\(\int \log (a \sin ^n(x)) \, dx\) [163]

Optimal result

Integrand size = 7, antiderivative size = 52 \[ \int \log \left (a \sin ^n(x)\right ) \, dx=\frac {1}{2} i n x^2-n x \log \left (1-e^{2 i x}\right )+x \log \left (a \sin ^n(x)\right )+\frac {1}{2} i n \operatorname {PolyLog}\left (2,e^{2 i x}\right ) \] Output:

1/2*I*n*x^2-n*x*ln(1-exp(2*I*x))+x*ln(a*sin(x)^n)+1/2*I*n*polylog(2,exp(2* 
I*x))
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00 \[ \int \log \left (a \sin ^n(x)\right ) \, dx=\frac {1}{2} i n x^2-n x \log \left (1-e^{2 i x}\right )+x \log \left (a \sin ^n(x)\right )+\frac {1}{2} i n \operatorname {PolyLog}\left (2,e^{2 i x}\right ) \] Input:

Integrate[Log[a*Sin[x]^n],x]
 

Output:

(I/2)*n*x^2 - n*x*Log[1 - E^((2*I)*x)] + x*Log[a*Sin[x]^n] + (I/2)*n*PolyL 
og[2, E^((2*I)*x)]
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.13, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.286, Rules used = {3028, 27, 3042, 25, 4200, 25, 2620, 2715, 2838}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \log \left (a \sin ^n(x)\right ) \, dx\)

\(\Big \downarrow \) 3028

\(\displaystyle x \log \left (a \sin ^n(x)\right )-\int n x \cot (x)dx\)

\(\Big \downarrow \) 27

\(\displaystyle x \log \left (a \sin ^n(x)\right )-n \int x \cot (x)dx\)

\(\Big \downarrow \) 3042

\(\displaystyle x \log \left (a \sin ^n(x)\right )-n \int -x \tan \left (x+\frac {\pi }{2}\right )dx\)

\(\Big \downarrow \) 25

\(\displaystyle n \int x \tan \left (x+\frac {\pi }{2}\right )dx+x \log \left (a \sin ^n(x)\right )\)

\(\Big \downarrow \) 4200

\(\displaystyle x \log \left (a \sin ^n(x)\right )+n \left (\frac {i x^2}{2}-2 i \int -\frac {e^{2 i x} x}{1-e^{2 i x}}dx\right )\)

\(\Big \downarrow \) 25

\(\displaystyle x \log \left (a \sin ^n(x)\right )+n \left (2 i \int \frac {e^{2 i x} x}{1-e^{2 i x}}dx+\frac {i x^2}{2}\right )\)

\(\Big \downarrow \) 2620

\(\displaystyle x \log \left (a \sin ^n(x)\right )+n \left (2 i \left (\frac {1}{2} i x \log \left (1-e^{2 i x}\right )-\frac {1}{2} i \int \log \left (1-e^{2 i x}\right )dx\right )+\frac {i x^2}{2}\right )\)

\(\Big \downarrow \) 2715

\(\displaystyle x \log \left (a \sin ^n(x)\right )+n \left (2 i \left (\frac {1}{2} i x \log \left (1-e^{2 i x}\right )-\frac {1}{4} \int e^{-2 i x} \log \left (1-e^{2 i x}\right )de^{2 i x}\right )+\frac {i x^2}{2}\right )\)

\(\Big \downarrow \) 2838

\(\displaystyle x \log \left (a \sin ^n(x)\right )+n \left (2 i \left (\frac {1}{4} \operatorname {PolyLog}\left (2,e^{2 i x}\right )+\frac {1}{2} i x \log \left (1-e^{2 i x}\right )\right )+\frac {i x^2}{2}\right )\)

Input:

Int[Log[a*Sin[x]^n],x]
 

Output:

x*Log[a*Sin[x]^n] + n*((I/2)*x^2 + (2*I)*((I/2)*x*Log[1 - E^((2*I)*x)] + P 
olyLog[2, E^((2*I)*x)]/4))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2620

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] - Si 
mp[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 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[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 2838

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 3028

Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[x*(D[u, 
 x]/u), x], x] /; InverseFunctionFreeQ[u, x]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4200

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol 
] :> Simp[I*((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[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]
 

Maple [F]

\[\int \ln \left (a \sin \left (x \right )^{n}\right )d x\]

Input:

int(ln(a*sin(x)^n),x)
 

Output:

int(ln(a*sin(x)^n),x)
 

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (37) = 74\).

