Optimal. Leaf size=159 \[ \frac {2 b^6 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2}}+\frac {\left (15 b^5-a \left (8 a^4-26 a^2 b^2+33 b^4\right ) \cosh (x)\right ) \text {csch}(x)}{15 \left (a^2-b^2\right )^3}+\frac {\left (5 b^3+a \left (4 a^2-9 b^2\right ) \cosh (x)\right ) \text {csch}^3(x)}{15 \left (a^2-b^2\right )^2}+\frac {(b-a \cosh (x)) \text {csch}^5(x)}{5 \left (a^2-b^2\right )} \]
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Rubi [A]
time = 0.34, antiderivative size = 159, 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 = {2775, 2945, 12,
2738, 214} \begin {gather*} \frac {\text {csch}^5(x) (b-a \cosh (x))}{5 \left (a^2-b^2\right )}+\frac {\text {csch}^3(x) \left (a \left (4 a^2-9 b^2\right ) \cosh (x)+5 b^3\right )}{15 \left (a^2-b^2\right )^2}+\frac {\text {csch}(x) \left (15 b^5-a \left (8 a^4-26 a^2 b^2+33 b^4\right ) \cosh (x)\right )}{15 \left (a^2-b^2\right )^3}+\frac {2 b^6 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 214
Rule 2738
Rule 2775
Rule 2945
Rubi steps
\begin {align*} \int \frac {\text {csch}^6(x)}{a+b \cosh (x)} \, dx &=\frac {(b-a \cosh (x)) \text {csch}^5(x)}{5 \left (a^2-b^2\right )}+\frac {\int \frac {\left (-4 a^2+5 b^2-4 a b \cosh (x)\right ) \text {csch}^4(x)}{a+b \cosh (x)} \, dx}{5 \left (a^2-b^2\right )}\\ &=\frac {\left (5 b^3+a \left (4 a^2-9 b^2\right ) \cosh (x)\right ) \text {csch}^3(x)}{15 \left (a^2-b^2\right )^2}+\frac {(b-a \cosh (x)) \text {csch}^5(x)}{5 \left (a^2-b^2\right )}+\frac {\int \frac {\left (8 a^4-18 a^2 b^2+15 b^4+2 a b \left (4 a^2-9 b^2\right ) \cosh (x)\right ) \text {csch}^2(x)}{a+b \cosh (x)} \, dx}{15 \left (a^2-b^2\right )^2}\\ &=\frac {\left (15 b^5-a \left (8 a^4-26 a^2 b^2+33 b^4\right ) \cosh (x)\right ) \text {csch}(x)}{15 \left (a^2-b^2\right )^3}+\frac {\left (5 b^3+a \left (4 a^2-9 b^2\right ) \cosh (x)\right ) \text {csch}^3(x)}{15 \left (a^2-b^2\right )^2}+\frac {(b-a \cosh (x)) \text {csch}^5(x)}{5 \left (a^2-b^2\right )}+\frac {\int \frac {15 b^6}{a+b \cosh (x)} \, dx}{15 \left (a^2-b^2\right )^3}\\ &=\frac {\left (15 b^5-a \left (8 a^4-26 a^2 b^2+33 b^4\right ) \cosh (x)\right ) \text {csch}(x)}{15 \left (a^2-b^2\right )^3}+\frac {\left (5 b^3+a \left (4 a^2-9 b^2\right ) \cosh (x)\right ) \text {csch}^3(x)}{15 \left (a^2-b^2\right )^2}+\frac {(b-a \cosh (x)) \text {csch}^5(x)}{5 \left (a^2-b^2\right )}+\frac {b^6 \int \frac {1}{a+b \cosh (x)} \, dx}{\left (a^2-b^2\right )^3}\\ &=\frac {\left (15 b^5-a \left (8 a^4-26 a^2 b^2+33 b^4\right ) \cosh (x)\right ) \text {csch}(x)}{15 \left (a^2-b^2\right )^3}+\frac {\left (5 b^3+a \left (4 a^2-9 b^2\right ) \cosh (x)\right ) \text {csch}^3(x)}{15 \left (a^2-b^2\right )^2}+\frac {(b-a \cosh (x)) \text {csch}^5(x)}{5 \left (a^2-b^2\right )}+\frac {\left (2 b^6\right ) \text {Subst}\left (\int \frac {1}{a+b-(a-b) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{\left (a^2-b^2\right )^3}\\ &=\frac {2 b^6 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2}}+\frac {\left (15 b^5-a \left (8 a^4-26 a^2 b^2+33 b^4\right ) \cosh (x)\right ) \text {csch}(x)}{15 \left (a^2-b^2\right )^3}+\frac {\left (5 b^3+a \left (4 a^2-9 b^2\right ) \cosh (x)\right ) \text {csch}^3(x)}{15 \left (a^2-b^2\right )^2}+\frac {(b-a \cosh (x)) \text {csch}^5(x)}{5 \left (a^2-b^2\right )}\\ \end {align*}
