Optimal. Leaf size=178 \[ \frac {\log (\cosh (x))}{a}+\frac {\left (8 a^2+21 a b+15 b^2\right ) \log (1-\text {sech}(x))}{16 (a+b)^3}+\frac {\left (8 a^2-21 a b+15 b^2\right ) \log (1+\text {sech}(x))}{16 (a-b)^3}-\frac {b^6 \log (a+b \text {sech}(x))}{a \left (a^2-b^2\right )^3}-\frac {1}{16 (a+b) (1-\text {sech}(x))^2}-\frac {5 a+7 b}{16 (a+b)^2 (1-\text {sech}(x))}-\frac {1}{16 (a-b) (1+\text {sech}(x))^2}-\frac {5 a-7 b}{16 (a-b)^2 (1+\text {sech}(x))} \]
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Rubi [A]
time = 0.23, antiderivative size = 178, 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 = {3970, 908}
\begin {gather*} \frac {\left (8 a^2+21 a b+15 b^2\right ) \log (1-\text {sech}(x))}{16 (a+b)^3}+\frac {\left (8 a^2-21 a b+15 b^2\right ) \log (\text {sech}(x)+1)}{16 (a-b)^3}-\frac {b^6 \log (a+b \text {sech}(x))}{a \left (a^2-b^2\right )^3}-\frac {5 a+7 b}{16 (a+b)^2 (1-\text {sech}(x))}-\frac {5 a-7 b}{16 (a-b)^2 (\text {sech}(x)+1)}-\frac {1}{16 (a+b) (1-\text {sech}(x))^2}-\frac {1}{16 (a-b) (\text {sech}(x)+1)^2}+\frac {\log (\cosh (x))}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 908
Rule 3970
Rubi steps
\begin {align*} \int \frac {\coth ^5(x)}{a+b \text {sech}(x)} \, dx &=-\left (b^6 \text {Subst}\left (\int \frac {1}{x (a+x) \left (b^2-x^2\right )^3} \, dx,x,b \text {sech}(x)\right )\right )\\ &=-\left (b^6 \text {Subst}\left (\int \left (\frac {1}{8 b^4 (a+b) (b-x)^3}+\frac {5 a+7 b}{16 b^5 (a+b)^2 (b-x)^2}+\frac {8 a^2+21 a b+15 b^2}{16 b^6 (a+b)^3 (b-x)}+\frac {1}{a b^6 x}+\frac {1}{a (a-b)^3 (a+b)^3 (a+x)}+\frac {1}{8 b^4 (-a+b) (b+x)^3}+\frac {-5 a+7 b}{16 (a-b)^2 b^5 (b+x)^2}+\frac {8 a^2-21 a b+15 b^2}{16 b^6 (-a+b)^3 (b+x)}\right ) \, dx,x,b \text {sech}(x)\right )\right )\\ &=\frac {\log (\cosh (x))}{a}+\frac {\left (8 a^2+21 a b+15 b^2\right ) \log (1-\text {sech}(x))}{16 (a+b)^3}+\frac {\left (8 a^2-21 a b+15 b^2\right ) \log (1+\text {sech}(x))}{16 (a-b)^3}-\frac {b^6 \log (a+b \text {sech}(x))}{a \left (a^2-b^2\right )^3}-\frac {1}{16 (a+b) (1-\text {sech}(x))^2}-\frac {5 a+7 b}{16 (a+b)^2 (1-\text {sech}(x))}-\frac {1}{16 (a-b) (1+\text {sech}(x))^2}-\frac {5 a-7 b}{16 (a-b)^2 (1+\text {sech}(x))}\\ \end {align*}
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Mathematica [A]
time = 0.68, size = 167, normalized size = 0.94 \begin {gather*} \frac {1}{64} \left (-\frac {2 (7 a+9 b) \text {csch}^2\left (\frac {x}{2}\right )}{(a+b)^2}-\frac {\text {csch}^4\left (\frac {x}{2}\right )}{a+b}-\frac {8 \left (8 b^6 \log (b+a \cosh (x))+a \left (-8 a \left (a^4-3 a^2 b^2+3 b^4\right ) \log (\sinh (x))+b \left (3 a^4-10 a^2 b^2+15 b^4\right ) \log \left (\tanh \left (\frac {x}{2}\right )\right )\right )\right )}{a (a-b)^3 (a+b)^3}+\frac {2 (7 a-9 b) \text {sech}^2\left (\frac {x}{2}\right )}{(a-b)^2}-\frac {\text {sech}^4\left (\frac {x}{2}\right )}{a-b}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.26, size = 162, normalized size = 0.91
method | result | size |
default | \(-\frac {\left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-b \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+6 a -8 b \right )^{2}}{64 \left (a -b \right )^{3}}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}-\frac {1}{64 \left (a +b \right ) \tanh \left (\frac {x}{2}\right )^{4}}-\frac {6 a +8 b}{32 \left (a +b \right )^{2} \tanh \left (\frac {x}{2}\right )^{2}}+\frac {\left (16 a^{2}+42 a b +30 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{16 \left (a +b \right )^{3}}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a}-\frac {b^{6} \ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-b \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+a +b \right )}{\left (a -b \right )^{3} \left (a +b \right )^{3} a}\) | \(162\) |
