Optimal. Leaf size=88 \[ -\frac {a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b}}\right )}{b^3 \sqrt {a+b}}+\frac {x \left (8 a^2-4 a b+3 b^2\right )}{8 b^3}-\frac {(4 a-3 b) \sinh (x) \cosh (x)}{8 b^2}+\frac {\sinh (x) \cosh ^3(x)}{4 b} \]
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Rubi [A] time = 0.21, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3187, 470, 578, 522, 206, 208} \[ \frac {x \left (8 a^2-4 a b+3 b^2\right )}{8 b^3}-\frac {a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b}}\right )}{b^3 \sqrt {a+b}}-\frac {(4 a-3 b) \sinh (x) \cosh (x)}{8 b^2}+\frac {\sinh (x) \cosh ^3(x)}{4 b} \]
Antiderivative was successfully verified.
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Rule 206
Rule 208
Rule 470
Rule 522
Rule 578
Rule 3187
Rubi steps
\begin {align*} \int \frac {\cosh ^6(x)}{a+b \cosh ^2(x)} \, dx &=-\operatorname {Subst}\left (\int \frac {x^6}{\left (1-x^2\right )^3 \left (a-(a+b) x^2\right )} \, dx,x,\coth (x)\right )\\ &=\frac {\cosh ^3(x) \sinh (x)}{4 b}-\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (3 a+(a-3 b) x^2\right )}{\left (1-x^2\right )^2 \left (a+(-a-b) x^2\right )} \, dx,x,\coth (x)\right )}{4 b}\\ &=-\frac {(4 a-3 b) \cosh (x) \sinh (x)}{8 b^2}+\frac {\cosh ^3(x) \sinh (x)}{4 b}+\frac {\operatorname {Subst}\left (\int \frac {-a (4 a-3 b)+\left (-4 a^2+a b-3 b^2\right ) x^2}{\left (1-x^2\right ) \left (a+(-a-b) x^2\right )} \, dx,x,\coth (x)\right )}{8 b^2}\\ &=-\frac {(4 a-3 b) \cosh (x) \sinh (x)}{8 b^2}+\frac {\cosh ^3(x) \sinh (x)}{4 b}-\frac {a^3 \operatorname {Subst}\left (\int \frac {1}{a+(-a-b) x^2} \, dx,x,\coth (x)\right )}{b^3}+\frac {\left (8 a^2-4 a b+3 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\coth (x)\right )}{8 b^3}\\ &=\frac {\left (8 a^2-4 a b+3 b^2\right ) x}{8 b^3}-\frac {a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b}}\right )}{b^3 \sqrt {a+b}}-\frac {(4 a-3 b) \cosh (x) \sinh (x)}{8 b^2}+\frac {\cosh ^3(x) \sinh (x)}{4 b}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 76, normalized size = 0.86 \[ \frac {-\frac {32 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b}}\right )}{\sqrt {a+b}}+4 x \left (8 a^2-4 a b+3 b^2\right )-8 b (a-b) \sinh (2 x)+b^2 \sinh (4 x)}{32 b^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 1245, normalized size = 14.15 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 150, normalized size = 1.70 \[ -\frac {a^{3} \arctan \left (\frac {b e^{\left (2 \, x\right )} + 2 \, a + b}{2 \, \sqrt {-a^{2} - a b}}\right )}{\sqrt {-a^{2} - a b} b^{3}} + \frac {b e^{\left (4 \, x\right )} - 8 \, a e^{\left (2 \, x\right )} + 8 \, b e^{\left (2 \, x\right )}}{64 \, b^{2}} + \frac {{\left (8 \, a^{2} - 4 \, a b + 3 \, b^{2}\right )} x}{8 \, b^{3}} - \frac {{\left (48 \, a^{2} e^{\left (4 \, x\right )} - 24 \, a b e^{\left (4 \, x\right )} + 18 \, b^{2} e^{\left (4 \, x\right )} - 8 \, a b e^{\left (2 \, x\right )} + 8 \, b^{2} e^{\left (2 \, x\right )} + b^{2}\right )} e^{\left (-4 \, x\right )}}{64 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.11, size = 326, normalized size = 3.70 \[ \frac {1}{4 b \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}+\frac {1}{2 b \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {a}{2 b^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {7}{8 b \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {a}{2 b^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {5}{8 b \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right ) a^{2}}{b^{3}}+\frac {a \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2 b^{2}}-\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{8 b}-\frac {1}{4 b \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}+\frac {1}{2 b \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {a}{2 b^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {7}{8 b \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {a}{2 b^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {5}{8 b \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right ) a^{2}}{b^{3}}-\frac {a \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2 b^{2}}+\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{8 b}+\frac {a^{\frac {5}{2}} \ln \left (-\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+2 \sqrt {a}\, \tanh \left (\frac {x}{2}\right )-\sqrt {a +b}\right )}{2 b^{3} \sqrt {a +b}}-\frac {a^{\frac {5}{2}} \ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+2 \sqrt {a}\, \tanh \left (\frac {x}{2}\right )+\sqrt {a +b}\right )}{2 b^{3} \sqrt {a +b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.50, size = 651, normalized size = 7.40 \[ -\frac {15 \, {\left (2 \, a + b\right )} \log \left (\frac {b e^{\left (2 \, x\right )} + 2 \, a + b - 2 \, \sqrt {{\left (a + b\right )} a}}{b e^{\left (2 \, x\right )} + 2 \, a + b + 2 \, \sqrt {{\left (a + b\right )} a}}\right )}{64 \, \sqrt {{\left (a + b\right )} a} b} - \frac {5 \, \log \left (\frac {b e^{\left (-2 \, x\right )} + 2 \, a + b - 2 \, \sqrt {{\left (a + b\right )} a}}{b e^{\left (-2 \, x\right )} + 2 \, a + b + 2 \, \sqrt {{\left (a + b\right )} a}}\right )}{32 \, \sqrt {{\left (a + b\right )} a}} - \frac {3 \, {\left (2 \, a + b\right )} x}{2 \, b^{2}} + \frac {15 \, x}{16 \, b} - \frac {{\left (4 \, {\left (2 \, a + b\right )} e^{\left (-2 \, x\right )} - b\right )} e^{\left (4 \, x\right )}}{64 \, b^{2}} + \frac {3 \, e^{\left (2 \, x\right )}}{16 \, b} - \frac {3 \, e^{\left (-2 \, x\right )}}{16 \, b} + \frac {{\left (4 \, {\left (2 \, a + b\right )} e^{\left (2 \, x\right )} - b\right )} e^{\left (-4 \, x\right )}}{64 \, b^{2}} + \frac {3 \, {\left (2 \, a + b\right )} \log \left (b e^{\left (4 \, x\right )} + 2 \, {\left (2 \, a + b\right )} e^{\left (2 \, x\right )} + b\right )}{16 \, b^{2}} - \frac {3 \, {\left (2 \, a + b\right )} \log \left (2 \, {\left (2 \, a + b\right )} e^{\left (-2 \, x\right )} + b e^{\left (-4 \, x\right )} + b\right )}{16 \, b^{2}} + \frac {3 \, {\left (8 \, a^{2} + 8 \, a b + b^{2}\right )} \log \left (\frac {b e^{\left (2 \, x\right )} + 2 \, a + b - 2 \, \sqrt {{\left (a + b\right )} a}}{b e^{\left (2 \, x\right )} + 2 \, a + b + 2 \, \sqrt {{\left (a + b\right )} a}}\right )}{64 \, \sqrt {{\left (a + b\right )} a} b^{2}} - \frac {3 \, {\left (8 \, a^{2} + 8 \, a b + b^{2}\right )} \log \left (\frac {b e^{\left (-2 \, x\right )} + 2 \, a + b - 2 \, \sqrt {{\left (a + b\right )} a}}{b e^{\left (-2 \, x\right )} + 2 \, a + b + 2 \, \sqrt {{\left (a + b\right )} a}}\right )}{64 \, \sqrt {{\left (a + b\right )} a} b^{2}} + \frac {{\left (16 \, a^{2} + 16 \, a b + 3 \, b^{2}\right )} x}{8 \, b^{3}} - \frac {{\left (16 \, a^{2} + 16 \, a b + 3 \, b^{2}\right )} \log \left (b e^{\left (4 \, x\right )} + 2 \, {\left (2 \, a + b\right )} e^{\left (2 \, x\right )} + b\right )}{64 \, b^{3}} + \frac {{\left (16 \, a^{2} + 16 \, a b + 3 \, b^{2}\right )} \log \left (2 \, {\left (2 \, a + b\right )} e^{\left (-2 \, x\right )} + b e^{\left (-4 \, x\right )} + b\right )}{64 \, b^{3}} - \frac {{\left (32 \, a^{3} + 48 \, a^{2} b + 18 \, a b^{2} + b^{3}\right )} \log \left (\frac {b e^{\left (2 \, x\right )} + 2 \, a + b - 2 \, \sqrt {{\left (a + b\right )} a}}{b e^{\left (2 \, x\right )} + 2 \, a + b + 2 \, \sqrt {{\left (a + b\right )} a}}\right )}{128 \, \sqrt {{\left (a + b\right )} a} b^{3}} + \frac {{\left (32 \, a^{3} + 48 \, a^{2} b + 18 \, a b^{2} + b^{3}\right )} \log \left (\frac {b e^{\left (-2 \, x\right )} + 2 \, a + b - 2 \, \sqrt {{\left (a + b\right )} a}}{b e^{\left (-2 \, x\right )} + 2 \, a + b + 2 \, \sqrt {{\left (a + b\right )} a}}\right )}{128 \, \sqrt {{\left (a + b\right )} a} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.33, size = 178, normalized size = 2.02 \[ \frac {{\mathrm {e}}^{4\,x}}{64\,b}-\frac {{\mathrm {e}}^{-4\,x}}{64\,b}+\frac {x\,\left (8\,a^2-4\,a\,b+3\,b^2\right )}{8\,b^3}+\frac {{\mathrm {e}}^{-2\,x}\,\left (a-b\right )}{8\,b^2}-\frac {{\mathrm {e}}^{2\,x}\,\left (a-b\right )}{8\,b^2}+\frac {a^{5/2}\,\ln \left (\frac {4\,a^3\,{\mathrm {e}}^{2\,x}}{b^4}-\frac {2\,a^{5/2}\,\left (b+2\,a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )}{b^4\,\sqrt {a+b}}\right )}{2\,b^3\,\sqrt {a+b}}-\frac {a^{5/2}\,\ln \left (\frac {4\,a^3\,{\mathrm {e}}^{2\,x}}{b^4}+\frac {2\,a^{5/2}\,\left (b+2\,a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )}{b^4\,\sqrt {a+b}}\right )}{2\,b^3\,\sqrt {a+b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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