Optimal. Leaf size=124 \[ -\frac {3 \sqrt {a \cosh ^2(e+f x)} \text {csch}(e+f x) \text {sech}(e+f x)}{f}-\frac {\sqrt {a \cosh ^2(e+f x)} \text {csch}^3(e+f x) \text {sech}(e+f x)}{f}-\frac {\sqrt {a \cosh ^2(e+f x)} \text {csch}^5(e+f x) \text {sech}(e+f x)}{5 f}+\frac {\sqrt {a \cosh ^2(e+f x)} \tanh (e+f x)}{f} \]
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Rubi [A]
time = 0.09, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3255, 3286,
2670, 276} \begin {gather*} \frac {\tanh (e+f x) \sqrt {a \cosh ^2(e+f x)}}{f}-\frac {\text {csch}^5(e+f x) \text {sech}(e+f x) \sqrt {a \cosh ^2(e+f x)}}{5 f}-\frac {\text {csch}^3(e+f x) \text {sech}(e+f x) \sqrt {a \cosh ^2(e+f x)}}{f}-\frac {3 \text {csch}(e+f x) \text {sech}(e+f x) \sqrt {a \cosh ^2(e+f x)}}{f} \end {gather*}
Antiderivative was successfully verified.
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Rule 276
Rule 2670
Rule 3255
Rule 3286
Rubi steps
\begin {align*} \int \coth ^6(e+f x) \sqrt {a+a \sinh ^2(e+f x)} \, dx &=\int \sqrt {a \cosh ^2(e+f x)} \coth ^6(e+f x) \, dx\\ &=\left (\sqrt {a \cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \int \cosh (e+f x) \coth ^6(e+f x) \, dx\\ &=-\frac {\left (i \sqrt {a \cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {\left (1-x^2\right )^3}{x^6} \, dx,x,-i \sinh (e+f x)\right )}{f}\\ &=-\frac {\left (i \sqrt {a \cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \left (-1+\frac {1}{x^6}-\frac {3}{x^4}+\frac {3}{x^2}\right ) \, dx,x,-i \sinh (e+f x)\right )}{f}\\ &=-\frac {3 \sqrt {a \cosh ^2(e+f x)} \text {csch}(e+f x) \text {sech}(e+f x)}{f}-\frac {\sqrt {a \cosh ^2(e+f x)} \text {csch}^3(e+f x) \text {sech}(e+f x)}{f}-\frac {\sqrt {a \cosh ^2(e+f x)} \text {csch}^5(e+f x) \text {sech}(e+f x)}{5 f}+\frac {\sqrt {a \cosh ^2(e+f x)} \tanh (e+f x)}{f}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 67, normalized size = 0.54 \begin {gather*} \frac {\sqrt {a \cosh ^2(e+f x)} (-182+235 \cosh (2 (e+f x))-90 \cosh (4 (e+f x))+5 \cosh (6 (e+f x))) \text {csch}^5(e+f x) \text {sech}(e+f x)}{160 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.32, size = 65, normalized size = 0.52
method | result | size |
default | \(\frac {\cosh \left (f x +e \right ) a \left (5 \left (\sinh ^{6}\left (f x +e \right )\right )-15 \left (\sinh ^{4}\left (f x +e \right )\right )-5 \left (\sinh ^{2}\left (f x +e \right )\right )-1\right )}{5 \sinh \left (f x +e \right )^{5} \sqrt {a \left (\cosh ^{2}\left (f x +e \right )\right )}\, f}\) | \(65\) |
risch | \(\frac {\sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, {\mathrm e}^{2 f x +2 e}}{2 f \left ({\mathrm e}^{2 f x +2 e}+1\right )}-\frac {\sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}}{2 f \left ({\mathrm e}^{2 f x +2 e}+1\right )}-\frac {2 \left (15 \,{\mathrm e}^{8 f x +8 e}-40 \,{\mathrm e}^{6 f x +6 e}+66 \,{\mathrm e}^{4 f x +4 e}-40 \,{\mathrm e}^{2 f x +2 e}+15\right ) \sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, {\mathrm e}^{2 f x +2 e}}{5 \left ({\mathrm e}^{2 f x +2 e}-1\right )^{5} f \left ({\mathrm e}^{2 f x +2 e}+1\right )}\) | \(211\) |
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. 1126 vs.