Time = 0.10 (sec) , antiderivative size = 115, normalized size of antiderivative = 2.21 \[ \int \log \left (a \sin ^n(x)\right ) \, dx=-\frac {1}{2} \, n x \log \left (\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) - \frac {1}{2} \, n x \log \left (\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) - \frac {1}{2} \, n x \log \left (-\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) - \frac {1}{2} \, n x \log \left (-\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) + n x \log \left (\sin \left (x\right )\right ) + \frac {1}{2} i \, n {\rm Li}_2\left (\cos \left (x\right ) + i \, \sin \left (x\right )\right ) - \frac {1}{2} i \, n {\rm Li}_2\left (\cos \left (x\right ) - i \, \sin \left (x\right )\right ) - \frac {1}{2} i \, n {\rm Li}_2\left (-\cos \left (x\right ) + i \, \sin \left (x\right )\right ) + \frac {1}{2} i \, n {\rm Li}_2\left (-\cos \left (x\right ) - i \, \sin \left (x\right )\right ) + x \log \left (a\right ) \] Input:

integrate(log(a*sin(x)^n),x, algorithm="fricas")
 

Output:

-1/2*n*x*log(cos(x) + I*sin(x) + 1) - 1/2*n*x*log(cos(x) - I*sin(x) + 1) - 
 1/2*n*x*log(-cos(x) + I*sin(x) + 1) - 1/2*n*x*log(-cos(x) - I*sin(x) + 1) 
 + n*x*log(sin(x)) + 1/2*I*n*dilog(cos(x) + I*sin(x)) - 1/2*I*n*dilog(cos( 
x) - I*sin(x)) - 1/2*I*n*dilog(-cos(x) + I*sin(x)) + 1/2*I*n*dilog(-cos(x) 
 - I*sin(x)) + x*log(a)
 

Sympy [F]

\[ \int \log \left (a \sin ^n(x)\right ) \, dx=\int \log {\left (a \sin ^{n}{\left (x \right )} \right )}\, dx \] Input:

integrate(ln(a*sin(x)**n),x)
 

Output:

Integral(log(a*sin(x)**n), x)
 

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (37) = 74\).

Time = 0.24 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.75 \[ \int \log \left (a \sin ^n(x)\right ) \, dx=-\frac {1}{2} \, {\left (-i \, x^{2} + 2 i \, x \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right ) - 2 i \, x \arctan \left (\sin \left (x\right ), -\cos \left (x\right ) + 1\right ) + x \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) + x \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) - 2 i \, {\rm Li}_2\left (-e^{\left (i \, x\right )}\right ) - 2 i \, {\rm Li}_2\left (e^{\left (i \, x\right )}\right )\right )} n + x \log \left (a \sin \left (x\right )^{n}\right ) \] Input:

integrate(log(a*sin(x)^n),x, algorithm="maxima")
 

Output:

-1/2*(-I*x^2 + 2*I*x*arctan2(sin(x), cos(x) + 1) - 2*I*x*arctan2(sin(x), - 
cos(x) + 1) + x*log(cos(x)^2 + sin(x)^2 + 2*cos(x) + 1) + x*log(cos(x)^2 + 
 sin(x)^2 - 2*cos(x) + 1) - 2*I*dilog(-e^(I*x)) - 2*I*dilog(e^(I*x)))*n + 
x*log(a*sin(x)^n)
 

Giac [F]

\[ \int \log \left (a \sin ^n(x)\right ) \, dx=\int { \log \left (a \sin \left (x\right )^{n}\right ) \,d x } \] Input:

integrate(log(a*sin(x)^n),x, algorithm="giac")
 

Output:

integrate(log(a*sin(x)^n), x)
 

Mupad [F(-1)]

Timed out. \[ \int \log \left (a \sin ^n(x)\right ) \, dx=\int \ln \left (a\,{\sin \left (x\right )}^n\right ) \,d x \] Input:

int(log(a*sin(x)^n),x)
 

Output:

int(log(a*sin(x)^n), x)
 

Reduce [F]

\[ \int \log \left (a \sin ^n(x)\right ) \, dx=\int \mathrm {log}\left (\sin \left (x \right )^{n} a \right )d x \] Input:

int(log(a*sin(x)^n),x)
 

Output:

int(log(sin(x)**n*a),x)