Integrand size = 7, antiderivative size = 44 \[ \int \log \left (a \cosh ^n(x)\right ) \, dx=\frac {n x^2}{2}-n x \log \left (1+e^{2 x}\right )+x \log \left (a \cosh ^n(x)\right )-\frac {1}{2} n \operatorname {PolyLog}\left (2,-e^{2 x}\right ) \] Output:
1/2*n*x^2-n*x*ln(1+exp(2*x))+x*ln(a*cosh(x)^n)-1/2*n*polylog(2,-exp(2*x))
Time = 0.04 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.98 \[ \int \log \left (a \cosh ^n(x)\right ) \, dx=\frac {1}{2} \left (-x \left (n x+2 n \log \left (1+e^{-2 x}\right )-2 \log \left (a \cosh ^n(x)\right )\right )+n \operatorname {PolyLog}\left (2,-e^{-2 x}\right )\right ) \] Input:
Integrate[Log[a*Cosh[x]^n],x]
Output:
(-(x*(n*x + 2*n*Log[1 + E^(-2*x)] - 2*Log[a*Cosh[x]^n])) + n*PolyLog[2, -E ^(-2*x)])/2
Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.27, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.143, Rules used = {3028, 27, 3042, 26, 4201, 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.
\(\displaystyle \int \log \left (a \cosh ^n(x)\right ) \, dx\) |
\(\Big \downarrow \) 3028 |
\(\displaystyle x \log \left (a \cosh ^n(x)\right )-\int n x \tanh (x)dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle x \log \left (a \cosh ^n(x)\right )-n \int x \tanh (x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle x \log \left (a \cosh ^n(x)\right )-n \int -i x \tan (i x)dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle x \log \left (a \cosh ^n(x)\right )+i n \int x \tan (i x)dx\) |
\(\Big \downarrow \) 4201 |
\(\displaystyle x \log \left (a \cosh ^n(x)\right )+i n \left (2 i \int \frac {e^{2 x} x}{1+e^{2 x}}dx-\frac {i x^2}{2}\right )\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle x \log \left (a \cosh ^n(x)\right )+i n \left (2 i \left (\frac {1}{2} x \log \left (e^{2 x}+1\right )-\frac {1}{2} \int \log \left (1+e^{2 x}\right )dx\right )-\frac {i x^2}{2}\right )\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle x \log \left (a \cosh ^n(x)\right )+i n \left (2 i \left (\frac {1}{2} x \log \left (e^{2 x}+1\right )-\frac {1}{4} \int e^{-2 x} \log \left (1+e^{2 x}\right )de^{2 x}\right )-\frac {i x^2}{2}\right )\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle x \log \left (a \cosh ^n(x)\right )+i n \left (2 i \left (\frac {1}{4} \operatorname {PolyLog}\left (2,-e^{2 x}\right )+\frac {1}{2} x \log \left (e^{2 x}+1\right )\right )-\frac {i x^2}{2}\right )\) |
Input:
Int[Log[a*Cosh[x]^n],x]
Output:
x*Log[a*Cosh[x]^n] + I*n*((-1/2*I)*x^2 + (2*I)*((x*Log[1 + E^(2*x)])/2 + P olyLog[2, -E^(2*x)]/4))
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
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]
Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[x*(D[u, x]/u), x], x] /; InverseFunctionFreeQ[u, x]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(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)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
\[\int \ln \left (a \cosh \left (x \right )^{n}\right )d x\]
Input:
int(ln(a*cosh(x)^n),x)
Output:
int(ln(a*cosh(x)^n),x)
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.66 \[ \int \log \left (a \cosh ^n(x)\right ) \, dx=\frac {1}{2} \, n x^{2} - n x \log \left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right ) + 1\right ) - n x \log \left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right ) + 1\right ) + n x \log \left (\cosh \left (x\right )\right ) - n {\rm Li}_2\left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right )\right ) - n {\rm Li}_2\left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right )\right ) + x \log \left (a\right ) \] Input:
integrate(log(a*cosh(x)^n),x, algorithm="fricas")
Output:
1/2*n*x^2 - n*x*log(I*cosh(x) + I*sinh(x) + 1) - n*x*log(-I*cosh(x) - I*si nh(x) + 1) + n*x*log(cosh(x)) - n*dilog(I*cosh(x) + I*sinh(x)) - n*dilog(- I*cosh(x) - I*sinh(x)) + x*log(a)
\[ \int \log \left (a \cosh ^n(x)\right ) \, dx=\int \log {\left (a \cosh ^{n}{\left (x \right )} \right )}\, dx \] Input:
integrate(ln(a*cosh(x)**n),x)
Output:
Integral(log(a*cosh(x)**n), x)
Time = 0.15 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.82 \[ \int \log \left (a \cosh ^n(x)\right ) \, dx=\frac {1}{2} \, {\left (x^{2} - 2 \, x \log \left (e^{\left (2 \, x\right )} + 1\right ) - {\rm Li}_2\left (-e^{\left (2 \, x\right )}\right )\right )} n + x \log \left (a \cosh \left (x\right )^{n}\right ) \] Input:
integrate(log(a*cosh(x)^n),x, algorithm="maxima")
Output:
1/2*(x^2 - 2*x*log(e^(2*x) + 1) - dilog(-e^(2*x)))*n + x*log(a*cosh(x)^n)
\[ \int \log \left (a \cosh ^n(x)\right ) \, dx=\int { \log \left (a \cosh \left (x\right )^{n}\right ) \,d x } \] Input:
integrate(log(a*cosh(x)^n),x, algorithm="giac")
Output:
integrate(log(a*cosh(x)^n), x)
Timed out. \[ \int \log \left (a \cosh ^n(x)\right ) \, dx=\int \ln \left (a\,{\mathrm {cosh}\left (x\right )}^n\right ) \,d x \] Input:
int(log(a*cosh(x)^n),x)
Output:
int(log(a*cosh(x)^n), x)
\[ \int \log \left (a \cosh ^n(x)\right ) \, dx=\int \mathrm {log}\left (\cosh \left (x \right )^{n} a \right )d x \] Input:
int(log(a*cosh(x)^n),x)
Output:
int(log(cosh(x)**n*a),x)