\(\int \frac {x}{a+b \arcsin (c x)} \, dx\) [159]

Optimal result

Integrand size = 12, antiderivative size = 63 \[ \int \frac {x}{a+b \arcsin (c x)} \, dx=-\frac {\operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {2 a}{b}\right )}{2 b c^2}+\frac {\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{2 b c^2} \]

[Out]

1/2*cos(2*a/b)*Si(2*(a+b*arcsin(c*x))/b)/b/c^2-1/2*Ci(2*(a+b*arcsin(c*x))/b)*sin(2*a/b)/b/c^2

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4731, 4491, 12, 3384, 3380, 3383} \[ \int \frac {x}{a+b \arcsin (c x)} \, dx=\frac {\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{2 b c^2}-\frac {\sin \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{2 b c^2} \]

[In]

Int[x/(a + b*ArcSin[c*x]),x]

[Out]

-1/2*(CosIntegral[(2*(a + b*ArcSin[c*x]))/b]*Sin[(2*a)/b])/(b*c^2) + (Cos[(2*a)/b]*SinIntegral[(2*(a + b*ArcSi
n[c*x]))/b])/(2*b*c^2)

Rule 12

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

Rule 3380

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3383

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 4491

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 4731

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/(b*c^(m + 1)), Subst[Int[x^n*Sin[-
a/b + x/b]^m*Cos[-a/b + x/b], x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right ) \sin \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{b c^2} \\ & = -\frac {\text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 x} \, dx,x,a+b \arcsin (c x)\right )}{b c^2} \\ & = -\frac {\text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{2 b c^2} \\ & = \frac {\cos \left (\frac {2 a}{b}\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{2 b c^2}-\frac {\sin \left (\frac {2 a}{b}\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{2 b c^2} \\ & = -\frac {\operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {2 a}{b}\right )}{2 b c^2}+\frac {\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{2 b c^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.89 \[ \int \frac {x}{a+b \arcsin (c x)} \, dx=\frac {-\operatorname {CosIntegral}\left (\frac {2 a}{b}+2 \arcsin (c x)\right ) \sin \left (\frac {2 a}{b}\right )+\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \arcsin (c x)\right )}{2 b c^2} \]

[In]

Integrate[x/(a + b*ArcSin[c*x]),x]

[Out]

(-(CosIntegral[(2*a)/b + 2*ArcSin[c*x]]*Sin[(2*a)/b]) + Cos[(2*a)/b]*SinIntegral[(2*a)/b + 2*ArcSin[c*x]])/(2*
b*c^2)

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.92

method	result	size
derivativedivides	\(\frac {\frac {\operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right )}{2 b}-\frac {\operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right )}{2 b}}{c^{2}}\)	\(58\)
default	\(\frac {\frac {\operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right )}{2 b}-\frac {\operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right )}{2 b}}{c^{2}}\)	\(58\)

[In]

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

[Out]

1/c^2*(1/2*Si(2*arcsin(c*x)+2*a/b)*cos(2*a/b)/b-1/2*Ci(2*arcsin(c*x)+2*a/b)*sin(2*a/b)/b)

Fricas [F]

\[ \int \frac {x}{a+b \arcsin (c x)} \, dx=\int { \frac {x}{b \arcsin \left (c x\right ) + a} \,d x } \]

[In]

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

[Out]

integral(x/(b*arcsin(c*x) + a), x)

Sympy [F]

\[ \int \frac {x}{a+b \arcsin (c x)} \, dx=\int \frac {x}{a + b \operatorname {asin}{\left (c x \right )}}\, dx \]

[In]

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

[Out]

Integral(x/(a + b*asin(c*x)), x)

Maxima [F]

\[ \int \frac {x}{a+b \arcsin (c x)} \, dx=\int { \frac {x}{b \arcsin \left (c x\right ) + a} \,d x } \]

[In]

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

[Out]

integrate(x/(b*arcsin(c*x) + a), x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.37 \[ \int \frac {x}{a+b \arcsin (c x)} \, dx=-\frac {\cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{b c^{2}} + \frac {\cos \left (\frac {a}{b}\right )^{2} \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{b c^{2}} - \frac {\operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{2 \, b c^{2}} \]

[In]

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

[Out]

-cos(a/b)*cos_integral(2*a/b + 2*arcsin(c*x))*sin(a/b)/(b*c^2) + cos(a/b)^2*sin_integral(2*a/b + 2*arcsin(c*x)
)/(b*c^2) - 1/2*sin_integral(2*a/b + 2*arcsin(c*x))/(b*c^2)

Mupad [F(-1)]

Timed out. \[ \int \frac {x}{a+b \arcsin (c x)} \, dx=\int \frac {x}{a+b\,\mathrm {asin}\left (c\,x\right )} \,d x \]

[In]

int(x/(a + b*asin(c*x)),x)

[Out]

int(x/(a + b*asin(c*x)), x)