\(\int \frac {a+b \arcsin (c x^n)}{x} \, dx\) [383]

Optimal result

Integrand size = 14, antiderivative size = 75 \[ \int \frac {a+b \arcsin \left (c x^n\right )}{x} \, dx=-\frac {i b \arcsin \left (c x^n\right )^2}{2 n}+\frac {b \arcsin \left (c x^n\right ) \log \left (1-e^{2 i \arcsin \left (c x^n\right )}\right )}{n}+a \log (x)-\frac {i b \operatorname {PolyLog}\left (2,e^{2 i \arcsin \left (c x^n\right )}\right )}{2 n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6874, 4914, 3798, 2221, 2317, 2438} \[ \int \frac {a+b \arcsin \left (c x^n\right )}{x} \, dx=a \log (x)-\frac {i b \operatorname {PolyLog}\left (2,e^{2 i \arcsin \left (c x^n\right )}\right )}{2 n}-\frac {i b \arcsin \left (c x^n\right )^2}{2 n}+\frac {b \arcsin \left (c x^n\right ) \log \left (1-e^{2 i \arcsin \left (c x^n\right )}\right )}{n} \]

[In]

Int[(a + b*ArcSin[c*x^n])/x,x]

[Out]

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

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 4914

Int[ArcSin[(a_.)*(x_)^(p_)]^(n_.)/(x_), x_Symbol] :> Dist[1/p, Subst[Int[x^n*Cot[x], x], x, ArcSin[a*x^p]], x]
/; FreeQ[{a, p}, x] && IGtQ[n, 0]

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

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(157\) vs. \(2(75)=150\).

Time = 0.15 (sec) , antiderivative size = 157, normalized size of antiderivative = 2.09 \[ \int \frac {a+b \arcsin \left (c x^n\right )}{x} \, dx=a \log (x)+b \arcsin \left (c x^n\right ) \log (x)-\frac {b c \left (\log (x) \log \left (\sqrt {-c^2} x^n+\sqrt {1-c^2 x^{2 n}}\right )+\frac {i \left (i \text {arcsinh}\left (\sqrt {-c^2} x^n\right ) \log \left (1-e^{-2 \text {arcsinh}\left (\sqrt {-c^2} x^n\right )}\right )-\frac {1}{2} i \left (-\text {arcsinh}\left (\sqrt {-c^2} x^n\right )^2+\operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}\left (\sqrt {-c^2} x^n\right )}\right )\right )\right )}{n}\right )}{\sqrt {-c^2}} \]

[In]

Integrate[(a + b*ArcSin[c*x^n])/x,x]

[Out]

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

Maple [A] (verified)

Time = 0.97 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.91

method	result	size
parts	\(a \ln \left (x \right )+\frac {b \left (-\frac {i \arcsin \left (c \,x^{n}\right )^{2}}{2}+\arcsin \left (c \,x^{n}\right ) \ln \left (1+i c \,x^{n}+\sqrt {1-c^{2} x^{2 n}}\right )-i \operatorname {polylog}\left (2, -i c \,x^{n}-\sqrt {1-c^{2} x^{2 n}}\right )+\arcsin \left (c \,x^{n}\right ) \ln \left (1-i c \,x^{n}-\sqrt {1-c^{2} x^{2 n}}\right )-i \operatorname {polylog}\left (2, i c \,x^{n}+\sqrt {1-c^{2} x^{2 n}}\right )\right )}{n}\)	\(143\)
derivativedivides	\(\frac {a \ln \left (c \,x^{n}\right )+b \left (-\frac {i \arcsin \left (c \,x^{n}\right )^{2}}{2}+\arcsin \left (c \,x^{n}\right ) \ln \left (1+i c \,x^{n}+\sqrt {1-c^{2} x^{2 n}}\right )-i \operatorname {polylog}\left (2, -i c \,x^{n}-\sqrt {1-c^{2} x^{2 n}}\right )+\arcsin \left (c \,x^{n}\right ) \ln \left (1-i c \,x^{n}-\sqrt {1-c^{2} x^{2 n}}\right )-i \operatorname {polylog}\left (2, i c \,x^{n}+\sqrt {1-c^{2} x^{2 n}}\right )\right )}{n}\)	\(148\)
default	\(\frac {a \ln \left (c \,x^{n}\right )+b \left (-\frac {i \arcsin \left (c \,x^{n}\right )^{2}}{2}+\arcsin \left (c \,x^{n}\right ) \ln \left (1+i c \,x^{n}+\sqrt {1-c^{2} x^{2 n}}\right )-i \operatorname {polylog}\left (2, -i c \,x^{n}-\sqrt {1-c^{2} x^{2 n}}\right )+\arcsin \left (c \,x^{n}\right ) \ln \left (1-i c \,x^{n}-\sqrt {1-c^{2} x^{2 n}}\right )-i \operatorname {polylog}\left (2, i c \,x^{n}+\sqrt {1-c^{2} x^{2 n}}\right )\right )}{n}\)	\(148\)

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {a+b \arcsin \left (c x^n\right )}{x} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

Sympy [F]

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

[In]

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

[Out]

Integral((a + b*asin(c*x**n))/x, x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

integrate((b*arcsin(c*x^n) + a)/x, x)

Mupad [F(-1)]

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

[In]

int((a + b*asin(c*x^n))/x,x)

[Out]

int((a + b*asin(c*x^n))/x, x)