\(\int \frac {x^3}{\arccos (a x)^{7/2}} \, dx\) [114]

Optimal result

Integrand size = 12, antiderivative size = 190 \[ \int \frac {x^3}{\arccos (a x)^{7/2}} \, dx=\frac {2 x^3 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}-\frac {4 x^2}{5 a^2 \arccos (a x)^{3/2}}+\frac {16 x^4}{15 \arccos (a x)^{3/2}}+\frac {16 x \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\arccos (a x)}}-\frac {128 x^3 \sqrt {1-a^2 x^2}}{15 a \sqrt {\arccos (a x)}}+\frac {32 \sqrt {2 \pi } \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{15 a^4}+\frac {16 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{15 a^4} \]

[Out]

-4/5*x^2/a^2/arccos(a*x)^(3/2)+16/15*x^4/arccos(a*x)^(3/2)+16/15*FresnelC(2*arccos(a*x)^(1/2)/Pi^(1/2))*Pi^(1/
2)/a^4+32/15*FresnelC(2*2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^4+2/5*x^3*(-a^2*x^2+1)^(1/2)/a/
arccos(a*x)^(5/2)+16/5*x*(-a^2*x^2+1)^(1/2)/a^3/arccos(a*x)^(1/2)-128/15*x^3*(-a^2*x^2+1)^(1/2)/a/arccos(a*x)^
(1/2)

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {4730, 4808, 4728, 3385, 3433} \[ \int \frac {x^3}{\arccos (a x)^{7/2}} \, dx=\frac {32 \sqrt {2 \pi } \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{15 a^4}+\frac {16 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{15 a^4}-\frac {4 x^2}{5 a^2 \arccos (a x)^{3/2}}-\frac {128 x^3 \sqrt {1-a^2 x^2}}{15 a \sqrt {\arccos (a x)}}+\frac {2 x^3 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}+\frac {16 x \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\arccos (a x)}}+\frac {16 x^4}{15 \arccos (a x)^{3/2}} \]

[In]

Int[x^3/ArcCos[a*x]^(7/2),x]

[Out]

(2*x^3*Sqrt[1 - a^2*x^2])/(5*a*ArcCos[a*x]^(5/2)) - (4*x^2)/(5*a^2*ArcCos[a*x]^(3/2)) + (16*x^4)/(15*ArcCos[a*
x]^(3/2)) + (16*x*Sqrt[1 - a^2*x^2])/(5*a^3*Sqrt[ArcCos[a*x]]) - (128*x^3*Sqrt[1 - a^2*x^2])/(15*a*Sqrt[ArcCos
[a*x]]) + (32*Sqrt[2*Pi]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(15*a^4) + (16*Sqrt[Pi]*FresnelC[(2*Sqrt[Ar
cCos[a*x]])/Sqrt[Pi]])/(15*a^4)

Rule 3385

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[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 3433

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 4728

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(-x^m)*Sqrt[1 - c^2*x^2]*((a + b*Arc
Cos[c*x])^(n + 1)/(b*c*(n + 1))), x] - Dist[1/(b^2*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[x^(n + 1), C
os[-a/b + x/b]^(m - 1)*(m - (m + 1)*Cos[-a/b + x/b]^2), x], x], x, a + b*ArcCos[c*x]], x] /; FreeQ[{a, b, c},
x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

Rule 4730

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(-x^m)*Sqrt[1 - c^2*x^2]*((a + b*Arc
Cos[c*x])^(n + 1)/(b*c*(n + 1))), x] + (-Dist[c*((m + 1)/(b*(n + 1))), Int[x^(m + 1)*((a + b*ArcCos[c*x])^(n +
1)/Sqrt[1 - c^2*x^2]), x], x] + Dist[m/(b*c*(n + 1)), Int[x^(m - 1)*((a + b*ArcCos[c*x])^(n + 1)/Sqrt[1 - c^2
*x^2]), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]

