3.29 \(\int \cosh (x) \log ^2(\cosh ^2(x)+\sinh (x)) \, dx\)

Optimal. Leaf size=395 \[ -\left (1+i \sqrt{3}\right ) \text{PolyLog}\left (2,-\frac{2 i \sinh (x)-\sqrt{3}+i}{2 \sqrt{3}}\right )-\left (1-i \sqrt{3}\right ) \text{PolyLog}\left (2,\frac{2 i \sinh (x)+\sqrt{3}+i}{2 \sqrt{3}}\right )+8 \sinh (x)+\sinh (x) \log ^2\left (\sinh ^2(x)+\sinh (x)+1\right )-\frac{1}{2} \left (1-i \sqrt{3}\right ) \log ^2\left (2 \sinh (x)-i \sqrt{3}+1\right )-\frac{1}{2} \left (1+i \sqrt{3}\right ) \log ^2\left (2 \sinh (x)+i \sqrt{3}+1\right )+\left (1-i \sqrt{3}\right ) \log \left (\sinh ^2(x)+\sinh (x)+1\right ) \log \left (2 \sinh (x)-i \sqrt{3}+1\right )+\left (1+i \sqrt{3}\right ) \log \left (2 \sinh (x)+i \sqrt{3}+1\right ) \log \left (\sinh ^2(x)+\sinh (x)+1\right )-2 \log \left (\sinh ^2(x)+\sinh (x)+1\right )-4 \sinh (x) \log \left (\sinh ^2(x)+\sinh (x)+1\right )-\left (1-i \sqrt{3}\right ) \log \left (-\frac{i \left (2 \sinh (x)+i \sqrt{3}+1\right )}{2 \sqrt{3}}\right ) \log \left (2 \sinh (x)-i \sqrt{3}+1\right )-\left (1+i \sqrt{3}\right ) \log \left (\frac{i \left (2 \sinh (x)-i \sqrt{3}+1\right )}{2 \sqrt{3}}\right ) \log \left (2 \sinh (x)+i \sqrt{3}+1\right )-4 \sqrt{3} \tan ^{-1}\left (\frac{2 \sinh (x)+1}{\sqrt{3}}\right ) \]

[Out]

-4*Sqrt[3]*ArcTan[(1 + 2*Sinh[x])/Sqrt[3]] - ((1 - I*Sqrt[3])*Log[1 - I*Sqrt[3] + 2*Sinh[x]]^2)/2 - (1 + I*Sqr
t[3])*Log[((I/2)*(1 - I*Sqrt[3] + 2*Sinh[x]))/Sqrt[3]]*Log[1 + I*Sqrt[3] + 2*Sinh[x]] - ((1 + I*Sqrt[3])*Log[1
+ I*Sqrt[3] + 2*Sinh[x]]^2)/2 - (1 - I*Sqrt[3])*Log[1 - I*Sqrt[3] + 2*Sinh[x]]*Log[((-I/2)*(1 + I*Sqrt[3] + 2
*Sinh[x]))/Sqrt[3]] - 2*Log[1 + Sinh[x] + Sinh[x]^2] + (1 - I*Sqrt[3])*Log[1 - I*Sqrt[3] + 2*Sinh[x]]*Log[1 +
Sinh[x] + Sinh[x]^2] + (1 + I*Sqrt[3])*Log[1 + I*Sqrt[3] + 2*Sinh[x]]*Log[1 + Sinh[x] + Sinh[x]^2] - (1 + I*Sq
rt[3])*PolyLog[2, -(I - Sqrt[3] + (2*I)*Sinh[x])/(2*Sqrt[3])] - (1 - I*Sqrt[3])*PolyLog[2, (I + Sqrt[3] + (2*I
)*Sinh[x])/(2*Sqrt[3])] + 8*Sinh[x] - 4*Log[1 + Sinh[x] + Sinh[x]^2]*Sinh[x] + Log[1 + Sinh[x] + Sinh[x]^2]^2*
Sinh[x]

