Optimal. Leaf size=154 \[ -\frac {a \left (8 a^4-20 a^2 b^2+15 b^4\right ) x}{8 b^6}+\frac {2 (a-b)^{5/2} (a+b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{b^6}+\frac {\left (8 \left (a^2-b^2\right )^2-a b \left (4 a^2-7 b^2\right ) \cosh (x)\right ) \sinh (x)}{8 b^5}+\frac {\left (4 \left (a^2-b^2\right )-3 a b \cosh (x)\right ) \sinh ^3(x)}{12 b^3}+\frac {\sinh ^5(x)}{5 b} \]
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Rubi [A]
time = 0.30, antiderivative size = 154, 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 = {2774, 2944,
2814, 2738, 214} \begin {gather*} \frac {\sinh (x) \left (8 \left (a^2-b^2\right )^2-a b \left (4 a^2-7 b^2\right ) \cosh (x)\right )}{8 b^5}+\frac {\sinh ^3(x) \left (4 \left (a^2-b^2\right )-3 a b \cosh (x)\right )}{12 b^3}-\frac {a x \left (8 a^4-20 a^2 b^2+15 b^4\right )}{8 b^6}+\frac {2 (a-b)^{5/2} (a+b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{b^6}+\frac {\sinh ^5(x)}{5 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 2738
Rule 2774
Rule 2814
Rule 2944
Rubi steps
\begin {align*} \int \frac {\sinh ^6(x)}{a+b \cosh (x)} \, dx &=\frac {\sinh ^5(x)}{5 b}+\frac {\int \frac {(-b-a \cosh (x)) \sinh ^4(x)}{a+b \cosh (x)} \, dx}{b}\\ &=\frac {\left (4 \left (a^2-b^2\right )-3 a b \cosh (x)\right ) \sinh ^3(x)}{12 b^3}+\frac {\sinh ^5(x)}{5 b}-\frac {\int \frac {\left (b \left (a^2-4 b^2\right )+a \left (4 a^2-7 b^2\right ) \cosh (x)\right ) \sinh ^2(x)}{a+b \cosh (x)} \, dx}{4 b^3}\\ &=\frac {\left (8 \left (a^2-b^2\right )^2-a b \left (4 a^2-7 b^2\right ) \cosh (x)\right ) \sinh (x)}{8 b^5}+\frac {\left (4 \left (a^2-b^2\right )-3 a b \cosh (x)\right ) \sinh ^3(x)}{12 b^3}+\frac {\sinh ^5(x)}{5 b}+\frac {\int \frac {-b \left (4 a^4-9 a^2 b^2+8 b^4\right )-a \left (8 a^4-20 a^2 b^2+15 b^4\right ) \cosh (x)}{a+b \cosh (x)} \, dx}{8 b^5}\\ &=-\frac {a \left (8 a^4-20 a^2 b^2+15 b^4\right ) x}{8 b^6}+\frac {\left (8 \left (a^2-b^2\right )^2-a b \left (4 a^2-7 b^2\right ) \cosh (x)\right ) \sinh (x)}{8 b^5}+\frac {\left (4 \left (a^2-b^2\right )-3 a b \cosh (x)\right ) \sinh ^3(x)}{12 b^3}+\frac {\sinh ^5(x)}{5 b}+\frac {\left (a^2-b^2\right )^3 \int \frac {1}{a+b \cosh (x)} \, dx}{b^6}\\ &=-\frac {a \left (8 a^4-20 a^2 b^2+15 b^4\right ) x}{8 b^6}+\frac {\left (8 \left (a^2-b^2\right )^2-a b \left (4 a^2-7 b^2\right ) \cosh (x)\right ) \sinh (x)}{8 b^5}+\frac {\left (4 \left (a^2-b^2\right )-3 a b \cosh (x)\right ) \sinh ^3(x)}{12 b^3}+\frac {\sinh ^5(x)}{5 b}+\frac {\left (2 \left (a^2-b^2\right )^3\right ) \text {Subst}\left (\int \frac {1}{a+b-(a-b) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{b^6}\\ &=-\frac {a \left (8 a^4-20 a^2 b^2+15 b^4\right ) x}{8 b^6}+\frac {2 (a-b)^{5/2} (a+b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{b^6}+\frac {\left (8 \left (a^2-b^2\right )^2-a b \left (4 a^2-7 b^2\right ) \cosh (x)\right ) \sinh (x)}{8 b^5}+\frac {\left (4 \left (a^2-b^2\right )-3 a b \cosh (x)\right ) \sinh ^3(x)}{12 b^3}+\frac {\sinh ^5(x)}{5 b}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 154, normalized size = 1.00 \begin {gather*} \frac {-60 a \left (8 a^4-20 a^2 b^2+15 b^4\right ) x+960 \left (-a^2+b^2\right )^{5/2} \text {ArcTan}\left (\frac {(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2+b^2}}\right )+60 b \left (8 a^4-18 a^2 b^2+11 b^4\right ) \sinh (x)-120 a b^2 \left (a^2-2 b^2\right ) \sinh (2 x)-10 b^3 \left (-4 a^2+7 b^2\right ) \sinh (3 x)-15 a b^4 \sinh (4 x)+6 b^5 \sinh (5 x)}{480 b^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(408\) vs.
