Optimal. Leaf size=83 \[ \frac {\left (a^2-b^2\right )^2 \log (a+b \tanh (x))}{b^5}-\frac {a \left (a^2-2 b^2\right ) \tanh (x)}{b^4}+\frac {\left (a^2-2 b^2\right ) \tanh ^2(x)}{2 b^3}-\frac {a \tanh ^3(x)}{3 b^2}+\frac {\tanh ^4(x)}{4 b} \]
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Rubi [A] time = 0.11, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3506, 697} \[ \frac {\left (a^2-2 b^2\right ) \tanh ^2(x)}{2 b^3}-\frac {a \left (a^2-2 b^2\right ) \tanh (x)}{b^4}+\frac {\left (a^2-b^2\right )^2 \log (a+b \tanh (x))}{b^5}-\frac {a \tanh ^3(x)}{3 b^2}+\frac {\tanh ^4(x)}{4 b} \]
Antiderivative was successfully verified.
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Rule 697
Rule 3506
Rubi steps
\begin {align*} \int \frac {\text {sech}^6(x)}{a+b \tanh (x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1-\frac {x^2}{b^2}\right )^2}{a+x} \, dx,x,b \tanh (x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {-a^3+2 a b^2}{b^4}-\frac {\left (-a^2+2 b^2\right ) x}{b^4}-\frac {a x^2}{b^4}+\frac {x^3}{b^4}+\frac {\left (-a^2+b^2\right )^2}{b^4 (a+x)}\right ) \, dx,x,b \tanh (x)\right )}{b}\\ &=\frac {\left (a^2-b^2\right )^2 \log (a+b \tanh (x))}{b^5}-\frac {a \left (a^2-2 b^2\right ) \tanh (x)}{b^4}+\frac {\left (a^2-2 b^2\right ) \tanh ^2(x)}{2 b^3}-\frac {a \tanh ^3(x)}{3 b^2}+\frac {\tanh ^4(x)}{4 b}\\ \end {align*}
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Mathematica [A] time = 0.41, size = 92, normalized size = 1.11 \[ \frac {-4 \left (a b \left (3 a^2-5 b^2\right ) \tanh (x)+3 \left (a^2-b^2\right )^2 (\log (\cosh (x))-\log (a \cosh (x)+b \sinh (x)))\right )+\text {sech}^2(x) \left (-6 a^2 b^2+4 a b^3 \tanh (x)+6 b^4\right )+3 b^4 \text {sech}^4(x)}{12 b^5} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.59, size = 1827, normalized size = 22.01 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 316, normalized size = 3.81 \[ \frac {{\left (a^{5} + a^{4} b - 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + b^{5}\right )} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a b^{5} + b^{6}} - \frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (e^{\left (2 \, x\right )} + 1\right )}{b^{5}} + \frac {25 \, a^{4} e^{\left (8 \, x\right )} - 50 \, a^{2} b^{2} e^{\left (8 \, x\right )} + 25 \, b^{4} e^{\left (8 \, x\right )} + 100 \, a^{4} e^{\left (6 \, x\right )} + 24 \, a^{3} b e^{\left (6 \, x\right )} - 224 \, a^{2} b^{2} e^{\left (6 \, x\right )} - 24 \, a b^{3} e^{\left (6 \, x\right )} + 124 \, b^{4} e^{\left (6 \, x\right )} + 150 \, a^{4} e^{\left (4 \, x\right )} + 72 \, a^{3} b e^{\left (4 \, x\right )} - 348 \, a^{2} b^{2} e^{\left (4 \, x\right )} - 120 \, a b^{3} e^{\left (4 \, x\right )} + 246 \, b^{4} e^{\left (4 \, x\right )} + 100 \, a^{4} e^{\left (2 \, x\right )} + 72 \, a^{3} b e^{\left (2 \, x\right )} - 224 \, a^{2} b^{2} e^{\left (2 \, x\right )} - 136 \, a b^{3} e^{\left (2 \, x\right )} + 124 \, b^{4} e^{\left (2 \, x\right )} + 25 \, a^{4} + 24 \, a^{3} b - 50 \, a^{2} b^{2} - 40 \, a b^{3} + 25 \, b^{4}}{12 \, b^{5} {\left (e^{\left (2 \, x\right )} + 1\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.15, size = 438, normalized size = 5.28 \[ \frac {\ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+2 \tanh \left (\frac {x}{2}\right ) b +a \right ) a^{4}}{b^{5}}-\frac {2 \ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+2 \tanh \left (\frac {x}{2}\right ) b +a \right ) a^{2}}{b^{3}}+\frac {\ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+2 \tanh \left (\frac {x}{2}\right ) b +a \right )}{b}-\frac {2 \left (\tanh ^{7}\left (\frac {x}{2}\right )\right ) a^{3}}{b^{4} \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}+\frac {4 \left (\tanh ^{7}\left (\frac {x}{2}\right )\right ) a}{b^{2} \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}+\frac {2 \left (\tanh ^{6}\left (\frac {x}{2}\right )\right ) a^{2}}{b^{3} \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}-\frac {4 \left (\tanh ^{6}\left (\frac {x}{2}\right )\right )}{b \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}-\frac {6 \left (\tanh ^{5}\left (\frac {x}{2}\right )\right ) a^{3}}{b^{4} \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}+\frac {28 \left (\tanh ^{5}\left (\frac {x}{2}\right )\right ) a}{3 b^{2} \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}+\frac {4 \left (\tanh ^{4}\left (\frac {x}{2}\right )\right ) a^{2}}{b^{3} \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}-\frac {4 \left (\tanh ^{4}\left (\frac {x}{2}\right )\right )}{b \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}-\frac {6 \left (\tanh ^{3}\left (\frac {x}{2}\right )\right ) a^{3}}{b^{4} \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}+\frac {28 \left (\tanh ^{3}\left (\frac {x}{2}\right )\right ) a}{3 b^{2} \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}+\frac {2 \left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) a^{2}}{b^{3} \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}-\frac {4 \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{b \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}-\frac {2 \tanh \left (\frac {x}{2}\right ) a^{3}}{b^{4} \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}+\frac {4 \tanh \left (\frac {x}{2}\right ) a}{b^{2} \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}-\frac {\ln \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right ) a^{4}}{b^{5}}+\frac {2 \ln \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right ) a^{2}}{b^{3}}-\frac {\ln \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 204, normalized size = 2.46 \[ -\frac {2 \, {\left (3 \, a^{3} - 5 \, a b^{2} + {\left (9 \, a^{3} + 3 \, a^{2} b - 17 \, a b^{2} - 3 \, b^{3}\right )} e^{\left (-2 \, x\right )} + 3 \, {\left (3 \, a^{3} + 2 \, a^{2} b - 5 \, a b^{2} - 4 \, b^{3}\right )} e^{\left (-4 \, x\right )} + 3 \, {\left (a^{3} + a^{2} b - a b^{2} - b^{3}\right )} e^{\left (-6 \, x\right )}\right )}}{3 \, {\left (4 \, b^{4} e^{\left (-2 \, x\right )} + 6 \, b^{4} e^{\left (-4 \, x\right )} + 4 \, b^{4} e^{\left (-6 \, x\right )} + b^{4} e^{\left (-8 \, x\right )} + b^{4}\right )}} + \frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} - a - b\right )}{b^{5}} - \frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (e^{\left (-2 \, x\right )} + 1\right )}{b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.29, size = 169, normalized size = 2.04 \[ \frac {4}{b\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}+\frac {2\,{\left (a-b\right )}^2}{b^3\,\left (2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1\right )}+\frac {8\,\left (a-3\,b\right )}{3\,b^2\,\left (3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1\right )}+\frac {2\,\left (a+b\right )\,{\left (a-b\right )}^2}{b^4\,\left ({\mathrm {e}}^{2\,x}+1\right )}+\frac {\ln \left (a-b+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )\,{\left (a+b\right )}^2\,{\left (a-b\right )}^2}{b^5}-\frac {\ln \left ({\mathrm {e}}^{2\,x}+1\right )\,{\left (a+b\right )}^2\,{\left (a-b\right )}^2}{b^5} \]
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 \[ \int \frac {\operatorname {sech}^{6}{\relax (x )}}{a + b \tanh {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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