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Mathematica [A]
time = 1.25, size = 201, normalized size = 1.26 \begin {gather*} \frac {1}{480} \left (\frac {960 b^6 \text {ArcTan}\left (\frac {(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{7/2}}-\frac {2 \left (64 a^2+183 a b+149 b^2\right ) \coth \left (\frac {x}{2}\right )}{(a+b)^3}-\frac {8 (19 a-29 b) \text {csch}^3(x) \sinh ^4\left (\frac {x}{2}\right )}{(a-b)^2}-\frac {96 \text {csch}^5(x) \sinh ^6\left (\frac {x}{2}\right )}{a-b}+\frac {(19 a+29 b) \text {csch}^4\left (\frac {x}{2}\right ) \sinh (x)}{2 (a+b)^2}-\frac {3 \text {csch}^6\left (\frac {x}{2}\right ) \sinh (x)}{2 (a+b)}-\frac {2 \left (64 a^2-183 a b+149 b^2\right ) \tanh \left (\frac {x}{2}\right )}{(a-b)^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.68, size = 213, normalized size = 1.34
method | result | size |
default | \(-\frac {\frac {a^{2} \left (\tanh ^{5}\left (\frac {x}{2}\right )\right )}{5}-\frac {2 a b \left (\tanh ^{5}\left (\frac {x}{2}\right )\right )}{5}+\frac {b^{2} \left (\tanh ^{5}\left (\frac {x}{2}\right )\right )}{5}-\frac {5 a^{2} \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{3}+4 a b \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )-\frac {7 b^{2} \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{3}+10 a^{2} \tanh \left (\frac {x}{2}\right )-28 a b \tanh \left (\frac {x}{2}\right )+22 b^{2} \tanh \left (\frac {x}{2}\right )}{32 \left (a -b \right )^{3}}-\frac {1}{160 \left (a +b \right ) \tanh \left (\frac {x}{2}\right )^{5}}-\frac {-5 a -7 b}{96 \left (a +b \right )^{2} \tanh \left (\frac {x}{2}\right )^{3}}-\frac {10 a^{2}+28 a b +22 b^{2}}{32 \left (a +b \right )^{3} \tanh \left (\frac {x}{2}\right )}+\frac {2 b^{6} \arctanh \left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a -b \right )^{3} \left (a +b \right )^{3} \sqrt {\left (a +b \right ) \left (a -b \right )}}\) | \(213\) |
risch | \(-\frac {2 \left (-15 b^{5} {\mathrm e}^{9 x}+15 a \,b^{4} {\mathrm e}^{8 x}-20 a^{2} b^{3} {\mathrm e}^{7 x}+80 b^{5} {\mathrm e}^{7 x}+30 a^{3} b^{2} {\mathrm e}^{6 x}-90 a \,b^{4} {\mathrm e}^{6 x}-48 a^{4} b \,{\mathrm e}^{5 x}+136 a^{2} b^{3} {\mathrm e}^{5 x}-178 b^{5} {\mathrm e}^{5 x}+80 a^{5} {\mathrm e}^{4 x}-230 a^{3} b^{2} {\mathrm e}^{4 x}+240 a \,b^{4} {\mathrm e}^{4 x}-20 a^{2} b^{3} {\mathrm e}^{3 x}+80 b^{5} {\mathrm e}^{3 x}-40 a^{5} {\mathrm e}^{2 x}+130 a^{3} b^{2} {\mathrm e}^{2 x}-150 a \,b^{4} {\mathrm e}^{2 x}-15 b^{5} {\mathrm e}^{x}+8 a^{5}-26 a^{3} b^{2}+33 a \,b^{4}\right )}{15 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \left ({\mathrm e}^{2 x}-1\right )^{5}}+\frac {b^{6} \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}-b^{2}}-a^{2}+b^{2}}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3}}-\frac {b^{6} \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}-b^{2}}+a^{2}-b^{2}}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3}}\) | \(379\) |
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. 3156 vs.