Rule 4808

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] + Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {2 x^3 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}-\frac {6 \int \frac {x^2}{\sqrt {1-a^2 x^2} \arccos (a x)^{5/2}} \, dx}{5 a}+\frac {1}{5} (8 a) \int \frac {x^4}{\sqrt {1-a^2 x^2} \arccos (a x)^{5/2}} \, dx \\ & = \frac {2 x^3 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}-\frac {4 x^2}{5 a^2 \arccos (a x)^{3/2}}+\frac {16 x^4}{15 \arccos (a x)^{3/2}}-\frac {64}{15} \int \frac {x^3}{\arccos (a x)^{3/2}} \, dx+\frac {8 \int \frac {x}{\arccos (a x)^{3/2}} \, dx}{5 a^2} \\ & = \frac {2 x^3 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}-\frac {4 x^2}{5 a^2 \arccos (a x)^{3/2}}+\frac {16 x^4}{15 \arccos (a x)^{3/2}}+\frac {16 x \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\arccos (a x)}}-\frac {128 x^3 \sqrt {1-a^2 x^2}}{15 a \sqrt {\arccos (a x)}}-\frac {16 \text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{5 a^4}-\frac {128 \text {Subst}\left (\int \left (-\frac {\cos (2 x)}{2 \sqrt {x}}-\frac {\cos (4 x)}{2 \sqrt {x}}\right ) \, dx,x,\arccos (a x)\right )}{15 a^4} \\ & = \frac {2 x^3 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}-\frac {4 x^2}{5 a^2 \arccos (a x)^{3/2}}+\frac {16 x^4}{15 \arccos (a x)^{3/2}}+\frac {16 x \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\arccos (a x)}}-\frac {128 x^3 \sqrt {1-a^2 x^2}}{15 a \sqrt {\arccos (a x)}}+\frac {64 \text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{15 a^4}+\frac {64 \text {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{15 a^4}-\frac {32 \text {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{5 a^4} \\ & = \frac {2 x^3 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}-\frac {4 x^2}{5 a^2 \arccos (a x)^{3/2}}+\frac {16 x^4}{15 \arccos (a x)^{3/2}}+\frac {16 x \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\arccos (a x)}}-\frac {128 x^3 \sqrt {1-a^2 x^2}}{15 a \sqrt {\arccos (a x)}}-\frac {16 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{5 a^4}+\frac {128 \text {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{15 a^4}+\frac {128 \text {Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{15 a^4} \\ & = \frac {2 x^3 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}-\frac {4 x^2}{5 a^2 \arccos (a x)^{3/2}}+\frac {16 x^4}{15 \arccos (a x)^{3/2}}+\frac {16 x \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\arccos (a x)}}-\frac {128 x^3 \sqrt {1-a^2 x^2}}{15 a \sqrt {\arccos (a x)}}+\frac {32 \sqrt {2 \pi } \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{15 a^4}+\frac {16 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{15 a^4} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.09 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.39 \[ \int \frac {x^3}{\arccos (a x)^{7/2}} \, dx=-\frac {-4 e^{-2 i \arccos (a x)} \left (1+e^{4 i \arccos (a x)} (1+4 i \arccos (a x))-4 i \arccos (a x)\right ) \arccos (a x)+\frac {16 \sqrt {2} \arccos (a x)^3 \Gamma \left (\frac {1}{2},-2 i \arccos (a x)\right )}{\sqrt {-i \arccos (a x)}}+16 i \sqrt {2} (i \arccos (a x))^{5/2} \Gamma \left (\frac {1}{2},2 i \arccos (a x)\right )-2 \arccos (a x) \left (2 e^{4 i \arccos (a x)} (1+8 i \arccos (a x))+32 (-i \arccos (a x))^{3/2} \Gamma \left (\frac {1}{2},-4 i \arccos (a x)\right )+2 e^{-4 i \arccos (a x)} \left (1-8 i \arccos (a x)+16 e^{4 i \arccos (a x)} (i \arccos (a x))^{3/2} \Gamma \left (\frac {1}{2},4 i \arccos (a x)\right )\right )\right )-6 \sin (2 \arccos (a x))-3 \sin (4 \arccos (a x))}{60 a^4 \arccos (a x)^{5/2}} \]

[In]

Integrate[x^3/ArcCos[a*x]^(7/2),x]

