\(\int \frac {x^4}{\arccos (a x)^{7/2}} \, dx\) [113]

Optimal result

Integrand size = 12, antiderivative size = 264 \[ \int \frac {x^4}{\arccos (a x)^{7/2}} \, dx=\frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}-\frac {16 x^3}{15 a^2 \arccos (a x)^{3/2}}+\frac {4 x^5}{3 \arccos (a x)^{3/2}}+\frac {32 x^2 \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\arccos (a x)}}-\frac {40 x^4 \sqrt {1-a^2 x^2}}{3 a \sqrt {\arccos (a x)}}+\frac {\sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{15 a^5}+\frac {5 \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{a^5}-\frac {8 \sqrt {6 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{5 a^5}+\frac {5 \sqrt {\frac {5 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )}{3 a^5} \]

[Out]

-16/15*x^3/a^2/arccos(a*x)^(3/2)+4/3*x^5/arccos(a*x)^(3/2)+9/10*FresnelC(6^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*6
^(1/2)*Pi^(1/2)/a^5+1/15*FresnelC(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^5+5/6*FresnelC(10^(1/
2)/Pi^(1/2)*arccos(a*x)^(1/2))*10^(1/2)*Pi^(1/2)/a^5+2/5*x^4*(-a^2*x^2+1)^(1/2)/a/arccos(a*x)^(5/2)+32/5*x^2*(
-a^2*x^2+1)^(1/2)/a^3/arccos(a*x)^(1/2)-40/3*x^4*(-a^2*x^2+1)^(1/2)/a/arccos(a*x)^(1/2)

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {4730, 4808, 4728, 3385, 3433} \[ \int \frac {x^4}{\arccos (a x)^{7/2}} \, dx=\frac {\sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{15 a^5}-\frac {8 \sqrt {6 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{5 a^5}+\frac {5 \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{a^5}+\frac {5 \sqrt {\frac {5 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )}{3 a^5}-\frac {16 x^3}{15 a^2 \arccos (a x)^{3/2}}-\frac {40 x^4 \sqrt {1-a^2 x^2}}{3 a \sqrt {\arccos (a x)}}+\frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}+\frac {32 x^2 \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\arccos (a x)}}+\frac {4 x^5}{3 \arccos (a x)^{3/2}} \]

[In]

Int[x^4/ArcCos[a*x]^(7/2),x]

[Out]

(2*x^4*Sqrt[1 - a^2*x^2])/(5*a*ArcCos[a*x]^(5/2)) - (16*x^3)/(15*a^2*ArcCos[a*x]^(3/2)) + (4*x^5)/(3*ArcCos[a*
x]^(3/2)) + (32*x^2*Sqrt[1 - a^2*x^2])/(5*a^3*Sqrt[ArcCos[a*x]]) - (40*x^4*Sqrt[1 - a^2*x^2])/(3*a*Sqrt[ArcCos
[a*x]]) + (Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(15*a^5) + (5*Sqrt[(3*Pi)/2]*FresnelC[Sqrt[6/Pi]
*Sqrt[ArcCos[a*x]]])/a^5 - (8*Sqrt[6*Pi]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/(5*a^5) + (5*Sqrt[(5*Pi)/2]*F
resnelC[Sqrt[10/Pi]*Sqrt[ArcCos[a*x]]])/(3*a^5)

