Optimal. Leaf size=395 \[ -\left (1+i \sqrt{3}\right ) \text{PolyLog}\left (2,-\frac{2 i \sinh (x)-\sqrt{3}+i}{2 \sqrt{3}}\right )-\left (1-i \sqrt{3}\right ) \text{PolyLog}\left (2,\frac{2 i \sinh (x)+\sqrt{3}+i}{2 \sqrt{3}}\right )+8 \sinh (x)+\sinh (x) \log ^2\left (\sinh ^2(x)+\sinh (x)+1\right )-\frac{1}{2} \left (1-i \sqrt{3}\right ) \log ^2\left (2 \sinh (x)-i \sqrt{3}+1\right )-\frac{1}{2} \left (1+i \sqrt{3}\right ) \log ^2\left (2 \sinh (x)+i \sqrt{3}+1\right )+\left (1-i \sqrt{3}\right ) \log \left (\sinh ^2(x)+\sinh (x)+1\right ) \log \left (2 \sinh (x)-i \sqrt{3}+1\right )+\left (1+i \sqrt{3}\right ) \log \left (2 \sinh (x)+i \sqrt{3}+1\right ) \log \left (\sinh ^2(x)+\sinh (x)+1\right )-2 \log \left (\sinh ^2(x)+\sinh (x)+1\right )-4 \sinh (x) \log \left (\sinh ^2(x)+\sinh (x)+1\right )-\left (1-i \sqrt{3}\right ) \log \left (-\frac{i \left (2 \sinh (x)+i \sqrt{3}+1\right )}{2 \sqrt{3}}\right ) \log \left (2 \sinh (x)-i \sqrt{3}+1\right )-\left (1+i \sqrt{3}\right ) \log \left (\frac{i \left (2 \sinh (x)-i \sqrt{3}+1\right )}{2 \sqrt{3}}\right ) \log \left (2 \sinh (x)+i \sqrt{3}+1\right )-4 \sqrt{3} \tan ^{-1}\left (\frac{2 \sinh (x)+1}{\sqrt{3}}\right ) \]
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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.
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Rule 4358
Rule 2523
Rule 2528
Rule 773
Rule 634
Rule 618
Rule 204
Rule 628
Rule 2524
Rule 2418
Rule 2390
Rule 2301
Rule 2394
Rule 2393
Rule 2391
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.
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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.
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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.
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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.
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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.
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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.
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