3.377 \(\int \frac{a+b \sin ^{-1}(\frac{c}{x})}{x^4} \, dx\)

Optimal. Leaf size=62 \[ -\frac{a+b \sin ^{-1}\left (\frac{c}{x}\right )}{3 x^3}+\frac{b \left (1-\frac{c^2}{x^2}\right )^{3/2}}{9 c^3}-\frac{b \sqrt{1-\frac{c^2}{x^2}}}{3 c^3} \]

[Out]

-(b*Sqrt[1 - c^2/x^2])/(3*c^3) + (b*(1 - c^2/x^2)^(3/2))/(9*c^3) - (a + b*ArcSin[c/x])/(3*x^3)

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Rubi [A]  time = 0.0474299, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {4842, 12, 266, 43} \[ -\frac{a+b \sin ^{-1}\left (\frac{c}{x}\right )}{3 x^3}+\frac{b \left (1-\frac{c^2}{x^2}\right )^{3/2}}{9 c^3}-\frac{b \sqrt{1-\frac{c^2}{x^2}}}{3 c^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSin[c/x])/x^4,x]

[Out]

-(b*Sqrt[1 - c^2/x^2])/(3*c^3) + (b*(1 - c^2/x^2)^(3/2))/(9*c^3) - (a + b*ArcSin[c/x])/(3*x^3)

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 266

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{a+b \sin ^{-1}\left (\frac{c}{x}\right )}{x^4} \, dx &=-\frac{a+b \sin ^{-1}\left (\frac{c}{x}\right )}{3 x^3}-\frac{1}{3} b \int \frac{c}{\sqrt{1-\frac{c^2}{x^2}} x^5} \, dx\\ &=-\frac{a+b \sin ^{-1}\left (\frac{c}{x}\right )}{3 x^3}-\frac{1}{3} (b c) \int \frac{1}{\sqrt{1-\frac{c^2}{x^2}} x^5} \, dx\\ &=-\frac{a+b \sin ^{-1}\left (\frac{c}{x}\right )}{3 x^3}+\frac{1}{6} (b c) \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-c^2 x}} \, dx,x,\frac{1}{x^2}\right )\\ &=-\frac{a+b \sin ^{-1}\left (\frac{c}{x}\right )}{3 x^3}+\frac{1}{6} (b c) \operatorname{Subst}\left (\int \left (\frac{1}{c^2 \sqrt{1-c^2 x}}-\frac{\sqrt{1-c^2 x}}{c^2}\right ) \, dx,x,\frac{1}{x^2}\right )\\ &=-\frac{b \sqrt{1-\frac{c^2}{x^2}}}{3 c^3}+\frac{b \left (1-\frac{c^2}{x^2}\right )^{3/2}}{9 c^3}-\frac{a+b \sin ^{-1}\left (\frac{c}{x}\right )}{3 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0553837, size = 60, normalized size = 0.97 \[ -\frac{a}{3 x^3}+b \left (-\frac{2}{9 c^3}-\frac{1}{9 c x^2}\right ) \sqrt{\frac{x^2-c^2}{x^2}}-\frac{b \sin ^{-1}\left (\frac{c}{x}\right )}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSin[c/x])/x^4,x]

[Out]

-a/(3*x^3) + b*(-2/(9*c^3) - 1/(9*c*x^2))*Sqrt[(-c^2 + x^2)/x^2] - (b*ArcSin[c/x])/(3*x^3)

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Maple [A]  time = 0.006, size = 67, normalized size = 1.1 \begin{align*} -{\frac{1}{{c}^{3}} \left ({\frac{a{c}^{3}}{3\,{x}^{3}}}+b \left ({\frac{{c}^{3}}{3\,{x}^{3}}\arcsin \left ({\frac{c}{x}} \right ) }+{\frac{{c}^{2}}{9\,{x}^{2}}\sqrt{1-{\frac{{c}^{2}}{{x}^{2}}}}}+{\frac{2}{9}\sqrt{1-{\frac{{c}^{2}}{{x}^{2}}}}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsin(c/x))/x^4,x)

[Out]

-1/c^3*(1/3*c^3/x^3*a+b*(1/3*arcsin(c/x)*c^3/x^3+1/9*c^2/x^2*(1-c^2/x^2)^(1/2)+2/9*(1-c^2/x^2)^(1/2)))

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Maxima [A]  time = 1.43452, size = 78, normalized size = 1.26 \begin{align*} \frac{1}{9} \,{\left (c{\left (\frac{{\left (-\frac{c^{2}}{x^{2}} + 1\right )}^{\frac{3}{2}}}{c^{4}} - \frac{3 \, \sqrt{-\frac{c^{2}}{x^{2}} + 1}}{c^{4}}\right )} - \frac{3 \, \arcsin \left (\frac{c}{x}\right )}{x^{3}}\right )} b - \frac{a}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c/x))/x^4,x, algorithm="maxima")

[Out]

1/9*(c*((-c^2/x^2 + 1)^(3/2)/c^4 - 3*sqrt(-c^2/x^2 + 1)/c^4) - 3*arcsin(c/x)/x^3)*b - 1/3*a/x^3

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Fricas [A]  time = 2.26782, size = 126, normalized size = 2.03 \begin{align*} -\frac{3 \, b c^{3} \arcsin \left (\frac{c}{x}\right ) + 3 \, a c^{3} +{\left (b c^{2} x + 2 \, b x^{3}\right )} \sqrt{-\frac{c^{2} - x^{2}}{x^{2}}}}{9 \, c^{3} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c/x))/x^4,x, algorithm="fricas")

[Out]

-1/9*(3*b*c^3*arcsin(c/x) + 3*a*c^3 + (b*c^2*x + 2*b*x^3)*sqrt(-(c^2 - x^2)/x^2))/(c^3*x^3)

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Sympy [A]  time = 5.5698, size = 114, normalized size = 1.84 \begin{align*} - \frac{a}{3 x^{3}} - \frac{b c \left (\begin{cases} \frac{\sqrt{-1 + \frac{x^{2}}{c^{2}}}}{3 c x^{3}} + \frac{2 \sqrt{-1 + \frac{x^{2}}{c^{2}}}}{3 c^{3} x} & \text{for}\: \frac{\left |{x^{2}}\right |}{\left |{c^{2}}\right |} > 1 \\\frac{i \sqrt{1 - \frac{x^{2}}{c^{2}}}}{3 c x^{3}} + \frac{2 i \sqrt{1 - \frac{x^{2}}{c^{2}}}}{3 c^{3} x} & \text{otherwise} \end{cases}\right )}{3} - \frac{b \operatorname{asin}{\left (\frac{c}{x} \right )}}{3 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asin(c/x))/x**4,x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \arcsin \left (\frac{c}{x}\right ) + a}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c/x))/x^4,x, algorithm="giac")

[Out]

integrate((b*arcsin(c/x) + a)/x^4, x)