\(\int \frac {(a+b \arcsin (c x))^2}{x^2} \, dx\) [152]

Optimal result

Integrand size = 14, antiderivative size = 81 \[ \int \frac {(a+b \arcsin (c x))^2}{x^2} \, dx=-\frac {(a+b \arcsin (c x))^2}{x}-4 b c (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )+2 i b^2 c \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-2 i b^2 c \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right ) \]

[Out]

-(a+b*arcsin(c*x))^2/x-4*b*c*(a+b*arcsin(c*x))*arctanh(I*c*x+(-c^2*x^2+1)^(1/2))+2*I*b^2*c*polylog(2,-I*c*x-(-
c^2*x^2+1)^(1/2))-2*I*b^2*c*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {4723, 4803, 4268, 2317, 2438} \[ \int \frac {(a+b \arcsin (c x))^2}{x^2} \, dx=-4 b c \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {(a+b \arcsin (c x))^2}{x}+2 i b^2 c \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-2 i b^2 c \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right ) \]

[In]

Int[(a + b*ArcSin[c*x])^2/x^2,x]

[Out]

-((a + b*ArcSin[c*x])^2/x) - 4*b*c*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])] + (2*I)*b^2*c*PolyLog[2, -E^
(I*ArcSin[c*x])] - (2*I)*b^2*c*PolyLog[2, E^(I*ArcSin[c*x])]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 4268

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*
x))]/f), x] + (-Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Dist[d*(m/f), Int[(c +
d*x)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IGtQ[m, 0]

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]

Rule 4803

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[(1/c^(m
+ 1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]], Subst[Int[(a + b*x)^n*Sin[x]^m, x], x, ArcSin[c*x]], x] /; Free
Q[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b \arcsin (c x))^2}{x}+(2 b c) \int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}} \, dx \\ & = -\frac {(a+b \arcsin (c x))^2}{x}+(2 b c) \text {Subst}(\int (a+b x) \csc (x) \, dx,x,\arcsin (c x)) \\ & = -\frac {(a+b \arcsin (c x))^2}{x}-4 b c (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )-\left (2 b^2 c\right ) \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\arcsin (c x)\right )+\left (2 b^2 c\right ) \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\arcsin (c x)\right ) \\ & = -\frac {(a+b \arcsin (c x))^2}{x}-4 b c (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )+\left (2 i b^2 c\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )-\left (2 i b^2 c\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \arcsin (c x)}\right ) \\ & = -\frac {(a+b \arcsin (c x))^2}{x}-4 b c (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )+2 i b^2 c \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-2 i b^2 c \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.56 \[ \int \frac {(a+b \arcsin (c x))^2}{x^2} \, dx=-\frac {a^2+2 a b \left (\arcsin (c x)+c x \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )-i b^2 \left (i \arcsin (c x) \left (\arcsin (c x)+2 c x \left (-\log \left (1-e^{i \arcsin (c x)}\right )+\log \left (1+e^{i \arcsin (c x)}\right )\right )\right )+2 c x \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-2 c x \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right )}{x} \]

[In]

Integrate[(a + b*ArcSin[c*x])^2/x^2,x]

[Out]

-((a^2 + 2*a*b*(ArcSin[c*x] + c*x*ArcTanh[Sqrt[1 - c^2*x^2]]) - I*b^2*(I*ArcSin[c*x]*(ArcSin[c*x] + 2*c*x*(-Lo
g[1 - E^(I*ArcSin[c*x])] + Log[1 + E^(I*ArcSin[c*x])])) + 2*c*x*PolyLog[2, -E^(I*ArcSin[c*x])] - 2*c*x*PolyLog
[2, E^(I*ArcSin[c*x])]))/x)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 165, normalized size of antiderivative = 2.04

method	result	size
parts	\(-\frac {a^{2}}{x}+b^{2} c \left (-\frac {\arcsin \left (c x \right )^{2}}{c x}+2 \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \operatorname {dilog}\left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )\right )+2 a b c \left (-\frac {\arcsin \left (c x \right )}{c x}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\)	\(165\)
derivativedivides	\(c \left (-\frac {a^{2}}{c x}+b^{2} \left (-\frac {\arcsin \left (c x \right )^{2}}{c x}+2 \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \operatorname {dilog}\left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )\right )+2 a b \left (-\frac {\arcsin \left (c x \right )}{c x}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\right )\)	\(168\)
default	\(c \left (-\frac {a^{2}}{c x}+b^{2} \left (-\frac {\arcsin \left (c x \right )^{2}}{c x}+2 \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \operatorname {dilog}\left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )\right )+2 a b \left (-\frac {\arcsin \left (c x \right )}{c x}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\right )\)	\(168\)

[In]

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

[Out]

-a^2/x+b^2*c*(-1/c/x*arcsin(c*x)^2+2*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))-2*arcsin(c*x)*ln(1+I*c*x+(-c^2
*x^2+1)^(1/2))+2*I*dilog(1+I*c*x+(-c^2*x^2+1)^(1/2))-2*I*dilog(1-I*c*x-(-c^2*x^2+1)^(1/2)))+2*a*b*c*(-1/c/x*ar
csin(c*x)-arctanh(1/(-c^2*x^2+1)^(1/2)))

Fricas [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{x^2} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \]

[In]

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

[Out]

integral((b^2*arcsin(c*x)^2 + 2*a*b*arcsin(c*x) + a^2)/x^2, x)

Sympy [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{x^2} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{x^2} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \]

[In]

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

[Out]

-2*(c*log(2*sqrt(-c^2*x^2 + 1)/abs(x) + 2/abs(x)) + arcsin(c*x)/x)*a*b - (2*c*x*integrate(sqrt(c*x + 1)*sqrt(-
c*x + 1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/(c^2*x^3 - x), x) + arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x +
1))^2)*b^2/x - a^2/x

Giac [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{x^2} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{x^2} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x^2} \,d x \]

[In]

int((a + b*asin(c*x))^2/x^2,x)

[Out]

int((a + b*asin(c*x))^2/x^2, x)