\(\int \frac {a+b \arcsin (c x)}{x^2} \, dx\) [145]

Optimal result

Integrand size = 12, antiderivative size = 33 \[ \int \frac {a+b \arcsin (c x)}{x^2} \, dx=-\frac {a+b \arcsin (c x)}{x}-b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4723, 272, 65, 214} \[ \int \frac {a+b \arcsin (c x)}{x^2} \, dx=-\frac {a+b \arcsin (c x)}{x}-b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right ) \]

[In]

Int[(a + b*ArcSin[c*x])/x^2,x]

[Out]

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

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 272

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 4723

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

Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \arcsin (c x)}{x}+(b c) \int \frac {1}{x \sqrt {1-c^2 x^2}} \, dx \\ & = -\frac {a+b \arcsin (c x)}{x}+\frac {1}{2} (b c) \text {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right ) \\ & = -\frac {a+b \arcsin (c x)}{x}-\frac {b \text {Subst}\left (\int \frac {1}{\frac {1}{c^2}-\frac {x^2}{c^2}} \, dx,x,\sqrt {1-c^2 x^2}\right )}{c} \\ & = -\frac {a+b \arcsin (c x)}{x}-b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {a+b \arcsin (c x)}{x^2} \, dx=-\frac {a}{x}-\frac {b \arcsin (c x)}{x}-b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right ) \]

[In]

Integrate[(a + b*ArcSin[c*x])/x^2,x]

[Out]

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

Maple [A] (verified)

Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18

method	result	size
parts	\(-\frac {a}{x}+b c \left (-\frac {\arcsin \left (c x \right )}{c x}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\)	\(39\)
derivativedivides	\(c \left (-\frac {a}{c x}+b \left (-\frac {\arcsin \left (c x \right )}{c x}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\right )\)	\(43\)
default	\(c \left (-\frac {a}{c x}+b \left (-\frac {\arcsin \left (c x \right )}{c x}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\right )\)	\(43\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.67 \[ \int \frac {a+b \arcsin (c x)}{x^2} \, dx=-\frac {b c x \log \left (\sqrt {-c^{2} x^{2} + 1} + 1\right ) - b c x \log \left (\sqrt {-c^{2} x^{2} + 1} - 1\right ) + 2 \, b \arcsin \left (c x\right ) + 2 \, a}{2 \, x} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.96 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18 \[ \int \frac {a+b \arcsin (c x)}{x^2} \, dx=- \frac {a}{x} + b c \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{c x} \right )} & \text {for}\: \frac {1}{\left |{c^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{c x} \right )} & \text {otherwise} \end {cases}\right ) - \frac {b \operatorname {asin}{\left (c x \right )}}{x} \]

[In]

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

[Out]

-a/x + b*c*Piecewise((-acosh(1/(c*x)), 1/Abs(c**2*x**2) > 1), (I*asin(1/(c*x)), True)) - b*asin(c*x)/x

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.42 \[ \int \frac {a+b \arcsin (c x)}{x^2} \, dx=-{\left (c \log \left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {\arcsin \left (c x\right )}{x}\right )} b - \frac {a}{x} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 325 vs. \(2 (31) = 62\).

Time = 0.30 (sec) , antiderivative size = 325, normalized size of antiderivative = 9.85 \[ \int \frac {a+b \arcsin (c x)}{x^2} \, dx=-\frac {\sqrt {-c^{2} x^{2} + 1} b c^{2} x \arcsin \left (c x\right )}{2 \, {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{2}} - \frac {b c^{2} x \arcsin \left (c x\right )}{2 \, {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{2}} - \frac {\sqrt {-c^{2} x^{2} + 1} a c^{2} x}{2 \, {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} b c \log \left ({\left | c \right |} {\left | x \right |}\right )}{\sqrt {-c^{2} x^{2} + 1} + 1} - \frac {\sqrt {-c^{2} x^{2} + 1} b c \log \left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}{\sqrt {-c^{2} x^{2} + 1} + 1} - \frac {a c^{2} x}{2 \, {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{2}} + \frac {b c \log \left ({\left | c \right |} {\left | x \right |}\right )}{\sqrt {-c^{2} x^{2} + 1} + 1} - \frac {b c \log \left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}{\sqrt {-c^{2} x^{2} + 1} + 1} - \frac {\sqrt {-c^{2} x^{2} + 1} b \arcsin \left (c x\right )}{2 \, x} - \frac {b \arcsin \left (c x\right )}{2 \, x} - \frac {\sqrt {-c^{2} x^{2} + 1} a}{2 \, x} - \frac {a}{2 \, x} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.03 \[ \int \frac {a+b \arcsin (c x)}{x^2} \, dx=-\frac {a}{x}-\frac {b\,\mathrm {asin}\left (c\,x\right )}{x}-b\,c\,\mathrm {atanh}\left (\frac {1}{\sqrt {1-c^2\,x^2}}\right ) \]

[In]

int((a + b*asin(c*x))/x^2,x)

[Out]

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