Optimal. Leaf size=435 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt [5]{b}-\sqrt [5]{a} \tanh \left (\frac {x}{2}\right )}{\sqrt {a^{2/5}+b^{2/5}}}\right )}{5 a^{4/5} \sqrt {a^{2/5}+b^{2/5}}}+\frac {2 (-1)^{9/10} \tanh ^{-1}\left (\frac {(-1)^{9/10} \left (\sqrt [5]{-1} \sqrt [5]{b}+\sqrt [5]{a} \tanh \left (\frac {x}{2}\right )\right )}{\sqrt {-(-1)^{4/5} a^{2/5}+\sqrt [5]{-1} b^{2/5}}}\right )}{5 a^{4/5} \sqrt {-(-1)^{4/5} a^{2/5}+\sqrt [5]{-1} b^{2/5}}}+\frac {2 \sqrt [5]{-1} \tanh ^{-1}\left (\frac {\sqrt [5]{b}+\sqrt [5]{-1} \sqrt [5]{a} \tanh \left (\frac {x}{2}\right )}{\sqrt {(-1)^{2/5} a^{2/5}+b^{2/5}}}\right )}{5 a^{4/5} \sqrt {(-1)^{2/5} a^{2/5}+b^{2/5}}}+\frac {2 (-1)^{9/10} \tanh ^{-1}\left (\frac {(-1)^{3/10} \left (\sqrt [5]{b}+(-1)^{3/5} \sqrt [5]{a} \tanh \left (\frac {x}{2}\right )\right )}{\sqrt {-(-1)^{4/5} a^{2/5}+(-1)^{3/5} b^{2/5}}}\right )}{5 a^{4/5} \sqrt {-(-1)^{4/5} a^{2/5}+(-1)^{3/5} b^{2/5}}}-\frac {2 (-1)^{9/10} \tanh ^{-1}\left (\frac {i \sqrt [5]{b}-(-1)^{9/10} \sqrt [5]{a} \tanh \left (\frac {x}{2}\right )}{\sqrt {-(-1)^{4/5} a^{2/5}-b^{2/5}}}\right )}{5 a^{4/5} \sqrt {-(-1)^{4/5} a^{2/5}-b^{2/5}}} \]
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Rubi [A]
time = 0.68, antiderivative size = 435, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3292, 2739,
632, 212, 210} \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt [5]{b}-\sqrt [5]{a} \tanh \left (\frac {x}{2}\right )}{\sqrt {a^{2/5}+b^{2/5}}}\right )}{5 a^{4/5} \sqrt {a^{2/5}+b^{2/5}}}+\frac {2 (-1)^{9/10} \tanh ^{-1}\left (\frac {(-1)^{9/10} \left (\sqrt [5]{a} \tanh \left (\frac {x}{2}\right )+\sqrt [5]{-1} \sqrt [5]{b}\right )}{\sqrt {\sqrt [5]{-1} b^{2/5}-(-1)^{4/5} a^{2/5}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{-1} b^{2/5}-(-1)^{4/5} a^{2/5}}}+\frac {2 \sqrt [5]{-1} \tanh ^{-1}\left (\frac {\sqrt [5]{-1} \sqrt [5]{a} \tanh \left (\frac {x}{2}\right )+\sqrt [5]{b}}{\sqrt {(-1)^{2/5} a^{2/5}+b^{2/5}}}\right )}{5 a^{4/5} \sqrt {(-1)^{2/5} a^{2/5}+b^{2/5}}}+\frac {2 (-1)^{9/10} \tanh ^{-1}\left (\frac {(-1)^{3/10} \left ((-1)^{3/5} \sqrt [5]{a} \tanh \left (\frac {x}{2}\right )+\sqrt [5]{b}\right )}{\sqrt {(-1)^{3/5} b^{2/5}-(-1)^{4/5} a^{2/5}}}\right )}{5 a^{4/5} \sqrt {(-1)^{3/5} b^{2/5}-(-1)^{4/5} a^{2/5}}}-\frac {2 (-1)^{9/10} \tanh ^{-1}\left (\frac {-(-1)^{9/10} \sqrt [5]{a} \tanh \left (\frac {x}{2}\right )+i \sqrt [5]{b}}{\sqrt {-(-1)^{4/5} a^{2/5}-b^{2/5}}}\right )}{5 a^{4/5} \sqrt {-(-1)^{4/5} a^{2/5}-b^{2/5}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 212
Rule 632
Rule 2739
Rule 3292
Rubi steps
\begin {align*} \int \frac {1}{a+b \sinh ^5(x)} \, dx &=\int \left (-\frac {(-1)^{9/10}}{5 a^{4/5} \left (-(-1)^{9/10} \sqrt [5]{a}-i \sqrt [5]{b} \sinh (x)\right )}-\frac {(-1)^{9/10}}{5 a^{4/5} \left (-(-1)^{9/10} \sqrt [5]{a}-\sqrt [10]{-1} \sqrt [5]{b} \sinh (x)\right )}-\frac {(-1)^{9/10}}{5 a^{4/5} \left (-(-1)^{9/10} \sqrt [5]{a}+(-1)^{3/10} \sqrt [5]{b} \sinh (x)\right )}-\frac {(-1)^{9/10}}{5 a^{4/5} \left (-(-1)^{9/10} \sqrt [5]{a}+(-1)^{7/10} \sqrt [5]{b} \sinh (x)\right )}-\frac {(-1)^{9/10}}{5 a^{4/5} \left (-(-1)^{9/10} \sqrt [5]{a}-(-1)^{9/10} \sqrt [5]{b} \sinh (x)\right )}\right ) \, dx\\ &=-\frac {(-1)^{9/10} \int \frac {1}{-(-1)^{9/10} \sqrt [5]{a}-i \sqrt [5]{b} \sinh (x)} \, dx}{5 a^{4/5}}-\frac {(-1)^{9/10} \int \frac {1}{-(-1)^{9/10} \sqrt [5]{a}-\sqrt [10]{-1} \sqrt [5]{b} \sinh (x)} \, dx}{5 a^{4/5}}-\frac {(-1)^{9/10} \int \frac {1}{-(-1)^{9/10} \sqrt [5]{a}+(-1)^{3/10} \sqrt [5]{b} \sinh (x)} \, dx}{5 a^{4/5}}-\frac {(-1)^{9/10} \int \frac {1}{-(-1)^{9/10} \sqrt [5]{a}+(-1)^{7/10} \sqrt [5]{b} \sinh (x)} \, dx}{5 a^{4/5}}-\frac {(-1)^{9/10} \int \frac {1}{-(-1)^{9/10} \sqrt [5]{a}-(-1)^{9/10} \sqrt [5]{b} \sinh (x)} \, dx}{5 a^{4/5}}\\ &=-\frac {\left (2 (-1)^{9/10}\right ) \text {Subst}\left (\int \frac {1}{-(-1)^{9/10} \sqrt [5]{a}-2 i \sqrt [5]{b} x+(-1)^{9/10} \sqrt [5]{a} x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{5 a^{4/5}}-\frac {\left (2 (-1)^{9/10}\right ) \text {Subst}\left (\int \frac {1}{-(-1)^{9/10} \sqrt [5]{a}-2 \sqrt [10]{-1} \sqrt [5]{b} x+(-1)^{9/10} \sqrt [5]{a} x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{5 a^{4/5}}-\frac {\left (2 (-1)^{9/10}\right ) \text {Subst}\left (\int \frac {1}{-(-1)^{9/10} \sqrt [5]{a}+2 (-1)^{3/10} \sqrt [5]{b} x+(-1)^{9/10} \sqrt [5]{a} x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{5 a^{4/5}}-\frac {\left (2 (-1)^{9/10}\right ) \text {Subst}\left (\int \frac {1}{-(-1)^{9/10} \sqrt [5]{a}+2 (-1)^{7/10} \sqrt [5]{b} x+(-1)^{9/10} \sqrt [5]{a} x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{5 a^{4/5}}-\frac {\left (2 (-1)^{9/10}\right ) \text {Subst}\left (\int \frac {1}{-(-1)^{9/10} \sqrt [5]{a}-2 (-1)^{9/10} \sqrt [5]{b} x+(-1)^{9/10} \sqrt [5]{a} x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{5 a^{4/5}}\\ &=\frac {\left (4 (-1)^{9/10}\right ) \text {Subst}\left (\int \frac {1}{-4 (-1)^{4/5} \left (a^{2/5}+b^{2/5}\right )-x^2} \, dx,x,-2 (-1)^{9/10} \sqrt [5]{b}+2 (-1)^{9/10} \sqrt [5]{a} \tanh \left (\frac {x}{2}\right )\right )}{5 a^{4/5}}+\frac {\left (4 (-1)^{9/10}\right ) \text {Subst}\left (\int \frac {1}{-4 (-1)^{2/5} \left ((-1)^{2/5} a^{2/5}+b^{2/5}\right )-x^2} \, dx,x,2 (-1)^{7/10} \sqrt [5]{b}+2 (-1)^{9/10} \sqrt [5]{a} \tanh \left (\frac {x}{2}\right )\right )}{5 a^{4/5}}+\frac {\left (4 (-1)^{9/10}\right ) \text {Subst}\left (\int \frac {1}{-4 \left ((-1)^{4/5} a^{2/5}+b^{2/5}\right )-x^2} \, dx,x,-2 i \sqrt [5]{b}+2 (-1)^{9/10} \sqrt [5]{a} \tanh \left (\frac {x}{2}\right )\right )}{5 a^{4/5}}+\frac {\left (4 (-1)^{9/10}\right ) \text {Subst}\left (\int \frac {1}{-4 \left ((-1)^{4/5} a^{2/5}-\sqrt [5]{-1} b^{2/5}\right )-x^2} \, dx,x,-2 \sqrt [10]{-1} \sqrt [5]{b}+2 (-1)^{9/10} \sqrt [5]{a} \tanh \left (\frac {x}{2}\right )\right )}{5 a^{4/5}}+\frac {\left (4 (-1)^{9/10}\right ) \text {Subst}\left (\int \frac {1}{-4 \left ((-1)^{4/5} a^{2/5}-(-1)^{3/5} b^{2/5}\right )-x^2} \, dx,x,2 (-1)^{3/10} \sqrt [5]{b}+2 (-1)^{9/10} \sqrt [5]{a} \tanh \left (\frac {x}{2}\right )\right )}{5 a^{4/5}}\\ &=-\frac {2 (-1)^{7/10} \tan ^{-1}\left (\frac {i \sqrt [5]{b}+(-1)^{7/10} \sqrt [5]{a} \tanh \left (\frac {x}{2}\right )}{\sqrt {(-1)^{2/5} a^{2/5}+b^{2/5}}}\right )}{5 a^{4/5} \sqrt {(-1)^{2/5} a^{2/5}+b^{2/5}}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [5]{b}-\sqrt [5]{a} \tanh \left (\frac {x}{2}\right )}{\sqrt {a^{2/5}+b^{2/5}}}\right )}{5 a^{4/5} \sqrt {a^{2/5}+b^{2/5}}}-\frac {2 (-1)^{9/10} \tanh ^{-1}\left (\frac {i \sqrt [5]{b}-(-1)^{9/10} \sqrt [5]{a} \tanh \left (\frac {x}{2}\right )}{\sqrt {-(-1)^{4/5} a^{2/5}-b^{2/5}}}\right )}{5 a^{4/5} \sqrt {-(-1)^{4/5} a^{2/5}-b^{2/5}}}-\frac {2 (-1)^{9/10} \tanh ^{-1}\left (\frac {\sqrt [10]{-1} \sqrt [5]{b}-(-1)^{9/10} \sqrt [5]{a} \tanh \left (\frac {x}{2}\right )}{\sqrt {-(-1)^{4/5} a^{2/5}+\sqrt [5]{-1} b^{2/5}}}\right )}{5 a^{4/5} \sqrt {-(-1)^{4/5} a^{2/5}+\sqrt [5]{-1} b^{2/5}}}+\frac {2 (-1)^{9/10} \tanh ^{-1}\left (\frac {(-1)^{3/10} \sqrt [5]{b}+(-1)^{9/10} \sqrt [5]{a} \tanh \left (\frac {x}{2}\right )}{\sqrt {-(-1)^{4/5} a^{2/5}+(-1)^{3/5} b^{2/5}}}\right )}{5 a^{4/5} \sqrt {-(-1)^{4/5} a^{2/5}+(-1)^{3/5} b^{2/5}}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.22, size = 141, normalized size = 0.32 \begin {gather*} \frac {8}{5} \text {RootSum}\left [-b+5 b \text {$\#$1}^2-10 b \text {$\#$1}^4+32 a \text {$\#$1}^5+10 b \text {$\#$1}^6-5 b \text {$\#$1}^8+b \text {$\#$1}^{10}\&,\frac {x \text {$\#$1}^3+2 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right ) \text {$\#$1}^3}{b-4 b \text {$\#$1}^2+16 a \text {$\#$1}^3+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 1.35, size = 113, normalized size = 0.26
method | result | size |
default | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{10}-5 a \,\textit {\_Z}^{8}+10 a \,\textit {\_Z}^{6}-32 b \,\textit {\_Z}^{5}-10 a \,\textit {\_Z}^{4}+5 a \,\textit {\_Z}^{2}-a \right )}{\sum }\frac {\left (-\textit {\_R}^{8}+4 \textit {\_R}^{6}-6 \textit {\_R}^{4}+4 \textit {\_R}^{2}-1\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{9} a -4 \textit {\_R}^{7} a +6 \textit {\_R}^{5} a -16 \textit {\_R}^{4} b -4 \textit {\_R}^{3} a +\textit {\_R} a}\right )}{5}\) | \(113\) |
risch | \(\munderset {\textit {\_R} =\RootOf \left (-1+\left (9765625 a^{10}+9765625 a^{8} b^{2}\right ) \textit {\_Z}^{10}-1953125 a^{8} \textit {\_Z}^{8}+156250 a^{6} \textit {\_Z}^{6}-6250 a^{4} \textit {\_Z}^{4}+125 \textit {\_Z}^{2} a^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{x}+\left (-\frac {11718750 a^{10}}{b}-11718750 a^{8} b \right ) \textit {\_R}^{9}+\left (\frac {1171875 a^{9}}{b}+1171875 a^{7} b \right ) \textit {\_R}^{8}+\left (\frac {2109375 a^{8}}{b}-234375 a^{6} b \right ) \textit {\_R}^{7}+\left (-\frac {218750 a^{7}}{b}+15625 a^{5} b \right ) \textit {\_R}^{6}+\left (-\frac {143750 a^{6}}{b}-3125 a^{4} b \right ) \textit {\_R}^{5}+\frac {15625 a^{5} \textit {\_R}^{4}}{b}+\frac {4375 a^{4} \textit {\_R}^{3}}{b}-\frac {500 a^{3} \textit {\_R}^{2}}{b}-\frac {50 a^{2} \textit {\_R}}{b}+\frac {6 a}{b}\right )\) | \(206\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \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 {1}{a + b \sinh ^{5}{\left (x \right )}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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