________________________________________________________________________________________

Rubi [A]  time = 0.536977, antiderivative size = 395, normalized size of antiderivative = 1., number of steps used = 28, number of rules used = 15, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.154, Rules used = {4358, 2523, 2528, 773, 634, 618, 204, 628, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ -\left (1+i \sqrt{3}\right ) \text{PolyLog}\left (2,-\frac{2 i \sinh (x)-\sqrt{3}+i}{2 \sqrt{3}}\right )-\left (1-i \sqrt{3}\right ) \text{PolyLog}\left (2,\frac{2 i \sinh (x)+\sqrt{3}+i}{2 \sqrt{3}}\right )+8 \sinh (x)+\sinh (x) \log ^2\left (\sinh ^2(x)+\sinh (x)+1\right )-\frac{1}{2} \left (1-i \sqrt{3}\right ) \log ^2\left (2 \sinh (x)-i \sqrt{3}+1\right )-\frac{1}{2} \left (1+i \sqrt{3}\right ) \log ^2\left (2 \sinh (x)+i \sqrt{3}+1\right )+\left (1-i \sqrt{3}\right ) \log \left (\sinh ^2(x)+\sinh (x)+1\right ) \log \left (2 \sinh (x)-i \sqrt{3}+1\right )+\left (1+i \sqrt{3}\right ) \log \left (2 \sinh (x)+i \sqrt{3}+1\right ) \log \left (\sinh ^2(x)+\sinh (x)+1\right )-2 \log \left (\sinh ^2(x)+\sinh (x)+1\right )-4 \sinh (x) \log \left (\sinh ^2(x)+\sinh (x)+1\right )-\left (1-i \sqrt{3}\right ) \log \left (-\frac{i \left (2 \sinh (x)+i \sqrt{3}+1\right )}{2 \sqrt{3}}\right ) \log \left (2 \sinh (x)-i \sqrt{3}+1\right )-\left (1+i \sqrt{3}\right ) \log \left (\frac{i \left (2 \sinh (x)-i \sqrt{3}+1\right )}{2 \sqrt{3}}\right ) \log \left (2 \sinh (x)+i \sqrt{3}+1\right )-4 \sqrt{3} \tan ^{-1}\left (\frac{2 \sinh (x)+1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]*Log[Cosh[x]^2 + Sinh[x]]^2,x]

[Out]

-4*Sqrt[3]*ArcTan[(1 + 2*Sinh[x])/Sqrt[3]] - ((1 - I*Sqrt[3])*Log[1 - I*Sqrt[3] + 2*Sinh[x]]^2)/2 - (1 + I*Sqr
t[3])*Log[((I/2)*(1 - I*Sqrt[3] + 2*Sinh[x]))/Sqrt[3]]*Log[1 + I*Sqrt[3] + 2*Sinh[x]] - ((1 + I*Sqrt[3])*Log[1
+ I*Sqrt[3] + 2*Sinh[x]]^2)/2 - (1 - I*Sqrt[3])*Log[1 - I*Sqrt[3] + 2*Sinh[x]]*Log[((-I/2)*(1 + I*Sqrt[3] + 2
*Sinh[x]))/Sqrt[3]] - 2*Log[1 + Sinh[x] + Sinh[x]^2] + (1 - I*Sqrt[3])*Log[1 - I*Sqrt[3] + 2*Sinh[x]]*Log[1 +
Sinh[x] + Sinh[x]^2] + (1 + I*Sqrt[3])*Log[1 + I*Sqrt[3] + 2*Sinh[x]]*Log[1 + Sinh[x] + Sinh[x]^2] - (1 + I*Sq
rt[3])*PolyLog[2, -(I - Sqrt[3] + (2*I)*Sinh[x])/(2*Sqrt[3])] - (1 - I*Sqrt[3])*PolyLog[2, (I + Sqrt[3] + (2*I
)*Sinh[x])/(2*Sqrt[3])] + 8*Sinh[x] - 4*Log[1 + Sinh[x] + Sinh[x]^2]*Sinh[x] + Log[1 + Sinh[x] + Sinh[x]^2]^2*
Sinh[x]

Rule 4358

Int[Cosh[(c_.)*((a_.) + (b_.)*(x_))]*(u_), x_Symbol] :> With[{d = FreeFactors[Sinh[c*(a + b*x)], x]}, Dist[d/(
b*c), Subst[Int[SubstFor[1, Sinh[c*(a + b*x)]/d, u, x], x], x, Sinh[c*(a + b*x)]/d], x] /; FunctionOfQ[Sinh[c*
(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x]

Rule 2523

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Log[c*RFx^p])^n, x] - Dist[b*n*p
, Int[SimplifyIntegrand[(x*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, p}, x] &
& RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rule 773