\(2 (125) = 250\).
time = 0.55, size = 1126, normalized size = 9.08 \begin {gather*} -\frac {105 \, \sqrt {a} \log \left (e^{\left (-f x - e\right )} + 1\right ) - 105 \, \sqrt {a} \log \left (e^{\left (-f x - e\right )} - 1\right ) - \frac {2 \, {\left (375 \, \sqrt {a} e^{\left (-f x - e\right )} - 790 \, \sqrt {a} e^{\left (-3 \, f x - 3 \, e\right )} + 896 \, \sqrt {a} e^{\left (-5 \, f x - 5 \, e\right )} - 490 \, \sqrt {a} e^{\left (-7 \, f x - 7 \, e\right )} + 105 \, \sqrt {a} e^{\left (-9 \, f x - 9 \, e\right )}\right )}}{5 \, e^{\left (-2 \, f x - 2 \, e\right )} - 10 \, e^{\left (-4 \, f x - 4 \, e\right )} + 10 \, e^{\left (-6 \, f x - 6 \, e\right )} - 5 \, e^{\left (-8 \, f x - 8 \, e\right )} + e^{\left (-10 \, f x - 10 \, e\right )} - 1}}{320 \, f} + \frac {105 \, \sqrt {a} \log \left (e^{\left (-f x - e\right )} + 1\right ) - 105 \, \sqrt {a} \log \left (e^{\left (-f x - e\right )} - 1\right ) + \frac {2 \, {\left (105 \, \sqrt {a} e^{\left (-f x - e\right )} - 490 \, \sqrt {a} e^{\left (-3 \, f x - 3 \, e\right )} + 896 \, \sqrt {a} e^{\left (-5 \, f x - 5 \, e\right )} - 790 \, \sqrt {a} e^{\left (-7 \, f x - 7 \, e\right )} + 375 \, \sqrt {a} e^{\left (-9 \, f x - 9 \, e\right )}\right )}}{5 \, e^{\left (-2 \, f x - 2 \, e\right )} - 10 \, e^{\left (-4 \, f x - 4 \, e\right )} + 10 \, e^{\left (-6 \, f x - 6 \, e\right )} - 5 \, e^{\left (-8 \, f x - 8 \, e\right )} + e^{\left (-10 \, f x - 10 \, e\right )} - 1}}{320 \, f} + \frac {15 \, \sqrt {a} \log \left (e^{\left (-f x - e\right )} + 1\right ) - 15 \, \sqrt {a} \log \left (e^{\left (-f x - e\right )} - 1\right ) + \frac {2 \, {\left (15 \, \sqrt {a} e^{\left (-f x - e\right )} + 250 \, \sqrt {a} e^{\left (-3 \, f x - 3 \, e\right )} - 128 \, \sqrt {a} e^{\left (-5 \, f x - 5 \, e\right )} + 70 \, \sqrt {a} e^{\left (-7 \, f x - 7 \, e\right )} - 15 \, \sqrt {a} e^{\left (-9 \, f x - 9 \, e\right )}\right )}}{5 \, e^{\left (-2 \, f x - 2 \, e\right )} - 10 \, e^{\left (-4 \, f x - 4 \, e\right )} + 10 \, e^{\left (-6 \, f x - 6 \, e\right )} - 5 \, e^{\left (-8 \, f x - 8 \, e\right )} + e^{\left (-10 \, f x - 10 \, e\right )} - 1}}{256 \, f} - \frac {15 \, \sqrt {a} \log \left (e^{\left (-f x - e\right )} + 1\right ) - 15 \, \sqrt {a} \log \left (e^{\left (-f x - e\right )} - 1\right ) + \frac {2 \, {\left (15 \, \sqrt {a} e^{\left (-f x - e\right )} - 70 \, \sqrt {a} e^{\left (-3 \, f x - 3 \, e\right )} + 128 \, \sqrt {a} e^{\left (-5 \, f x - 5 \, e\right )} - 250 \, \sqrt {a} e^{\left (-7 \, f x - 7 \, e\right )} - 15 \, \sqrt {a} e^{\left (-9 \, f x - 9 \, e\right )}\right )}}{5 \, e^{\left (-2 \, f x - 2 \, e\right )} - 10 \, e^{\left (-4 \, f x - 4 \, e\right )} + 10 \, e^{\left (-6 \, f x - 6 \, e\right )} - 5 \, e^{\left (-8 \, f x - 8 \, e\right )} + e^{\left (-10 \, f x - 10 \, e\right )} - 1}}{256 \, f} + \frac {2 \, \sqrt {a} e^{\left (-5 \, f x - 5 \, e\right )}}{f {\left (5 \, e^{\left (-2 \, f x - 2 \, e\right )} - 10 \, e^{\left (-4 \, f x - 4 \, e\right )} + 10 \, e^{\left (-6 \, f x - 6 \, e\right )} - 5 \, e^{\left (-8 \, f x - 8 \, e\right )} + e^{\left (-10 \, f x - 10 \, e\right )} - 1\right )}} - \frac {2895 \, \sqrt {a} e^{\left (-2 \, f x - 2 \, e\right )} - 7110 \, \sqrt {a} e^{\left (-4 \, f x - 4 \, e\right )} + 8064 \, \sqrt {a} e^{\left (-6 \, f x - 6 \, e\right )} - 4410 \, \sqrt {a} e^{\left (-8 \, f x - 8 \, e\right )} + 945 \, \sqrt {a} e^{\left (-10 \, f x - 10 \, e\right )} - 320 \, \sqrt {a}}{640 \, f {\left (e^{\left (-f x - e\right )} - 5 \, e^{\left (-3 \, f x - 3 \, e\right )} + 10 \, e^{\left (-5 \, f x - 5 \, e\right )} - 10 \, e^{\left (-7 \, f x - 7 \, e\right )} + 5 \, e^{\left (-9 \, f x - 9 \, e\right )} - e^{\left (-11 \, f x - 11 \, e\right )}\right )}} + \frac {945 \, \sqrt {a} e^{\left (-f x - e\right )} - 4410 \, \sqrt {a} e^{\left (-3 \, f x - 3 \, e\right )} + 8064 \, \sqrt {a} e^{\left (-5 \, f x - 5 \, e\right )} - 7110 \, \sqrt {a} e^{\left (-7 \, f x - 7 \, e\right )} + 2895 \, \sqrt {a} e^{\left (-9 \, f x - 9 \, e\right )} - 320 \, \sqrt {a} e^{\left (-11 \, f x - 11 \, e\right )}}{640 \, f {\left (5 \, e^{\left (-2 \, f x - 2 \, e\right )} - 10 \, e^{\left (-4 \, f x - 4 \, e\right )} + 10 \, e^{\left (-6 \, f x - 6 \, e\right )} - 5 \, e^{\left (-8 \, f x - 8 \, e\right )} + e^{\left (-10 \, f x - 10 \, e\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. 1696 vs.
\(2 (114) = 228\).
time = 0.50, size = 1696, normalized size = 13.68 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 96, normalized size = 0.77 \begin {gather*} -\frac {\sqrt {a} {\left (\frac {4 \, {\left (15 \, {\left (e^{\left (f x + e\right )} - e^{\left (-f x - e\right )}\right )}^{4} + 20 \, {\left (e^{\left (f x + e\right )} - e^{\left (-f x - e\right )}\right )}^{2} + 16\right )}}{{\left (e^{\left (f x + e\right )} - e^{\left (-f x - e\right )}\right )}^{5}} - 5 \, e^{\left (f x + e\right )} + 5 \, e^{\left (-f x - e\right )}\right )}}{10 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.91, size = 427, normalized size = 3.44 \begin {gather*} -\frac {\left (\frac {1}{f}-\frac {{\mathrm {e}}^{2\,e+2\,f\,x}}{f}\right )\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{{\mathrm {e}}^{2\,e+2\,f\,x}+1}-\frac {12\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{f\,\left ({\mathrm {e}}^{2\,e+2\,f\,x}-1\right )\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}-\frac {16\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}-1\right )}^2\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}-\frac {144\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{5\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}-1\right )}^3\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}-\frac {128\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{5\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}-1\right )}^4\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}-\frac {64\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{5\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}-1\right )}^5\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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