Optimal. Leaf size=175 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {\sqrt [3]{a}-\sqrt [3]{b}} \tanh (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-\sqrt [3]{b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} \tanh (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} \tanh (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}}} \]
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Rubi [A] time = 0.28, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3211, 3181, 206} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {\sqrt [3]{a}-\sqrt [3]{b}} \tanh (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-\sqrt [3]{b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} \tanh (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} \tanh (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 3181
Rule 3211
Rubi steps
\begin {align*} \int \frac {1}{a+b \sinh ^6(x)} \, dx &=\frac {\int \frac {1}{1+\frac {\sqrt [3]{b} \sinh ^2(x)}{\sqrt [3]{a}}} \, dx}{3 a}+\frac {\int \frac {1}{1-\frac {\sqrt [3]{-1} \sqrt [3]{b} \sinh ^2(x)}{\sqrt [3]{a}}} \, dx}{3 a}+\frac {\int \frac {1}{1+\frac {(-1)^{2/3} \sqrt [3]{b} \sinh ^2(x)}{\sqrt [3]{a}}} \, dx}{3 a}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{1-\left (1-\frac {\sqrt [3]{b}}{\sqrt [3]{a}}\right ) x^2} \, dx,x,\tanh (x)\right )}{3 a}+\frac {\operatorname {Subst}\left (\int \frac {1}{1-\left (1+\frac {\sqrt [3]{-1} \sqrt [3]{b}}{\sqrt [3]{a}}\right ) x^2} \, dx,x,\tanh (x)\right )}{3 a}+\frac {\operatorname {Subst}\left (\int \frac {1}{1-\left (1-\frac {(-1)^{2/3} \sqrt [3]{b}}{\sqrt [3]{a}}\right ) x^2} \, dx,x,\tanh (x)\right )}{3 a}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {\sqrt [3]{a}-\sqrt [3]{b}} \tanh (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-\sqrt [3]{b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} \tanh (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} \tanh (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}}}\\ \end {align*}
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Mathematica [C] time = 0.18, size = 134, normalized size = 0.77 \[ \frac {16}{3} \text {RootSum}\left [\text {$\#$1}^6 b-6 \text {$\#$1}^5 b+15 \text {$\#$1}^4 b+64 \text {$\#$1}^3 a-20 \text {$\#$1}^3 b+15 \text {$\#$1}^2 b-6 \text {$\#$1} b+b\& ,\frac {\text {$\#$1}^2 x+\text {$\#$1}^2 \log (-\text {$\#$1} \sinh (x)+\text {$\#$1} \cosh (x)-\sinh (x)-\cosh (x))}{\text {$\#$1}^5 b-5 \text {$\#$1}^4 b+10 \text {$\#$1}^3 b+32 \text {$\#$1}^2 a-10 \text {$\#$1}^2 b+5 \text {$\#$1} b-b}\& \right ] \]
Antiderivative was successfully verified.
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fricas [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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giac [A] time = 0.35, size = 1, normalized size = 0.01 \[ 0 \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.06, size = 128, normalized size = 0.73 \[ \frac {\left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{12}-6 a \,\textit {\_Z}^{10}+15 a \,\textit {\_Z}^{8}+\left (-20 a +64 b \right ) \textit {\_Z}^{6}+15 a \,\textit {\_Z}^{4}-6 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (-\textit {\_R}^{10}+5 \textit {\_R}^{8}-10 \textit {\_R}^{6}+10 \textit {\_R}^{4}-5 \textit {\_R}^{2}+1\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{11} a -5 \textit {\_R}^{9} a +10 \textit {\_R}^{7} a -10 \textit {\_R}^{5} a +32 \textit {\_R}^{5} b +5 \textit {\_R}^{3} a -\textit {\_R} a}\right )}{6} \]
