Optimal. Leaf size=494 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}+\sqrt [5]{b}} \tanh \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-\sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-\sqrt [5]{b}} \sqrt {\sqrt [5]{a}+\sqrt [5]{b}}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}} \tanh \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}} \tanh \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}} \tanh \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}} \tanh \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}}} \]
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Rubi [A] time = 0.63, antiderivative size = 494, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3213, 2659, 208} \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}+\sqrt [5]{b}} \tanh \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-\sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-\sqrt [5]{b}} \sqrt {\sqrt [5]{a}+\sqrt [5]{b}}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}} \tanh \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}} \tanh \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}} \tanh \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}} \tanh \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}}} \]
Antiderivative was successfully verified.
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Rule 208
Rule 2659
Rule 3213
Rubi steps
\begin {align*} \int \frac {1}{a-b \cosh ^5(x)} \, dx &=\int \left (\frac {1}{5 a^{4/5} \left (\sqrt [5]{a}-\sqrt [5]{b} \cosh (x)\right )}+\frac {1}{5 a^{4/5} \left (\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b} \cosh (x)\right )}+\frac {1}{5 a^{4/5} \left (\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b} \cosh (x)\right )}+\frac {1}{5 a^{4/5} \left (\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b} \cosh (x)\right )}+\frac {1}{5 a^{4/5} \left (\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b} \cosh (x)\right )}\right ) \, dx\\ &=\frac {\int \frac {1}{\sqrt [5]{a}-\sqrt [5]{b} \cosh (x)} \, dx}{5 a^{4/5}}+\frac {\int \frac {1}{\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b} \cosh (x)} \, dx}{5 a^{4/5}}+\frac {\int \frac {1}{\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b} \cosh (x)} \, dx}{5 a^{4/5}}+\frac {\int \frac {1}{\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b} \cosh (x)} \, dx}{5 a^{4/5}}+\frac {\int \frac {1}{\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b} \cosh (x)} \, dx}{5 a^{4/5}}\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {1}{\sqrt [5]{a}-\sqrt [5]{b}-\left (\sqrt [5]{a}+\sqrt [5]{b}\right ) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{5 a^{4/5}}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}-\left (\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}\right ) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{5 a^{4/5}}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}-\left (\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}\right ) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{5 a^{4/5}}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}-\left (\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}\right ) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{5 a^{4/5}}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}-\left (\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}\right ) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{5 a^{4/5}}\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}+\sqrt [5]{b}} \tanh \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-\sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-\sqrt [5]{b}} \sqrt {\sqrt [5]{a}+\sqrt [5]{b}}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}} \tanh \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}} \tanh \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}} \tanh \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}} \tanh \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}}}\\ \end {align*}
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Mathematica [C] time = 0.27, size = 139, normalized size = 0.28 \[ -\frac {8}{5} \text {RootSum}\left [\text {$\#$1}^{10} b+5 \text {$\#$1}^8 b+10 \text {$\#$1}^6 b-32 \text {$\#$1}^5 a+10 \text {$\#$1}^4 b+5 \text {$\#$1}^2 b+b\& ,\frac {\text {$\#$1}^3 x+2 \text {$\#$1}^3 \log \left (-\text {$\#$1} \sinh \left (\frac {x}{2}\right )+\text {$\#$1} \cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )-\cosh \left (\frac {x}{2}\right )\right )}{\text {$\#$1}^8 b+4 \text {$\#$1}^6 b+6 \text {$\#$1}^4 b-16 \text {$\#$1}^3 a+4 \text {$\#$1}^2 b+b}\& \right ] \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {1}{b \cosh \relax (x)^{5} - a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.09, size = 150, normalized size = 0.30 \[ \frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (a +b \right ) \textit {\_Z}^{10}+\left (-5 a +5 b \right ) \textit {\_Z}^{8}+\left (10 a +10 b \right ) \textit {\_Z}^{6}+\left (-10 a +10 b \right ) \textit {\_Z}^{4}+\left (5 a +5 b \right ) \textit {\_Z}^{2}-a +b \right )}{\sum }\frac {\left (-\textit {\_R}^{8}+4 \textit {\_R}^{6}-6 \textit {\_R}^{4}+4 \textit {\_R}^{2}-1\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{9} a +\textit {\_R}^{9} b -4 \textit {\_R}^{7} a +4 \textit {\_R}^{7} b +6 \textit {\_R}^{5} a +6 \textit {\_R}^{5} b -4 \textit {\_R}^{3} a +4 \textit {\_R}^{3} b +\textit {\_R} a +\textit {\_R} b}\right )}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {1}{b \cosh \relax (x)^{5} - a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.00 \[ \text {Hanged} \]
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 \[ \int \frac {1}{a - b \cosh ^{5}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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