Optimal. Leaf size=49 \[ -\frac {3 \text {ArcTan}(\sinh (a+b x))}{2 b}-\frac {3 \text {csch}(a+b x)}{2 b}+\frac {\text {csch}(a+b x) \text {sech}^2(a+b x)}{2 b} \]
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Rubi [A]
time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2701, 294, 327,
213} \begin {gather*} -\frac {3 \text {ArcTan}(\sinh (a+b x))}{2 b}-\frac {3 \text {csch}(a+b x)}{2 b}+\frac {\text {csch}(a+b x) \text {sech}^2(a+b x)}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 213
Rule 294
Rule 327
Rule 2701
Rubi steps
\begin {align*} \int \text {csch}^2(a+b x) \text {sech}^3(a+b x) \, dx &=-\frac {i \text {Subst}\left (\int \frac {x^4}{\left (-1+x^2\right )^2} \, dx,x,-i \text {csch}(a+b x)\right )}{b}\\ &=\frac {\text {csch}(a+b x) \text {sech}^2(a+b x)}{2 b}-\frac {(3 i) \text {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,-i \text {csch}(a+b x)\right )}{2 b}\\ &=-\frac {3 \text {csch}(a+b x)}{2 b}+\frac {\text {csch}(a+b x) \text {sech}^2(a+b x)}{2 b}-\frac {(3 i) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,-i \text {csch}(a+b x)\right )}{2 b}\\ &=-\frac {3 \tan ^{-1}(\sinh (a+b x))}{2 b}-\frac {3 \text {csch}(a+b x)}{2 b}+\frac {\text {csch}(a+b x) \text {sech}^2(a+b x)}{2 b}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.01, size = 29, normalized size = 0.59 \begin {gather*} -\frac {\text {csch}(a+b x) \, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};-\sinh ^2(a+b x)\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.92, size = 95, normalized size = 1.94
method | result | size |
risch | \(-\frac {{\mathrm e}^{b x +a} \left (3 \,{\mathrm e}^{4 b x +4 a}+2 \,{\mathrm e}^{2 b x +2 a}+3\right )}{b \left ({\mathrm e}^{2 b x +2 a}-1\right ) \left ({\mathrm e}^{2 b x +2 a}+1\right )^{2}}+\frac {3 i \ln \left ({\mathrm e}^{b x +a}-i\right )}{2 b}-\frac {3 i \ln \left ({\mathrm e}^{b x +a}+i\right )}{2 b}\) | \(95\) |
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. 90 vs.
\(2 (43) = 86\).
time = 0.47, size = 90, normalized size = 1.84 \begin {gather*} \frac {3 \, \arctan \left (e^{\left (-b x - a\right )}\right )}{b} - \frac {3 \, e^{\left (-b x - a\right )} + 2 \, e^{\left (-3 \, b x - 3 \, a\right )} + 3 \, e^{\left (-5 \, b x - 5 \, a\right )}}{b {\left (e^{\left (-2 \, b x - 2 \, a\right )} - e^{\left (-4 \, b x - 4 \, a\right )} - e^{\left (-6 \, b x - 6 \, 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. 511 vs.
\(2 (43) = 86\).
time = 0.42, size = 511, normalized size = 10.43 \begin {gather*} -\frac {3 \, \cosh \left (b x + a\right )^{5} + 15 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} + 3 \, \sinh \left (b x + a\right )^{5} + 2 \, {\left (15 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{3} + 2 \, \cosh \left (b x + a\right )^{3} + 6 \, {\left (5 \, \cosh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} + 3 \, {\left (\cosh \left (b x + a\right )^{6} + 6 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + \sinh \left (b x + a\right )^{6} + {\left (15 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{4} + \cosh \left (b x + a\right )^{4} + 4 \, {\left (5 \, \cosh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + {\left (15 \, \cosh \left (b x + a\right )^{4} + 6 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{2} - \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{5} + 2 \, \cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) - 1\right )} \arctan \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + 3 \, {\left (5 \, \cosh \left (b x + a\right )^{4} + 2 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right ) + 3 \, \cosh \left (b x + a\right )}{b \cosh \left (b x + a\right )^{6} + 6 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + b \sinh \left (b x + a\right )^{6} + b \cosh \left (b x + a\right )^{4} + {\left (15 \, b \cosh \left (b x + a\right )^{2} + b\right )} \sinh \left (b x + a\right )^{4} + 4 \, {\left (5 \, b \cosh \left (b x + a\right )^{3} + b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} - b \cosh \left (b x + a\right )^{2} + {\left (15 \, b \cosh \left (b x + a\right )^{4} + 6 \, b \cosh \left (b x + a\right )^{2} - b\right )} \sinh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b \cosh \left (b x + a\right )^{5} + 2 \, b \cosh \left (b x + a\right )^{3} - b \cosh \left (b x + a\right )\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}^{2}{\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. 102 vs.
\(2 (43) = 86\).
time = 0.41, size = 102, normalized size = 2.08 \begin {gather*} -\frac {3 \, \pi + \frac {4 \, {\left (3 \, {\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{2} + 8\right )}}{{\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{3} + 4 \, e^{\left (b x + a\right )} - 4 \, e^{\left (-b x - a\right )}} + 6 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-b x - a\right )}\right )}{4 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 107, normalized size = 2.18 \begin {gather*} \frac {2\,{\mathrm {e}}^{a+b\,x}}{b\,\left (2\,{\mathrm {e}}^{2\,a+2\,b\,x}+{\mathrm {e}}^{4\,a+4\,b\,x}+1\right )}-\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {b^2}}{b}\right )}{\sqrt {b^2}}-\frac {2\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )}-\frac {{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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