\(\int \log (a \sinh ^2(x)) \, dx\) [202]

Optimal result

Integrand size = 7, antiderivative size = 35 \[ \int \log \left (a \sinh ^2(x)\right ) \, dx=x^2-2 x \log \left (1-e^{2 x}\right )+x \log \left (a \sinh ^2(x)\right )-\operatorname {PolyLog}\left (2,e^{2 x}\right ) \]

[Out]

x^2-2*x*ln(1-exp(2*x))+x*ln(a*sinh(x)^2)-polylog(2,exp(2*x))

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {2628, 12, 3797, 2221, 2317, 2438} \[ \int \log \left (a \sinh ^2(x)\right ) \, dx=x \log \left (a \sinh ^2(x)\right )-\operatorname {PolyLog}\left (2,e^{2 x}\right )+x^2-2 x \log \left (1-e^{2 x}\right ) \]

[In]

Int[Log[a*Sinh[x]^2],x]

[Out]

x^2 - 2*x*Log[1 - E^(2*x)] + x*Log[a*Sinh[x]^2] - PolyLog[2, E^(2*x)]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 2628

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

Rule 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(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)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = x \log \left (a \sinh ^2(x)\right )-\int 2 x \coth (x) \, dx \\ & = x \log \left (a \sinh ^2(x)\right )-2 \int x \coth (x) \, dx \\ & = x^2+x \log \left (a \sinh ^2(x)\right )+4 \int \frac {e^{2 x} x}{1-e^{2 x}} \, dx \\ & = x^2-2 x \log \left (1-e^{2 x}\right )+x \log \left (a \sinh ^2(x)\right )+2 \int \log \left (1-e^{2 x}\right ) \, dx \\ & = x^2-2 x \log \left (1-e^{2 x}\right )+x \log \left (a \sinh ^2(x)\right )+\text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 x}\right ) \\ & = x^2-2 x \log \left (1-e^{2 x}\right )+x \log \left (a \sinh ^2(x)\right )-\text {Li}_2\left (e^{2 x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94 \[ \int \log \left (a \sinh ^2(x)\right ) \, dx=x \left (-x-2 \log \left (1-e^{-2 x}\right )+\log \left (a \sinh ^2(x)\right )\right )+\operatorname {PolyLog}\left (2,e^{-2 x}\right ) \]

[In]

Integrate[Log[a*Sinh[x]^2],x]

[Out]

x*(-x - 2*Log[1 - E^(-2*x)] + Log[a*Sinh[x]^2]) + PolyLog[2, E^(-2*x)]

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 3.76 (sec) , antiderivative size = 454, normalized size of antiderivative = 12.97

method	result	size
risch	\(\ln \left (a \right ) x -\frac {i \pi {\operatorname {csgn}\left (i a \left (-1+{\mathrm e}^{2 x}\right )^{2} {\mathrm e}^{-2 x}\right )}^{3} x}{2}-2 x \ln \left (2\right )+2 \operatorname {dilog}\left ({\mathrm e}^{x}\right )+x^{2}+\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-2 x} \left (-1+{\mathrm e}^{2 x}\right )^{2}\right ) {\operatorname {csgn}\left (i a \left (-1+{\mathrm e}^{2 x}\right )^{2} {\mathrm e}^{-2 x}\right )}^{2} x}{2}-i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{2 x}\right )^{2} x +\frac {i \pi \operatorname {csgn}\left (i {\mathrm e}^{x}\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{2 x}\right ) x}{2}+i \pi \,\operatorname {csgn}\left (i \left (-1+{\mathrm e}^{2 x}\right )\right ) {\operatorname {csgn}\left (i \left (-1+{\mathrm e}^{2 x}\right )^{2}\right )}^{2} x -\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-2 x} \left (-1+{\mathrm e}^{2 x}\right )^{2}\right ) \operatorname {csgn}\left (i a \left (-1+{\mathrm e}^{2 x}\right )^{2} {\mathrm e}^{-2 x}\right ) \operatorname {csgn}\left (i a \right ) x}{2}-\frac {i \pi {\operatorname {csgn}\left (i \left (-1+{\mathrm e}^{2 x}\right )\right )}^{2} \operatorname {csgn}\left (i \left (-1+{\mathrm e}^{2 x}\right )^{2}\right ) x}{2}-2 \operatorname {dilog}\left (1+{\mathrm e}^{x}\right )-\frac {i \pi {\operatorname {csgn}\left (i \left (-1+{\mathrm e}^{2 x}\right )^{2}\right )}^{3} x}{2}-\frac {i \pi \operatorname {csgn}\left (i {\mathrm e}^{-2 x} \left (-1+{\mathrm e}^{2 x}\right )^{2}\right )^{3} x}{2}+\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-2 x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-2 x} \left (-1+{\mathrm e}^{2 x}\right )^{2}\right )^{2} x}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (-1+{\mathrm e}^{2 x}\right )^{2}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-2 x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-2 x} \left (-1+{\mathrm e}^{2 x}\right )^{2}\right ) x}{2}+\frac {i \pi \operatorname {csgn}\left (i {\mathrm e}^{2 x}\right )^{3} x}{2}+\frac {i \pi \,\operatorname {csgn}\left (i \left (-1+{\mathrm e}^{2 x}\right )^{2}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-2 x} \left (-1+{\mathrm e}^{2 x}\right )^{2}\right )^{2} x}{2}+\frac {i \pi {\operatorname {csgn}\left (i a \left (-1+{\mathrm e}^{2 x}\right )^{2} {\mathrm e}^{-2 x}\right )}^{2} \operatorname {csgn}\left (i a \right ) x}{2}-2 \ln \left ({\mathrm e}^{x}\right ) \ln \left (1+{\mathrm e}^{x}\right )-2 x \ln \left ({\mathrm e}^{x}\right )+2 \ln \left ({\mathrm e}^{x}\right ) \ln \left (-1+{\mathrm e}^{2 x}\right )\)	\(454\)

