3.51 \(\int \frac {x^6}{\sin ^{-1}(a x)^2} \, dx\)

Optimal. Leaf size=83 \[ -\frac {5 \text {Si}\left (\sin ^{-1}(a x)\right )}{64 a^7}+\frac {27 \text {Si}\left (3 \sin ^{-1}(a x)\right )}{64 a^7}-\frac {25 \text {Si}\left (5 \sin ^{-1}(a x)\right )}{64 a^7}+\frac {7 \text {Si}\left (7 \sin ^{-1}(a x)\right )}{64 a^7}-\frac {x^6 \sqrt {1-a^2 x^2}}{a \sin ^{-1}(a x)} \]

[Out]

-5/64*Si(arcsin(a*x))/a^7+27/64*Si(3*arcsin(a*x))/a^7-25/64*Si(5*arcsin(a*x))/a^7+7/64*Si(7*arcsin(a*x))/a^7-x
^6*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)

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Rubi [A]  time = 0.07, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4631, 3299} \[ -\frac {5 \text {Si}\left (\sin ^{-1}(a x)\right )}{64 a^7}+\frac {27 \text {Si}\left (3 \sin ^{-1}(a x)\right )}{64 a^7}-\frac {25 \text {Si}\left (5 \sin ^{-1}(a x)\right )}{64 a^7}+\frac {7 \text {Si}\left (7 \sin ^{-1}(a x)\right )}{64 a^7}-\frac {x^6 \sqrt {1-a^2 x^2}}{a \sin ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

Int[x^6/ArcSin[a*x]^2,x]

[Out]

-((x^6*Sqrt[1 - a^2*x^2])/(a*ArcSin[a*x])) - (5*SinIntegral[ArcSin[a*x]])/(64*a^7) + (27*SinIntegral[3*ArcSin[
a*x]])/(64*a^7) - (25*SinIntegral[5*ArcSin[a*x]])/(64*a^7) + (7*SinIntegral[7*ArcSin[a*x]])/(64*a^7)

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

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^6}{\sin ^{-1}(a x)^2} \, dx &=-\frac {x^6 \sqrt {1-a^2 x^2}}{a \sin ^{-1}(a x)}+\frac {\operatorname {Subst}\left (\int \left (-\frac {5 \sin (x)}{64 x}+\frac {27 \sin (3 x)}{64 x}-\frac {25 \sin (5 x)}{64 x}+\frac {7 \sin (7 x)}{64 x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a^7}\\ &=-\frac {x^6 \sqrt {1-a^2 x^2}}{a \sin ^{-1}(a x)}-\frac {5 \operatorname {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{64 a^7}+\frac {7 \operatorname {Subst}\left (\int \frac {\sin (7 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{64 a^7}-\frac {25 \operatorname {Subst}\left (\int \frac {\sin (5 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{64 a^7}+\frac {27 \operatorname {Subst}\left (\int \frac {\sin (3 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{64 a^7}\\ &=-\frac {x^6 \sqrt {1-a^2 x^2}}{a \sin ^{-1}(a x)}-\frac {5 \text {Si}\left (\sin ^{-1}(a x)\right )}{64 a^7}+\frac {27 \text {Si}\left (3 \sin ^{-1}(a x)\right )}{64 a^7}-\frac {25 \text {Si}\left (5 \sin ^{-1}(a x)\right )}{64 a^7}+\frac {7 \text {Si}\left (7 \sin ^{-1}(a x)\right )}{64 a^7}\\ \end {align*}

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Mathematica [A]  time = 0.25, size = 86, normalized size = 1.04 \[ -\frac {64 a^6 x^6 \sqrt {1-a^2 x^2}+5 \sin ^{-1}(a x) \text {Si}\left (\sin ^{-1}(a x)\right )-27 \sin ^{-1}(a x) \text {Si}\left (3 \sin ^{-1}(a x)\right )+25 \sin ^{-1}(a x) \text {Si}\left (5 \sin ^{-1}(a x)\right )-7 \sin ^{-1}(a x) \text {Si}\left (7 \sin ^{-1}(a x)\right )}{64 a^7 \sin ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6/ArcSin[a*x]^2,x]

[Out]

