3.93 \(\int \frac {x^3}{\sqrt {\cos ^{-1}(a x)}} \, dx\)

Optimal. Leaf size=65 \[ -\frac {\sqrt {\frac {\pi }{2}} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{8 a^4}-\frac {\sqrt {\pi } S\left (\frac {2 \sqrt {\cos ^{-1}(a x)}}{\sqrt {\pi }}\right )}{4 a^4} \]

[Out]

-1/16*FresnelS(2*2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^4-1/4*FresnelS(2*arccos(a*x)^(1/2)/Pi^
(1/2))*Pi^(1/2)/a^4

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Rubi [A]  time = 0.08, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4636, 4406, 3305, 3351} \[ -\frac {\sqrt {\frac {\pi }{2}} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{8 a^4}-\frac {\sqrt {\pi } S\left (\frac {2 \sqrt {\cos ^{-1}(a x)}}{\sqrt {\pi }}\right )}{4 a^4} \]

Antiderivative was successfully verified.

[In]

Int[x^3/Sqrt[ArcCos[a*x]],x]

[Out]

-(Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(8*a^4) - (Sqrt[Pi]*FresnelS[(2*Sqrt[ArcCos[a*x]])/Sqrt
[Pi]])/(4*a^4)

Rule 3305

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

Rule 3351

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rule 4406

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(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]
&& IGtQ[p, 0]

Rule 4636

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

Rubi steps

\begin {align*} \int \frac {x^3}{\sqrt {\cos ^{-1}(a x)}} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\cos ^3(x) \sin (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{a^4}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {\sin (2 x)}{4 \sqrt {x}}+\frac {\sin (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{a^4}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\sin (4 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{8 a^4}-\frac {\operatorname {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{4 a^4}\\ &=-\frac {\operatorname {Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{4 a^4}-\frac {\operatorname {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{2 a^4}\\ &=-\frac {\sqrt {\frac {\pi }{2}} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{8 a^4}-\frac {\sqrt {\pi } S\left (\frac {2 \sqrt {\cos ^{-1}(a x)}}{\sqrt {\pi }}\right )}{4 a^4}\\ \end {align*}

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Mathematica [C]  time = 0.09, size = 130, normalized size = 2.00 \[ -\frac {-2 \sqrt {2} \sqrt {-i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},-2 i \cos ^{-1}(a x)\right )-2 \sqrt {2} \sqrt {i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},2 i \cos ^{-1}(a x)\right )-\sqrt {-i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},-4 i \cos ^{-1}(a x)\right )-\sqrt {i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},4 i \cos ^{-1}(a x)\right )}{32 a^4 \sqrt {\cos ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^3/Sqrt[ArcCos[a*x]],x]

[Out]

-1/32*(-2*Sqrt[2]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-2*I)*ArcCos[a*x]] - 2*Sqrt[2]*Sqrt[I*ArcCos[a*x]]*Gamma[
1/2, (2*I)*ArcCos[a*x]] - Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-4*I)*ArcCos[a*x]] - Sqrt[I*ArcCos[a*x]]*Gamma[1/
2, (4*I)*ArcCos[a*x]])/(a^4*Sqrt[ArcCos[a*x]])

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/arccos(a*x)^(1/2),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 3.60, size = 113, normalized size = 1.74 \[ \frac {\sqrt {2} \sqrt {\pi } i \operatorname {erf}\left (\sqrt {2} {\left (i - 1\right )} \sqrt {\arccos \left (a x\right )}\right )}{32 \, a^{4} {\left (i - 1\right )}} + \frac {\sqrt {\pi } i \operatorname {erf}\left ({\left (i - 1\right )} \sqrt {\arccos \left (a x\right )}\right )}{8 \, a^{4} {\left (i - 1\right )}} - \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} {\left (i + 1\right )} \sqrt {\arccos \left (a x\right )}\right )}{32 \, a^{4} {\left (i - 1\right )}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-{\left (i + 1\right )} \sqrt {\arccos \left (a x\right )}\right )}{8 \, a^{4} {\left (i - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/arccos(a*x)^(1/2),x, algorithm="giac")

[Out]

1/32*sqrt(2)*sqrt(pi)*i*erf(sqrt(2)*(i - 1)*sqrt(arccos(a*x)))/(a^4*(i - 1)) + 1/8*sqrt(pi)*i*erf((i - 1)*sqrt
(arccos(a*x)))/(a^4*(i - 1)) - 1/32*sqrt(2)*sqrt(pi)*erf(-sqrt(2)*(i + 1)*sqrt(arccos(a*x)))/(a^4*(i - 1)) - 1
/8*sqrt(pi)*erf(-(i + 1)*sqrt(arccos(a*x)))/(a^4*(i - 1))

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maple [A]  time = 0.19, size = 43, normalized size = 0.66 \[ -\frac {\sqrt {\pi }\, \left (\sqrt {2}\, \mathrm {S}\left (\frac {2 \sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )+4 \,\mathrm {S}\left (\frac {2 \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )\right )}{16 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/arccos(a*x)^(1/2),x)

[Out]

-1/16/a^4*Pi^(1/2)*(2^(1/2)*FresnelS(2*2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))+4*FresnelS(2*arccos(a*x)^(1/2)/Pi^(
1/2)))

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/arccos(a*x)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^3}{\sqrt {\mathrm {acos}\left (a\,x\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/acos(a*x)^(1/2),x)

[Out]

int(x^3/acos(a*x)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\sqrt {\operatorname {acos}{\left (a x \right )}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/acos(a*x)**(1/2),x)

[Out]

Integral(x**3/sqrt(acos(a*x)), x)

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