Optimal. Leaf size=40 \[ \frac {\log (\tanh (a+b x))}{b}-\frac {\tanh ^2(a+b x)}{b}+\frac {\tanh ^4(a+b x)}{4 b} \]
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Rubi [A]
time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2700, 272, 45}
\begin {gather*} \frac {\tanh ^4(a+b x)}{4 b}-\frac {\tanh ^2(a+b x)}{b}+\frac {\log (\tanh (a+b x))}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 2700
Rubi steps
\begin {align*} \int \text {csch}(a+b x) \text {sech}^5(a+b x) \, dx &=\frac {\text {Subst}\left (\int \frac {\left (1+x^2\right )^2}{x} \, dx,x,i \tanh (a+b x)\right )}{b}\\ &=\frac {\text {Subst}\left (\int \frac {(1+x)^2}{x} \, dx,x,-\tanh ^2(a+b x)\right )}{2 b}\\ &=\frac {\text {Subst}\left (\int \left (2+\frac {1}{x}+x\right ) \, dx,x,-\tanh ^2(a+b x)\right )}{2 b}\\ &=\frac {\log (\tanh (a+b x))}{b}-\frac {\tanh ^2(a+b x)}{b}+\frac {\tanh ^4(a+b x)}{4 b}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 46, normalized size = 1.15 \begin {gather*} -\frac {4 \log (\cosh (a+b x))-4 \log (\sinh (a+b x))-2 \text {sech}^2(a+b x)-\text {sech}^4(a+b x)}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.73, size = 33, normalized size = 0.82
method | result | size |
derivativedivides | \(\frac {\frac {1}{4 \cosh \left (b x +a \right )^{4}}+\frac {1}{2 \cosh \left (b x +a \right )^{2}}+\ln \left (\tanh \left (b x +a \right )\right )}{b}\) | \(33\) |
default | \(\frac {\frac {1}{4 \cosh \left (b x +a \right )^{4}}+\frac {1}{2 \cosh \left (b x +a \right )^{2}}+\ln \left (\tanh \left (b x +a \right )\right )}{b}\) | \(33\) |
risch | \(\frac {2 \,{\mathrm e}^{2 b x +2 a} \left ({\mathrm e}^{4 b x +4 a}+4 \,{\mathrm e}^{2 b x +2 a}+1\right )}{b \left ({\mathrm e}^{2 b x +2 a}+1\right )^{4}}-\frac {\ln \left ({\mathrm e}^{2 b x +2 a}+1\right )}{b}+\frac {\ln \left ({\mathrm e}^{2 b x +2 a}-1\right )}{b}\) | \(84\) |
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. 131 vs.
\(2 (38) = 76\).
time = 0.47, size = 131, normalized size = 3.28 \begin {gather*} \frac {\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} + \frac {\log \left (e^{\left (-b x - a\right )} - 1\right )}{b} - \frac {\log \left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )}{b} + \frac {2 \, {\left (e^{\left (-2 \, b x - 2 \, a\right )} + 4 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )}\right )}}{b {\left (4 \, e^{\left (-2 \, b x - 2 \, a\right )} + 6 \, e^{\left (-4 \, b x - 4 \, a\right )} + 4 \, e^{\left (-6 \, b x - 6 \, a\right )} + e^{\left (-8 \, b x - 8 \, a\right )} + 1\right )}} \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. 1073 vs.
\(2 (38) = 76\).