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 \[ \int \frac {1}{b \sinh \relax (x)^{6} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 58.56, size = 857, normalized size = 4.90 \[ \sum _{k=1}^6\ln \left (\mathrm {root}\left (46656\,a^5\,b\,d^6-46656\,a^6\,d^6+3888\,a^4\,d^4-108\,a^2\,d^2+1,d,k\right )\,\left (\mathrm {root}\left (46656\,a^5\,b\,d^6-46656\,a^6\,d^6+3888\,a^4\,d^4-108\,a^2\,d^2+1,d,k\right )\,\left (\mathrm {root}\left (46656\,a^5\,b\,d^6-46656\,a^6\,d^6+3888\,a^4\,d^4-108\,a^2\,d^2+1,d,k\right )\,\left (\mathrm {root}\left (46656\,a^5\,b\,d^6-46656\,a^6\,d^6+3888\,a^4\,d^4-108\,a^2\,d^2+1,d,k\right )\,\left (\frac {1459166279268040704\,\left (327680\,a^7\,{\mathrm {e}}^{2\,x}+298496\,a^6\,b-65536\,a^7+158\,a^2\,b^5-91315\,a^3\,b^4+348176\,a^4\,b^3-489952\,a^5\,b^2-196\,a^2\,b^5\,{\mathrm {e}}^{2\,x}+274019\,a^3\,b^4\,{\mathrm {e}}^{2\,x}-1132876\,a^4\,b^3\,{\mathrm {e}}^{2\,x}+1770440\,a^5\,b^2\,{\mathrm {e}}^{2\,x}-1239040\,a^6\,b\,{\mathrm {e}}^{2\,x}\right )}{b^{10}\,{\left (a-b\right )}^3}+\frac {\mathrm {root}\left (46656\,a^5\,b\,d^6-46656\,a^6\,d^6+3888\,a^4\,d^4-108\,a^2\,d^2+1,d,k\right )\,\left (262144\,a^7\,{\mathrm {e}}^{2\,x}+203520\,a^6\,b-65536\,a^7-453\,a^3\,b^4+72022\,a^4\,b^3-209472\,a^5\,b^2+630\,a^3\,b^4\,{\mathrm {e}}^{2\,x}-254512\,a^4\,b^3\,{\mathrm {e}}^{2\,x}+767508\,a^5\,b^2\,{\mathrm {e}}^{2\,x}-775680\,a^6\,b\,{\mathrm {e}}^{2\,x}\right )\,17509995351216488448}{b^{10}\,{\left (a-b\right )}^2}\right )-\frac {486388759756013568\,\left (655360\,a^5\,{\mathrm {e}}^{2\,x}+9\,a\,b^4+370176\,a^4\,b-196608\,a^5-24408\,a^2\,b^3-149088\,a^3\,b^2+63676\,a^2\,b^3\,{\mathrm {e}}^{2\,x}+526248\,a^3\,b^2\,{\mathrm {e}}^{2\,x}-10\,a\,b^4\,{\mathrm {e}}^{2\,x}-1245184\,a^4\,b\,{\mathrm {e}}^{2\,x}\right )}{b^{10}\,{\left (a-b\right )}^2}\right )-\frac {40532396646334464\,\left (655360\,a^5\,{\mathrm {e}}^{2\,x}+b^5\,{\mathrm {e}}^{2\,x}+24677\,a\,b^4+773120\,a^4\,b-262144\,a^5-b^5+198071\,a^2\,b^3-733696\,a^3\,b^2-477713\,a^2\,b^3\,{\mathrm {e}}^{2\,x}+1770640\,a^3\,b^2\,{\mathrm {e}}^{2\,x}-53861\,a\,b^4\,{\mathrm {e}}^{2\,x}-1894400\,a^4\,b\,{\mathrm {e}}^{2\,x}\right )}{b^{10}\,{\left (a-b\right )}^3}\right )+\frac {13510798882111488\,\left (655360\,a^3\,{\mathrm {e}}^{2\,x}-11382\,b^3\,{\mathrm {e}}^{2\,x}-144416\,a\,b^2+269056\,a^2\,b-131072\,a^3+6459\,b^3+677524\,a\,b^2\,{\mathrm {e}}^{2\,x}-1321472\,a^2\,b\,{\mathrm {e}}^{2\,x}\right )}{b^{10}\,{\left (a-b\right )}^2}\right )-\frac {1125899906842624\,\left (851968\,a^4\,{\mathrm {e}}^{2\,x}+6006\,b^4\,{\mathrm {e}}^{2\,x}+211497\,a\,b^3+597504\,a^3\,b-196608\,a^4-3840\,b^4-608544\,a^2\,b^2+2562504\,a^2\,b^2\,{\mathrm {e}}^{2\,x}-864565\,a\,b^3\,{\mathrm {e}}^{2\,x}-2555904\,a^3\,b\,{\mathrm {e}}^{2\,x}\right )}{b^{10}\,{\left (a-b\right )}^2\,\left (a\,b-a^2\right )}\right )\,\mathrm {root}\left (46656\,a^5\,b\,d^6-46656\,a^6\,d^6+3888\,a^4\,d^4-108\,a^2\,d^2+1,d,k\right ) \]
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 \[ \int \frac {1}{a + b \sinh ^{6}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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