Int[(((d_.) + (e_.)*(x_))*((f_) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*g*x)/
c, x] + Dist[1/c, Int[(c*d*f - a*e*g + (c*e*f + c*d*g - b*e*g)*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c,
d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 2524

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[d + e*x]*(a + b
*Log[c*RFx^p])^n)/e, x] - Dist[(b*n*p)/e, Int[(Log[d + e*x]*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x
] /; FreeQ[{a, b, c, d, e, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 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 \cosh (x) \log ^2\left (\cosh ^2(x)+\sinh (x)\right ) \, dx &=\operatorname{Subst}\left (\int \log ^2\left (1+x+x^2\right ) \, dx,x,\sinh (x)\right )\\ &=\log ^2\left (1+\sinh (x)+\sinh ^2(x)\right ) \sinh (x)-2 \operatorname{Subst}\left (\int \frac{x (1+2 x) \log \left (1+x+x^2\right )}{1+x+x^2} \, dx,x,\sinh (x)\right )\\ &=\log ^2\left (1+\sinh (x)+\sinh ^2(x)\right ) \sinh (x)-2 \operatorname{Subst}\left (\int \left (2 \log \left (1+x+x^2\right )-\frac{(2+x) \log \left (1+x+x^2\right )}{1+x+x^2}\right ) \, dx,x,\sinh (x)\right )\\ &=\log ^2\left (1+\sinh (x)+\sinh ^2(x)\right ) \sinh (x)+2 \operatorname{Subst}\left (\int \frac{(2+x) \log \left (1+x+x^2\right )}{1+x+x^2} \, dx,x,\sinh (x)\right )-4 \operatorname{Subst}\left (\int \log \left (1+x+x^2\right ) \, dx,x,\sinh (x)\right )\\ &=-4 \log \left (1+\sinh (x)+\sinh ^2(x)\right ) \sinh (x)+\log ^2\left (1+\sinh (x)+\sinh ^2(x)\right ) \sinh (x)+2 \operatorname{Subst}\left (\int \left (\frac{\left (1-i \sqrt{3}\right ) \log \left (1+x+x^2\right )}{1-i \sqrt{3}+2 x}+\frac{\left (1+i \sqrt{3}\right ) \log \left (1+x+x^2\right )}{1+i \sqrt{3}+2 x}\right ) \, dx,x,\sinh (x)\right )+4 \operatorname{Subst}\left (\int \frac{x (1+2 x)}{1+x+x^2} \, dx,x,\sinh (x)\right )\\ &=8 \sinh (x)-4 \log \left (1+\sinh (x)+\sinh ^2(x)\right ) \sinh (x)+\log ^2\left (1+\sinh (x)+\sinh ^2(x)\right ) \sinh (x)+4 \operatorname{Subst}\left (\int \frac{-2-x}{1+x+x^2} \, dx,x,\sinh (x)\right )+\left (2 \left (1-i \sqrt{3}\right )\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+x+x^2\right )}{1-i \sqrt{3}+2 x} \, dx,x,\sinh (x)\right )+\left (2 \left (1+i \sqrt{3}\right )\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+x+x^2\right )}{1+i \sqrt{3}+2 x} \, dx,x,\sinh (x)\right )\\ &=\left (1-i \sqrt{3}\right ) \log \left (1-i \sqrt{3}+2 \sinh (x)\right ) \log \left (1+\sinh (x)+\sinh ^2(x)\right )+\left (1+i \sqrt{3}\right ) \log \left (1+i \sqrt{3}+2 \sinh (x)\right ) \log \left (1+\sinh (x)+\sinh ^2(x)\right )+8 \sinh (x)-4 \log \left (1+\sinh (x)+\sinh ^2(x)\right ) \sinh (x)+\log ^2\left (1+\sinh (x)+\sinh ^2(x)\right ) \sinh (x)-2 \operatorname{Subst}\left (\int \frac{1+2 x}{1+x+x^2} \, dx,x,\sinh (x)\right )-6 \operatorname{Subst}\left (\int \frac{1}{1+x+x^2} \, dx,x,\sinh (x)\right )+\left (-1-i \sqrt{3}\right ) \operatorname{Subst}\left (\int \frac{(1+2 x) \log \left (1+i \sqrt{3}+2 x\right )}{1+x+x^2} \, dx,x,\sinh (x)\right )+\left (-1+i \sqrt{3}\right ) \operatorname{Subst}\left (\int \frac{(1+2 