Optimal. Leaf size=47 \[ -\frac {1}{3 b (\tanh (a+b x)+1)}-\frac {4 \tan ^{-1}\left (\frac {1-2 \tanh (a+b x)}{\sqrt {3}}\right )}{3 \sqrt {3} b} \]
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Rubi [A] time = 0.33, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2074, 618, 204} \[ -\frac {1}{3 b (\tanh (a+b x)+1)}-\frac {4 \tan ^{-1}\left (\frac {1-2 \tanh (a+b x)}{\sqrt {3}}\right )}{3 \sqrt {3} b} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 2074
Rubi steps
\begin {align*} \int \frac {\cosh ^3(a+b x)-\sinh ^3(a+b x)}{\cosh ^3(a+b x)+\sinh ^3(a+b x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1+x+x^2}{1+x+x^3+x^4} \, dx,x,\tanh (a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{3 (1+x)^2}+\frac {2}{3 \left (1-x+x^2\right )}\right ) \, dx,x,\tanh (a+b x)\right )}{b}\\ &=-\frac {1}{3 b (1+\tanh (a+b x))}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\tanh (a+b x)\right )}{3 b}\\ &=-\frac {1}{3 b (1+\tanh (a+b x))}-\frac {4 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \tanh (a+b x)\right )}{3 b}\\ &=-\frac {4 \tan ^{-1}\left (\frac {1-2 \tanh (a+b x)}{\sqrt {3}}\right )}{3 \sqrt {3} b}-\frac {1}{3 b (1+\tanh (a+b x))}\\ \end {align*}
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Mathematica [B] time = 1.35, size = 115, normalized size = 2.45 \[ \frac {(\sinh (a+b x)-\cosh (a+b x)) \left (\cosh (a+b x) \left (8 \sqrt {3} \tan ^{-1}\left (\frac {\text {sech}(b x) (\cosh (2 a+b x)-2 \sinh (2 a+b x))}{\sqrt {3}}\right )+3\right )+\sinh (a+b x) \left (8 \sqrt {3} \tan ^{-1}\left (\frac {\text {sech}(b x) (\cosh (2 a+b x)-2 \sinh (2 a+b x))}{\sqrt {3}}\right )-3\right )\right )}{18 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 127, normalized size = 2.70 \[ -\frac {8 \, {\left (\sqrt {3} \cosh \left (b x + a\right )^{2} + 2 \, \sqrt {3} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sqrt {3} \sinh \left (b x + a\right )^{2}\right )} \arctan \left (-\frac {\sqrt {3} \cosh \left (b x + a\right ) + \sqrt {3} \sinh \left (b x + a\right )}{3 \, {\left (\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right )}}\right ) + 3}{18 \, {\left (b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 37, normalized size = 0.79 \[ \frac {8 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} e^{\left (2 \, b x + 2 \, a\right )}\right ) - 3 \, e^{\left (-2 \, b x - 2 \, a\right )}}{18 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.12, size = 120, normalized size = 2.55 \[ \frac {2 i \sqrt {3}\, \ln \left (\tanh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )+\left (-i \sqrt {3}-1\right ) \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )}{9 b}-\frac {2 i \sqrt {3}\, \ln \left (\tanh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )+\left (i \sqrt {3}-1\right ) \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )}{9 b}-\frac {2}{3 b \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )^{2}}+\frac {2}{3 b \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.46, size = 93, normalized size = 1.98 \[ \frac {4 \, {\left (\sqrt {3} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (2 \, \sqrt {3} e^{\left (-b x - a\right )} + 3^{\frac {1}{4}} \sqrt {2}\right )}\right ) - \sqrt {3} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (2 \, \sqrt {3} e^{\left (-b x - a\right )} - 3^{\frac {1}{4}} \sqrt {2}\right )}\right )\right )}}{9 \, b} - \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{6 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.82, size = 48, normalized size = 1.02 \[ \frac {4\,\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}\,\sqrt {b^2}}{3\,b}\right )}{9\,\sqrt {b^2}}-\frac {{\mathrm {e}}^{-2\,a-2\,b\,x}}{6\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 15.24, size = 197, normalized size = 4.19 \[ \begin {cases} - x & \text {for}\: a = \log {\left (- i e^{- b x} \right )} \vee a = \log {\left (i e^{- b x} \right )} \\\frac {x \left (- \sinh ^{3}{\relax (a )} + \cosh ^{3}{\relax (a )}\right )}{\sinh ^{3}{\relax (a )} + \cosh ^{3}{\relax (a )}} & \text {for}\: b = 0 \\\frac {4 \sqrt {3} \sinh {\left (a + b x \right )} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sinh {\left (a + b x \right )}}{3 \cosh {\left (a + b x \right )}} - \frac {\sqrt {3}}{3} \right )}}{9 b \sinh {\left (a + b x \right )} + 9 b \cosh {\left (a + b x \right )}} + \frac {3 \sinh {\left (a + b x \right )}}{9 b \sinh {\left (a + b x \right )} + 9 b \cosh {\left (a + b x \right )}} + \frac {4 \sqrt {3} \cosh {\left (a + b x \right )} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sinh {\left (a + b x \right )}}{3 \cosh {\left (a + b x \right )}} - \frac {\sqrt {3}}{3} \right )}}{9 b \sinh {\left (a + b x \right )} + 9 b \cosh {\left (a + b x \right )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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