risch | \(\frac {x}{a}-\frac {x \,a^{2}}{a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}}+\frac {21 x a b}{8 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}-\frac {15 x \,b^{2}}{8 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}-\frac {x \,a^{2}}{a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}}-\frac {21 x a b}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {15 x \,b^{2}}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {2 x \,b^{6}}{a \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}-\frac {\left (-5 a^{2} b \,{\mathrm e}^{6 x}+9 b^{3} {\mathrm e}^{6 x}+16 a^{3} {\mathrm e}^{5 x}-24 a \,b^{2} {\mathrm e}^{5 x}-3 a^{2} b \,{\mathrm e}^{4 x}-b^{3} {\mathrm e}^{4 x}-16 a^{3} {\mathrm e}^{3 x}+32 a \,b^{2} {\mathrm e}^{3 x}-3 a^{2} b \,{\mathrm e}^{2 x}-b^{3} {\mathrm e}^{2 x}+16 a^{3} {\mathrm e}^{x}-24 a \,b^{2} {\mathrm e}^{x}-5 a^{2} b +9 b^{3}\right ) {\mathrm e}^{x}}{4 \left (a^{2}-b^{2}\right )^{2} \left ({\mathrm e}^{2 x}-1\right )^{4}}+\frac {\ln \left ({\mathrm e}^{x}+1\right ) a^{2}}{a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}}-\frac {21 \ln \left ({\mathrm e}^{x}+1\right ) a b}{8 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}+\frac {15 \ln \left ({\mathrm e}^{x}+1\right ) b^{2}}{8 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}+\frac {\ln \left ({\mathrm e}^{x}-1\right ) a^{2}}{a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}}+\frac {21 \ln \left ({\mathrm e}^{x}-1\right ) a b}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {15 \ln \left ({\mathrm e}^{x}-1\right ) b^{2}}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {b^{6} \ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}+1\right )}{a \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}\) | \(592\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 366 vs.
\(2 (162) = 324\).
time = 0.30, size = 366, normalized size = 2.06 \begin {gather*} -\frac {b^{6} \log \left (2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} + a\right )}{a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}} + \frac {{\left (8 \, a^{2} - 21 \, a b + 15 \, b^{2}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{8 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}} + \frac {{\left (8 \, a^{2} + 21 \, a b + 15 \, b^{2}\right )} \log \left (e^{\left (-x\right )} - 1\right )}{8 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} + \frac {{\left (5 \, a^{2} b - 9 \, b^{3}\right )} e^{\left (-x\right )} - 8 \, {\left (2 \, a^{3} - 3 \, a b^{2}\right )} e^{\left (-2 \, x\right )} + {\left (3 \, a^{2} b + b^{3}\right )} e^{\left (-3 \, x\right )} + 16 \, {\left (a^{3} - 2 \, a b^{2}\right )} e^{\left (-4 \, x\right )} + {\left (3 \, a^{2} b + b^{3}\right )} e^{\left (-5 \, x\right )} - 8 \, {\left (2 \, a^{3} - 3 \, a b^{2}\right )} e^{\left (-6 \, x\right )} + {\left (5 \, a^{2} b - 9 \, b^{3}\right )} e^{\left (-7 \, x\right )}}{4 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4} - 4 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-2 \, x\right )} + 6 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-4 \, x\right )} - 4 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-6 \, x\right )} + {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-8 \, x\right )}\right )}} + \frac {x}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 5181 vs.
\(2 (162) = 324\).
time = 0.49, size = 5181, normalized size = 29.11 \begin {gather*} \text {Too large to display} \end {gather*}
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 \begin {gather*} \int \frac {\coth ^{5}{\left (x \right )}}{a + b \operatorname {sech}{\left (x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 380 vs.
