{\sqrt {1-(a+b x)^2} \sqrt {\arccos (a+b x)}}d(a+b x)+(a+b x) \sqrt {\arccos (a+b x)}\right )-\sqrt {1-(a+b x)^2} \arccos (a+b x)^{3/2}\right )+(a+b x) \arccos (a+b x)^{5/2}}{b}\) |
\(\Big \downarrow \) 5225 |
\(\displaystyle \frac {\frac {5}{2} \left (-\frac {3}{2} \left ((a+b x) \sqrt {\arccos (a+b x)}-\frac {1}{2} \int \frac {a+b x}{\sqrt {\arccos (a+b x)}}d\arccos (a+b x)\right )-\sqrt {1-(a+b x)^2} \arccos (a+b x)^{3/2}\right )+(a+b x) \arccos (a+b x)^{5/2}}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {5}{2} \left (-\frac {3}{2} \left ((a+b x) \sqrt {\arccos (a+b x)}-\frac {1}{2} \int \frac {\sin \left (\arccos (a+b x)+\frac {\pi }{2}\right )}{\sqrt {\arccos (a+b x)}}d\arccos (a+b x)\right )-\sqrt {1-(a+b x)^2} \arccos (a+b x)^{3/2}\right )+(a+b x) \arccos (a+b x)^{5/2}}{b}\) |
\(\Big \downarrow \) 3785 |
\(\displaystyle \frac {\frac {5}{2} \left (-\frac {3}{2} \left ((a+b x) \sqrt {\arccos (a+b x)}-\int (a+b x)d\sqrt {\arccos (a+b x)}\right )-\sqrt {1-(a+b x)^2} \arccos (a+b x)^{3/2}\right )+(a+b x) \arccos (a+b x)^{5/2}}{b}\) |
\(\Big \downarrow \) 3833 |
\(\displaystyle \frac {\frac {5}{2} \left (-\frac {3}{2} \left ((a+b x) \sqrt {\arccos (a+b x)}-\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a+b x)}\right )\right )-\sqrt {1-(a+b x)^2} \arccos (a+b x)^{3/2}\right )+(a+b x) \arccos (a+b x)^{5/2}}{b}\) |
((a + b*x)*ArcCos[a + b*x]^(5/2) + (5*(-(Sqrt[1 - (a + b*x)^2]*ArcCos[a + b*x]^(3/2)) - (3*((a + b*x)*Sqrt[ArcCos[a + b*x]] - Sqrt[Pi/2]*FresnelC[Sq rt[2/Pi]*Sqrt[ArcCos[a + b*x]]]))/2))/2)/b
3.1.37.3.1 Defintions of rubi rules used
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[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[x*(a + b*Ar cCos[c*x])^n, x] + Simp[b*c*n Int[x*((a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ .), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] I nt[(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] && GtQ[n, 0] && NeQ[p, -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[((a_.) + ArcCos[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[1/d Subst[Int[(a + b*ArcCos[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, n}, x]
Time = 0.95 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.26
method | result | size |
default | \(-\frac {\sqrt {2}\, \left (-4 \arccos \left (b x +a \right )^{\frac {5}{2}} \sqrt {2}\, \sqrt {\pi }\, b x -4 \arccos \left (b x +a \right )^{\frac {5}{2}} \sqrt {2}\, \sqrt {\pi }\, a +10 \arccos \left (b x +a \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}+15 \sqrt {2}\, \sqrt {\arccos \left (b x +a \right )}\, \sqrt {\pi }\, b x +15 \sqrt {2}\, \sqrt {\arccos \left (b x +a \right )}\, \sqrt {\pi }\, a -15 \pi \,\operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (b x +a \right )}}{\sqrt {\pi }}\right )\right )}{8 b \sqrt {\pi }}\) | \(140\) |
-1/8/b*2^(1/2)*(-4*arccos(b*x+a)^(5/2)*2^(1/2)*Pi^(1/2)*b*x-4*arccos(b*x+a )^(5/2)*2^(1/2)*Pi^(1/2)*a+10*arccos(b*x+a)^(3/2)*2^(1/2)*Pi^(1/2)*(-b^2*x ^2-2*a*b*x-a^2+1)^(1/2)+15*2^(1/2)*arccos(b*x+a)^(1/2)*Pi^(1/2)*b*x+15*2^( 1/2)*arccos(b*x+a)^(1/2)*Pi^(1/2)*a-15*Pi*FresnelC(2^(1/2)/Pi^(1/2)*arccos (b*x+a)^(1/2)))/Pi^(1/2)
Exception generated. \[ \int \arccos (a+b x)^{5/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \arccos (a+b x)^{5/2} \, dx=\int \operatorname {acos}^{\frac {5}{2}}{\left (a + b x \right )}\, dx \]
Exception generated. \[ \int \arccos (a+b x)^{5/2} \, dx=\text {Exception raised: RuntimeError} \]
Result contains complex when optimal does not.
Time = 0.41 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.65 \[ \int \arccos (a+b x)^{5/2} \, dx=\frac {\arccos \left (b x + a\right )^{\frac {5}{2}} e^{\left (i \, \arccos \left (b x + a\right )\right )}}{2 \, b} + \frac {\arccos \left (b x + a\right )^{\frac {5}{2}} e^{\left (-i \, \arccos \left (b x + a\right )\right )}}{2 \, b} + \frac {5 i \, \arccos \left (b x + a\right )^{\frac {3}{2}} e^{\left (i \, \arccos \left (b x + a\right )\right )}}{4 \, b} - \frac {5 i \, \arccos \left (b x + a\right )^{\frac {3}{2}} e^{\left (-i \, \arccos \left (b x + a\right )\right )}}{4 \, b} - \frac {\left (15 i + 15\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arccos \left (b x + a\right )}\right )}{32 \, b} + \frac {\left (15 i - 15\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arccos \left (b x + a\right )}\right )}{32 \, b} - \frac {15 \, \sqrt {\arccos \left (b x + a\right )} e^{\left (i \, \arccos \left (b x + a\right )\right )}}{8 \, b} - \frac {15 \, \sqrt {\arccos \left (b x + a\right )} e^{\left (-i \, \arccos \left (b x + a\right )\right )}}{8 \, b} \]
1/2*arccos(b*x + a)^(5/2)*e^(I*arccos(b*x + a))/b + 1/2*arccos(b*x + a)^(5 /2)*e^(-I*arccos(b*x + a))/b + 5/4*I*arccos(b*x + a)^(3/2)*e^(I*arccos(b*x + a))/b - 5/4*I*arccos(b*x + a)^(3/2)*e^(-I*arccos(b*x + a))/b - (15/32*I + 15/32)*sqrt(2)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(2)*sqrt(arccos(b*x + a)) )/b + (15/32*I - 15/32)*sqrt(2)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(2)*sqrt(a rccos(b*x + a)))/b - 15/8*sqrt(arccos(b*x + a))*e^(I*arccos(b*x + a))/b - 15/8*sqrt(arccos(b*x + a))*e^(-I*arccos(b*x + a))/b
Timed out. \[ \int \arccos (a+b x)^{5/2} \, dx=\int {\mathrm {acos}\left (a+b\,x\right )}^{5/2} \,d x \]