Optimal. Leaf size=101 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt [4]{a} \tanh (x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{2 a^{3/4} \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{a} \tanh (x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{2 a^{3/4} \sqrt{\sqrt{a}+\sqrt{b}}} \]
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Rubi [A] time = 0.12548, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {3209, 1166, 208} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt [4]{a} \tanh (x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{2 a^{3/4} \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{a} \tanh (x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{2 a^{3/4} \sqrt{\sqrt{a}+\sqrt{b}}} \]
Antiderivative was successfully verified.
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Rule 3209
Rule 1166
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{a-b \cosh ^4(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1-x^2}{a-2 a x^2+(a-b) x^4} \, dx,x,\coth (x)\right )\\ &=\frac{1}{2} \left (-1-\frac{\sqrt{b}}{\sqrt{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{-a-\sqrt{a} \sqrt{b}+(a-b) x^2} \, dx,x,\coth (x)\right )+\frac{1}{2} \left (-1+\frac{\sqrt{b}}{\sqrt{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{-a+\sqrt{a} \sqrt{b}+(a-b) x^2} \, dx,x,\coth (x)\right )\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{a} \tanh (x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{2 a^{3/4} \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{a} \tanh (x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{2 a^{3/4} \sqrt{\sqrt{a}+\sqrt{b}}}\\ \end{align*}
Mathematica [A] time = 0.191286, size = 109, normalized size = 1.08 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a} \tanh (x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right )}{2 \sqrt{a} \sqrt{\sqrt{a} \sqrt{b}+a}}-\frac{\tan ^{-1}\left (\frac{\sqrt{a} \tanh (x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right )}{2 \sqrt{a} \sqrt{\sqrt{a} \sqrt{b}-a}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.023, size = 127, normalized size = 1.3 \begin{align*}{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ( \left ( a-b \right ){{\it \_Z}}^{8}+ \left ( -4\,a-4\,b \right ){{\it \_Z}}^{6}+ \left ( 6\,a-6\,b \right ){{\it \_Z}}^{4}+ \left ( -4\,a-4\,b \right ){{\it \_Z}}^{2}+a-b \right ) }{\frac{-{{\it \_R}}^{6}+3\,{{\it \_R}}^{4}-3\,{{\it \_R}}^{2}+1}{{{\it \_R}}^{7}a-{{\it \_R}}^{7}b-3\,{{\it \_R}}^{5}a-3\,{{\it \_R}}^{5}b+3\,{{\it \_R}}^{3}a-3\,{{\it \_R}}^{3}b-{\it \_R}\,a-{\it \_R}\,b}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -{\it \_R} \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{1}{b \cosh \left (x\right )^{4} - a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.70433, size = 1706, normalized size = 16.89 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{1}{b \cosh \left (x\right )^{4} - a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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