Optimal. Leaf size=30 \[ \frac {\log (\sinh (2 a+2 b x))}{b^2}-\frac {2 x \coth (2 a+2 b x)}{b} \]
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Rubi [A] time = 0.06, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5461, 4184, 3475} \[ \frac {\log (\sinh (2 a+2 b x))}{b^2}-\frac {2 x \coth (2 a+2 b x)}{b} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 4184
Rule 5461
Rubi steps
\begin {align*} \int x \text {csch}^2(a+b x) \text {sech}^2(a+b x) \, dx &=4 \int x \text {csch}^2(2 a+2 b x) \, dx\\ &=-\frac {2 x \coth (2 a+2 b x)}{b}+\frac {2 \int \coth (2 a+2 b x) \, dx}{b}\\ &=-\frac {2 x \coth (2 a+2 b x)}{b}+\frac {\log (\sinh (2 a+2 b x))}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 26, normalized size = 0.87 \[ \frac {\log (\sinh (2 (a+b x)))-2 b x \coth (2 (a+b x))}{b^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 292, normalized size = 9.73 \[ -\frac {4 \, b x \cosh \left (b x + a\right )^{4} + 16 \, b x \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right ) + 24 \, b x \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} + 16 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + 4 \, b x \sinh \left (b x + a\right )^{4} - {\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right ) + 6 \, \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} - 1\right )} \log \left (\frac {4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )}{\cosh \left (b x + a\right )^{2} - 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2}}\right )}{b^{2} \cosh \left (b x + a\right )^{4} + 4 \, b^{2} \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right ) + 6 \, b^{2} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} + 4 \, b^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b^{2} \sinh \left (b x + a\right )^{4} - b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 72, normalized size = 2.40 \[ -\frac {4 \, b x e^{\left (4 \, b x + 4 \, a\right )} - e^{\left (4 \, b x + 4 \, a\right )} \log \left (e^{\left (4 \, b x + 4 \, a\right )} - 1\right ) + \log \left (e^{\left (4 \, b x + 4 \, a\right )} - 1\right )}{b^{2} e^{\left (4 \, b x + 4 \, a\right )} - b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.53, size = 62, normalized size = 2.07 \[ -\frac {4 x}{b}-\frac {4 a}{b^{2}}-\frac {4 x}{b \left ({\mathrm e}^{2 b x +2 a}-1\right ) \left (1+{\mathrm e}^{2 b x +2 a}\right )}+\frac {\ln \left ({\mathrm e}^{4 b x +4 a}-1\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 87, normalized size = 2.90 \[ -\frac {4 \, x e^{\left (4 \, b x + 4 \, a\right )}}{b e^{\left (4 \, b x + 4 \, a\right )} - b} + \frac {\log \left ({\left (e^{\left (b x + a\right )} + 1\right )} e^{\left (-a\right )}\right )}{b^{2}} + \frac {\log \left ({\left (e^{\left (b x + a\right )} - 1\right )} e^{\left (-a\right )}\right )}{b^{2}} + \frac {\log \left ({\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )} e^{\left (-2 \, a\right )}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 43, normalized size = 1.43 \[ \frac {\ln \left ({\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{4\,b\,x}-1\right )}{b^2}-\frac {4\,x}{b}-\frac {4\,x}{b\,\left ({\mathrm {e}}^{4\,a+4\,b\,x}-1\right )} \]
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 \[ \int x \operatorname {csch}^{2}{\left (a + b x \right )} \operatorname {sech}^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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