Optimal. Leaf size=31 \[ -\frac {\text {csch}^3(a+b x)}{3 b}-\frac {\text {csch}^5(a+b x)}{5 b} \]
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Rubi [A]
time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2686, 14}
\begin {gather*} -\frac {\text {csch}^5(a+b x)}{5 b}-\frac {\text {csch}^3(a+b x)}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2686
Rubi steps
\begin {align*} \int \coth ^3(a+b x) \text {csch}^3(a+b x) \, dx &=-\frac {i \text {Subst}\left (\int x^2 \left (-1+x^2\right ) \, dx,x,-i \text {csch}(a+b x)\right )}{b}\\ &=-\frac {i \text {Subst}\left (\int \left (-x^2+x^4\right ) \, dx,x,-i \text {csch}(a+b x)\right )}{b}\\ &=-\frac {\text {csch}^3(a+b x)}{3 b}-\frac {\text {csch}^5(a+b x)}{5 b}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 31, normalized size = 1.00 \begin {gather*} -\frac {\text {csch}^3(a+b x)}{3 b}-\frac {\text {csch}^5(a+b x)}{5 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.86, size = 34, normalized size = 1.10
method | result | size |
derivativedivides | \(\frac {-\frac {\cosh ^{2}\left (b x +a \right )}{3 \sinh \left (b x +a \right )^{5}}+\frac {2}{15 \sinh \left (b x +a \right )^{5}}}{b}\) | \(34\) |
default | \(\frac {-\frac {\cosh ^{2}\left (b x +a \right )}{3 \sinh \left (b x +a \right )^{5}}+\frac {2}{15 \sinh \left (b x +a \right )^{5}}}{b}\) | \(34\) |
risch | \(-\frac {8 \,{\mathrm e}^{3 b x +3 a} \left (5 \,{\mathrm e}^{4 b x +4 a}+2 \,{\mathrm e}^{2 b x +2 a}+5\right )}{15 b \left ({\mathrm e}^{2 b x +2 a}-1\right )^{5}}\) | \(52\) |
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. 214 vs.
\(2 (27) = 54\).
time = 0.26, size = 214, normalized size = 6.90 \begin {gather*} \frac {8 \, e^{\left (-3 \, b x - 3 \, a\right )}}{3 \, b {\left (5 \, e^{\left (-2 \, b x - 2 \, a\right )} - 10 \, e^{\left (-4 \, b x - 4 \, a\right )} + 10 \, e^{\left (-6 \, b x - 6 \, a\right )} - 5 \, e^{\left (-8 \, b x - 8 \, a\right )} + e^{\left (-10 \, b x - 10 \, a\right )} - 1\right )}} + \frac {16 \, e^{\left (-5 \, b x - 5 \, a\right )}}{15 \, b {\left (5 \, e^{\left (-2 \, b x - 2 \, a\right )} - 10 \, e^{\left (-4 \, b x - 4 \, a\right )} + 10 \, e^{\left (-6 \, b x - 6 \, a\right )} - 5 \, e^{\left (-8 \, b x - 8 \, a\right )} + e^{\left (-10 \, b x - 10 \, a\right )} - 1\right )}} + \frac {8 \, e^{\left (-7 \, b x - 7 \, a\right )}}{3 \, b {\left (5 \, e^{\left (-2 \, b x - 2 \, a\right )} - 10 \, e^{\left (-4 \, b x - 4 \, a\right )} + 10 \, e^{\left (-6 \, b x - 6 \, a\right )} - 5 \, e^{\left (-8 \, b x - 8 \, a\right )} + e^{\left (-10 \, b x - 10 \, 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. 343 vs.
\(2 (27) = 54\).
time = 0.36, size = 343, normalized size = 11.06 \begin {gather*} -\frac {8 \, {\left (5 \, \cosh \left (b x + a\right )^{4} + 20 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + 5 \, \sinh \left (b x + a\right )^{4} + 2 \, {\left (15 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right )^{2} + 4 \, {\left (5 \, \cosh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 5\right )}}{15 \, {\left (b \cosh \left (b x + a\right )^{7} + 7 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{6} + b \sinh \left (b x + a\right )^{7} - 5 \, b \cosh \left (b x + a\right )^{5} + {\left (21 \, b \cosh \left (b x + a\right )^{2} - 5 \, b\right )} \sinh \left (b x + a\right )^{5} + 5 \, {\left (7 \, b \cosh \left (b x + a\right )^{3} - 5 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{4} + 9 \, b \cosh \left (b x + a\right )^{3} + {\left (35 \, b \cosh \left (b x + a\right )^{4} - 50 \, b \cosh \left (b x + a\right )^{2} + 11 \, b\right )} \sinh \left (b x + a\right )^{3} + {\left (21 \, b \cosh \left (b x + a\right )^{5} - 50 \, b \cosh \left (b x + a\right )^{3} + 27 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} - 5 \, b \cosh \left (b x + a\right ) + {\left (7 \, b \cosh \left (b x + a\right )^{6} - 25 \, b \cosh \left (b x + a\right )^{4} + 33 \, b \cosh \left (b x + a\right )^{2} - 15 \, b\right )} \sinh \left (b x + a\right )\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 ^{3}{\left (a + b x \right )} \operatorname {csch}^{3}{\left (a + b x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 49, normalized size = 1.58 \begin {gather*} -\frac {8 \, {\left (5 \, {\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{2} + 12\right )}}{15 \, b {\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.51, size = 252, normalized size = 8.13 \begin {gather*} -\frac {\frac {4\,{\mathrm {e}}^{a+b\,x}}{5\,b}+\frac {12\,{\mathrm {e}}^{3\,a+3\,b\,x}}{5\,b}+\frac {12\,{\mathrm {e}}^{5\,a+5\,b\,x}}{5\,b}+\frac {4\,{\mathrm {e}}^{7\,a+7\,b\,x}}{5\,b}}{5\,{\mathrm {e}}^{2\,a+2\,b\,x}-10\,{\mathrm {e}}^{4\,a+4\,b\,x}+10\,{\mathrm {e}}^{6\,a+6\,b\,x}-5\,{\mathrm {e}}^{8\,a+8\,b\,x}+{\mathrm {e}}^{10\,a+10\,b\,x}-1}-\frac {28\,{\mathrm {e}}^{a+b\,x}}{15\,b\,\left ({\mathrm {e}}^{4\,a+4\,b\,x}-2\,{\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}-\frac {64\,{\mathrm {e}}^{a+b\,x}}{15\,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 {16\,{\mathrm {e}}^{a+b\,x}}{5\,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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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