\(\int \frac {a+b \arcsin (c x)}{x} \, dx\) [144]

Optimal result

Integrand size = 12, antiderivative size = 63 \[ \int \frac {a+b \arcsin (c x)}{x} \, dx=-\frac {i (a+b \arcsin (c x))^2}{2 b}+(a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )-\frac {1}{2} i b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {4721, 3798, 2221, 2317, 2438} \[ \int \frac {a+b \arcsin (c x)}{x} \, dx=-\frac {i (a+b \arcsin (c x))^2}{2 b}+\log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{2} i b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) \]

[In]

Int[(a + b*ArcSin[c*x])/x,x]

[Out]

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

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

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 3798

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Simp[I*((c + d*x)^(m + 1)/(d*(
m + 1))), x] - Dist[2*I, Int[(c + d*x)^m*E^(2*I*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x))))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 4721

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

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83 \[ \int \frac {a+b \arcsin (c x)}{x} \, dx=b \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )+a \log (x)-\frac {1}{2} i b \left (\arcsin (c x)^2+\operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right ) \]

[In]

Integrate[(a + b*ArcSin[c*x])/x,x]

[Out]

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

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.87

method	result	size
parts	\(a \ln \left (x \right )+b \left (-\frac {i \arcsin \left (c x \right )^{2}}{2}+\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\)	\(118\)
derivativedivides	\(a \ln \left (c x \right )+b \left (-\frac {i \arcsin \left (c x \right )^{2}}{2}+\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\)	\(120\)
default	\(a \ln \left (c x \right )+b \left (-\frac {i \arcsin \left (c x \right )^{2}}{2}+\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\)	\(120\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral((b*arcsin(c*x) + a)/x, x)

Sympy [F]

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

[In]

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

[Out]

Integral((a + b*asin(c*x))/x, x)

Maxima [F]

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

[In]

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

[Out]

b*integrate(arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/x, x) + a*log(x)

Giac [F]

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

[In]

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

[Out]

integrate((b*arcsin(c*x) + a)/x, x)

Mupad [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.76 \[ \int \frac {a+b \arcsin (c x)}{x} \, dx=a\,\ln \left (x\right )-\frac {b\,\mathrm {polylog}\left (2,{\mathrm {e}}^{\mathrm {asin}\left (c\,x\right )\,2{}\mathrm {i}}\right )\,1{}\mathrm {i}}{2}-\frac {b\,{\mathrm {asin}\left (c\,x\right )}^2\,1{}\mathrm {i}}{2}+b\,\ln \left (1-{\mathrm {e}}^{\mathrm {asin}\left (c\,x\right )\,2{}\mathrm {i}}\right )\,\mathrm {asin}\left (c\,x\right ) \]

[In]

int((a + b*asin(c*x))/x,x)

[Out]

a*log(x) - (b*polylog(2, exp(asin(c*x)*2i))*1i)/2 - (b*asin(c*x)^2*1i)/2 + b*log(1 - exp(asin(c*x)*2i))*asin(c
*x)