Optimal. Leaf size=23 \[ \frac {\text {csch}^3(x)}{3 a}-\frac {\coth ^3(x)}{3 a} \]
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Rubi [A] time = 0.14, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3872, 2839, 2606, 30, 2607} \[ \frac {\text {csch}^3(x)}{3 a}-\frac {\coth ^3(x)}{3 a} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2606
Rule 2607
Rule 2839
Rule 3872
Rubi steps
\begin {align*} \int \frac {\text {csch}^2(x)}{a+a \text {sech}(x)} \, dx &=-\int \frac {\coth (x) \text {csch}(x)}{-a-a \cosh (x)} \, dx\\ &=\frac {\int \coth ^2(x) \text {csch}^2(x) \, dx}{a}-\frac {\int \coth (x) \text {csch}^3(x) \, dx}{a}\\ &=-\frac {i \operatorname {Subst}\left (\int x^2 \, dx,x,i \coth (x)\right )}{a}-\frac {i \operatorname {Subst}\left (\int x^2 \, dx,x,-i \text {csch}(x)\right )}{a}\\ &=-\frac {\coth ^3(x)}{3 a}+\frac {\text {csch}^3(x)}{3 a}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 25, normalized size = 1.09 \[ -\frac {(2 \cosh (x)+\cosh (2 x)+3) \text {csch}(x)}{6 a (\cosh (x)+1)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.38, size = 71, normalized size = 3.09 \[ -\frac {4 \, {\left (2 \, \cosh \relax (x) + \sinh \relax (x) + 1\right )}}{3 \, {\left (a \cosh \relax (x)^{3} + a \sinh \relax (x)^{3} + 2 \, a \cosh \relax (x)^{2} + {\left (3 \, a \cosh \relax (x) + 2 \, a\right )} \sinh \relax (x)^{2} - a \cosh \relax (x) + {\left (3 \, a \cosh \relax (x)^{2} + 4 \, a \cosh \relax (x) + a\right )} \sinh \relax (x) - 2 \, a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 31, normalized size = 1.35 \[ -\frac {1}{2 \, a {\left (e^{x} - 1\right )}} + \frac {3 \, e^{\left (2 \, x\right )} + 1}{6 \, a {\left (e^{x} + 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 23, normalized size = 1.00 \[ \frac {-\frac {\left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{3}-\frac {1}{\tanh \left (\frac {x}{2}\right )}}{4 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 90, normalized size = 3.91 \[ -\frac {4 \, e^{\left (-x\right )}}{3 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-3 \, x\right )} - a e^{\left (-4 \, x\right )} + a\right )}} - \frac {2 \, e^{\left (-2 \, x\right )}}{2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-3 \, x\right )} - a e^{\left (-4 \, x\right )} + a} - \frac {2}{3 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-3 \, x\right )} - a e^{\left (-4 \, x\right )} + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.35, size = 91, normalized size = 3.96 \[ \frac {\frac {{\mathrm {e}}^{2\,x}}{6\,a}+\frac {1}{6\,a}-\frac {{\mathrm {e}}^x}{3\,a}}{3\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x+1}-\frac {\frac {1}{6\,a}-\frac {{\mathrm {e}}^x}{6\,a}}{{\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^x+1}-\frac {1}{2\,a\,\left ({\mathrm {e}}^x-1\right )}+\frac {1}{6\,a\,\left ({\mathrm {e}}^x+1\right )} \]
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 \[ \frac {\int \frac {\operatorname {csch}^{2}{\relax (x )}}{\operatorname {sech}{\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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