\(2 (144) = 288\).
time = 0.49, size = 6381, normalized size = 40.13 \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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {csch}^{6}{\left (x \right )}}{a + b \cosh {\left (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. 303 vs.
\(2 (144) = 288\).
time = 0.42, size = 303, normalized size = 1.91 \begin {gather*} \frac {2 \, b^{6} \arctan \left (\frac {b e^{x} + a}{\sqrt {-a^{2} + b^{2}}}\right )}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \sqrt {-a^{2} + b^{2}}} + \frac {2 \, {\left (15 \, b^{5} e^{\left (9 \, x\right )} - 15 \, a b^{4} e^{\left (8 \, x\right )} + 20 \, a^{2} b^{3} e^{\left (7 \, x\right )} - 80 \, b^{5} e^{\left (7 \, x\right )} - 30 \, a^{3} b^{2} e^{\left (6 \, x\right )} + 90 \, a b^{4} e^{\left (6 \, x\right )} + 48 \, a^{4} b e^{\left (5 \, x\right )} - 136 \, a^{2} b^{3} e^{\left (5 \, x\right )} + 178 \, b^{5} e^{\left (5 \, x\right )} - 80 \, a^{5} e^{\left (4 \, x\right )} + 230 \, a^{3} b^{2} e^{\left (4 \, x\right )} - 240 \, a b^{4} e^{\left (4 \, x\right )} + 20 \, a^{2} b^{3} e^{\left (3 \, x\right )} - 80 \, b^{5} e^{\left (3 \, x\right )} + 40 \, a^{5} e^{\left (2 \, x\right )} - 130 \, a^{3} b^{2} e^{\left (2 \, x\right )} + 150 \, a b^{4} e^{\left (2 \, x\right )} + 15 \, b^{5} e^{x} - 8 \, a^{5} + 26 \, a^{3} b^{2} - 33 \, a b^{4}\right )}}{15 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} {\left (e^{\left (2 \, x\right )} - 1\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.61, size = 1031, normalized size = 6.48 \begin {gather*} \frac {\frac {16\,\left (a\,b^2-a^3\right )}{{\left (a^2-b^2\right )}^2}+\frac {64\,{\mathrm {e}}^x\,\left (a^2\,b-b^3\right )}{5\,{\left (a^2-b^2\right )}^2}}{6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1}-\frac {\frac {2\,a\,b^4}{{\left (a^2-b^2\right )}^3}-\frac {2\,b^5\,{\mathrm {e}}^x}{{\left (a^2-b^2\right )}^3}}{{\mathrm {e}}^{2\,x}-1}-\frac {\frac {32\,a}{5\,\left (a^2-b^2\right )}-\frac {32\,b\,{\mathrm {e}}^x}{5\,\left (a^2-b^2\right )}}{5\,{\mathrm {e}}^{2\,x}-10\,{\mathrm {e}}^{4\,x}+10\,{\mathrm {e}}^{6\,x}-5\,{\mathrm {e}}^{8\,x}+{\mathrm {e}}^{10\,x}-1}+\frac {\frac {8\,\left (3\,a\,b^2-4\,a^3\right )}{3\,{\left (a^2-b^2\right )}^2}+\frac {8\,{\mathrm {e}}^x\,\left (12\,a^2\,b-7\,b^3\right )}{15\,{\left (a^2-b^2\right )}^2}}{3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1}+\frac {\frac {4\,\left (a\,b^4-a^3\,b^2\right )}{{\left (a^2-b^2\right )}^3}-\frac {8\,{\mathrm {e}}^x\,\left (b^5-a^2\,b^3\right )}{3\,{\left (a^2-b^2\right )}^3}}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1}-\frac {2\,\mathrm {atan}\left (\left ({\mathrm {e}}^x\,\left (\frac {2\,b^4}{{\left (a^2-b^2\right )}^3\,\sqrt {b^{12}}\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}+\frac {2\,a\,\left (a^7\,\sqrt {b^{12}}+3\,a^3\,b^4\,\sqrt {b^{12}}-3\,a^5\,b^2\,\sqrt {b^{12}}-a\,b^6\,\sqrt {b^{12}}\right )}{b^8\,\sqrt {-{\left (a^2-b^2\right )}^7}\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )\,\sqrt {-a^{14}+7\,a^{12}\,b^2-21\,a^{10}\,b^4+35\,a^8\,b^6-35\,a^6\,b^8+21\,a^4\,b^{10}-7\,a^2\,b^{12}+b^{14}}}\right )-\frac {2\,a\,\left (b^7\,\sqrt {b^{12}}-3\,a^2\,b^5\,\sqrt {b^{12}}+3\,a^4\,b^3\,\sqrt {b^{12}}-a^6\,b\,\sqrt {b^{12}}\right )}{b^8\,\sqrt {-{\left (a^2-b^2\right )}^7}\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )\,\sqrt {-a^{14}+7\,a^{12}\,b^2-21\,a^{10}\,b^4+35\,a^8\,b^6-35\,a^6\,b^8+21\,a^4\,b^{10}-7\,a^2\,b^{12}+b^{14}}}\right )\,\left (\frac {b^7\,\sqrt {-a^{14}+7\,a^{12}\,b^2-21\,a^{10}\,b^4+35\,a^8\,b^6-35\,a^6\,b^8+21\,a^4\,b^{10}-7\,a^2\,b^{12}+b^{14}}}{2}-\frac {a^6\,b\,\sqrt {-a^{14}+7\,a^{12}\,b^2-21\,a^{10}\,b^4+35\,a^8\,b^6-35\,a^6\,b^8+21\,a^4\,b^{10}-7\,a^2\,b^{12}+b^{14}}}{2}-\frac {3\,a^2\,b^5\,\sqrt {-a^{14}+7\,a^{12}\,b^2-21\,a^{10}\,b^4+35\,a^8\,b^6-35\,a^6\,b^8+21\,a^4\,b^{10}-7\,a^2\,b^{12}+b^{14}}}{2}+\frac {3\,a^4\,b^3\,\sqrt {-a^{14}+7\,a^{12}\,b^2-21\,a^{10}\,b^4+35\,a^8\,b^6-35\,a^6\,b^8+21\,a^4\,b^{10}-7\,a^2\,b^{12}+b^{14}}}{2}\right )\right )\,\sqrt {b^{12}}}{\sqrt {-a^{14}+7\,a^{12}\,b^2-21\,a^{10}\,b^4+35\,a^8\,b^6-35\,a^6\,b^8+21\,a^4\,b^{10}-7\,a^2\,b^{12}+b^{14}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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