x) \log \left (1-i \sqrt{3}+2 x\right )}{1+x+x^2} \, dx,x,\sinh (x)\right )\\ &=-2 \log \left (1+\sinh (x)+\sinh ^2(x)\right )+\left (1-i \sqrt{3}\right ) \log \left (1-i \sqrt{3}+2 \sinh (x)\right ) \log \left (1+\sinh (x)+\sinh ^2(x)\right )+\left (1+i \sqrt{3}\right ) \log \left (1+i \sqrt{3}+2 \sinh (x)\right ) \log \left (1+\sinh (x)+\sinh ^2(x)\right )+8 \sinh (x)-4 \log \left (1+\sinh (x)+\sinh ^2(x)\right ) \sinh (x)+\log ^2\left (1+\sinh (x)+\sinh ^2(x)\right ) \sinh (x)+12 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 \sinh (x)\right )+\left (-1-i \sqrt{3}\right ) \operatorname{Subst}\left (\int \left (\frac{2 \log \left (1+i \sqrt{3}+2 x\right )}{1-i \sqrt{3}+2 x}+\frac{2 \log \left (1+i \sqrt{3}+2 x\right )}{1+i \sqrt{3}+2 x}\right ) \, dx,x,\sinh (x)\right )+\left (-1+i \sqrt{3}\right ) \operatorname{Subst}\left (\int \left (\frac{2 \log \left (1-i \sqrt{3}+2 x\right )}{1-i \sqrt{3}+2 x}+\frac{2 \log \left (1-i \sqrt{3}+2 x\right )}{1+i \sqrt{3}+2 x}\right ) \, dx,x,\sinh (x)\right )\\ &=-4 \sqrt{3} \tan ^{-1}\left (\frac{1+2 \sinh (x)}{\sqrt{3}}\right )-2 \log \left (1+\sinh (x)+\sinh ^2(x)\right )+\left (1-i \sqrt{3}\right ) \log \left (1-i \sqrt{3}+2 \sinh (x)\right ) \log \left (1+\sinh (x)+\sinh ^2(x)\right )+\left (1+i \sqrt{3}\right ) \log \left (1+i \sqrt{3}+2 \sinh (x)\right ) \log \left (1+\sinh (x)+\sinh ^2(x)\right )+8 \sinh (x)-4 \log \left (1+\sinh (x)+\sinh ^2(x)\right ) \sinh (x)+\log ^2\left (1+\sinh (x)+\sinh ^2(x)\right ) \sinh (x)-\left (2 \left (1-i \sqrt{3}\right )\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-i \sqrt{3}+2 x\right )}{1-i \sqrt{3}+2 x} \, dx,x,\sinh (x)\right )-\left (2 \left (1-i \sqrt{3}\right )\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-i \sqrt{3}+2 x\right )}{1+i \sqrt{3}+2 x} \, dx,x,\sinh (x)\right )-\left (2 \left (1+i \sqrt{3}\right )\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+i \sqrt{3}+2 x\right )}{1-i \sqrt{3}+2 x} \, dx,x,\sinh (x)\right )-\left (2 \left (1+i \sqrt{3}\right )\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+i \sqrt{3}+2 x\right )}{1+i \sqrt{3}+2 x} \, dx,x,\sinh (x)\right )\\ &=-4 \sqrt{3} \tan ^{-1}\left (\frac{1+2 \sinh (x)}{\sqrt{3}}\right )-\left (1+i \sqrt{3}\right ) \log \left (\frac{i \left (1-i \sqrt{3}+2 \sinh (x)\right )}{2 \sqrt{3}}\right ) \log \left (1+i \sqrt{3}+2 \sinh (x)\right )-\left (1-i \sqrt{3}\right ) \log \left (1-i \sqrt{3}+2 \sinh (x)\right ) \log \left (-\frac{i \left (1+i \sqrt{3}+2 \sinh (x)\right )}{2 \sqrt{3}}\right )-2 \log \left (1+\sinh (x)+\sinh ^2(x)\right )+\left (1-i \sqrt{3}\right ) \log \left (1-i \sqrt{3}+2 \sinh (x)\right ) \log \left (1+\sinh (x)+\sinh ^2(x)\right )+\left (1+i \sqrt{3}\right ) \log \left (1+i \sqrt{3}+2 \sinh (x)\right ) \log \left (1+\sinh (x)+\sinh ^2(x)\right )+8 \sinh (x)-4 \log \left (1+\sinh (x)+\sinh ^2(x)\right ) \sinh (x)+\log ^2\left (1+\sinh (x)+\sinh ^2(x)\right ) \sinh (x)-\left (1-i \sqrt{3}\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1-i \sqrt{3}+2 \sinh (x)\right )+\left (2 \left (1-i \sqrt{3}\right )\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2 \left (1+i \sqrt{3}+2 x\right )}{-2 \left (1-i \sqrt{3}\right )+2 \left (1+i \sqrt{3}\right )}\right )}{1-i \sqrt{3}+2 x} \, dx,x,\sinh (x)\right )-\left (1+i \sqrt{3}\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1+i \sqrt{3}+2 \sinh (x)\right )+\left (2 \left (1+i \sqrt{3}\right )\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2 \left (1-i \sqrt{3}+2 x\right )}{2 \left (1-i \sqrt{3}\right )-2 \left (1+i \sqrt{3}\right )}\right )}{1+i \sqrt{3}+2 x} \, dx,x,\sinh (x)\right )\\ &=-4 \sqrt{3} \tan ^{-1}\left (\frac{1+2 \sinh (x)}{\sqrt{3}}\right )-\frac{1}{2} \left (1-i \sqrt{3}\right ) \log ^2\left (1-i \sqrt{3}+2 \sinh (x)\right )-\left (1+i \sqrt{3}\right ) \log \left (\frac{i \left (1-i \sqrt{3}+2 \sinh (x)\right )}{2 \sqrt{3}}\right ) \log \left (1+i \sqrt{3}+2 \sinh (x)\right )-\frac{1}{2} \left (1+i \sqrt{3}\right ) \log ^2\left (1+i \sqrt{3}+2 \sinh (x)\right )-\left (1-i \sqrt{3}\right ) \log \left (1-i \sqrt{3}+2 \sinh (x)\right ) \log \left (-\frac{i \left (1+i \sqrt{3}+2 \sinh (x)\right )}{2 \sqrt{3}}\right )-2 \log \left (1+\sinh (x)+\sinh ^2(x)\right )+\left (1-i \sqrt{3}\right ) \log \left (1-i \sqrt{3}+2 \sinh (x)\right ) \log \left (1+\sinh (x)+\sinh ^2(x)\right )+\left (1+i \sqrt{3}\right ) \log \left (1+i \sqrt{3}+2 \sinh (x)\right ) \log \left (1+\sinh (x)+\sinh ^2(x)\right )+8 \sinh (x)-4 \log \left (1+\sinh (x)+\sinh ^2(x)\right ) \sinh (x)+\log ^2\left (1+\sinh (x)+\sinh ^2(x)\right ) \sinh (x)+\left (1-i \sqrt{3}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 x}{-2 \left (1-i \sqrt{3}\right )+2 \left (1+i \sqrt{3}\right )}\right )}{x} \, dx,x,1-i \sqrt{3}+2 \sinh (x)\right )+\left (1+i \sqrt{3}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 x}{2 \left (1-i \sqrt{3}\right )-2 \left (1+i \sqrt{3}\right )}\right )}{x} \, dx,x,1+i \sqrt{3}+2 \sinh (x)\right )\\ &=-4 \sqrt{3} \tan ^{-1}\left (\frac{1+2 \sinh (x)}{\sqrt{3}}\right )-\frac{1}{2} \left (1-i \sqrt{3}\right ) \log ^2\left (1-i \sqrt{3}+2 \sinh (x)\right )-\left (1+i \sqrt{3}\right ) \log \left (\frac{i \left (1-i \sqrt{3}+2 \sinh (x)\right )}{2 \sqrt{3}}\right ) \log \left (1+i \sqrt{3}+2 \sinh (x)\right )-\frac{1}{2} \left (1+i \sqrt{3}\right ) \log ^2\left (1+i \sqrt{3}+2 \sinh (x)\right )-\left (1-i \sqrt{3}\right ) \log \left (1-i \sqrt{3}+2 \sinh (x)\right ) \log \left (-\frac{i \left (1+i \sqrt{3}+2 \sinh (x)\right )}{2 \sqrt{3}}\right )-2 \log \left (1+\sinh (x)+\sinh ^2(x)\right )+\left (1-i \sqrt{3}\right ) \log \left (1-i \sqrt{3}+2 \sinh (x)\right ) \log \left (1+\sinh (x)+\sinh ^2(x)\right )+\left (1+i \sqrt{3}\right ) \log \left (1+i \sqrt{3}+2 \sinh (x)\right ) \log \left (1+\sinh (x)+\sinh ^2(x)\right )-\left (1-i \sqrt{3}\right ) \text{Li}_2\left (\frac{i \left (1-i \sqrt{3}+2 \sinh (x)\right )}{2 \sqrt{3}}\right )-\left (1+i \sqrt{3}\right ) \text{Li}_2\left (-\frac{i \left (1+i \sqrt{3}+2 \sinh (x)\right )}{2 \sqrt{3}}\right )+8 \sinh (x)-4 \log \left (1+\sinh (x)+\sinh ^2(x)\right ) \sinh (x)+\log ^2\left (1+\sinh (x)+\sinh ^2(x)\right ) \sinh (x)\\ \end{align*}