\(2(135)=270\).
time = 0.48, size = 409, normalized size = 2.66
method | result | size |
default | \(-\frac {2 \left (-a^{6}+3 a^{4} b^{2}-3 a^{2} b^{4}+b^{6}\right ) \arctanh \left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{b^{6} \sqrt {\left (a +b \right ) \left (a -b \right )}}-\frac {1}{5 b \left (\tanh \left (\frac {x}{2}\right )-1\right )^{5}}-\frac {2 b +a}{4 b^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}-\frac {4 a^{2}+6 a b -b^{2}}{12 b^{3} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {4 a^{3}+4 a^{2} b -5 a \,b^{2}-5 b^{3}}{8 b^{4} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {8 a^{4}+4 a^{3} b -16 a^{2} b^{2}-7 a \,b^{3}+8 b^{4}}{8 b^{5} \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {a \left (8 a^{4}-20 a^{2} b^{2}+15 b^{4}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{8 b^{6}}-\frac {1}{5 b \left (\tanh \left (\frac {x}{2}\right )+1\right )^{5}}-\frac {-2 b -a}{4 b^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {4 a^{2}+6 a b -b^{2}}{12 b^{3} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {-4 a^{3}-4 a^{2} b +5 a \,b^{2}+5 b^{3}}{8 b^{4} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {8 a^{4}+4 a^{3} b -16 a^{2} b^{2}-7 a \,b^{3}+8 b^{4}}{8 b^{5} \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {a \left (8 a^{4}-20 a^{2} b^{2}+15 b^{4}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{8 b^{6}}\) | \(409\) |
risch | \(-\frac {15 a x}{8 b^{2}}+\frac {{\mathrm e}^{3 x} a^{2}}{24 b^{3}}-\frac {a^{3} {\mathrm e}^{2 x}}{8 b^{4}}+\frac {a \,{\mathrm e}^{2 x}}{4 b^{2}}-\frac {a^{5} x}{b^{6}}-\frac {a \,{\mathrm e}^{4 x}}{64 b^{2}}+\frac {11 \,{\mathrm e}^{x}}{16 b}+\frac {5 a^{3} x}{2 b^{4}}+\frac {a \,{\mathrm e}^{-4 x}}{64 b^{2}}+\frac {{\mathrm e}^{x} a^{4}}{2 b^{5}}-\frac {9 \,{\mathrm e}^{x} a^{2}}{8 b^{3}}-\frac {{\mathrm e}^{-x} a^{4}}{2 b^{5}}+\frac {9 \,{\mathrm e}^{-x} a^{2}}{8 b^{3}}+\frac {a^{3} {\mathrm e}^{-2 x}}{8 b^{4}}-\frac {a \,{\mathrm e}^{-2 x}}{4 b^{2}}-\frac {{\mathrm e}^{-3 x} a^{2}}{24 b^{3}}+\frac {\sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{x}-\frac {-a +\sqrt {a^{2}-b^{2}}}{b}\right )}{b^{2}}-\frac {\sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{x}+\frac {a +\sqrt {a^{2}-b^{2}}}{b}\right )}{b^{2}}-\frac {{\mathrm e}^{-5 x}}{160 b}+\frac {{\mathrm e}^{5 x}}{160 b}-\frac {7 \,{\mathrm e}^{3 x}}{96 b}+\frac {7 \,{\mathrm e}^{-3 x}}{96 b}-\frac {11 \,{\mathrm e}^{-x}}{16 b}+\frac {\sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{x}-\frac {-a +\sqrt {a^{2}-b^{2}}}{b}\right ) a^{4}}{b^{6}}-\frac {2 \sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{x}-\frac {-a +\sqrt {a^{2}-b^{2}}}{b}\right ) a^{2}}{b^{4}}-\frac {\sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{x}+\frac {a +\sqrt {a^{2}-b^{2}}}{b}\right ) a^{4}}{b^{6}}+\frac {2 \sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{x}+\frac {a +\sqrt {a^{2}-b^{2}}}{b}\right ) a^{2}}{b^{4}}\) | \(452\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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. 1422 vs.
