3.383 \(\int \frac{a+b \sin ^{-1}(c x^n)}{x} \, dx\)

Optimal. Leaf size=75 \[ -\frac{i b \text{PolyLog}\left (2,e^{2 i \sin ^{-1}\left (c x^n\right )}\right )}{2 n}+a \log (x)-\frac{i b \sin ^{-1}\left (c x^n\right )^2}{2 n}+\frac{b \sin ^{-1}\left (c x^n\right ) \log \left (1-e^{2 i \sin ^{-1}\left (c x^n\right )}\right )}{n} \]

[Out]

((-I/2)*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*Poly
Log[2, E^((2*I)*ArcSin[c*x^n])])/n

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Rubi [A]  time = 0.104086, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {6742, 4830, 3717, 2190, 2279, 2391} \[ -\frac{i b \text{PolyLog}\left (2,e^{2 i \sin ^{-1}\left (c x^n\right )}\right )}{2 n}+a \log (x)-\frac{i b \sin ^{-1}\left (c x^n\right )^2}{2 n}+\frac{b \sin ^{-1}\left (c x^n\right ) \log \left (1-e^{2 i \sin ^{-1}\left (c x^n\right )}\right )}{n} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSin[c*x^n])/x,x]

[Out]

((-I/2)*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*Poly
Log[2, E^((2*I)*ArcSin[c*x^n])])/n

Rule 6742

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

Rule 4830

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 3717

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 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2279

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 2391

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]

Rubi steps

\begin{align*} \int \frac{a+b \sin ^{-1}\left (c x^n\right )}{x} \, dx &=\int \left (\frac{a}{x}+\frac{b \sin ^{-1}\left (c x^n\right )}{x}\right ) \, dx\\ &=a \log (x)+b \int \frac{\sin ^{-1}\left (c x^n\right )}{x} \, dx\\ &=a \log (x)+\frac{b \operatorname{Subst}\left (\int x \cot (x) \, dx,x,\sin ^{-1}\left (c x^n\right )\right )}{n}\\ &=-\frac{i b \sin ^{-1}\left (c x^n\right )^2}{2 n}+a \log (x)-\frac{(2 i b) \operatorname{Subst}\left (\int \frac{e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\sin ^{-1}\left (c x^n\right )\right )}{n}\\ &=-\frac{i b \sin ^{-1}\left (c x^n\right )^2}{2 n}+\frac{b \sin ^{-1}\left (c x^n\right ) \log \left (1-e^{2 i \sin ^{-1}\left (c x^n\right )}\right )}{n}+a \log (x)-\frac{b \operatorname{Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}\left (c x^n\right )\right )}{n}\\ &=-\frac{i b \sin ^{-1}\left (c x^n\right )^2}{2 n}+\frac{b \sin ^{-1}\left (c x^n\right ) \log \left (1-e^{2 i \sin ^{-1}\left (c x^n\right )}\right )}{n}+a \log (x)+\frac{(i b) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}\left (c x^n\right )}\right )}{2 n}\\ &=-\frac{i b \sin ^{-1}\left (c x^n\right )^2}{2 n}+\frac{b \sin ^{-1}\left (c x^n\right ) \log \left (1-e^{2 i \sin ^{-1}\left (c x^n\right )}\right )}{n}+a \log (x)-\frac{i b \text{Li}_2\left (e^{2 i \sin ^{-1}\left (c x^n\right )}\right )}{2 n}\\ \end{align*}

Mathematica [B]  time = 0.17181, size = 157, normalized size = 2.09 \[ -\frac{b c \left (\log (x) \log \left (\sqrt{-c^2} x^n+\sqrt{1-c^2 x^{2 n}}\right )+\frac{i \left (i \sinh ^{-1}\left (\sqrt{-c^2} x^n\right ) \log \left (1-e^{-2 \sinh ^{-1}\left (\sqrt{-c^2} x^n\right )}\right )-\frac{1}{2} i \left (\text{PolyLog}\left (2,e^{-2 \sinh ^{-1}\left (\sqrt{-c^2} x^n\right )}\right )-\sinh ^{-1}\left (\sqrt{-c^2} x^n\right )^2\right )\right )}{n}\right )}{\sqrt{-c^2}}+a \log (x)+b \log (x) \sin ^{-1}\left (c x^n\right ) \]

Warning: Unable to verify antiderivative.

[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]

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Maple [A]  time = 0.004, size = 164, normalized size = 2.2 \begin{align*}{\frac{a\ln \left ( c{x}^{n} \right ) }{n}}-{\frac{{\frac{i}{2}}b \left ( \arcsin \left ( c{x}^{n} \right ) \right ) ^{2}}{n}}+{\frac{b\arcsin \left ( c{x}^{n} \right ) }{n}\ln \left ( 1+ic{x}^{n}+\sqrt{1-{c}^{2} \left ({x}^{n} \right ) ^{2}} \right ) }+{\frac{b\arcsin \left ( c{x}^{n} \right ) }{n}\ln \left ( 1-ic{x}^{n}-\sqrt{1-{c}^{2} \left ({x}^{n} \right ) ^{2}} \right ) }-{\frac{ib}{n}{\it polylog} \left ( 2,-ic{x}^{n}-\sqrt{1-{c}^{2} \left ({x}^{n} \right ) ^{2}} \right ) }-{\frac{ib}{n}{\it polylog} \left ( 2,ic{x}^{n}+\sqrt{1-{c}^{2} \left ({x}^{n} \right ) ^{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\left (c n \int \frac{\sqrt{c x^{n} + 1} \sqrt{-c x^{n} + 1} x^{n} \log \left (x\right )}{c^{2} x x^{2 \, n} - x}\,{d x} + \arctan \left (c x^{n}, \sqrt{c x^{n} + 1} \sqrt{-c x^{n} + 1}\right ) \log \left (x\right )\right )} b + a \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \operatorname{asin}{\left (c x^{n} \right )}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*asin(c*x**n))/x, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \arcsin \left (c x^{n}\right ) + a}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arcsin(c*x^n) + a)/x, x)