3.349 \(\int \frac{a+b \sin ^{-1}(c x^2)}{x^9} \, dx\)

Optimal. Leaf size=66 \[ -\frac{a+b \sin ^{-1}\left (c x^2\right )}{8 x^8}-\frac{b c^3 \sqrt{1-c^2 x^4}}{12 x^2}-\frac{b c \sqrt{1-c^2 x^4}}{24 x^6} \]

[Out]

-(b*c*Sqrt[1 - c^2*x^4])/(24*x^6) - (b*c^3*Sqrt[1 - c^2*x^4])/(12*x^2) - (a + b*ArcSin[c*x^2])/(8*x^8)

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Rubi [A]  time = 0.0372158, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {4842, 12, 271, 264} \[ -\frac{a+b \sin ^{-1}\left (c x^2\right )}{8 x^8}-\frac{b c^3 \sqrt{1-c^2 x^4}}{12 x^2}-\frac{b c \sqrt{1-c^2 x^4}}{24 x^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSin[c*x^2])/x^9,x]

[Out]

-(b*c*Sqrt[1 - c^2*x^4])/(24*x^6) - (b*c^3*Sqrt[1 - c^2*x^4])/(12*x^2) - (a + b*ArcSin[c*x^2])/(8*x^8)

Rule 4842

Int[((a_.) + ArcSin[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(a + b*ArcSin[
u]))/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[((c + d*x)^(m + 1)*D[u, x])/Sqrt[1 - u^2], x]
, x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] &&  !FunctionOfQ[(c + d*x)^(
m + 1), u, x] &&  !FunctionOfExponentialQ[u, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{a+b \sin ^{-1}\left (c x^2\right )}{x^9} \, dx &=-\frac{a+b \sin ^{-1}\left (c x^2\right )}{8 x^8}+\frac{1}{8} b \int \frac{2 c}{x^7 \sqrt{1-c^2 x^4}} \, dx\\ &=-\frac{a+b \sin ^{-1}\left (c x^2\right )}{8 x^8}+\frac{1}{4} (b c) \int \frac{1}{x^7 \sqrt{1-c^2 x^4}} \, dx\\ &=-\frac{b c \sqrt{1-c^2 x^4}}{24 x^6}-\frac{a+b \sin ^{-1}\left (c x^2\right )}{8 x^8}+\frac{1}{6} \left (b c^3\right ) \int \frac{1}{x^3 \sqrt{1-c^2 x^4}} \, dx\\ &=-\frac{b c \sqrt{1-c^2 x^4}}{24 x^6}-\frac{b c^3 \sqrt{1-c^2 x^4}}{12 x^2}-\frac{a+b \sin ^{-1}\left (c x^2\right )}{8 x^8}\\ \end{align*}

Mathematica [A]  time = 0.0350782, size = 60, normalized size = 0.91 \[ \frac{1}{2} b \left (-\frac{c \sqrt{1-c^2 x^4} \left (2 c^2 x^4+1\right )}{12 x^6}-\frac{\sin ^{-1}\left (c x^2\right )}{4 x^8}\right )-\frac{a}{8 x^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSin[c*x^2])/x^9,x]

[Out]

-a/(8*x^8) + (b*(-(c*Sqrt[1 - c^2*x^4]*(1 + 2*c^2*x^4))/(12*x^6) - ArcSin[c*x^2]/(4*x^8)))/2

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Maple [A]  time = 0.011, size = 64, normalized size = 1. \begin{align*} -{\frac{a}{8\,{x}^{8}}}+b \left ( -{\frac{\arcsin \left ( c{x}^{2} \right ) }{8\,{x}^{8}}}+{\frac{c \left ( c{x}^{2}-1 \right ) \left ( c{x}^{2}+1 \right ) \left ( 2\,{c}^{2}{x}^{4}+1 \right ) }{24\,{x}^{6}}{\frac{1}{\sqrt{-{c}^{2}{x}^{4}+1}}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsin(c*x^2))/x^9,x)

[Out]

-1/8*a/x^8+b*(-1/8/x^8*arcsin(c*x^2)+1/24*c*(c*x^2-1)*(c*x^2+1)*(2*c^2*x^4+1)/x^6/(-c^2*x^4+1)^(1/2))

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Maxima [A]  time = 1.43331, size = 82, normalized size = 1.24 \begin{align*} -\frac{1}{24} \,{\left (c{\left (\frac{3 \, \sqrt{-c^{2} x^{4} + 1} c^{2}}{x^{2}} + \frac{{\left (-c^{2} x^{4} + 1\right )}^{\frac{3}{2}}}{x^{6}}\right )} + \frac{3 \, \arcsin \left (c x^{2}\right )}{x^{8}}\right )} b - \frac{a}{8 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x^2))/x^9,x, algorithm="maxima")

[Out]

-1/24*(c*(3*sqrt(-c^2*x^4 + 1)*c^2/x^2 + (-c^2*x^4 + 1)^(3/2)/x^6) + 3*arcsin(c*x^2)/x^8)*b - 1/8*a/x^8

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Fricas [A]  time = 3.07533, size = 123, normalized size = 1.86 \begin{align*} \frac{3 \, a x^{8} - 3 \, b \arcsin \left (c x^{2}\right ) -{\left (2 \, b c^{3} x^{6} + b c x^{2}\right )} \sqrt{-c^{2} x^{4} + 1} - 3 \, a}{24 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x^2))/x^9,x, algorithm="fricas")