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^4 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}-\frac {8 \int \frac {x^3}{\sqrt {1-a^2 x^2} \arccos (a x)^{5/2}} \, dx}{5 a}+(2 a) \int \frac {x^5}{\sqrt {1-a^2 x^2} \arccos (a x)^{5/2}} \, dx \\ & = \frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}-\frac {16 x^3}{15 a^2 \arccos (a x)^{3/2}}+\frac {4 x^5}{3 \arccos (a x)^{3/2}}-\frac {20}{3} \int \frac {x^4}{\arccos (a x)^{3/2}} \, dx+\frac {16 \int \frac {x^2}{\arccos (a x)^{3/2}} \, dx}{5 a^2} \\ & = \frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}-\frac {16 x^3}{15 a^2 \arccos (a x)^{3/2}}+\frac {4 x^5}{3 \arccos (a x)^{3/2}}+\frac {32 x^2 \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\arccos (a x)}}-\frac {40 x^4 \sqrt {1-a^2 x^2}}{3 a \sqrt {\arccos (a x)}}+\frac {32 \text {Subst}\left (\int \left (-\frac {\cos (x)}{4 \sqrt {x}}-\frac {3 \cos (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\arccos (a x)\right )}{5 a^5}-\frac {40 \text {Subst}\left (\int \left (-\frac {\cos (x)}{8 \sqrt {x}}-\frac {9 \cos (3 x)}{16 \sqrt {x}}-\frac {5 \cos (5 x)}{16 \sqrt {x}}\right ) \, dx,x,\arccos (a x)\right )}{3 a^5} \\ & = \frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}-\frac {16 x^3}{15 a^2 \arccos (a x)^{3/2}}+\frac {4 x^5}{3 \arccos (a x)^{3/2}}+\frac {32 x^2 \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\arccos (a x)}}-\frac {40 x^4 \sqrt {1-a^2 x^2}}{3 a \sqrt {\arccos (a x)}}-\frac {8 \text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{5 a^5}+\frac {5 \text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{3 a^5}+\frac {25 \text {Subst}\left (\int \frac {\cos (5 x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{6 a^5}-\frac {24 \text {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{5 a^5}+\frac {15 \text {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{2 a^5} \\ & = \frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}-\frac {16 x^3}{15 a^2 \arccos (a x)^{3/2}}+\frac {4 x^5}{3 \arccos (a x)^{3/2}}+\frac {32 x^2 \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\arccos (a x)}}-\frac {40 x^4 \sqrt {1-a^2 x^2}}{3 a \sqrt {\arccos (a x)}}-\frac {16 \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{5 a^5}+\frac {10 \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{3 a^5}+\frac {25 \text {Subst}\left (\int \cos \left (5 x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{3 a^5}-\frac {48 \text {Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{5 a^5}+\frac {15 \text {Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{a^5} \\ & = \frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}-\frac {16 x^3}{15 a^2 \arccos (a x)^{3/2}}+\frac {4 x^5}{3 \arccos (a x)^{3/2}}+\frac {32 x^2 \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\arccos (a x)}}-\frac {40 x^4 \sqrt {1-a^2 x^2}}{3 a \sqrt {\arccos (a x)}}+\frac {\sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{15 a^5}+\frac {5 \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{a^5}-\frac {8 \sqrt {6 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{5 a^5}+\frac {5 \sqrt {\frac {5 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )}{3 a^5} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 7.37 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.58 \[ \int \frac {x^4}{\arccos (a x)^{7/2}} \, dx=-\frac {2 \left (-6 \sqrt {1-a^2 x^2}-2 i e^{i \arccos (a x)} \arccos (a x) (-i+2 \arccos (a x))-4 (-i \arccos (a x))^{3/2} \arccos (a x) \Gamma \left (\frac {1}{2},-i \arccos (a x)\right )+e^{-i \arccos (a x)} \arccos (a x) \left (-2+4 i \arccos (a x)-4 e^{i \arccos (a x)} (i \arccos (a x))^{3/2} \Gamma \left (\frac {1}{2},i \arccos (a x)\right )\right )\right )-5 \arccos (a x) \left (2 e^{5 i \arccos (a x)} (1+10 i \arccos (a x))+20 \sqrt {5} (-i \arccos (a x))^{3/2} \Gamma \left (\frac {1}{2},-5 i \arccos (a x)\right )+e^{-5 i \arccos (a x)} \left (2-20 i \arccos (a x)+20 \sqrt {5} e^{5 i \arccos (a x)} (i \arccos (a x))^{3/2} \Gamma \left (\frac {1}{2},5 i \arccos (a x)\right )\right )\right )+9 \left (-2 \arccos (a x) \left (e^{3 i \arccos (a x)} (1+6 i \arccos (a x))+6 \sqrt {3} (-i \arccos (a x))^{3/2} \Gamma \left (\frac {1}{2},-3 i \arccos (a x)\right )+e^{-3 i \arccos (a x)} \left (1-6 i \arccos (a x)+6 \sqrt {3} e^{3 i \arccos (a x)} (i \arccos (a x))^{3/2} \Gamma \left (\frac {1}{2},3 i \arccos (a x)\right )\right )\right )-2 \sin (3 \arccos (a x))\right )-6 \sin (5 \arccos (a x))}{240 a^5 \arccos (a x)^{5/2}} \]

[In]

Integrate[x^4/ArcCos[a*x]^(7/2),x]

[Out]

