Optimal. Leaf size=46 \[ \frac {\text {csch}^4(x)}{4 a}+\frac {\tanh ^{-1}(\cosh (x))}{8 a}-\frac {\coth (x) \text {csch}^3(x)}{4 a}-\frac {\coth (x) \text {csch}(x)}{8 a} \]
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Rubi [A] time = 0.19, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {3872, 2835, 2606, 30, 2611, 3768, 3770} \[ \frac {\text {csch}^4(x)}{4 a}+\frac {\tanh ^{-1}(\cosh (x))}{8 a}-\frac {\coth (x) \text {csch}^3(x)}{4 a}-\frac {\coth (x) \text {csch}(x)}{8 a} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2606
Rule 2611
Rule 2835
Rule 3768
Rule 3770
Rule 3872
Rubi steps
\begin {align*} \int \frac {\text {csch}^3(x)}{a+a \text {sech}(x)} \, dx &=-\int \frac {\coth (x) \text {csch}^2(x)}{-a-a \cosh (x)} \, dx\\ &=\frac {\int \coth ^2(x) \text {csch}^3(x) \, dx}{a}-\frac {\int \coth (x) \text {csch}^4(x) \, dx}{a}\\ &=-\frac {\coth (x) \text {csch}^3(x)}{4 a}+\frac {\int \text {csch}^3(x) \, dx}{4 a}+\frac {\operatorname {Subst}\left (\int x^3 \, dx,x,-i \text {csch}(x)\right )}{a}\\ &=-\frac {\coth (x) \text {csch}(x)}{8 a}-\frac {\coth (x) \text {csch}^3(x)}{4 a}+\frac {\text {csch}^4(x)}{4 a}-\frac {\int \text {csch}(x) \, dx}{8 a}\\ &=\frac {\tanh ^{-1}(\cosh (x))}{8 a}-\frac {\coth (x) \text {csch}(x)}{8 a}-\frac {\coth (x) \text {csch}^3(x)}{4 a}+\frac {\text {csch}^4(x)}{4 a}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 59, normalized size = 1.28 \[ \frac {\cosh ^2\left (\frac {x}{2}\right ) \text {sech}(x) \left (-2 \text {csch}^2\left (\frac {x}{2}\right )+\text {sech}^4\left (\frac {x}{2}\right )-4 \log \left (\sinh \left (\frac {x}{2}\right )\right )+4 \log \left (\cosh \left (\frac {x}{2}\right )\right )\right )}{16 (a \text {sech}(x)+a)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 630, normalized size = 13.70 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.11, size = 90, normalized size = 1.96 \[ \frac {\log \left (e^{\left (-x\right )} + e^{x} + 2\right )}{16 \, a} - \frac {\log \left (e^{\left (-x\right )} + e^{x} - 2\right )}{16 \, a} + \frac {e^{\left (-x\right )} + e^{x} - 6}{16 \, a {\left (e^{\left (-x\right )} + e^{x} - 2\right )}} - \frac {3 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 12 \, e^{\left (-x\right )} + 12 \, e^{x} - 4}{32 \, a {\left (e^{\left (-x\right )} + e^{x} + 2\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 45, normalized size = 0.98 \[ \frac {\tanh ^{4}\left (\frac {x}{2}\right )}{32 a}-\frac {\tanh ^{2}\left (\frac {x}{2}\right )}{16 a}-\frac {1}{16 a \tanh \left (\frac {x}{2}\right )^{2}}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right )}{8 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 99, normalized size = 2.15 \[ -\frac {e^{\left (-x\right )} + 2 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-3 \, x\right )} + 2 \, e^{\left (-4 \, x\right )} + e^{\left (-5 \, x\right )}}{4 \, {\left (2 \, a e^{\left (-x\right )} - a e^{\left (-2 \, x\right )} - 4 \, a e^{\left (-3 \, x\right )} - a e^{\left (-4 \, x\right )} + 2 \, a e^{\left (-5 \, x\right )} + a e^{\left (-6 \, x\right )} + a\right )}} + \frac {\log \left (e^{\left (-x\right )} + 1\right )}{8 \, a} - \frac {\log \left (e^{\left (-x\right )} - 1\right )}{8 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.35, size = 121, normalized size = 2.63 \[ \frac {1}{2\,a\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^x+1\right )}-\frac {1}{4\,a\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^x+1\right )}+\frac {1}{2\,a\,\left (6\,{\mathrm {e}}^{2\,x}+4\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^x+1\right )}-\frac {1}{4\,a\,\left ({\mathrm {e}}^x-1\right )}+\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {-a^2}}{a}\right )}{4\,\sqrt {-a^2}}-\frac {1}{a\,\left (3\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x+1\right )} \]
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 \[ \frac {\int \frac {\operatorname {csch}^{3}{\relax (x )}}{\operatorname {sech}{\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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