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

Optimal. Leaf size=39 \[ -\frac{a}{x}-\frac{b \sqrt{1-\frac{c^2}{x^2}}}{c}-\frac{b \csc ^{-1}\left (\frac{x}{c}\right )}{x} \]

[Out]

-((b*Sqrt[1 - c^2/x^2])/c) - a/x - (b*ArcCsc[x/c])/x

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Rubi [A]  time = 0.0359881, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {6715, 4619, 261} \[ -\frac{a}{x}-\frac{b \sqrt{1-\frac{c^2}{x^2}}}{c}-\frac{b \csc ^{-1}\left (\frac{x}{c}\right )}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSin[c/x])/x^2,x]

[Out]

-((b*Sqrt[1 - c^2/x^2])/c) - a/x - (b*ArcCsc[x/c])/x

Rule 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rule 4619

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSin[c*x])^n, x] - Dist[b*c*n, Int[
(x*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 261

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

Rubi steps

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

Mathematica [A]  time = 0.0233979, size = 39, normalized size = 1. \[ -\frac{a}{x}-\frac{b \sqrt{1-\frac{c^2}{x^2}}}{c}-\frac{b \sin ^{-1}\left (\frac{c}{x}\right )}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSin[c/x])/x^2,x]

[Out]

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

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Maple [A]  time = 0.003, size = 39, normalized size = 1. \begin{align*} -{\frac{1}{c} \left ({\frac{ac}{x}}+b \left ({\frac{c}{x}\arcsin \left ({\frac{c}{x}} \right ) }+\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^2,x)

[Out]

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

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Maxima [A]  time = 1.40248, size = 50, normalized size = 1.28 \begin{align*} -\frac{b{\left (\frac{c \arcsin \left (\frac{c}{x}\right )}{x} + \sqrt{-\frac{c^{2}}{x^{2}} + 1}\right )}}{c} - \frac{a}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.1986, size = 82, normalized size = 2.1 \begin{align*} -\frac{b c \arcsin \left (\frac{c}{x}\right ) + b x \sqrt{-\frac{c^{2} - x^{2}}{x^{2}}} + a c}{c x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 2.59582, size = 32, normalized size = 0.82 \begin{align*} \begin{cases} - \frac{a}{x} - \frac{b \operatorname{asin}{\left (\frac{c}{x} \right )}}{x} - \frac{b \sqrt{- \frac{c^{2}}{x^{2}} + 1}}{c} & \text{for}\: c \neq 0 \\- \frac{a}{x} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a/x - b*asin(c/x)/x - b*sqrt(-c**2/x**2 + 1)/c, Ne(c, 0)), (-a/x, True))

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Giac [A]  time = 1.16384, size = 51, normalized size = 1.31 \begin{align*} -\frac{b{\left (\frac{c \arcsin \left (\frac{c}{x}\right )}{x} + \sqrt{-\frac{c^{2}}{x^{2}} + 1}\right )} + \frac{a c}{x}}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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