Optimal. Leaf size=43 \[ -\frac {\coth ^2(a+b x)}{2 b}-\frac {2 \log (\tanh (a+b x))}{b}+\frac {\tanh ^2(a+b x)}{2 b} \]
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Rubi [A]
time = 0.03, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2700, 272, 45}
\begin {gather*} \frac {\tanh ^2(a+b x)}{2 b}-\frac {\coth ^2(a+b x)}{2 b}-\frac {2 \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}^3(a+b x) \text {sech}^3(a+b x) \, dx &=-\frac {\text {Subst}\left (\int \frac {\left (1+x^2\right )^2}{x^3} \, dx,x,i \tanh (a+b x)\right )}{b}\\ &=-\frac {\text {Subst}\left (\int \frac {(1+x)^2}{x^2} \, dx,x,-\tanh ^2(a+b x)\right )}{2 b}\\ &=-\frac {\text {Subst}\left (\int \left (1+\frac {1}{x^2}+\frac {2}{x}\right ) \, dx,x,-\tanh ^2(a+b x)\right )}{2 b}\\ &=-\frac {\coth ^2(a+b x)}{2 b}-\frac {2 \log (\tanh (a+b x))}{b}+\frac {\tanh ^2(a+b x)}{2 b}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 47, normalized size = 1.09 \begin {gather*} 8 \left (-\frac {\text {csch}^2(a+b x)}{16 b}-\frac {\log (\tanh (a+b x))}{4 b}-\frac {\text {sech}^2(a+b x)}{16 b}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.25, size = 43, normalized size = 1.00
method | result | size |
derivativedivides | \(\frac {-\frac {1}{2 \sinh \left (b x +a \right )^{2} \cosh \left (b x +a \right )^{2}}-\frac {1}{\cosh \left (b x +a \right )^{2}}-2 \ln \left (\tanh \left (b x +a \right )\right )}{b}\) | \(43\) |
default | \(\frac {-\frac {1}{2 \sinh \left (b x +a \right )^{2} \cosh \left (b x +a \right )^{2}}-\frac {1}{\cosh \left (b x +a \right )^{2}}-2 \ln \left (\tanh \left (b x +a \right )\right )}{b}\) | \(43\) |
risch | \(-\frac {4 \,{\mathrm e}^{2 b x +2 a} \left ({\mathrm e}^{4 b x +4 a}+1\right )}{b \left ({\mathrm e}^{2 b x +2 a}+1\right )^{2} \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}+\frac {2 \ln \left ({\mathrm e}^{2 b x +2 a}+1\right )}{b}-\frac {2 \ln \left ({\mathrm e}^{2 b x +2 a}-1\right )}{b}\) | \(87\) |
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. 102 vs.
\(2 (39) = 78\).
time = 0.48, size = 102, normalized size = 2.37 \begin {gather*} -\frac {2 \, \log \left (e^{\left (-b x - a\right )} + 1\right )}{b} - \frac {2 \, \log \left (e^{\left (-b x - a\right )} - 1\right )}{b} + \frac {2 \, \log \left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )}{b} + \frac {4 \, {\left (e^{\left (-2 \, b x - 2 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )}\right )}}{b {\left (2 \, e^{\left (-4 \, b x - 4 \, 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. 774 vs.
\(2 (39) = 78\).
time = 0.40, size = 774, normalized size = 18.00 \begin {gather*} -\frac {2 \, {\left (2 \, \cosh \left (b x + a\right )^{6} + 40 \, \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{3} + 30 \, \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{4} + 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 )^{4} + 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} + 56 \, \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{5} + 28 \, \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{6} + 8 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{7} + \sinh \left (b x + a\right )^{8} + 2 \, {\left (35 \, \cosh \left (b x + a\right )^{4} - 1\right )} \sinh \left (b x + a\right )^{4} - 2 \, \cosh \left (b x + a\right )^{4} + 8 \, {\left (7 \, \cosh \left (b x + a\right )^{5} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 4 \, {\left (7 \, \cosh \left (b x + a\right )^{6} - 3 \, \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )^{2} + 8 \, {\left (\cosh \left (b x + a\right )^{7} - \cosh \left (b x + a\right )^{3}\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} + 56 \, \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{5} + 28 \, \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{6} + 8 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{7} + \sinh \left (b x + a\right )^{8} + 2 \, {\left (35 \, \cosh \left (b x + a\right )^{4} - 1\right )} \sinh \left (b x + a\right )^{4} - 2 \, \cosh \left (b x + a\right )^{4} + 8 \, {\left (7 \, \cosh \left (b x + a\right )^{5} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 4 \, {\left (7 \, \cosh \left (b x + a\right )^{6} - 3 \, \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )^{2} + 8 \, {\left (\cosh \left (b x + a\right )^{7} - \cosh \left (b x + a\right )^{3}\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} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )\right )}}{b \cosh \left (b x + a\right )^{8} + 56 \, b \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{5} + 28 \, b \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{6} + 8 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{7} + b \sinh \left (b x + a\right )^{8} - 2 \, b \cosh \left (b x + a\right )^{4} + 2 \, {\left (35 \, b \cosh \left (b x + a\right )^{4} - b\right )} \sinh \left (b x + a\right )^{4} + 8 \, {\left (7 \, b \cosh \left (b x + a\right )^{5} - b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 4 \, {\left (7 \, b \cosh \left (b x + a\right )^{6} - 3 \, b \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )^{2} + 8 \, {\left (b \cosh \left (b x + a\right )^{7} - b \cosh \left (b x + a\right )^{3}\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}^{3}{\left (a + b x \right )} \operatorname {sech}^{3}{\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. 96 vs.
\(2 (39) = 78\).
time = 0.41, size = 96, normalized size = 2.23 \begin {gather*} -\frac {\frac {4 \, {\left (e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (-2 \, b x - 2 \, a\right )}\right )}}{{\left (e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (-2 \, b x - 2 \, a\right )}\right )}^{2} - 4} - \log \left (e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (-2 \, b x - 2 \, a\right )} + 2\right ) + \log \left (e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (-2 \, b x - 2 \, a\right )} - 2\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.44, size = 96, normalized size = 2.23 \begin {gather*} \frac {4\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}\,\sqrt {-b^2}}{b}\right )}{\sqrt {-b^2}}-\frac {4\,{\mathrm {e}}^{2\,a+2\,b\,x}}{b\,\left ({\mathrm {e}}^{4\,a+4\,b\,x}-1\right )}-\frac {8\,{\mathrm {e}}^{2\,a+2\,b\,x}}{b\,\left ({\mathrm {e}}^{8\,a+8\,b\,x}-2\,{\mathrm {e}}^{4\,a+4\,b\,x}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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