Optimal. Leaf size=395 \[ -\left (1+i \sqrt {3}\right ) \operatorname {PolyLog}\left (2,-\frac {2 i \sinh (x)-\sqrt {3}+i}{2 \sqrt {3}}\right )-\left (1-i \sqrt {3}\right ) \operatorname {PolyLog}\left (2,\frac {2 i \sinh (x)+\sqrt {3}+i}{2 \sqrt {3}}\right )+8 \sinh (x)+\sinh (x) \log ^2\left (\sinh ^2(x)+\sinh (x)+1\right )-\frac {1}{2} \left (1-i \sqrt {3}\right ) \log ^2\left (2 \sinh (x)-i \sqrt {3}+1\right )-\frac {1}{2} \left (1+i \sqrt {3}\right ) \log ^2\left (2 \sinh (x)+i \sqrt {3}+1\right )+\left (1-i \sqrt {3}\right ) \log \left (\sinh ^2(x)+\sinh (x)+1\right ) \log \left (2 \sinh (x)-i \sqrt {3}+1\right )+\left (1+i \sqrt {3}\right ) \log \left (2 \sinh (x)+i \sqrt {3}+1\right ) \log \left (\sinh ^2(x)+\sinh (x)+1\right )-2 \log \left (\sinh ^2(x)+\sinh (x)+1\right )-4 \sinh (x) \log \left (\sinh ^2(x)+\sinh (x)+1\right )-\left (1-i \sqrt {3}\right ) \log \left (-\frac {i \left (2 \sinh (x)+i \sqrt {3}+1\right )}{2 \sqrt {3}}\right ) \log \left (2 \sinh (x)-i \sqrt {3}+1\right )-\left (1+i \sqrt {3}\right ) \log \left (\frac {i \left (2 \sinh (x)-i \sqrt {3}+1\right )}{2 \sqrt {3}}\right ) \log \left (2 \sinh (x)+i \sqrt {3}+1\right )-4 \sqrt {3} \tan ^{-1}\left (\frac {2 \sinh (x)+1}{\sqrt {3}}\right ) \]
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Rubi [A] time = 0.54, antiderivative size = 395, normalized size of antiderivative = 1.00, 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.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 773
Rule 2301
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2418
Rule 2523
Rule 2524
Rule 2528
Rule 4358
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*}
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Mathematica [A] time = 0.25, size = 347, normalized size = 0.88 \[ -\frac {1}{2} i \left (\sqrt {3}-i\right ) \left (2 \operatorname {PolyLog}\left (2,\frac {-2 i \sinh (x)+\sqrt {3}-i}{2 \sqrt {3}}\right )+\log \left (2 \sinh (x)+i \sqrt {3}+1\right ) \left (2 \log \left (\frac {2 i \sinh (x)+\sqrt {3}+i}{2 \sqrt {3}}\right )+\log \left (2 \sinh (x)+i \sqrt {3}+1\right )\right )\right )+\frac {1}{2} i \left (\sqrt {3}+i\right ) \left (2 \operatorname {PolyLog}\left (2,\frac {2 i \sinh (x)+\sqrt {3}+i}{2 \sqrt {3}}\right )+\log \left (2 \sinh (x)-i \sqrt {3}+1\right ) \left (2 \log \left (\frac {-2 i \sinh (x)+\sqrt {3}-i}{2 \sqrt {3}}\right )+\log \left (2 \sinh (x)-i \sqrt {3}+1\right )\right )\right )+8 \sinh (x)+\sinh (x) \log ^2\left (\sinh ^2(x)+\sinh (x)+1\right )+\left (1-i \sqrt {3}\right ) \log \left (2 \sinh (x)-i \sqrt {3}+1\right ) \log \left (\sinh ^2(x)+\sinh (x)+1\right )+\left (1+i \sqrt {3}\right ) \log \left (2 \sinh (x)+i \sqrt {3}+1\right ) \log \left (\sinh ^2(x)+\sinh (x)+1\right )-4 \sinh (x) \log \left (\sinh ^2(x)+\sinh (x)+1\right )-2 \log \left (\sinh ^2(x)+\sinh (x)+1\right )-4 \sqrt {3} \tan ^{-1}\left (\frac {2 \sinh (x)+1}{\sqrt {3}}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\cosh \relax (x) \log \left (\cosh \relax (x)^{2} + \sinh \relax (x)\right )^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cosh \relax (x) \log \left (\cosh \relax (x)^{2} + \sinh \relax (x)\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 6.56, size = 0, normalized size = 0.00 \[ \int \cosh \relax (x ) \ln \left (\cosh ^{2}\relax (x )+\sinh \relax (x )\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \mathrm {cosh}\relax (x)\,{\ln \left ({\mathrm {cosh}\relax (x)}^2+\mathrm {sinh}\relax (x)\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \log {\left (\sinh {\relax (x )} + \cosh ^{2}{\relax (x )} \right )}^{2} \cosh {\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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