Integrand size = 12, antiderivative size = 159 \[ \int \cosh (x) \log ^2\left (1+\cosh ^2(x)\right ) \, dx=-8 \sqrt {2} \arctan \left (\frac {\sinh (x)}{\sqrt {2}}\right )+4 i \sqrt {2} \arctan \left (\frac {\sinh (x)}{\sqrt {2}}\right )^2+8 \sqrt {2} \arctan \left (\frac {\sinh (x)}{\sqrt {2}}\right ) \log \left (\frac {2 \sqrt {2}}{\sqrt {2}+i \sinh (x)}\right )+4 \sqrt {2} \arctan \left (\frac {\sinh (x)}{\sqrt {2}}\right ) \log \left (2+\sinh ^2(x)\right )+4 i \sqrt {2} \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {2}}{\sqrt {2}+i \sinh (x)}\right )+8 \sinh (x)-4 \log \left (2+\sinh ^2(x)\right ) \sinh (x)+\log ^2\left (2+\sinh ^2(x)\right ) \sinh (x) \]
8*sinh(x)-4*ln(2+sinh(x)^2)*sinh(x)+ln(2+sinh(x)^2)^2*sinh(x)-8*arctan(1/2 *sinh(x)*2^(1/2))*2^(1/2)+4*I*arctan(1/2*sinh(x)*2^(1/2))^2*2^(1/2)+4*arct an(1/2*sinh(x)*2^(1/2))*ln(2+sinh(x)^2)*2^(1/2)+8*arctan(1/2*sinh(x)*2^(1/ 2))*ln(2*2^(1/2)/(I*sinh(x)+2^(1/2)))*2^(1/2)+4*I*polylog(2,1-2*2^(1/2)/(I *sinh(x)+2^(1/2)))*2^(1/2)
Time = 0.07 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.77 \[ \int \cosh (x) \log ^2\left (1+\cosh ^2(x)\right ) \, dx=4 \sqrt {2} \arctan \left (\frac {\sinh (x)}{\sqrt {2}}\right ) \left (-2+i \arctan \left (\frac {\sinh (x)}{\sqrt {2}}\right )+2 \log \left (\frac {4 i}{2 i-\sqrt {2} \sinh (x)}\right )+\log \left (2+\sinh ^2(x)\right )\right )+4 i \sqrt {2} \operatorname {PolyLog}\left (2,\frac {2 i+\sqrt {2} \sinh (x)}{-2 i+\sqrt {2} \sinh (x)}\right )+\left (8-4 \log \left (2+\sinh ^2(x)\right )+\log ^2\left (2+\sinh ^2(x)\right )\right ) \sinh (x) \]
4*Sqrt[2]*ArcTan[Sinh[x]/Sqrt[2]]*(-2 + I*ArcTan[Sinh[x]/Sqrt[2]] + 2*Log[ (4*I)/(2*I - Sqrt[2]*Sinh[x])] + Log[2 + Sinh[x]^2]) + (4*I)*Sqrt[2]*PolyL og[2, (2*I + Sqrt[2]*Sinh[x])/(-2*I + Sqrt[2]*Sinh[x])] + (8 - 4*Log[2 + S inh[x]^2] + Log[2 + Sinh[x]^2]^2)*Sinh[x]
Time = 0.40 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4858, 2900, 2926, 2009}
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.
| \(\displaystyle \int \cosh (x) \log ^2\left (\cosh ^2(x)+1\right ) \, dx\) |
| \(\Big \downarrow \) 4858 |
| \(\displaystyle \int \log ^2\left (\sinh ^2(x)+2\right )d\sinh (x)\) |
| \(\Big \downarrow \) 2900 |
| \(\displaystyle \sinh (x) \log ^2\left (\sinh ^2(x)+2\right )-4 \int \frac {\log \left (\sinh ^2(x)+2\right ) \sinh ^2(x)}{\sinh ^2(x)+2}d\sinh (x)\) |
| \(\Big \downarrow \) 2926 |
| \(\displaystyle \sinh (x) \log ^2\left (\sinh ^2(x)+2\right )-4 \int \left (\log \left (\sinh ^2(x)+2\right )-\frac {2 \log \left (\sinh ^2(x)+2\right )}{\sinh ^2(x)+2}\right )d\sinh (x)\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \sinh (x) \log ^2\left (\sinh ^2(x)+2\right )-4 \left (-i \sqrt {2} \arctan \left (\frac {\sinh (x)}{\sqrt {2}}\right )^2+2 \sqrt {2} \arctan \left (\frac {\sinh (x)}{\sqrt {2}}\right )-\sqrt {2} \arctan \left (\frac {\sinh (x)}{\sqrt {2}}\right ) \log \left (\sinh ^2(x)+2\right )-2 \sqrt {2} \arctan \left (\frac {\sinh (x)}{\sqrt {2}}\right ) \log \left (\frac {2 \sqrt {2}}{\sqrt {2}+i \sinh (x)}\right )-i \sqrt {2} \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {2}}{i \sinh (x)+\sqrt {2}}\right )-2 \sinh (x)+\sinh (x) \log \left (\sinh ^2(x)+2\right )\right )\) |
