Integrand size = 13, antiderivative size = 395 \[ \int \cosh (x) \log ^2\left (\cosh ^2(x)+\sinh (x)\right ) \, dx=-4 \sqrt {3} \arctan \left (\frac {1+2 \sinh (x)}{\sqrt {3}}\right )-\frac {1}{2} \left (1-i \sqrt {3}\right ) \log ^2\left (1-i \sqrt {3}+2 \sinh (x)\right )-\left (1+i \sqrt {3}\right ) \log \left (\frac {i \left (1-i \sqrt {3}+2 \sinh (x)\right )}{2 \sqrt {3}}\right ) \log \left (1+i \sqrt {3}+2 \sinh (x)\right )-\frac {1}{2} \left (1+i \sqrt {3}\right ) \log ^2\left (1+i \sqrt {3}+2 \sinh (x)\right )-\left (1-i \sqrt {3}\right ) \log \left (1-i \sqrt {3}+2 \sinh (x)\right ) \log \left (-\frac {i \left (1+i \sqrt {3}+2 \sinh (x)\right )}{2 \sqrt {3}}\right )-2 \log \left (1+\sinh (x)+\sinh ^2(x)\right )+\left (1-i \sqrt {3}\right ) \log \left (1-i \sqrt {3}+2 \sinh (x)\right ) \log \left (1+\sinh (x)+\sinh ^2(x)\right )+\left (1+i \sqrt {3}\right ) \log \left (1+i \sqrt {3}+2 \sinh (x)\right ) \log \left (1+\sinh (x)+\sinh ^2(x)\right )-\left (1+i \sqrt {3}\right ) \operatorname {PolyLog}\left (2,-\frac {i-\sqrt {3}+2 i \sinh (x)}{2 \sqrt {3}}\right )-\left (1-i \sqrt {3}\right ) \operatorname {PolyLog}\left (2,\frac {i+\sqrt {3}+2 i \sinh (x)}{2 \sqrt {3}}\right )+8 \sinh (x)-4 \log \left (1+\sinh (x)+\sinh ^2(x)\right ) \sinh (x)+\log ^2\left (1+\sinh (x)+\sinh ^2(x)\right ) \sinh (x) \]
-2*ln(1+sinh(x)+sinh(x)^2)+8*sinh(x)-4*ln(1+sinh(x)+sinh(x)^2)*sinh(x)+ln( 1+sinh(x)+sinh(x)^2)^2*sinh(x)+ln(1+sinh(x)+sinh(x)^2)*ln(1+2*sinh(x)-I*3^ (1/2))*(1-I*3^(1/2))-1/2*ln(1+2*sinh(x)-I*3^(1/2))^2*(1-I*3^(1/2))-ln(1+2* sinh(x)-I*3^(1/2))*ln(-1/6*I*(1+2*sinh(x)+I*3^(1/2))*3^(1/2))*(1-I*3^(1/2) )-polylog(2,1/6*(I+2*I*sinh(x)+3^(1/2))*3^(1/2))*(1-I*3^(1/2))+ln(1+sinh(x )+sinh(x)^2)*ln(1+2*sinh(x)+I*3^(1/2))*(1+I*3^(1/2))-1/2*ln(1+2*sinh(x)+I* 3^(1/2))^2*(1+I*3^(1/2))-ln(1+2*sinh(x)+I*3^(1/2))*ln(1/6*I*(1+2*sinh(x)-I *3^(1/2))*3^(1/2))*(1+I*3^(1/2))-polylog(2,1/6*(-I-2*I*sinh(x)+3^(1/2))*3^ (1/2))*(1+I*3^(1/2))-4*arctan(1/3*(1+2*sinh(x))*3^(1/2))*3^(1/2)
Time = 0.16 (sec) , antiderivative size = 389, normalized size of antiderivative = 0.98 \[ \int \cosh (x) \log ^2\left (\cosh ^2(x)+\sinh (x)\right ) \, dx=-4 \sqrt {3} \arctan \left (\frac {1+2 \sinh (x)}{\sqrt {3}}\right )+i \left (i+\sqrt {3}\right ) \log \left (\frac {-i+\sqrt {3}-2 i \sinh (x)}{2 \sqrt {3}}\right ) \log \left (1-i \sqrt {3}+2 \sinh (x)\right )+\frac {1}{2} i \left (i+\sqrt {3}\right ) \log ^2\left (1-i \sqrt {3}+2 \sinh (x)\right )-\left (1+i \sqrt {3}\right ) \log \left (\frac {i+\sqrt {3}+2 i \sinh (x)}{2 \sqrt {3}}\right ) \log \left (1+i \sqrt {3}+2 \sinh (x)\right )-\frac {1}{2} \left (1+i \sqrt {3}\right ) \log ^2\left (1+i \sqrt {3}+2 \sinh (x)\right )-2 \log \left (1+\sinh (x)+\sinh ^2(x)\right )+\left (1-i \sqrt {3}\right ) \log \left (1-i \sqrt {3}+2 \sinh (x)\right ) \log \left (1+\sinh (x)+\sinh ^2(x)\right )+\left (1+i \sqrt {3}\right ) \log \left (1+i \sqrt {3}+2 \sinh (x)\right ) \log \left (1+\sinh (x)+\sinh ^2(x)\right )-\left (1+i \sqrt {3}\right ) \operatorname {PolyLog}\left (2,\frac {-i+\sqrt {3}-2 i \sinh (x)}{2 \sqrt {3}}\right )+i \left (i+\sqrt {3}\right ) \operatorname {PolyLog}\left (2,\frac {i+\sqrt {3}+2 i \sinh (x)}{2 \sqrt {3}}\right )+8 \sinh (x)-4 \log \left (1+\sinh (x)+\sinh ^2(x)\right ) \sinh (x)+\log ^2\left (1+\sinh (x)+\sinh ^2(x)\right ) \sinh (x) \]
-4*Sqrt[3]*ArcTan[(1 + 2*Sinh[x])/Sqrt[3]] + I*(I + Sqrt[3])*Log[(-I + Sqr t[3] - (2*I)*Sinh[x])/(2*Sqrt[3])]*Log[1 - I*Sqrt[3] + 2*Sinh[x]] + (I/2)* (I + Sqrt[3])*Log[1 - I*Sqrt[3] + 2*Sinh[x]]^2 - (1 + I*Sqrt[3])*Log[(I + Sqrt[3] + (2*I)*Sinh[x])/(2*Sqrt[3])]*Log[1 + I*Sqrt[3] + 2*Sinh[x]] - ((1 + I*Sqrt[3])*Log[1 + I*Sqrt[3] + 2*Sinh[x]]^2)/2 - 2*Log[1 + Sinh[x] + Si nh[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*Sqrt[3])*PolyLog[2, (-I + Sqrt[3] - (2*I)*Sinh[x ])/(2*Sqrt[3])] + I*(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]
Time = 0.72 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {4858, 3003, 3008, 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.
| \(\displaystyle \int \cosh (x) \log ^2\left (\sinh (x)+\cosh ^2(x)\right ) \, dx\) |
| \(\Big \downarrow \) 4858 |
| \(\displaystyle \int \log ^2\left (\sinh ^2(x)+\sinh (x)+1\right )d\sinh (x)\) |
| \(\Big \downarrow \) 3003 |
| \(\displaystyle \sinh (x) \log ^2\left (\sinh ^2(x)+\sinh (x)+1\right )-2 \int \frac {\log \left (\sinh ^2(x)+\sinh (x)+1\right ) \sinh (x) (2 \sinh (x)+1)}{\sinh ^2(x)+\sinh (x)+1}d\sinh (x)\) |
| \(\Big \downarrow \) 3008 |
| \(\displaystyle \sinh (x) \log ^2\left (\sinh ^2(x)+\sinh (x)+1\right )-2 \int \left (2 \log \left (\sinh ^2(x)+\sinh (x)+1\right )-\frac {\log \left (\sinh ^2(x)+\sinh (x)+1\right ) (\sinh (x)+2)}{\sinh ^2(x)+\sinh (x)+1}\right )d\sinh (x)\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \sinh (x) \log ^2\left (\sinh ^2(x)+\sinh (x)+1\right )-2 \left (2 \sqrt {3} \arctan \left (\frac {2 \sinh (x)+1}{\sqrt {3}}\right )+\frac {1}{2} \left (1+i \sqrt {3}\right ) \operatorname {PolyLog}\left (2,-\frac {2 i \sinh (x)-\sqrt {3}+i}{2 \sqrt {3}}\right )+\frac {1}{2} \left (1-i \sqrt {3}\right ) \operatorname {PolyLog}\left (2,\frac {2 i \sinh (x)+\sqrt {3}+i}{2 \sqrt {3}}\right )-4 \sinh (x)+\frac {1}{4} \left (1-i \sqrt {3}\right ) \log ^2\left (2 \sinh (x)-i \sqrt {3}+1\right )+\frac {1}{4} \left (1+i \sqrt {3}\right ) \log ^2\left (2 \sinh (x)+i \sqrt {3}+1\right )-\frac {1}{2} \left (1-i \sqrt {3}\right ) \log \left (\sinh ^2(x)+\sinh (x)+1\right ) \log \left (2 \sinh (x)-i \sqrt {3}+1\right )-\frac {1}{2} \left (1+i \sqrt {3}\right ) \log \left (2 \sinh (x)+i \sqrt {3}+1\right ) \log \left (\sinh ^2(x)+\sinh (x)+1\right )+\log \left (\sinh ^2(x)+\sinh (x)+1\right )+2 \sinh (x) \log \left (\sinh ^2(x)+\sinh (x)+1\right )+\frac {1}{2} \left (1-i \sqrt {3}\right ) \log \left (-\frac {i \left (2 \sinh (x)+i \sqrt {3}+1\right )}{2 \sqrt {3}}\right ) \log \left (2 \sinh (x)-i \sqrt {3}+1\right )+\frac {1}{2} \left (1+i \sqrt {3}\right ) \log \left (\frac {i \left (2 \sinh (x)-i \sqrt {3}+1\right )}{2 \sqrt {3}}\right ) \log \left (2 \sinh (x)+i \sqrt {3}+1\right )\right )\) |
Log[1 + Sinh[x] + Sinh[x]^2]^2*Sinh[x] - 2*(2*Sqrt[3]*ArcTan[(1 + 2*Sinh[x ])/Sqrt[3]] + ((1 - I*Sqrt[3])*Log[1 - I*Sqrt[3] + 2*Sinh[x]]^2)/4 + ((1 + I*Sqrt[3])*Log[((I/2)*(1 - I*Sqrt[3] + 2*Sinh[x]))/Sqrt[3]]*Log[1 + I*Sqr t[3] + 2*Sinh[x]])/2 + ((1 + I*Sqrt[3])*Log[1 + I*Sqrt[3] + 2*Sinh[x]]^2)/ 4 + ((1 - I*Sqrt[3])*Log[1 - I*Sqrt[3] + 2*Sinh[x]]*Log[((-1/2*I)*(1 + I*S qrt[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])/2 - ((1 + I*Sqrt[3])*Log[1 + I*Sqrt[3] + 2*Sinh[x]]*Log[1 + Sinh[x] + Sinh[x]^ 2])/2 + ((1 + I*Sqrt[3])*PolyLog[2, -1/2*(I - Sqrt[3] + (2*I)*Sinh[x])/Sqr t[3]])/2 + ((1 - I*Sqrt[3])*PolyLog[2, (I + Sqrt[3] + (2*I)*Sinh[x])/(2*Sq rt[3])])/2 - 4*Sinh[x] + 2*Log[1 + Sinh[x] + Sinh[x]^2]*Sinh[x])
3.1.29.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Log[c*RFx^p])^n, x] - Simp[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] && Ra tionalFunctionQ[RFx, x] && IGtQ[n, 0]
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] && RationalFuncti onQ[RGx, x] && IGtQ[n, 0]
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]
\[\int \cosh \left (x \right ) \ln \left (\cosh ^{2}\left (x \right )+\sinh \left (x \right )\right )^{2}d x\]
\[ \int \cosh (x) \log ^2\left (\cosh ^2(x)+\sinh (x)\right ) \, dx=\int { \cosh \left (x\right ) \log \left (\cosh \left (x\right )^{2} + \sinh \left (x\right )\right )^{2} \,d x } \]
Timed out. \[ \int \cosh (x) \log ^2\left (\cosh ^2(x)+\sinh (x)\right ) \, dx=\text {Timed out} \]
\[ \int \cosh (x) \log ^2\left (\cosh ^2(x)+\sinh (x)\right ) \, dx=\int { \cosh \left (x\right ) \log \left (\cosh \left (x\right )^{2} + \sinh \left (x\right )\right )^{2} \,d x } \]
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 - integra te((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)*lo g(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)*l og(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*inte grate(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) +...
\[ \int \cosh (x) \log ^2\left (\cosh ^2(x)+\sinh (x)\right ) \, dx=\int { \cosh \left (x\right ) \log \left (\cosh \left (x\right )^{2} + \sinh \left (x\right )\right )^{2} \,d x } \]
Timed out. \[ \int \cosh (x) \log ^2\left (\cosh ^2(x)+\sinh (x)\right ) \, dx=\int \mathrm {cosh}\left (x\right )\,{\ln \left ({\mathrm {cosh}\left (x\right )}^2+\mathrm {sinh}\left (x\right )\right )}^2 \,d x \]