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

Optimal. Leaf size=75 \[ a \log (x)-\frac {i b \text {Li}_2\left (e^{2 i \sin ^{-1}\left (c x^n\right )}\right )}{2 n}-\frac {i b \sin ^{-1}\left (c x^n\right )^2}{2 n}+\frac {b \sin ^{-1}\left (c x^n\right ) \log \left (1-e^{2 i \sin ^{-1}\left (c x^n\right )}\right )}{n} \]

[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

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Rubi [A]  time = 0.10, antiderivative size = 75, normalized size of antiderivative = 1.00, 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 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]

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 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 6742

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

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*}

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Mathematica [B]  time = 0.21, size = 157, normalized size = 2.09 \[ a \log (x)-\frac {b c \left (\log (x) \log \left (\sqrt {1-c^2 x^{2 n}}+\sqrt {-c^2} x^n\right )+\frac {i \left (i \sinh ^{-1}\left (\sqrt {-c^2} x^n\right ) \log \left (1-e^{-2 \sinh ^{-1}\left (\sqrt {-c^2} x^n\right )}\right )-\frac {1}{2} i \left (\text {Li}_2\left (e^{-2 \sinh ^{-1}\left (\sqrt {-c^2} x^n\right )}\right )-\sinh ^{-1}\left (\sqrt {-c^2} x^n\right )^2\right )\right )}{n}\right )}{\sqrt {-c^2}}+b \log (x) \sin ^{-1}\left (c x^n\right ) \]

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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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \arcsin \left (c x^{n}\right ) + a}{x}\,{d x} \]

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)

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maple [A]  time = 0.01, size = 164, normalized size = 2.19 \[ \frac {a \ln \left (c \,x^{n}\right )}{n}-\frac {i b \arcsin \left (c \,x^{n}\right )^{2}}{2 n}+\frac {b \arcsin \left (c \,x^{n}\right ) \ln \left (1+i c \,x^{n}+\sqrt {1-c^{2} x^{2 n}}\right )}{n}+\frac {b \arcsin \left (c \,x^{n}\right ) \ln \left (1-i c \,x^{n}-\sqrt {1-c^{2} x^{2 n}}\right )}{n}-\frac {i b \polylog \left (2, -i c \,x^{n}-\sqrt {1-c^{2} x^{2 n}}\right )}{n}-\frac {i b \polylog \left (2, i c \,x^{n}+\sqrt {1-c^{2} x^{2 n}}\right )}{n} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ {\left (c n \int \frac {\sqrt {c x^{n} + 1} \sqrt {-c x^{n} + 1} x^{n} \log \relax (x)}{c^{2} x x^{2 \, n} - x}\,{d x} + \arctan \left (c x^{n}, \sqrt {c x^{n} + 1} \sqrt {-c x^{n} + 1}\right ) \log \relax (x)\right )} b + a \log \relax (x) \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {asin}\left (c\,x^n\right )}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asin(c*x^n))/x,x)

[Out]

int((a + b*asin(c*x^n))/x, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \operatorname {asin}{\left (c x^{n} \right )}}{x}\, dx \]

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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