Optimal. Leaf size=69 \[ \frac {\tanh ^{-1}\left (\sqrt {1-i} \tanh (x)\right )}{4 \sqrt {1-i}}+\frac {\tanh ^{-1}\left (\sqrt {1+i} \tanh (x)\right )}{4 \sqrt {1+i}}+\frac {\tanh ^{-1}\left (\sqrt {2} \tanh (x)\right )}{4 \sqrt {2}}+\frac {\tanh (x)}{4} \]
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Rubi [A] time = 0.08, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3211, 3181, 206, 3175, 3767, 8} \[ \frac {\tanh ^{-1}\left (\sqrt {1-i} \tanh (x)\right )}{4 \sqrt {1-i}}+\frac {\tanh ^{-1}\left (\sqrt {1+i} \tanh (x)\right )}{4 \sqrt {1+i}}+\frac {\tanh ^{-1}\left (\sqrt {2} \tanh (x)\right )}{4 \sqrt {2}}+\frac {\tanh (x)}{4} \]
Antiderivative was successfully verified.
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Rule 8
Rule 206
Rule 3175
Rule 3181
Rule 3211
Rule 3767
Rubi steps
\begin {align*} \int \frac {1}{1-\sinh ^8(x)} \, dx &=\frac {1}{4} \int \frac {1}{1-\sinh ^2(x)} \, dx+\frac {1}{4} \int \frac {1}{1-i \sinh ^2(x)} \, dx+\frac {1}{4} \int \frac {1}{1+i \sinh ^2(x)} \, dx+\frac {1}{4} \int \frac {1}{1+\sinh ^2(x)} \, dx\\ &=\frac {1}{4} \int \text {sech}^2(x) \, dx+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\tanh (x)\right )+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1-(1+i) x^2} \, dx,x,\tanh (x)\right )+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1-(1-i) x^2} \, dx,x,\tanh (x)\right )\\ &=\frac {\tanh ^{-1}\left (\sqrt {1-i} \tanh (x)\right )}{4 \sqrt {1-i}}+\frac {\tanh ^{-1}\left (\sqrt {1+i} \tanh (x)\right )}{4 \sqrt {1+i}}+\frac {\tanh ^{-1}\left (\sqrt {2} \tanh (x)\right )}{4 \sqrt {2}}+\frac {1}{4} i \operatorname {Subst}(\int 1 \, dx,x,-i \tanh (x))\\ &=\frac {\tanh ^{-1}\left (\sqrt {1-i} \tanh (x)\right )}{4 \sqrt {1-i}}+\frac {\tanh ^{-1}\left (\sqrt {1+i} \tanh (x)\right )}{4 \sqrt {1+i}}+\frac {\tanh ^{-1}\left (\sqrt {2} \tanh (x)\right )}{4 \sqrt {2}}+\frac {\tanh (x)}{4}\\ \end {align*}
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Mathematica [A] time = 0.50, size = 64, normalized size = 0.93 \[ \frac {1}{8} \left (\frac {2 \tanh ^{-1}\left (\sqrt {1-i} \tanh (x)\right )}{\sqrt {1-i}}+\frac {2 \tanh ^{-1}\left (\sqrt {1+i} \tanh (x)\right )}{\sqrt {1+i}}+\sqrt {2} \tanh ^{-1}\left (\sqrt {2} \tanh (x)\right )+2 \tanh (x)\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 1.55, size = 708, normalized size = 10.26 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [C] time = 0.05, size = 99, normalized size = 1.43 \[ \frac {\sqrt {2}\, \arctanh \left (\frac {\left (2 \tanh \left (\frac {x}{2}\right )-2\right ) \sqrt {2}}{4}\right )}{8}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (2 \textit {\_Z}^{4}-2 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (\tanh ^{2}\left (\frac {x}{2}\right )+\left (-4 \textit {\_R}^{3}+4 \textit {\_R} \right ) \tanh \left (\frac {x}{2}\right )+1\right )\right )}{8}+\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2 \tanh \left (\frac {x}{2}\right )+2\right ) \sqrt {2}}{4}\right )}{8}+\frac {\tanh \left (\frac {x}{2}\right )}{2 \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+2} \]
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 \[ -\frac {1}{16} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - e^{x} + 1}{\sqrt {2} + e^{x} - 1}\right ) + \frac {1}{16} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - e^{x} - 1}{\sqrt {2} + e^{x} + 1}\right ) - \frac {1}{2 \, {\left (e^{\left (2 \, x\right )} + 1\right )}} + 8 \, \int \frac {e^{\left (4 \, x\right )}}{e^{\left (8 \, x\right )} - 4 \, e^{\left (6 \, x\right )} + 22 \, e^{\left (4 \, x\right )} - 4 \, e^{\left (2 \, x\right )} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.81, size = 273, normalized size = 3.96 \[ \frac {\sqrt {2}\,\ln \left (582732658686033920\,{\mathrm {e}}^{2\,x}-70697326355677184\,\sqrt {2}+412054214575915008\,\sqrt {2}\,{\mathrm {e}}^{2\,x}-99981117754441728\right )}{16}-\frac {\sqrt {2}\,\ln \left (582732658686033920\,{\mathrm {e}}^{2\,x}+70697326355677184\,\sqrt {2}-412054214575915008\,\sqrt {2}\,{\mathrm {e}}^{2\,x}-99981117754441728\right )}{16}-\frac {1}{2\,\left ({\mathrm {e}}^{2\,x}+1\right )}-\frac {\sqrt {2}\,\sqrt {1-\mathrm {i}}\,\ln \left ({\mathrm {e}}^{2\,x}\,\left (155613434002538496+429723297714798592{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1-\mathrm {i}}\,\left (-54684829282729984+21956972328779776{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1-\mathrm {i}}\,{\mathrm {e}}^{2\,x}\,\left (12296353929494528-271474128182050816{}\mathrm {i}\right )+70836483296067584-69311013991743488{}\mathrm {i}\right )}{16}+\frac {\sqrt {2}\,\sqrt {1-\mathrm {i}}\,\ln \left ({\mathrm {e}}^{2\,x}\,\left (155613434002538496+429723297714798592{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1-\mathrm {i}}\,\left (54684829282729984-21956972328779776{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1-\mathrm {i}}\,{\mathrm {e}}^{2\,x}\,\left (-12296353929494528+271474128182050816{}\mathrm {i}\right )+70836483296067584-69311013991743488{}\mathrm {i}\right )}{16}-\frac {\sqrt {2}\,\sqrt {1+1{}\mathrm {i}}\,\ln \left ({\mathrm {e}}^{2\,x}\,\left (155613434002538496-429723297714798592{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1+1{}\mathrm {i}}\,\left (-54684829282729984-21956972328779776{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1+1{}\mathrm {i}}\,{\mathrm {e}}^{2\,x}\,\left (12296353929494528+271474128182050816{}\mathrm {i}\right )+70836483296067584+69311013991743488{}\mathrm {i}\right )}{16}+\frac {\sqrt {2}\,\sqrt {1+1{}\mathrm {i}}\,\ln \left ({\mathrm {e}}^{2\,x}\,\left (155613434002538496-429723297714798592{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1+1{}\mathrm {i}}\,\left (54684829282729984+21956972328779776{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1+1{}\mathrm {i}}\,{\mathrm {e}}^{2\,x}\,\left (-12296353929494528-271474128182050816{}\mathrm {i}\right )+70836483296067584+69311013991743488{}\mathrm {i}\right )}{16} \]
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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