Optimal. Leaf size=55 \[ -\frac {3 \tanh ^{-1}(\cosh (a+b x))}{8 b}-\frac {3 \coth (a+b x) \text {csch}(a+b x)}{8 b}-\frac {\coth ^3(a+b x) \text {csch}(a+b x)}{4 b} \]
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Rubi [A]
time = 0.05, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2691, 3855}
\begin {gather*} -\frac {3 \tanh ^{-1}(\cosh (a+b x))}{8 b}-\frac {\coth ^3(a+b x) \text {csch}(a+b x)}{4 b}-\frac {3 \coth (a+b x) \text {csch}(a+b x)}{8 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2691
Rule 3855
Rubi steps
\begin {align*} \int \coth ^4(a+b x) \text {csch}(a+b x) \, dx &=-\frac {\coth ^3(a+b x) \text {csch}(a+b x)}{4 b}+\frac {3}{4} \int \coth ^2(a+b x) \text {csch}(a+b x) \, dx\\ &=-\frac {3 \coth (a+b x) \text {csch}(a+b x)}{8 b}-\frac {\coth ^3(a+b x) \text {csch}(a+b x)}{4 b}+\frac {3}{8} \int \text {csch}(a+b x) \, dx\\ &=-\frac {3 \tanh ^{-1}(\cosh (a+b x))}{8 b}-\frac {3 \coth (a+b x) \text {csch}(a+b x)}{8 b}-\frac {\coth ^3(a+b x) \text {csch}(a+b x)}{4 b}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 95, normalized size = 1.73 \begin {gather*} -\frac {5 \text {csch}^2\left (\frac {1}{2} (a+b x)\right )}{32 b}-\frac {\text {csch}^4\left (\frac {1}{2} (a+b x)\right )}{64 b}+\frac {3 \log \left (\tanh \left (\frac {1}{2} (a+b x)\right )\right )}{8 b}-\frac {5 \text {sech}^2\left (\frac {1}{2} (a+b x)\right )}{32 b}+\frac {\text {sech}^4\left (\frac {1}{2} (a+b x)\right )}{64 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.50, size = 89, normalized size = 1.62
method | result | size |
risch | \(-\frac {{\mathrm e}^{b x +a} \left (5 \,{\mathrm e}^{6 b x +6 a}+3 \,{\mathrm e}^{4 b x +4 a}+3 \,{\mathrm e}^{2 b x +2 a}+5\right )}{4 b \left ({\mathrm e}^{2 b x +2 a}-1\right )^{4}}+\frac {3 \ln \left ({\mathrm e}^{b x +a}-1\right )}{8 b}-\frac {3 \ln \left ({\mathrm e}^{b x +a}+1\right )}{8 b}\) | \(89\) |
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. 133 vs.
\(2 (49) = 98\).
time = 0.27, size = 133, normalized size = 2.42 \begin {gather*} -\frac {3 \, \log \left (e^{\left (-b x - a\right )} + 1\right )}{8 \, b} + \frac {3 \, \log \left (e^{\left (-b x - a\right )} - 1\right )}{8 \, b} + \frac {5 \, e^{\left (-b x - a\right )} + 3 \, e^{\left (-3 \, b x - 3 \, a\right )} + 3 \, e^{\left (-5 \, b x - 5 \, a\right )} + 5 \, e^{\left (-7 \, b x - 7 \, a\right )}}{4 \, b {\left (4 \, e^{\left (-2 \, b x - 2 \, a\right )} - 6 \, e^{\left (-4 \, b x - 4 \, a\right )} + 4 \, e^{\left (-6 \, b x - 6 \, a\right )} - e^{\left (-8 \, b x - 8 \, 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. 1114 vs.
\(2 (49) = 98\).
