\(\int \frac {a+b \arcsin (c+d x^2)}{x} \, dx\) [389]

Optimal result

Integrand size = 16, antiderivative size = 214 \[ \int \frac {a+b \arcsin \left (c+d x^2\right )}{x} \, dx=-\frac {1}{4} i b \arcsin \left (c+d x^2\right )^2+\frac {1}{2} b \arcsin \left (c+d x^2\right ) \log \left (1-\frac {e^{i \arcsin \left (c+d x^2\right )}}{i c-\sqrt {1-c^2}}\right )+\frac {1}{2} b \arcsin \left (c+d x^2\right ) \log \left (1-\frac {e^{i \arcsin \left (c+d x^2\right )}}{i c+\sqrt {1-c^2}}\right )+a \log (x)-\frac {1}{2} i b \operatorname {PolyLog}\left (2,\frac {e^{i \arcsin \left (c+d x^2\right )}}{i c-\sqrt {1-c^2}}\right )-\frac {1}{2} i b \operatorname {PolyLog}\left (2,\frac {e^{i \arcsin \left (c+d x^2\right )}}{i c+\sqrt {1-c^2}}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {6874, 4889, 4825, 4617, 2221, 2317, 2438} \[ \int \frac {a+b \arcsin \left (c+d x^2\right )}{x} \, dx=a \log (x)-\frac {1}{2} i b \operatorname {PolyLog}\left (2,\frac {e^{i \arcsin \left (d x^2+c\right )}}{i c-\sqrt {1-c^2}}\right )-\frac {1}{2} i b \operatorname {PolyLog}\left (2,\frac {e^{i \arcsin \left (d x^2+c\right )}}{i c+\sqrt {1-c^2}}\right )+\frac {1}{2} b \arcsin \left (c+d x^2\right ) \log \left (1-\frac {e^{i \arcsin \left (c+d x^2\right )}}{-\sqrt {1-c^2}+i c}\right )+\frac {1}{2} b \arcsin \left (c+d x^2\right ) \log \left (1-\frac {e^{i \arcsin \left (c+d x^2\right )}}{\sqrt {1-c^2}+i c}\right )-\frac {1}{4} i b \arcsin \left (c+d x^2\right )^2 \]

[In]

Int[(a + b*ArcSin[c + d*x^2])/x,x]

[Out]

(-1/4*I)*b*ArcSin[c + d*x^2]^2 + (b*ArcSin[c + d*x^2]*Log[1 - E^(I*ArcSin[c + d*x^2])/(I*c - Sqrt[1 - c^2])])/
2 + (b*ArcSin[c + d*x^2]*Log[1 - E^(I*ArcSin[c + d*x^2])/(I*c + Sqrt[1 - c^2])])/2 + a*Log[x] - (I/2)*b*PolyLo
g[2, E^(I*ArcSin[c + d*x^2])/(I*c - Sqrt[1 - c^2])] - (I/2)*b*PolyLog[2, E^(I*ArcSin[c + d*x^2])/(I*c + Sqrt[1
- c^2])]

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 4617

Int[(Cos[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbol] :>
Simp[(-I)*((e + f*x)^(m + 1)/(b*f*(m + 1))), x] + (Dist[I, Int[(e + f*x)^m*(E^(I*(c + d*x))/(I*a - Rt[-a^2 + b
^2, 2] + b*E^(I*(c + d*x)))), x], x] + Dist[I, Int[(e + f*x)^m*(E^(I*(c + d*x))/(I*a + Rt[-a^2 + b^2, 2] + b*E
^(I*(c + d*x)))), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NegQ[a^2 - b^2]

Rule 4825

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

Rule 4889

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[I
nt[((d*e - c*f)/d + f*(x/d))^m*(a + b*ArcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.03 \[ \int \frac {a+b \arcsin \left (c+d x^2\right )}{x} \, dx=a \log (x)+\frac {1}{2} b \left (-\frac {1}{2} i \arcsin \left (c+d x^2\right )^2+\arcsin \left (c+d x^2\right ) \log \left (1+\frac {e^{i \arcsin \left (c+d x^2\right )}}{\left (-\frac {i c}{d}-\frac {\sqrt {1-c^2}}{d}\right ) d}\right )+\arcsin \left (c+d x^2\right ) \log \left (1+\frac {e^{i \arcsin \left (c+d x^2\right )}}{\left (-\frac {i c}{d}+\frac {\sqrt {1-c^2}}{d}\right ) d}\right )-i \operatorname {PolyLog}\left (2,-\frac {e^{i \arcsin \left (c+d x^2\right )}}{-i c+\sqrt {1-c^2}}\right )-i \operatorname {PolyLog}\left (2,\frac {e^{i \arcsin \left (c+d x^2\right )}}{i c+\sqrt {1-c^2}}\right )\right ) \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int((a + b*asin(c + d*x^2))/x,x)

[Out]

int((a + b*asin(c + d*x^2))/x, x)