\(\int \frac {a+b \arcsin (\frac {c}{x})}{x^2} \, dx\) [375]

Optimal result

Integrand size = 14, antiderivative size = 39 \[ \int \frac {a+b \arcsin \left (\frac {c}{x}\right )}{x^2} \, dx=-\frac {b \sqrt {1-\frac {c^2}{x^2}}}{c}-\frac {a}{x}-\frac {b \csc ^{-1}\left (\frac {x}{c}\right )}{x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6847, 4715, 267} \[ \int \frac {a+b \arcsin \left (\frac {c}{x}\right )}{x^2} \, dx=-\frac {a}{x}-\frac {b \sqrt {1-\frac {c^2}{x^2}}}{c}-\frac {b \csc ^{-1}\left (\frac {x}{c}\right )}{x} \]

[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 267

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]

Rule 4715

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 6847

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]

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int (a+b \arcsin (c x)) \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {a}{x}-b \text {Subst}\left (\int \arcsin (c x) \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {a}{x}-\frac {b \csc ^{-1}\left (\frac {x}{c}\right )}{x}+(b c) \text {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] (verified)

Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00 \[ \int \frac {a+b \arcsin \left (\frac {c}{x}\right )}{x^2} \, dx=-\frac {b \sqrt {1-\frac {c^2}{x^2}}}{c}-\frac {a}{x}-\frac {b \arcsin \left (\frac {c}{x}\right )}{x} \]

[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

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.97

method	result	size
parts	\(-\frac {a}{x}-\frac {b \left (\frac {c \arcsin \left (\frac {c}{x}\right )}{x}+\sqrt {1-\frac {c^{2}}{x^{2}}}\right )}{c}\)	\(38\)
derivativedivides	\(-\frac {\frac {c a}{x}+b \left (\frac {c \arcsin \left (\frac {c}{x}\right )}{x}+\sqrt {1-\frac {c^{2}}{x^{2}}}\right )}{c}\)	\(39\)
default	\(-\frac {\frac {c a}{x}+b \left (\frac {c \arcsin \left (\frac {c}{x}\right )}{x}+\sqrt {1-\frac {c^{2}}{x^{2}}}\right )}{c}\)	\(39\)

[In]

int((a+b*arcsin(c/x))/x^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.03 \[ \int \frac {a+b \arcsin \left (\frac {c}{x}\right )}{x^2} \, dx=-\frac {b c \arcsin \left (\frac {c}{x}\right ) + b x \sqrt {-\frac {c^{2} - x^{2}}{x^{2}}} + a c}{c x} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.82 \[ \int \frac {a+b \arcsin \left (\frac {c}{x}\right )}{x^2} \, dx=\begin {cases} - \frac {a}{x} - \frac {b \operatorname {asin}{\left (\frac {c}{x} \right )}}{x} - \frac {b \sqrt {- \frac {c^{2}}{x^{2}} + 1}}{c} & \text {for}\: c \neq 0 \\- \frac {a}{x} & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.95 \[ \int \frac {a+b \arcsin \left (\frac {c}{x}\right )}{x^2} \, dx=-\frac {b {\left (\frac {c \arcsin \left (\frac {c}{x}\right )}{x} + \sqrt {-\frac {c^{2}}{x^{2}} + 1}\right )}}{c} - \frac {a}{x} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.97 \[ \int \frac {a+b \arcsin \left (\frac {c}{x}\right )}{x^2} \, dx=-\frac {\frac {b c \arcsin \left (\frac {c}{x}\right )}{x} + b \sqrt {-\frac {c^{2}}{x^{2}} + 1} + \frac {a c}{x}}{c} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.95 \[ \int \frac {a+b \arcsin \left (\frac {c}{x}\right )}{x^2} \, dx=-\frac {a}{x}-\frac {b\,\sqrt {1-\frac {c^2}{x^2}}}{c}-\frac {b\,\mathrm {asin}\left (\frac {c}{x}\right )}{x} \]

[In]

int((a + b*asin(c/x))/x^2,x)

[Out]

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