3.100 \(\int \frac {x^5}{\sin ^{-1}(a x)^{3/2}} \, dx\)

Optimal. Leaf size=127 \[ -\frac {\sqrt {\frac {\pi }{2}} C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{a^6}+\frac {\sqrt {3 \pi } C\left (2 \sqrt {\frac {3}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{8 a^6}+\frac {5 \sqrt {\pi } C\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{8 a^6}-\frac {2 x^5 \sqrt {1-a^2 x^2}}{a \sqrt {\sin ^{-1}(a x)}} \]

[Out]

-1/2*FresnelC(2*2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^6+5/8*FresnelC(2*arcsin(a*x)^(1/2)/Pi^(
1/2))*Pi^(1/2)/a^6+1/8*FresnelC(2*3^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*3^(1/2)*Pi^(1/2)/a^6-2*x^5*(-a^2*x^2+1)^
(1/2)/a/arcsin(a*x)^(1/2)

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Rubi [A]  time = 0.11, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4631, 3304, 3352} \[ -\frac {\sqrt {\frac {\pi }{2}} \text {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{a^6}+\frac {\sqrt {3 \pi } \text {FresnelC}\left (2 \sqrt {\frac {3}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{8 a^6}+\frac {5 \sqrt {\pi } \text {FresnelC}\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{8 a^6}-\frac {2 x^5 \sqrt {1-a^2 x^2}}{a \sqrt {\sin ^{-1}(a x)}} \]

Antiderivative was successfully verified.

[In]

Int[x^5/ArcSin[a*x]^(3/2),x]

[Out]

(-2*x^5*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcSin[a*x]]) - (Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/a^6 +
(Sqrt[3*Pi]*FresnelC[2*Sqrt[3/Pi]*Sqrt[ArcSin[a*x]]])/(8*a^6) + (5*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sq
rt[Pi]])/(8*a^6)

Rule 3304

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 3352

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

Rule 4631

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

Rubi steps

\begin {align*} \int \frac {x^5}{\sin ^{-1}(a x)^{3/2}} \, dx &=-\frac {2 x^5 \sqrt {1-a^2 x^2}}{a \sqrt {\sin ^{-1}(a x)}}+\frac {2 \operatorname {Subst}\left (\int \left (\frac {5 \cos (2 x)}{16 \sqrt {x}}-\frac {\cos (4 x)}{2 \sqrt {x}}+\frac {3 \cos (6 x)}{16 \sqrt {x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a^6}\\ &=-\frac {2 x^5 \sqrt {1-a^2 x^2}}{a \sqrt {\sin ^{-1}(a x)}}+\frac {3 \operatorname {Subst}\left (\int \frac {\cos (6 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^6}+\frac {5 \operatorname {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^6}-\frac {\operatorname {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{a^6}\\ &=-\frac {2 x^5 \sqrt {1-a^2 x^2}}{a \sqrt {\sin ^{-1}(a x)}}+\frac {3 \operatorname {Subst}\left (\int \cos \left (6 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{4 a^6}+\frac {5 \operatorname {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{4 a^6}-\frac {2 \operatorname {Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{a^6}\\ &=-\frac {2 x^5 \sqrt {1-a^2 x^2}}{a \sqrt {\sin ^{-1}(a x)}}-\frac {\sqrt {\frac {\pi }{2}} C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{a^6}+\frac {\sqrt {3 \pi } C\left (2 \sqrt {\frac {3}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{8 a^6}+\frac {5 \sqrt {\pi } C\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{8 a^6}\\ \end {align*}

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Mathematica [C]  time = 0.14, size = 231, normalized size = 1.82 \[ -\frac {10 \sin \left (2 \sin ^{-1}(a x)\right )-8 \sin \left (4 \sin ^{-1}(a x)\right )+2 \sin \left (6 \sin ^{-1}(a x)\right )+5 i \sqrt {2} \sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},-2 i \sin ^{-1}(a x)\right )-5 i \sqrt {2} \sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},2 i \sin ^{-1}(a x)\right )-8 i \sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},-4 i \sin ^{-1}(a x)\right )+8 i \sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},4 i \sin ^{-1}(a x)\right )+i \sqrt {6} \sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},-6 i \sin ^{-1}(a x)\right )-i \sqrt {6} \sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},6 i \sin ^{-1}(a x)\right )}{32 a^6 \sqrt {\sin ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^5/ArcSin[a*x]^(3/2),x]

[Out]

-1/32*((5*I)*Sqrt[2]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-2*I)*ArcSin[a*x]] - (5*I)*Sqrt[2]*Sqrt[I*ArcSin[a*x]]
*Gamma[1/2, (2*I)*ArcSin[a*x]] - (8*I)*Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-4*I)*ArcSin[a*x]] + (8*I)*Sqrt[I*Ar
cSin[a*x]]*Gamma[1/2, (4*I)*ArcSin[a*x]] + I*Sqrt[6]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-6*I)*ArcSin[a*x]] - I
*Sqrt[6]*Sqrt[I*ArcSin[a*x]]*Gamma[1/2, (6*I)*ArcSin[a*x]] + 10*Sin[2*ArcSin[a*x]] - 8*Sin[4*ArcSin[a*x]] + 2*
Sin[6*ArcSin[a*x]])/(a^6*Sqrt[ArcSin[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^5/arcsin(a*x)^(3/2),x, algorithm="fricas")

[Out]

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

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giac [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^5/arcsin(a*x)^(3/2),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

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maple [A]  time = 0.11, size = 121, normalized size = 0.95 \[ -\frac {8 \FresnelC \left (\frac {2 \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }-2 \sqrt {\pi }\, \sqrt {3}\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {6}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {\arcsin \left (a x \right )}-10 \FresnelC \left (\frac {2 \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }+5 \sin \left (2 \arcsin \left (a x \right )\right )-4 \sin \left (4 \arcsin \left (a x \right )\right )+\sin \left (6 \arcsin \left (a x \right )\right )}{16 a^{6} \sqrt {\arcsin \left (a x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/arcsin(a*x)^(3/2),x)

[Out]

-1/16/a^6*(8*FresnelC(2*2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2)-2*Pi^(1/2)*3^(1
/2)*FresnelC(2^(1/2)/Pi^(1/2)*6^(1/2)*arcsin(a*x)^(1/2))*arcsin(a*x)^(1/2)-10*FresnelC(2*arcsin(a*x)^(1/2)/Pi^
(1/2))*arcsin(a*x)^(1/2)*Pi^(1/2)+5*sin(2*arcsin(a*x))-4*sin(4*arcsin(a*x))+sin(6*arcsin(a*x)))/arcsin(a*x)^(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^5/arcsin(a*x)^(3/2),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 = -0.01 \[ \int \frac {x^5}{{\mathrm {asin}\left (a\,x\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/asin(a*x)^(3/2),x)

[Out]

int(x^5/asin(a*x)^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{5}}{\operatorname {asin}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/asin(a*x)**(3/2),x)

[Out]

Integral(x**5/asin(a*x)**(3/2), x)

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