Mathematica [A]  time = 0.222666, size = 347, normalized size = 0.88 \[ -\frac{1}{2} i \left (\sqrt{3}-i\right ) \left (2 \text{PolyLog}\left (2,\frac{-2 i \sinh (x)+\sqrt{3}-i}{2 \sqrt{3}}\right )+\log \left (2 \sinh (x)+i \sqrt{3}+1\right ) \left (2 \log \left (\frac{2 i \sinh (x)+\sqrt{3}+i}{2 \sqrt{3}}\right )+\log \left (2 \sinh (x)+i \sqrt{3}+1\right )\right )\right )+\frac{1}{2} i \left (\sqrt{3}+i\right ) \left (2 \text{PolyLog}\left (2,\frac{2 i \sinh (x)+\sqrt{3}+i}{2 \sqrt{3}}\right )+\log \left (2 \sinh (x)-i \sqrt{3}+1\right ) \left (2 \log \left (\frac{-2 i \sinh (x)+\sqrt{3}-i}{2 \sqrt{3}}\right )+\log \left (2 \sinh (x)-i \sqrt{3}+1\right )\right )\right )+8 \sinh (x)+\sinh (x) \log ^2\left (\sinh ^2(x)+\sinh (x)+1\right )+\left (1-i \sqrt{3}\right ) \log \left (2 \sinh (x)-i \sqrt{3}+1\right ) \log \left (\sinh ^2(x)+\sinh (x)+1\right )+\left (1+i \sqrt{3}\right ) \log \left (2 \sinh (x)+i \sqrt{3}+1\right ) \log \left (\sinh ^2(x)+\sinh (x)+1\right )-4 \sinh (x) \log \left (\sinh ^2(x)+\sinh (x)+1\right )-2 \log \left (\sinh ^2(x)+\sinh (x)+1\right )-4 \sqrt{3} \tan ^{-1}\left (\frac{2 \sinh (x)+1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]*Log[Cosh[x]^2 + Sinh[x]]^2,x]