[In]

int(ln(a*sinh(x)^2),x,method=_RETURNVERBOSE)

[Out]

ln(a)*x-1/2*I*Pi*csgn(I*a*(-1+exp(2*x))^2*exp(-2*x))^3*x-2*x*ln(2)+2*dilog(exp(x))+x^2+1/2*I*Pi*csgn(I*exp(-2*
x)*(-1+exp(2*x))^2)*csgn(I*a*(-1+exp(2*x))^2*exp(-2*x))^2*x-I*Pi*csgn(I*exp(x))*csgn(I*exp(2*x))^2*x+1/2*I*Pi*
csgn(I*exp(x))^2*csgn(I*exp(2*x))*x+I*Pi*csgn(I*(-1+exp(2*x)))*csgn(I*(-1+exp(2*x))^2)^2*x-1/2*I*Pi*csgn(I*exp
(-2*x)*(-1+exp(2*x))^2)*csgn(I*a*(-1+exp(2*x))^2*exp(-2*x))*csgn(I*a)*x-1/2*I*Pi*csgn(I*(-1+exp(2*x)))^2*csgn(
I*(-1+exp(2*x))^2)*x-2*dilog(1+exp(x))-1/2*I*Pi*csgn(I*(-1+exp(2*x))^2)^3*x-1/2*I*Pi*csgn(I*exp(-2*x)*(-1+exp(
2*x))^2)^3*x+1/2*I*Pi*csgn(I*exp(-2*x))*csgn(I*exp(-2*x)*(-1+exp(2*x))^2)^2*x-1/2*I*Pi*csgn(I*(-1+exp(2*x))^2)
*csgn(I*exp(-2*x))*csgn(I*exp(-2*x)*(-1+exp(2*x))^2)*x+1/2*I*Pi*csgn(I*exp(2*x))^3*x+1/2*I*Pi*csgn(I*(-1+exp(2
*x))^2)*csgn(I*exp(-2*x)*(-1+exp(2*x))^2)^2*x+1/2*I*Pi*csgn(I*a*(-1+exp(2*x))^2*exp(-2*x))^2*csgn(I*a)*x-2*ln(
exp(x))*ln(1+exp(x))-2*x*ln(exp(x))+2*ln(exp(x))*ln(-1+exp(2*x))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (32) = 64\).

Time = 0.32 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.97 \[ \int \log \left (a \sinh ^2(x)\right ) \, dx=x^{2} + x \log \left (\frac {1}{2} \, a \cosh \left (x\right )^{2} + \frac {1}{2} \, a \sinh \left (x\right )^{2} - \frac {1}{2} \, a\right ) - 2 \, x \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) - 2 \, x \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right ) - 2 \, {\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - 2 \, {\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) \]

[In]

integrate(log(a*sinh(x)^2),x, algorithm="fricas")

[Out]

x^2 + x*log(1/2*a*cosh(x)^2 + 1/2*a*sinh(x)^2 - 1/2*a) - 2*x*log(cosh(x) + sinh(x) + 1) - 2*x*log(-cosh(x) - s
inh(x) + 1) - 2*dilog(cosh(x) + sinh(x)) - 2*dilog(-cosh(x) - sinh(x))

Sympy [F]

\[ \int \log \left (a \sinh ^2(x)\right ) \, dx=\int \log {\left (a \sinh ^{2}{\left (x \right )} \right )}\, dx \]

[In]

integrate(ln(a*sinh(x)**2),x)

[Out]

Integral(log(a*sinh(x)**2), x)

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.23 \[ \int \log \left (a \sinh ^2(x)\right ) \, dx=x^{2} + x \log \left (a \sinh \left (x\right )^{2}\right ) - 2 \, x \log \left (e^{x} + 1\right ) - 2 \, x \log \left (-e^{x} + 1\right ) - 2 \, {\rm Li}_2\left (-e^{x}\right ) - 2 \, {\rm Li}_2\left (e^{x}\right ) \]

[In]

integrate(log(a*sinh(x)^2),x, algorithm="maxima")

[Out]

x^2 + x*log(a*sinh(x)^2) - 2*x*log(e^x + 1) - 2*x*log(-e^x + 1) - 2*dilog(-e^x) - 2*dilog(e^x)

Giac [F]

\[ \int \log \left (a \sinh ^2(x)\right ) \, dx=\int { \log \left (a \sinh \left (x\right )^{2}\right ) \,d x } \]

[In]

integrate(log(a*sinh(x)^2),x, algorithm="giac")

[Out]

integrate(log(a*sinh(x)^2), x)

Mupad [F(-1)]

Timed out. \[ \int \log \left (a \sinh ^2(x)\right ) \, dx=\int \ln \left (a\,{\mathrm {sinh}\left (x\right )}^2\right ) \,d x \]

[In]

int(log(a*sinh(x)^2),x)

[Out]

int(log(a*sinh(x)^2), x)