Optimal. Leaf size=71 \[ \frac {\tanh ^{-1}\left (\sqrt {1+\sqrt [3]{-1}} \tanh (x)\right )}{3 \sqrt {1+\sqrt [3]{-1}}}+\frac {\tanh ^{-1}\left (\sqrt {1-(-1)^{2/3}} \tanh (x)\right )}{3 \sqrt {1-(-1)^{2/3}}}+\frac {\tanh (x)}{3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3211, 3181, 206, 3175, 3767, 8} \[ \frac {\tanh ^{-1}\left (\sqrt {1+\sqrt [3]{-1}} \tanh (x)\right )}{3 \sqrt {1+\sqrt [3]{-1}}}+\frac {\tanh ^{-1}\left (\sqrt {1-(-1)^{2/3}} \tanh (x)\right )}{3 \sqrt {1-(-1)^{2/3}}}+\frac {\tanh (x)}{3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 206
Rule 3175
Rule 3181
Rule 3211
Rule 3767
Rubi steps
\begin {align*} \int \frac {1}{1+\sinh ^6(x)} \, dx &=\frac {1}{3} \int \frac {1}{1+\sinh ^2(x)} \, dx+\frac {1}{3} \int \frac {1}{1-\sqrt [3]{-1} \sinh ^2(x)} \, dx+\frac {1}{3} \int \frac {1}{1+(-1)^{2/3} \sinh ^2(x)} \, dx\\ &=\frac {1}{3} \int \text {sech}^2(x) \, dx+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{1-\left (1+\sqrt [3]{-1}\right ) x^2} \, dx,x,\tanh (x)\right )+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{1-\left (1-(-1)^{2/3}\right ) x^2} \, dx,x,\tanh (x)\right )\\ &=\frac {\tanh ^{-1}\left (\sqrt {1+\sqrt [3]{-1}} \tanh (x)\right )}{3 \sqrt {1+\sqrt [3]{-1}}}+\frac {\tanh ^{-1}\left (\sqrt {1-(-1)^{2/3}} \tanh (x)\right )}{3 \sqrt {1-(-1)^{2/3}}}+\frac {1}{3} i \operatorname {Subst}(\int 1 \, dx,x,-i \tanh (x))\\ &=\frac {\tanh ^{-1}\left (\sqrt {1+\sqrt [3]{-1}} \tanh (x)\right )}{3 \sqrt {1+\sqrt [3]{-1}}}+\frac {\tanh ^{-1}\left (\sqrt {1-(-1)^{2/3}} \tanh (x)\right )}{3 \sqrt {1-(-1)^{2/3}}}+\frac {\tanh (x)}{3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.22, size = 87, normalized size = 1.23 \[ \frac {1}{18} \left (6 \tanh (x)+\sqrt [4]{-3} \left (\left (-3-i \sqrt {3}\right ) \tan ^{-1}\left (\frac {1}{2} \sqrt [4]{-3} \left (1+i \sqrt {3}\right ) \tanh (x)\right )-\left (\sqrt {3}+3 i\right ) \tan ^{-1}\left (\frac {1}{2} \sqrt [4]{-\frac {1}{3}} \left (3+i \sqrt {3}\right ) \tanh (x)\right )\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.08, size = 692, normalized size = 9.75 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.12, size = 10, normalized size = 0.14 \[ -\frac {2}{3 \, {\left (e^{\left (2 \, x\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.06, size = 61, normalized size = 0.86 \[ \frac {\left (\munderset {\textit {\_R} =\RootOf \left (3 \textit {\_Z}^{4}-3 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (\tanh ^{2}\left (\frac {x}{2}\right )+\left (-6 \textit {\_R}^{3}+6 \textit {\_R} \right ) \tanh \left (\frac {x}{2}\right )+1\right )\right )}{6}+\frac {2 \tanh \left (\frac {x}{2}\right )}{3 \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {2}{3 \, {\left (e^{\left (2 \, x\right )} + 1\right )}} - \int \frac {4 \, {\left (e^{\left (6 \, x\right )} - 10 \, e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}\right )}}{3 \, {\left (e^{\left (8 \, x\right )} - 8 \, e^{\left (6 \, x\right )} + 30 \, e^{\left (4 \, x\right )} - 8 \, e^{\left (2 \, x\right )} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.21, size = 325, normalized size = 4.58 \[ -\ln \left (\frac {1061158912\,{\mathrm {e}}^{2\,x}}{27}+\sqrt {\frac {1}{72}-\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}\,\left (\frac {548405248}{27}+\sqrt {\frac {1}{72}-\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}\,\left (\frac {3870294016}{9}+\sqrt {\frac {1}{72}-\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}\,\left (19788726272\,{\mathrm {e}}^{2\,x}-2864709632\right )-\frac {21515730944\,{\mathrm {e}}^{2\,x}}{9}\right )-\frac {2539651072\,{\mathrm {e}}^{2\,x}}{9}\right )-\frac {351797248}{81}\right )\,\sqrt {\frac {1}{72}-\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}-\ln \left (\frac {1061158912\,{\mathrm {e}}^{2\,x}}{27}+\sqrt {\frac {1}{72}+\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}\,\left (\frac {548405248}{27}+\sqrt {\frac {1}{72}+\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}\,\left (\frac {3870294016}{9}+\sqrt {\frac {1}{72}+\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}\,\left (19788726272\,{\mathrm {e}}^{2\,x}-2864709632\right )-\frac {21515730944\,{\mathrm {e}}^{2\,x}}{9}\right )-\frac {2539651072\,{\mathrm {e}}^{2\,x}}{9}\right )-\frac {351797248}{81}\right )\,\sqrt {\frac {1}{72}+\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}+\ln \left (\frac {1061158912\,{\mathrm {e}}^{2\,x}}{27}-\sqrt {\frac {1}{72}-\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}\,\left (\frac {548405248}{27}+\sqrt {\frac {1}{72}-\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}\,\left (\frac {21515730944\,{\mathrm {e}}^{2\,x}}{9}+\sqrt {\frac {1}{72}-\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}\,\left (19788726272\,{\mathrm {e}}^{2\,x}-2864709632\right )-\frac {3870294016}{9}\right )-\frac {2539651072\,{\mathrm {e}}^{2\,x}}{9}\right )-\frac {351797248}{81}\right )\,\sqrt {\frac {1}{72}-\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}+\ln \left (\frac {1061158912\,{\mathrm {e}}^{2\,x}}{27}-\sqrt {\frac {1}{72}+\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}\,\left (\frac {548405248}{27}+\sqrt {\frac {1}{72}+\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}\,\left (\frac {21515730944\,{\mathrm {e}}^{2\,x}}{9}+\sqrt {\frac {1}{72}+\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}\,\left (19788726272\,{\mathrm {e}}^{2\,x}-2864709632\right )-\frac {3870294016}{9}\right )-\frac {2539651072\,{\mathrm {e}}^{2\,x}}{9}\right )-\frac {351797248}{81}\right )\,\sqrt {\frac {1}{72}+\frac {\sqrt {3}\,1{}\mathrm {i}}{216}}-\frac {2}{3\,\left ({\mathrm {e}}^{2\,x}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________