[Out]

-4*Sqrt[3]*ArcTan[(1 + 2*Sinh[x])/Sqrt[3]] - 2*Log[1 + Sinh[x] + Sinh[x]^2] + (1 - I*Sqrt[3])*Log[1 - I*Sqrt[3
] + 2*Sinh[x]]*Log[1 + Sinh[x] + Sinh[x]^2] + (1 + I*Sqrt[3])*Log[1 + I*Sqrt[3] + 2*Sinh[x]]*Log[1 + Sinh[x] +
Sinh[x]^2] - (I/2)*(-I + Sqrt[3])*(Log[1 + I*Sqrt[3] + 2*Sinh[x]]*(2*Log[(I + Sqrt[3] + (2*I)*Sinh[x])/(2*Sqr
t[3])] + Log[1 + I*Sqrt[3] + 2*Sinh[x]]) + 2*PolyLog[2, (-I + Sqrt[3] - (2*I)*Sinh[x])/(2*Sqrt[3])]) + (I/2)*(
I + Sqrt[3])*(Log[1 - I*Sqrt[3] + 2*Sinh[x]]*(2*Log[(-I + Sqrt[3] - (2*I)*Sinh[x])/(2*Sqrt[3])] + Log[1 - I*Sq
rt[3] + 2*Sinh[x]]) + 2*PolyLog[2, (I + Sqrt[3] + (2*I)*Sinh[x])/(2*Sqrt[3])]) + 8*Sinh[x] - 4*Log[1 + Sinh[x]
+ Sinh[x]^2]*Sinh[x] + Log[1 + Sinh[x] + Sinh[x]^2]^2*Sinh[x]