\(2 (134) = 268\).
time = 0.40, size = 2913, normalized size = 18.92 \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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 266, normalized size = 1.73 \begin {gather*} \frac {6 \, b^{4} e^{\left (5 \, x\right )} - 15 \, a b^{3} e^{\left (4 \, x\right )} + 40 \, a^{2} b^{2} e^{\left (3 \, x\right )} - 70 \, b^{4} e^{\left (3 \, x\right )} - 120 \, a^{3} b e^{\left (2 \, x\right )} + 240 \, a b^{3} e^{\left (2 \, x\right )} + 480 \, a^{4} e^{x} - 1080 \, a^{2} b^{2} e^{x} + 660 \, b^{4} e^{x}}{960 \, b^{5}} - \frac {{\left (8 \, a^{5} - 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} x}{8 \, b^{6}} + \frac {{\left (15 \, a b^{4} e^{x} - 6 \, b^{5} - 60 \, {\left (8 \, a^{4} b - 18 \, a^{2} b^{3} + 11 \, b^{5}\right )} e^{\left (4 \, x\right )} + 120 \, {\left (a^{3} b^{2} - 2 \, a b^{4}\right )} e^{\left (3 \, x\right )} - 10 \, {\left (4 \, a^{2} b^{3} - 7 \, b^{5}\right )} e^{\left (2 \, x\right )}\right )} e^{\left (-5 \, x\right )}}{960 \, b^{6}} + \frac {2 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \arctan \left (\frac {b e^{x} + a}{\sqrt {-a^{2} + b^{2}}}\right )}{\sqrt {-a^{2} + b^{2}} b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.70, size = 348, normalized size = 2.26 \begin {gather*} \frac {{\mathrm {e}}^{5\,x}}{160\,b}-\frac {{\mathrm {e}}^{-5\,x}}{160\,b}-\frac {{\mathrm {e}}^{-2\,x}\,\left (2\,a\,b^2-a^3\right )}{8\,b^4}+\frac {{\mathrm {e}}^{2\,x}\,\left (2\,a\,b^2-a^3\right )}{8\,b^4}-\frac {x\,\left (8\,a^5-20\,a^3\,b^2+15\,a\,b^4\right )}{8\,b^6}+\frac {{\mathrm {e}}^x\,\left (8\,a^4-18\,a^2\,b^2+11\,b^4\right )}{16\,b^5}+\frac {a\,{\mathrm {e}}^{-4\,x}}{64\,b^2}-\frac {a\,{\mathrm {e}}^{4\,x}}{64\,b^2}-\frac {{\mathrm {e}}^{-x}\,\left (8\,a^4-18\,a^2\,b^2+11\,b^4\right )}{16\,b^5}-\frac {{\mathrm {e}}^{-3\,x}\,\left (4\,a^2-7\,b^2\right )}{96\,b^3}+\frac {{\mathrm {e}}^{3\,x}\,\left (4\,a^2-7\,b^2\right )}{96\,b^3}+\frac {\ln \left (-\frac {2\,{\mathrm {e}}^x\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}{b^7}-\frac {2\,{\left (a+b\right )}^{5/2}\,\left (b+a\,{\mathrm {e}}^x\right )\,{\left (a-b\right )}^{5/2}}{b^7}\right )\,{\left (a+b\right )}^{5/2}\,{\left (a-b\right )}^{5/2}}{b^6}-\frac {\ln \left (\frac {2\,{\left (a+b\right )}^{5/2}\,\left (b+a\,{\mathrm {e}}^x\right )\,{\left (a-b\right )}^{5/2}}{b^7}-\frac {2\,{\mathrm {e}}^x\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}{b^7}\right )\,{\left (a+b\right )}^{5/2}\,{\left (a-b\right )}^{5/2}}{b^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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