-1/64*(64*a^6*x^6*Sqrt[1 - a^2*x^2] + 5*ArcSin[a*x]*SinIntegral[ArcSin[a*x]] - 27*ArcSin[a*x]*SinIntegral[3*Ar
cSin[a*x]] + 25*ArcSin[a*x]*SinIntegral[5*ArcSin[a*x]] - 7*ArcSin[a*x]*SinIntegral[7*ArcSin[a*x]])/(a^7*ArcSin
[a*x])

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fricas [F]  time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{6}}{\arcsin \left (a x\right )^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/arcsin(a*x)^2,x, algorithm="fricas")

[Out]

integral(x^6/arcsin(a*x)^2, x)

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giac [B]  time = 0.15, size = 161, normalized size = 1.94 \[ -\frac {{\left (a^{2} x^{2} - 1\right )}^{3} \sqrt {-a^{2} x^{2} + 1}}{a^{7} \arcsin \left (a x\right )} - \frac {3 \, {\left (a^{2} x^{2} - 1\right )}^{2} \sqrt {-a^{2} x^{2} + 1}}{a^{7} \arcsin \left (a x\right )} + \frac {7 \, \operatorname {Si}\left (7 \, \arcsin \left (a x\right )\right )}{64 \, a^{7}} - \frac {25 \, \operatorname {Si}\left (5 \, \arcsin \left (a x\right )\right )}{64 \, a^{7}} + \frac {27 \, \operatorname {Si}\left (3 \, \arcsin \left (a x\right )\right )}{64 \, a^{7}} - \frac {5 \, \operatorname {Si}\left (\arcsin \left (a x\right )\right )}{64 \, a^{7}} + \frac {3 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{a^{7} \arcsin \left (a x\right )} - \frac {\sqrt {-a^{2} x^{2} + 1}}{a^{7} \arcsin \left (a x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/arcsin(a*x)^2,x, algorithm="giac")

[Out]

-(a^2*x^2 - 1)^3*sqrt(-a^2*x^2 + 1)/(a^7*arcsin(a*x)) - 3*(a^2*x^2 - 1)^2*sqrt(-a^2*x^2 + 1)/(a^7*arcsin(a*x))
+ 7/64*sin_integral(7*arcsin(a*x))/a^7 - 25/64*sin_integral(5*arcsin(a*x))/a^7 + 27/64*sin_integral(3*arcsin(
a*x))/a^7 - 5/64*sin_integral(arcsin(a*x))/a^7 + 3*(-a^2*x^2 + 1)^(3/2)/(a^7*arcsin(a*x)) - sqrt(-a^2*x^2 + 1)
/(a^7*arcsin(a*x))

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maple [A]  time = 0.07, size = 105, normalized size = 1.27 \[ \frac {-\frac {5 \sqrt {-a^{2} x^{2}+1}}{64 \arcsin \left (a x \right )}-\frac {5 \Si \left (\arcsin \left (a x \right )\right )}{64}+\frac {9 \cos \left (3 \arcsin \left (a x \right )\right )}{64 \arcsin \left (a x \right )}+\frac {27 \Si \left (3 \arcsin \left (a x \right )\right )}{64}-\frac {5 \cos \left (5 \arcsin \left (a x \right )\right )}{64 \arcsin \left (a x \right )}-\frac {25 \Si \left (5 \arcsin \left (a x \right )\right )}{64}+\frac {\cos \left (7 \arcsin \left (a x \right )\right )}{64 \arcsin \left (a x \right )}+\frac {7 \Si \left (7 \arcsin \left (a x \right )\right )}{64}}{a^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/arcsin(a*x)^2,x)

[Out]

1/a^7*(-5/64/arcsin(a*x)*(-a^2*x^2+1)^(1/2)-5/64*Si(arcsin(a*x))+9/64/arcsin(a*x)*cos(3*arcsin(a*x))+27/64*Si(
3*arcsin(a*x))-5/64/arcsin(a*x)*cos(5*arcsin(a*x))-25/64*Si(5*arcsin(a*x))+1/64/arcsin(a*x)*cos(7*arcsin(a*x))
+7/64*Si(7*arcsin(a*x)))

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/arcsin(a*x)^2,x, algorithm="maxima")

[Out]

Timed out

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^6}{{\mathrm {asin}\left (a\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/asin(a*x)^2,x)

[Out]

int(x^6/asin(a*x)^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{6}}{\operatorname {asin}^{2}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/asin(a*x)**2,x)

[Out]

Integral(x**6/asin(a*x)**2, x)

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