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Maple [F]  time = 3.147, size = 0, normalized size = 0. \begin{align*} \int \cosh \left ( x \right ) \left ( \ln \left ( \left ( \cosh \left ( x \right ) \right ) ^{2}+\sinh \left ( x \right ) \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)*ln(cosh(x)^2+sinh(x))^2,x)

[Out]

int(cosh(x)*ln(cosh(x)^2+sinh(x))^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(e^(2*x) - 1)*e^(-x)*log(e^(4*x) + 2*e^(3*x) + 2*e^(2*x) - 2*e^x + 1)^2 + 2*(2*x - e^(-x) - integrate((2*e
^(3*x) + 5*e^(2*x) + 6*e^x - 2)*e^x/(e^(4*x) + 2*e^(3*x) + 2*e^(2*x) - 2*e^x + 1), x))*log(2)^2 - 4*(x - integ
rate((e^(3*x) + 2*e^(2*x) + 2*e^x - 2)*e^x/(e^(4*x) + 2*e^(3*x) + 2*e^(2*x) - 2*e^x + 1), x))*log(2)^2 + 2*(e^
x - integrate((2*e^(3*x) + 2*e^(2*x) - 2*e^x + 1)*e^x/(e^(4*x) + 2*e^(3*x) + 2*e^(2*x) - 2*e^x + 1), x))*log(2
)^2 + 4*integrate(e^(4*x)/(e^(4*x) + 2*e^(3*x) + 2*e^(2*x) - 2*e^x + 1), x)*log(2)^2 + 6*integrate(e^(3*x)/(e^
(4*x) + 2*e^(3*x) + 2*e^(2*x) - 2*e^x + 1), x)*log(2)^2 + 6*integrate(e^x/(e^(4*x) + 2*e^(3*x) + 2*e^(2*x) - 2
*e^x + 1), x)*log(2)^2 + 4*integrate(x*e^(6*x)/(e^(5*x) + 2*e^(4*x) + 2*e^(3*x) - 2*e^(2*x) + e^x), x)*log(2)
+ 8*integrate(x*e^(5*x)/(e^(5*x) + 2*e^(4*x) + 2*e^(3*x) - 2*e^(2*x) + e^x), x)*log(2) + 12*integrate(x*e^(4*x
)/(e^(5*x) + 2*e^(4*x) + 2*e^(3*x) - 2*e^(2*x) + e^x), x)*log(2) + 12*integrate(x*e^(2*x)/(e^(5*x) + 2*e^(4*x)
+ 2*e^(3*x) - 2*e^(2*x) + e^x), x)*log(2) - 8*integrate(x*e^x/(e^(5*x) + 2*e^(4*x) + 2*e^(3*x) - 2*e^(2*x) +
e^x), x)*log(2) - 2*integrate(e^(6*x)*log(e^(4*x) + 2*e^(3*x) + 2*e^(2*x) - 2*e^x + 1)/(e^(5*x) + 2*e^(4*x) +
2*e^(3*x) - 2*e^(2*x) + e^x), x)*log(2) - 4*integrate(e^(5*x)*log(e^(4*x) + 2*e^(3*x) + 2*e^(2*x) - 2*e^x + 1)
/(e^(5*x) + 2*e^(4*x) + 2*e^(3*x) - 2*e^(2*x) + e^x), x)*log(2) - 6*integrate(e^(4*x)*log(e^(4*x) + 2*e^(3*x)
+ 2*e^(2*x) - 2*e^x + 1)/(e^(5*x) + 2*e^(4*x) + 2*e^(3*x) - 2*e^(2*x) + e^x), x)*log(2) - 6*integrate(e^(2*x)*
log(e^(4*x) + 2*e^(3*x) + 2*e^(2*x) - 2*e^x + 1)/(e^(5*x) + 2*e^(4*x) + 2*e^(3*x) - 2*e^(2*x) + e^x), x)*log(2
) + 4*integrate(e^x*log(e^(4*x) + 2*e^(3*x) + 2*e^(2*x) - 2*e^x + 1)/(e^(5*x) + 2*e^(4*x) + 2*e^(3*x) - 2*e^(2
*x) + e^x), x)*log(2) + 4*integrate(x/(e^(5*x) + 2*e^(4*x) + 2*e^(3*x) - 2*e^(2*x) + e^x), x)*log(2) - 2*integ
rate(log(e^(4*x) + 2*e^(3*x) + 2*e^(2*x) - 2*e^x + 1)/(e^(5*x) + 2*e^(4*x) + 2*e^(3*x) - 2*e^(2*x) + e^x), x)*
log(2) + 2*integrate(x^2*e^(6*x)/(e^(5*x) + 2*e^(4*x) + 2*e^(3*x) - 2*e^(2*x) + e^x), x) + 4*integrate(x^2*e^(
5*x)/(e^(5*x) + 2*e^(4*x) + 2*e^(3*x) - 2*e^(2*x) + e^x), x) + 6*integrate(x^2*e^(4*x)/(e^(5*x) + 2*e^(4*x) +
2*e^(3*x) - 2*e^(2*x) + e^x), x) + 6*integrate(x^2*e^(2*x)/(e^(5*x) + 2*e^(4*x) + 2*e^(3*x) - 2*e^(2*x) + e^x)
, x) - 4*integrate(x^2*e^x/(e^(5*x) + 2*e^(4*x) + 2*e^(3*x) - 2*e^(2*x) + e^x), x) - 2*integrate(x*e^(6*x)*log
(e^(4*x) + 2*e^(3*x) + 2*e^(2*x) - 2*e^x + 1)/(e^(5*x) + 2*e^(4*x) + 2*e^(3*x) - 2*e^(2*x) + e^x), x) - 4*inte
grate(x*e^(5*x)*log(e^(4*x) + 2*e^(3*x) + 2*e^(2*x) - 2*e^x + 1)/(e^(5*x) + 2*e^(4*x) + 2*e^(3*x) - 2*e^(2*x)
+ e^x), x) - 6*integrate(x*e^(4*x)*log(e^(4*x) + 2*e^(3*x) + 2*e^(2*x) - 2*e^x + 1)/(e^(5*x) + 2*e^(4*x) + 2*e
^(3*x) - 2*e^(2*x) + e^x), x) - 6*integrate(x*e^(2*x)*log(e^(4*x) + 2*e^(3*x) + 2*e^(2*x) - 2*e^x + 1)/(e^(5*x
) + 2*e^(4*x) + 2*e^(3*x) - 2*e^(2*x) + e^x), x) + 4*integrate(x*e^x*log(e^(4*x) + 2*e^(3*x) + 2*e^(2*x) - 2*e
^x + 1)/(e^(5*x) + 2*e^(4*x) + 2*e^(3*x) - 2*e^(2*x) + e^x), x) + 2*integrate(x^2/(e^(5*x) + 2*e^(4*x) + 2*e^(
3*x) - 2*e^(2*x) + e^x), x) - 2*integrate(x*log(e^(4*x) + 2*e^(3*x) + 2*e^(2*x) - 2*e^x + 1)/(e^(5*x) + 2*e^(4
*x) + 2*e^(3*x) - 2*e^(2*x) + e^x), x) - 4*integrate(e^(6*x)*log(e^(4*x) + 2*e^(3*x) + 2*e^(2*x) - 2*e^x + 1)/
(e^(5*x) + 2*e^(4*x) + 2*e^(3*x) - 2*e^(2*x) + e^x), x) - 6*integrate(e^(5*x)*log(e^(4*x) + 2*e^(3*x) + 2*e^(2
*x) - 2*e^x + 1)/(e^(5*x) + 2*e^(4*x) + 2*e^(3*x) - 2*e^(2*x) + e^x), x) + 8*integrate(e^(3*x)*log(e^(4*x) + 2
*e^(3*x) + 2*e^(2*x) - 2*e^x + 1)/(e^(5*x) + 2*e^(4*x) + 2*e^(3*x) - 2*e^(2*x) + e^x), x) + 4*integrate(e^(2*x
)*log(e^(4*x) + 2*e^(3*x) + 2*e^(2*x) - 2*e^x + 1)/(e^(5*x) + 2*e^(4*x) + 2*e^(3*x) - 2*e^(2*x) + e^x), x) - 2
*integrate(e^x*log(e^(4*x) + 2*e^(3*x) + 2*e^(2*x) - 2*e^x + 1)/(e^(5*x) + 2*e^(4*x) + 2*e^(3*x) - 2*e^(2*x) +
e^x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\cosh \left (x\right ) \log \left (\cosh \left (x\right )^{2} + \sinh \left (x\right )\right )^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(cosh(x)*log(cosh(x)^2 + sinh(x))^2, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*ln(cosh(x)**2+sinh(x))**2,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x