[Out]

-1/60*((-4*(1 + E^((4*I)*ArcCos[a*x])*(1 + (4*I)*ArcCos[a*x]) - (4*I)*ArcCos[a*x])*ArcCos[a*x])/E^((2*I)*ArcCo
s[a*x]) + (16*Sqrt[2]*ArcCos[a*x]^3*Gamma[1/2, (-2*I)*ArcCos[a*x]])/Sqrt[(-I)*ArcCos[a*x]] + (16*I)*Sqrt[2]*(I
*ArcCos[a*x])^(5/2)*Gamma[1/2, (2*I)*ArcCos[a*x]] - 2*ArcCos[a*x]*(2*E^((4*I)*ArcCos[a*x])*(1 + (8*I)*ArcCos[a
*x]) + 32*((-I)*ArcCos[a*x])^(3/2)*Gamma[1/2, (-4*I)*ArcCos[a*x]] + (2*(1 - (8*I)*ArcCos[a*x] + 16*E^((4*I)*Ar
cCos[a*x])*(I*ArcCos[a*x])^(3/2)*Gamma[1/2, (4*I)*ArcCos[a*x]]))/E^((4*I)*ArcCos[a*x])) - 6*Sin[2*ArcCos[a*x]]
- 3*Sin[4*ArcCos[a*x]])/(a^4*ArcCos[a*x]^(5/2))

Maple [A] (verified)

Time = 0.98 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.73

method	result	size
default	\(-\frac {-128 \sqrt {2}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {5}{2}}-64 \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {5}{2}}+32 \sin \left (2 \arccos \left (a x \right )\right ) \arccos \left (a x \right )^{2}+64 \sin \left (4 \arccos \left (a x \right )\right ) \arccos \left (a x \right )^{2}-8 \arccos \left (a x \right ) \cos \left (2 \arccos \left (a x \right )\right )-8 \arccos \left (a x \right ) \cos \left (4 \arccos \left (a x \right )\right )-6 \sin \left (2 \arccos \left (a x \right )\right )-3 \sin \left (4 \arccos \left (a x \right )\right )}{60 a^{4} \arccos \left (a x \right )^{\frac {5}{2}}}\)	\(139\)

[In]

int(x^3/arccos(a*x)^(7/2),x,method=_RETURNVERBOSE)

[Out]

-1/60/a^4*(-128*2^(1/2)*Pi^(1/2)*FresnelC(2*2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*arccos(a*x)^(5/2)-64*Pi^(1/2)*
FresnelC(2*arccos(a*x)^(1/2)/Pi^(1/2))*arccos(a*x)^(5/2)+32*sin(2*arccos(a*x))*arccos(a*x)^2+64*sin(4*arccos(a
*x))*arccos(a*x)^2-8*arccos(a*x)*cos(2*arccos(a*x))-8*arccos(a*x)*cos(4*arccos(a*x))-6*sin(2*arccos(a*x))-3*si
n(4*arccos(a*x)))/arccos(a*x)^(5/2)

Fricas [F(-2)]

Exception generated. \[ \int \frac {x^3}{\arccos (a x)^{7/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^3/arccos(a*x)^(7/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F]

\[ \int \frac {x^3}{\arccos (a x)^{7/2}} \, dx=\int \frac {x^{3}}{\operatorname {acos}^{\frac {7}{2}}{\left (a x \right )}}\, dx \]

[In]

integrate(x**3/acos(a*x)**(7/2),x)

[Out]

Integral(x**3/acos(a*x)**(7/2), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^3}{\arccos (a x)^{7/2}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(x^3/arccos(a*x)^(7/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [F(-2)]

Exception generated. \[ \int \frac {x^3}{\arccos (a x)^{7/2}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(x^3/arccos(a*x)^(7/2),x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3}{\arccos (a x)^{7/2}} \, dx=\int \frac {x^3}{{\mathrm {acos}\left (a\,x\right )}^{7/2}} \,d x \]

[In]

int(x^3/acos(a*x)^(7/2),x)

[Out]

int(x^3/acos(a*x)^(7/2), x)