-1/240*(2*(-6*Sqrt[1 - a^2*x^2] - (2*I)*E^(I*ArcCos[a*x])*ArcCos[a*x]*(-I + 2*ArcCos[a*x]) - 4*((-I)*ArcCos[a*
x])^(3/2)*ArcCos[a*x]*Gamma[1/2, (-I)*ArcCos[a*x]] + (ArcCos[a*x]*(-2 + (4*I)*ArcCos[a*x] - 4*E^(I*ArcCos[a*x]
)*(I*ArcCos[a*x])^(3/2)*Gamma[1/2, I*ArcCos[a*x]]))/E^(I*ArcCos[a*x])) - 5*ArcCos[a*x]*(2*E^((5*I)*ArcCos[a*x]
)*(1 + (10*I)*ArcCos[a*x]) + 20*Sqrt[5]*((-I)*ArcCos[a*x])^(3/2)*Gamma[1/2, (-5*I)*ArcCos[a*x]] + (2 - (20*I)*
ArcCos[a*x] + 20*Sqrt[5]*E^((5*I)*ArcCos[a*x])*(I*ArcCos[a*x])^(3/2)*Gamma[1/2, (5*I)*ArcCos[a*x]])/E^((5*I)*A
rcCos[a*x])) + 9*(-2*ArcCos[a*x]*(E^((3*I)*ArcCos[a*x])*(1 + (6*I)*ArcCos[a*x]) + 6*Sqrt[3]*((-I)*ArcCos[a*x])
^(3/2)*Gamma[1/2, (-3*I)*ArcCos[a*x]] + (1 - (6*I)*ArcCos[a*x] + 6*Sqrt[3]*E^((3*I)*ArcCos[a*x])*(I*ArcCos[a*x
])^(3/2)*Gamma[1/2, (3*I)*ArcCos[a*x]])/E^((3*I)*ArcCos[a*x])) - 2*Sin[3*ArcCos[a*x]]) - 6*Sin[5*ArcCos[a*x]])
/(a^5*ArcCos[a*x]^(5/2))

Maple [A] (verified)

Time = 1.07 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.85

method	result	size
default	\(-\frac {-100 \sqrt {2}\, \sqrt {\pi }\, \sqrt {5}\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {5}{2}}-108 \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {5}{2}}-8 \sqrt {2}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {5}{2}}+8 \arccos \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}+100 \arccos \left (a x \right )^{2} \sin \left (5 \arccos \left (a x \right )\right )+108 \arccos \left (a x \right )^{2} \sin \left (3 \arccos \left (a x \right )\right )-4 \arccos \left (a x \right ) a x -10 \arccos \left (a x \right ) \cos \left (5 \arccos \left (a x \right )\right )-18 \arccos \left (a x \right ) \cos \left (3 \arccos \left (a x \right )\right )-6 \sqrt {-a^{2} x^{2}+1}-3 \sin \left (5 \arccos \left (a x \right )\right )-9 \sin \left (3 \arccos \left (a x \right )\right )}{120 a^{5} \arccos \left (a x \right )^{\frac {5}{2}}}\)	\(225\)

[In]

int(x^4/arccos(a*x)^(7/2),x,method=_RETURNVERBOSE)

[Out]

-1/120/a^5*(-100*2^(1/2)*Pi^(1/2)*5^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*5^(1/2)*arccos(a*x)^(1/2))*arccos(a*x)^(5/
2)-108*2^(1/2)*Pi^(1/2)*3^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*3^(1/2)*arccos(a*x)^(1/2))*arccos(a*x)^(5/2)-8*2^(1/
2)*Pi^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*arccos(a*x)^(5/2)+8*arccos(a*x)^2*(-a^2*x^2+1)^(1/2)+
100*arccos(a*x)^2*sin(5*arccos(a*x))+108*arccos(a*x)^2*sin(3*arccos(a*x))-4*arccos(a*x)*a*x-10*arccos(a*x)*cos
(5*arccos(a*x))-18*arccos(a*x)*cos(3*arccos(a*x))-6*(-a^2*x^2+1)^(1/2)-3*sin(5*arccos(a*x))-9*sin(3*arccos(a*x
)))/arccos(a*x)^(5/2)

Fricas [F(-2)]

Exception generated. \[ \int \frac {x^4}{\arccos (a x)^{7/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^4/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]

\[ \int \frac {x^4}{\arccos (a x)^{7/2}} \, dx=\int \frac {x^{4}}{\operatorname {acos}^{\frac {7}{2}}{\left (a x \right )}}\, dx \]

[In]

integrate(x**4/acos(a*x)**(7/2),x)

[Out]

Integral(x**4/acos(a*x)**(7/2), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^4}{\arccos (a x)^{7/2}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(x^4/arccos(a*x)^(7/2),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {x^4}{\arccos (a x)^{7/2}} \, dx=\int { \frac {x^{4}}{\arccos \left (a x\right )^{\frac {7}{2}}} \,d x } \]

[In]

integrate(x^4/arccos(a*x)^(7/2),x, algorithm="giac")

[Out]

integrate(x^4/arccos(a*x)^(7/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4}{\arccos (a x)^{7/2}} \, dx=\int \frac {x^4}{{\mathrm {acos}\left (a\,x\right )}^{7/2}} \,d x \]

[In]

int(x^4/acos(a*x)^(7/2),x)

[Out]

int(x^4/acos(a*x)^(7/2), x)