[Out]

1/24*(3*a*x^8 - 3*b*arcsin(c*x^2) - (2*b*c^3*x^6 + b*c*x^2)*sqrt(-c^2*x^4 + 1) - 3*a)/x^8

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Sympy [A]  time = 11.542, size = 112, normalized size = 1.7 \begin{align*} - \frac{a}{8 x^{8}} + \frac{b c \left (\begin{cases} - \frac{i c^{2} \sqrt{c^{2} x^{4} - 1}}{3 x^{2}} - \frac{i \sqrt{c^{2} x^{4} - 1}}{6 x^{6}} & \text{for}\: \left |{c^{2} x^{4}}\right | > 1 \\- \frac{c^{2} \sqrt{- c^{2} x^{4} + 1}}{3 x^{2}} - \frac{\sqrt{- c^{2} x^{4} + 1}}{6 x^{6}} & \text{otherwise} \end{cases}\right )}{4} - \frac{b \operatorname{asin}{\left (c x^{2} \right )}}{8 x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asin(c*x**2))/x**9,x)

[Out]

-a/(8*x**8) + b*c*Piecewise((-I*c**2*sqrt(c**2*x**4 - 1)/(3*x**2) - I*sqrt(c**2*x**4 - 1)/(6*x**6), Abs(c**2*x
**4) > 1), (-c**2*sqrt(-c**2*x**4 + 1)/(3*x**2) - sqrt(-c**2*x**4 + 1)/(6*x**6), True))/4 - b*asin(c*x**2)/(8*
x**8)

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Giac [B]  time = 1.2306, size = 462, normalized size = 7. \begin{align*} -\frac{\frac{3 \, b c^{9} x^{8} \arcsin \left (c x^{2}\right )}{{\left (\sqrt{-c^{2} x^{4} + 1} + 1\right )}^{4}} + \frac{3 \, a c^{9} x^{8}}{{\left (\sqrt{-c^{2} x^{4} + 1} + 1\right )}^{4}} - \frac{2 \, b c^{8} x^{6}}{{\left (\sqrt{-c^{2} x^{4} + 1} + 1\right )}^{3}} + \frac{12 \, b c^{7} x^{4} \arcsin \left (c x^{2}\right )}{{\left (\sqrt{-c^{2} x^{4} + 1} + 1\right )}^{2}} + \frac{12 \, a c^{7} x^{4}}{{\left (\sqrt{-c^{2} x^{4} + 1} + 1\right )}^{2}} - \frac{18 \, b c^{6} x^{2}}{\sqrt{-c^{2} x^{4} + 1} + 1} + 18 \, b c^{5} \arcsin \left (c x^{2}\right ) + 18 \, a c^{5} + \frac{18 \, b c^{4}{\left (\sqrt{-c^{2} x^{4} + 1} + 1\right )}}{x^{2}} + \frac{12 \, b c^{3}{\left (\sqrt{-c^{2} x^{4} + 1} + 1\right )}^{2} \arcsin \left (c x^{2}\right )}{x^{4}} + \frac{12 \, a c^{3}{\left (\sqrt{-c^{2} x^{4} + 1} + 1\right )}^{2}}{x^{4}} + \frac{2 \, b c^{2}{\left (\sqrt{-c^{2} x^{4} + 1} + 1\right )}^{3}}{x^{6}} + \frac{3 \, b c{\left (\sqrt{-c^{2} x^{4} + 1} + 1\right )}^{4} \arcsin \left (c x^{2}\right )}{x^{8}} + \frac{3 \, a c{\left (\sqrt{-c^{2} x^{4} + 1} + 1\right )}^{4}}{x^{8}}}{384 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x^2))/x^9,x, algorithm="giac")

[Out]

-1/384*(3*b*c^9*x^8*arcsin(c*x^2)/(sqrt(-c^2*x^4 + 1) + 1)^4 + 3*a*c^9*x^8/(sqrt(-c^2*x^4 + 1) + 1)^4 - 2*b*c^
8*x^6/(sqrt(-c^2*x^4 + 1) + 1)^3 + 12*b*c^7*x^4*arcsin(c*x^2)/(sqrt(-c^2*x^4 + 1) + 1)^2 + 12*a*c^7*x^4/(sqrt(
-c^2*x^4 + 1) + 1)^2 - 18*b*c^6*x^2/(sqrt(-c^2*x^4 + 1) + 1) + 18*b*c^5*arcsin(c*x^2) + 18*a*c^5 + 18*b*c^4*(s
qrt(-c^2*x^4 + 1) + 1)/x^2 + 12*b*c^3*(sqrt(-c^2*x^4 + 1) + 1)^2*arcsin(c*x^2)/x^4 + 12*a*c^3*(sqrt(-c^2*x^4 +
1) + 1)^2/x^4 + 2*b*c^2*(sqrt(-c^2*x^4 + 1) + 1)^3/x^6 + 3*b*c*(sqrt(-c^2*x^4 + 1) + 1)^4*arcsin(c*x^2)/x^8 +
3*a*c*(sqrt(-c^2*x^4 + 1) + 1)^4/x^8)/c