time = 0.39, size = 1114, normalized size = 20.25 \begin {gather*} -\frac {10 \, \cosh \left (b x + a\right )^{7} + 70 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{6} + 10 \, \sinh \left (b x + a\right )^{7} + 6 \, {\left (35 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{5} + 6 \, \cosh \left (b x + a\right )^{5} + 10 \, {\left (35 \, \cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{4} + 2 \, {\left (175 \, \cosh \left (b x + a\right )^{4} + 30 \, \cosh \left (b x + a\right )^{2} + 3\right )} \sinh \left (b x + a\right )^{3} + 6 \, \cosh \left (b x + a\right )^{3} + 6 \, {\left (35 \, \cosh \left (b x + a\right )^{5} + 10 \, \cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} + 3 \, {\left (\cosh \left (b x + a\right )^{8} + 8 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{7} + \sinh \left (b x + a\right )^{8} + 4 \, {\left (7 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{6} - 4 \, \cosh \left (b x + a\right )^{6} + 8 \, {\left (7 \, \cosh \left (b x + a\right )^{3} - 3 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{5} + 2 \, {\left (35 \, \cosh \left (b x + a\right )^{4} - 30 \, \cosh \left (b x + a\right )^{2} + 3\right )} \sinh \left (b x + a\right )^{4} + 6 \, \cosh \left (b x + a\right )^{4} + 8 \, {\left (7 \, \cosh \left (b x + a\right )^{5} - 10 \, \cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 4 \, {\left (7 \, \cosh \left (b x + a\right )^{6} - 15 \, \cosh \left (b x + a\right )^{4} + 9 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{2} - 4 \, \cosh \left (b x + a\right )^{2} + 8 \, {\left (\cosh \left (b x + a\right )^{7} - 3 \, \cosh \left (b x + a\right )^{5} + 3 \, \cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - 3 \, {\left (\cosh \left (b x + a\right )^{8} + 8 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{7} + \sinh \left (b x + a\right )^{8} + 4 \, {\left (7 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{6} - 4 \, \cosh \left (b x + a\right )^{6} + 8 \, {\left (7 \, \cosh \left (b x + a\right )^{3} - 3 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{5} + 2 \, {\left (35 \, \cosh \left (b x + a\right )^{4} - 30 \, \cosh \left (b x + a\right )^{2} + 3\right )} \sinh \left (b x + a\right )^{4} + 6 \, \cosh \left (b x + a\right )^{4} + 8 \, {\left (7 \, \cosh \left (b x + a\right )^{5} - 10 \, \cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 4 \, {\left (7 \, \cosh \left (b x + a\right )^{6} - 15 \, \cosh \left (b x + a\right )^{4} + 9 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{2} - 4 \, \cosh \left (b x + a\right )^{2} + 8 \, {\left (\cosh \left (b x + a\right )^{7} - 3 \, \cosh \left (b x + a\right )^{5} + 3 \, \cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 2 \, {\left (35 \, \cosh \left (b x + a\right )^{6} + 15 \, \cosh \left (b x + a\right )^{4} + 9 \, \cosh \left (b x + a\right )^{2} + 5\right )} \sinh \left (b x + a\right ) + 10 \, \cosh \left (b x + a\right )}{8 \, {\left (b \cosh \left (b x + a\right )^{8} + 8 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{7} + b \sinh \left (b x + a\right )^{8} - 4 \, b \cosh \left (b x + a\right )^{6} + 4 \, {\left (7 \, b \cosh \left (b x + a\right )^{2} - b\right )} \sinh \left (b x + a\right )^{6} + 8 \, {\left (7 \, b \cosh \left (b x + a\right )^{3} - 3 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{5} + 6 \, b \cosh \left (b x + a\right )^{4} + 2 \, {\left (35 \, b \cosh \left (b x + a\right )^{4} - 30 \, b \cosh \left (b x + a\right )^{2} + 3 \, b\right )} \sinh \left (b x + a\right )^{4} + 8 \, {\left (7 \, b \cosh \left (b x + a\right )^{5} - 10 \, b \cosh \left (b x + a\right )^{3} + 3 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} - 4 \, b \cosh \left (b x + a\right )^{2} + 4 \, {\left (7 \, b \cosh \left (b x + a\right )^{6} - 15 \, b \cosh \left (b x + a\right )^{4} + 9 \, b \cosh \left (b x + a\right )^{2} - b\right )} \sinh \left (b x + a\right )^{2} + 8 \, {\left (b \cosh \left (b x + a\right )^{7} - 3 \, b \cosh \left (b x + a\right )^{5} + 3 \, b \cosh \left (b x + a\right )^{3} - b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + b\right )}} \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 \coth ^{4}{\left (a + b x \right )} \operatorname {csch}{\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. 110 vs.
\(2 (49) = 98\).
time = 0.41, size = 110, normalized size = 2.00 \begin {gather*} -\frac {\frac {4 \, {\left (5 \, {\left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}\right )}^{3} - 12 \, e^{\left (b x + a\right )} - 12 \, e^{\left (-b x - a\right )}\right )}}{{\left ({\left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}\right )}^{2} - 4\right )}^{2}} + 3 \, \log \left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} + 2\right ) - 3 \, \log \left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} - 2\right )}{16 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.41, size = 190, normalized size = 3.45 \begin {gather*} -\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^2}}{b}\right )}{4\,\sqrt {-b^2}}-\frac {9\,{\mathrm {e}}^{a+b\,x}}{2\,b\,\left ({\mathrm {e}}^{4\,a+4\,b\,x}-2\,{\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}-\frac {6\,{\mathrm {e}}^{a+b\,x}}{b\,\left (3\,{\mathrm {e}}^{2\,a+2\,b\,x}-3\,{\mathrm {e}}^{4\,a+4\,b\,x}+{\mathrm {e}}^{6\,a+6\,b\,x}-1\right )}-\frac {4\,{\mathrm {e}}^{a+b\,x}}{b\,\left (6\,{\mathrm {e}}^{4\,a+4\,b\,x}-4\,{\mathrm {e}}^{2\,a+2\,b\,x}-4\,{\mathrm {e}}^{6\,a+6\,b\,x}+{\mathrm {e}}^{8\,a+8\,b\,x}+1\right )}-\frac {5\,{\mathrm {e}}^{a+b\,x}}{4\,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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