Optimal. Leaf size=43 \[ \frac {\tanh ^2(a+b x)}{2 b}-\frac {\coth ^2(a+b x)}{2 b}-\frac {2 \log (\tanh (a+b x))}{b} \]
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Rubi [A] time = 0.04, 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 = {2620, 266, 43} \[ \frac {\tanh ^2(a+b x)}{2 b}-\frac {\coth ^2(a+b x)}{2 b}-\frac {2 \log (\tanh (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 2620
Rubi steps
\begin {align*} \int \text {csch}^3(a+b x) \text {sech}^3(a+b x) \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^2}{x^3} \, dx,x,i \tanh (a+b x)\right )}{b}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {(1+x)^2}{x^2} \, dx,x,-\tanh ^2(a+b x)\right )}{2 b}\\ &=-\frac {\operatorname {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 \[ 8 \left (-\frac {\text {csch}^2(a+b x)}{16 b}-\frac {\text {sech}^2(a+b x)}{16 b}-\frac {\log (\tanh (a+b x))}{4 b}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 774, normalized size = 18.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 96, normalized size = 2.23 \[ -\frac {\frac {4 \, {\left (e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (-2 \, b x - 2 \, a\right )}\right )}}{{\left (e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (-2 \, b x - 2 \, a\right )}\right )}^{2} - 4} - \log \left (e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (-2 \, b x - 2 \, a\right )} + 2\right ) + \log \left (e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (-2 \, b x - 2 \, a\right )} - 2\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 48, normalized size = 1.12 \[ -\frac {1}{2 b \sinh \left (b x +a \right )^{2} \cosh \left (b x +a \right )^{2}}-\frac {1}{b \cosh \left (b x +a \right )^{2}}-\frac {2 \ln \left (\tanh \left (b x +a \right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 102, normalized size = 2.37 \[ -\frac {2 \, \log \left (e^{\left (-b x - a\right )} + 1\right )}{b} - \frac {2 \, \log \left (e^{\left (-b x - a\right )} - 1\right )}{b} + \frac {2 \, \log \left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )}{b} + \frac {4 \, {\left (e^{\left (-2 \, b x - 2 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )}\right )}}{b {\left (2 \, e^{\left (-4 \, b x - 4 \, a\right )} - e^{\left (-8 \, b x - 8 \, a\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 96, normalized size = 2.23 \[ \frac {4\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}\,\sqrt {-b^2}}{b}\right )}{\sqrt {-b^2}}-\frac {4\,{\mathrm {e}}^{2\,a+2\,b\,x}}{b\,\left ({\mathrm {e}}^{4\,a+4\,b\,x}-1\right )}-\frac {8\,{\mathrm {e}}^{2\,a+2\,b\,x}}{b\,\left ({\mathrm {e}}^{8\,a+8\,b\,x}-2\,{\mathrm {e}}^{4\,a+4\,b\,x}+1\right )} \]
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 \[ \int \operatorname {csch}^{3}{\left (a + b x \right )} \operatorname {sech}^{3}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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