time = 0.38, size = 1073, normalized size = 26.82 \begin {gather*} \frac {2 \, \cosh \left (b x + a\right )^{6} + 12 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + 2 \, \sinh \left (b x + a\right )^{6} + 2 \, {\left (15 \, \cosh \left (b x + a\right )^{2} + 4\right )} \sinh \left (b x + a\right )^{4} + 8 \, \cosh \left (b x + a\right )^{4} + 8 \, {\left (5 \, \cosh \left (b x + a\right )^{3} + 4 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 2 \, {\left (15 \, \cosh \left (b x + a\right )^{4} + 24 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right )^{2} - {\left (\cosh \left (b x + a\right )^{8} + 8 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{7} + \sinh \left (b x + a\right )^{8} + 4 \, {\left (7 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{6} + 4 \, \cosh \left (b x + a\right )^{6} + 8 \, {\left (7 \, \cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{5} + 2 \, {\left (35 \, \cosh \left (b x + a\right )^{4} + 30 \, \cosh \left (b x + a\right )^{2} + 3\right )} \sinh \left (b x + a\right )^{4} + 6 \, \cosh \left (b x + a\right )^{4} + 8 \, {\left (7 \, \cosh \left (b x + a\right )^{5} + 10 \, \cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 4 \, {\left (7 \, \cosh \left (b x + a\right )^{6} + 15 \, \cosh \left (b x + a\right )^{4} + 9 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{2} + 4 \, \cosh \left (b x + a\right )^{2} + 8 \, {\left (\cosh \left (b x + a\right )^{7} + 3 \, \cosh \left (b x + a\right )^{5} + 3 \, \cosh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} \log \left (\frac {2 \, \cosh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right ) + {\left (\cosh \left (b x + a\right )^{8} + 8 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{7} + \sinh \left (b x + a\right )^{8} + 4 \, {\left (7 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{6} + 4 \, \cosh \left (b x + a\right )^{6} + 8 \, {\left (7 \, \cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{5} + 2 \, {\left (35 \, \cosh \left (b x + a\right )^{4} + 30 \, \cosh \left (b x + a\right )^{2} + 3\right )} \sinh \left (b x + a\right )^{4} + 6 \, \cosh \left (b x + a\right )^{4} + 8 \, {\left (7 \, \cosh \left (b x + a\right )^{5} + 10 \, \cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 4 \, {\left (7 \, \cosh \left (b x + a\right )^{6} + 15 \, \cosh \left (b x + a\right )^{4} + 9 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{2} + 4 \, \cosh \left (b x + a\right )^{2} + 8 \, {\left (\cosh \left (b x + a\right )^{7} + 3 \, \cosh \left (b x + a\right )^{5} + 3 \, \cosh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} \log \left (\frac {2 \, \sinh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right ) + 4 \, {\left (3 \, \cosh \left (b x + a\right )^{5} + 8 \, \cosh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{b \cosh \left (b x + a\right )^{8} + 8 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{7} + b \sinh \left (b x + a\right )^{8} + 4 \, b \cosh \left (b x + a\right )^{6} + 4 \, {\left (7 \, b \cosh \left (b x + a\right )^{2} + b\right )} \sinh \left (b x + a\right )^{6} + 8 \, {\left (7 \, b \cosh \left (b x + a\right )^{3} + 3 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{5} + 6 \, b \cosh \left (b x + a\right )^{4} + 2 \, {\left (35 \, b \cosh \left (b x + a\right )^{4} + 30 \, b \cosh \left (b x + a\right )^{2} + 3 \, b\right )} \sinh \left (b x + a\right )^{4} + 8 \, {\left (7 \, b \cosh \left (b x + a\right )^{5} + 10 \, b \cosh \left (b x + a\right )^{3} + 3 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 4 \, b \cosh \left (b x + a\right )^{2} + 4 \, {\left (7 \, b \cosh \left (b x + a\right )^{6} + 15 \, b \cosh \left (b x + a\right )^{4} + 9 \, b \cosh \left (b x + a\right )^{2} + b\right )} \sinh \left (b x + a\right )^{2} + 8 \, {\left (b \cosh \left (b x + a\right )^{7} + 3 \, b \cosh \left (b x + a\right )^{5} + 3 \, b \cosh \left (b x + a\right )^{3} + b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + b} \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 \operatorname {csch}{\left (a + b x \right )} \operatorname {sech}^{5}{\left (a + b 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. 122 vs.
\(2 (38) = 76\).
time = 0.40, size = 122, normalized size = 3.05 \begin {gather*} \frac {\frac {3 \, {\left (e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (-2 \, b x - 2 \, a\right )}\right )}^{2} + 20 \, e^{\left (2 \, b x + 2 \, a\right )} + 20 \, e^{\left (-2 \, b x - 2 \, a\right )} + 44}{{\left (e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (-2 \, b x - 2 \, a\right )} + 2\right )}^{2}} - 2 \, \log \left (e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (-2 \, b x - 2 \, a\right )} + 2\right ) + 2 \, \log \left (e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (-2 \, b x - 2 \, a\right )} - 2\right )}{4 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.46, size = 169, normalized size = 4.22 \begin {gather*} \frac {2}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}-\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}\,\sqrt {-b^2}}{b}\right )}{\sqrt {-b^2}}+\frac {2}{b\,\left (2\,{\mathrm {e}}^{2\,a+2\,b\,x}+{\mathrm {e}}^{4\,a+4\,b\,x}+1\right )}-\frac {8}{b\,\left (3\,{\mathrm {e}}^{2\,a+2\,b\,x}+3\,{\mathrm {e}}^{4\,a+4\,b\,x}+{\mathrm {e}}^{6\,a+6\,b\,x}+1\right )}+\frac {4}{b\,\left (4\,{\mathrm {e}}^{2\,a+2\,b\,x}+6\,{\mathrm {e}}^{4\,a+4\,b\,x}+4\,{\mathrm {e}}^{6\,a+6\,b\,x}+{\mathrm {e}}^{8\,a+8\,b\,x}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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