\(2 (162) = 324\).
time = 0.39, size = 380, normalized size = 2.13 \begin {gather*} -\frac {b^{6} \log \left ({\left | a {\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, b \right |}\right )}{a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}} + \frac {{\left (8 \, a^{2} - 21 \, a b + 15 \, b^{2}\right )} \log \left (e^{\left (-x\right )} + e^{x} + 2\right )}{16 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}} + \frac {{\left (8 \, a^{2} + 21 \, a b + 15 \, b^{2}\right )} \log \left (e^{\left (-x\right )} + e^{x} - 2\right )}{16 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} - \frac {3 \, a^{5} {\left (e^{\left (-x\right )} + e^{x}\right )}^{4} - 9 \, a^{3} b^{2} {\left (e^{\left (-x\right )} + e^{x}\right )}^{4} + 9 \, a b^{4} {\left (e^{\left (-x\right )} + e^{x}\right )}^{4} - 5 \, a^{4} b {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} + 14 \, a^{2} b^{3} {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} - 9 \, b^{5} {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} - 8 \, a^{5} {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 32 \, a^{3} b^{2} {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 48 \, a b^{4} {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 12 \, a^{4} b {\left (e^{\left (-x\right )} + e^{x}\right )} - 40 \, a^{2} b^{3} {\left (e^{\left (-x\right )} + e^{x}\right )} + 28 \, b^{5} {\left (e^{\left (-x\right )} + e^{x}\right )} - 16 \, a^{3} b^{2} + 64 \, a b^{4}}{4 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} {\left ({\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.75, size = 623, normalized size = 3.50 \begin {gather*} \frac {\ln \left ({\mathrm {e}}^x-1\right )\,\left (8\,a^2+21\,a\,b+15\,b^2\right )}{8\,a^3+24\,a^2\,b+24\,a\,b^2+8\,b^3}-\frac {\frac {2\,\left (4\,a^4-5\,a^2\,b^2\right )}{a\,{\left (a^2-b^2\right )}^2}-\frac {{\mathrm {e}}^x\,\left (9\,a^2\,b-13\,b^3\right )}{2\,{\left (a^2-b^2\right )}^2}}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1}-\frac {\frac {2\,\left (2\,a^6-5\,a^4\,b^2+3\,a^2\,b^4\right )}{a\,{\left (a^2-b^2\right )}^3}-\frac {{\mathrm {e}}^x\,\left (5\,a^4\,b-14\,a^2\,b^3+9\,b^5\right )}{4\,{\left (a^2-b^2\right )}^3}}{{\mathrm {e}}^{2\,x}-1}-\frac {\frac {8\,\left (a^4-a^2\,b^2\right )}{a\,{\left (a^2-b^2\right )}^2}-\frac {6\,{\mathrm {e}}^x\,\left (a^2\,b-b^3\right )}{{\left (a^2-b^2\right )}^2}}{3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1}-\frac {x}{a}-\frac {\frac {4\,a}{a^2-b^2}-\frac {4\,b\,{\mathrm {e}}^x}{a^2-b^2}}{6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1}+\frac {\ln \left ({\mathrm {e}}^x+1\right )\,\left (8\,a^2-21\,a\,b+15\,b^2\right )}{8\,a^3-24\,a^2\,b+24\,a\,b^2-8\,b^3}+\frac {b^6\,\ln \left (64\,a^{13}\,{\mathrm {e}}^{2\,x}+64\,a\,b^{12}+64\,a^{13}+159\,a^3\,b^{10}+492\,a^5\,b^8-1214\,a^7\,b^6+1020\,a^9\,b^4-393\,a^{11}\,b^2+128\,b^{13}\,{\mathrm {e}}^x+159\,a^3\,b^{10}\,{\mathrm {e}}^{2\,x}+492\,a^5\,b^8\,{\mathrm {e}}^{2\,x}-1214\,a^7\,b^6\,{\mathrm {e}}^{2\,x}+1020\,a^9\,b^4\,{\mathrm {e}}^{2\,x}-393\,a^{11}\,b^2\,{\mathrm {e}}^{2\,x}+128\,a^{12}\,b\,{\mathrm {e}}^x+64\,a\,b^{12}\,{\mathrm {e}}^{2\,x}+318\,a^2\,b^{11}\,{\mathrm {e}}^x+984\,a^4\,b^9\,{\mathrm {e}}^x-2428\,a^6\,b^7\,{\mathrm {e}}^x+2040\,a^8\,b^5\,{\mathrm {e}}^x-786\,a^{10}\,b^3\,{\mathrm {e}}^x\right )}{-a^7+3\,a^5\,b^2-3\,a^3\,b^4+a\,b^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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