Log[2 + Sinh[x]^2]^2*Sinh[x] - 4*(2*Sqrt[2]*ArcTan[Sinh[x]/Sqrt[2]] - I*Sq rt[2]*ArcTan[Sinh[x]/Sqrt[2]]^2 - 2*Sqrt[2]*ArcTan[Sinh[x]/Sqrt[2]]*Log[(2 *Sqrt[2])/(Sqrt[2] + I*Sinh[x])] - Sqrt[2]*ArcTan[Sinh[x]/Sqrt[2]]*Log[2 + Sinh[x]^2] - I*Sqrt[2]*PolyLog[2, 1 - (2*Sqrt[2])/(Sqrt[2] + I*Sinh[x])] - 2*Sinh[x] + Log[2 + Sinh[x]^2]*Sinh[x])
3.1.28.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_), x_Symbo l] :> Simp[x*(a + b*Log[c*(d + e*x^n)^p])^q, x] - Simp[b*e*n*p*q Int[x^n* ((a + b*Log[c*(d + e*x^n)^p])^(q - 1)/(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && IGtQ[q, 0] && (EqQ[q, 1] || IntegerQ[n])
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m _.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> Int[ExpandIntegrand[(a + b *Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x^s)^r, x], x] /; FreeQ[{a, b, c, d, e , f, g, m, n, p, q, r, s}, x] && IGtQ[q, 0] && IntegerQ[m] && IntegerQ[r] & & IntegerQ[s]
Int[Cosh[(c_.)*((a_.) + (b_.)*(x_))]*(u_), x_Symbol] :> With[{d = FreeFacto rs[Sinh[c*(a + b*x)], x]}, Simp[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]
\[\int \cosh \left (x \right ) \ln \left (1+\cosh ^{2}\left (x \right )\right )^{2}d x\]
\[ \int \cosh (x) \log ^2\left (1+\cosh ^2(x)\right ) \, dx=\int { \cosh \left (x\right ) \log \left (\cosh \left (x\right )^{2} + 1\right )^{2} \,d x } \]
\[ \int \cosh (x) \log ^2\left (1+\cosh ^2(x)\right ) \, dx=\int \log {\left (\cosh ^{2}{\left (x \right )} + 1 \right )}^{2} \cosh {\left (x \right )}\, dx \]
\[ \int \cosh (x) \log ^2\left (1+\cosh ^2(x)\right ) \, dx=\int { \cosh \left (x\right ) \log \left (\cosh \left (x\right )^{2} + 1\right )^{2} \,d x } \]
1/2*(e^(2*x) - 1)*e^(-x)*log(e^(4*x) + 6*e^(2*x) + 1)^2 - 2*(e^(-x) + inte grate((e^(2*x) + 6)*e^x/(e^(4*x) + 6*e^(2*x) + 1), x))*log(2)^2 + 2*(e^x - integrate((6*e^(2*x) + 1)*e^x/(e^(4*x) + 6*e^(2*x) + 1), x))*log(2)^2 + 1 4*integrate(e^(3*x)/(e^(4*x) + 6*e^(2*x) + 1), x)*log(2)^2 + 14*integrate( e^x/(e^(4*x) + 6*e^(2*x) + 1), x)*log(2)^2 + 4*integrate(x*e^(6*x)/(e^(5*x ) + 6*e^(3*x) + e^x), x)*log(2) + 28*integrate(x*e^(4*x)/(e^(5*x) + 6*e^(3 *x) + e^x), x)*log(2) + 28*integrate(x*e^(2*x)/(e^(5*x) + 6*e^(3*x) + e^x) , x)*log(2) - 2*integrate(e^(6*x)*log(e^(4*x) + 6*e^(2*x) + 1)/(e^(5*x) + 6*e^(3*x) + e^x), x)*log(2) - 14*integrate(e^(4*x)*log(e^(4*x) + 6*e^(2*x) + 1)/(e^(5*x) + 6*e^(3*x) + e^x), x)*log(2) - 14*integrate(e^(2*x)*log(e^ (4*x) + 6*e^(2*x) + 1)/(e^(5*x) + 6*e^(3*x) + e^x), x)*log(2) + 4*integrat e(x/(e^(5*x) + 6*e^(3*x) + e^x), x)*log(2) - 2*integrate(log(e^(4*x) + 6*e ^(2*x) + 1)/(e^(5*x) + 6*e^(3*x) + e^x), x)*log(2) + 2*integrate(x^2*e^(6* x)/(e^(5*x) + 6*e^(3*x) + e^x), x) + 14*integrate(x^2*e^(4*x)/(e^(5*x) + 6 *e^(3*x) + e^x), x) + 14*integrate(x^2*e^(2*x)/(e^(5*x) + 6*e^(3*x) + e^x) , x) - 2*integrate(x*e^(6*x)*log(e^(4*x) + 6*e^(2*x) + 1)/(e^(5*x) + 6*e^( 3*x) + e^x), x) - 14*integrate(x*e^(4*x)*log(e^(4*x) + 6*e^(2*x) + 1)/(e^( 5*x) + 6*e^(3*x) + e^x), x) - 14*integrate(x*e^(2*x)*log(e^(4*x) + 6*e^(2* x) + 1)/(e^(5*x) + 6*e^(3*x) + e^x), x) + 2*integrate(x^2/(e^(5*x) + 6*e^( 3*x) + e^x), x) - 2*integrate(x*log(e^(4*x) + 6*e^(2*x) + 1)/(e^(5*x) +...
\[ \int \cosh (x) \log ^2\left (1+\cosh ^2(x)\right ) \, dx=\int { \cosh \left (x\right ) \log \left (\cosh \left (x\right )^{2} + 1\right )^{2} \,d x } \]
Timed out. \[ \int \cosh (x) \log ^2\left (1+\cosh ^2(x)\right ) \, dx=\int {\ln \left ({\mathrm {cosh}\left (x\right )}^2+1\right )}^2\,\mathrm {cosh}\left (x\right ) \,d x \]