Integrand size = 21, antiderivative size = 106 \[ \int \frac {\tanh (x)}{\sqrt {a+b \coth ^2(x)+c \coth ^4(x)}} \, dx=-\frac {\text {arctanh}\left (\frac {2 a+b \coth ^2(x)}{2 \sqrt {a} \sqrt {a+b \coth ^2(x)+c \coth ^4(x)}}\right )}{2 \sqrt {a}}+\frac {\text {arctanh}\left (\frac {2 a+b+(b+2 c) \coth ^2(x)}{2 \sqrt {a+b+c} \sqrt {a+b \coth ^2(x)+c \coth ^4(x)}}\right )}{2 \sqrt {a+b+c}} \]
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Time = 0.17 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3782, 1265, 974, 738, 212} \[ \int \frac {\tanh (x)}{\sqrt {a+b \coth ^2(x)+c \coth ^4(x)}} \, dx=\frac {\text {arctanh}\left (\frac {2 a+(b+2 c) \coth ^2(x)+b}{2 \sqrt {a+b+c} \sqrt {a+b \coth ^2(x)+c \coth ^4(x)}}\right )}{2 \sqrt {a+b+c}}-\frac {\text {arctanh}\left (\frac {2 a+b \coth ^2(x)}{2 \sqrt {a} \sqrt {a+b \coth ^2(x)+c \coth ^4(x)}}\right )}{2 \sqrt {a}} \]
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Rule 212
Rule 738
Rule 974
Rule 1265
Rule 3782
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{x \left (1+x^2\right ) \sqrt {a-b x^2+c x^4}} \, dx,x,-i \coth (x)\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1}{x (1+x) \sqrt {a-b x+c x^2}} \, dx,x,-\coth ^2(x)\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{(-1-x) \sqrt {a-b x+c x^2}}+\frac {1}{x \sqrt {a-b x+c x^2}}\right ) \, dx,x,-\coth ^2(x)\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1}{(-1-x) \sqrt {a-b x+c x^2}} \, dx,x,-\coth ^2(x)\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{x \sqrt {a-b x+c x^2}} \, dx,x,-\coth ^2(x)\right ) \\ & = -\text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b \coth ^2(x)}{\sqrt {a+b \coth ^2(x)+c \coth ^4(x)}}\right )-\text {Subst}\left (\int \frac {1}{4 a+4 b+4 c-x^2} \, dx,x,\frac {-2 a-b+(-b-2 c) \coth ^2(x)}{\sqrt {a+b \coth ^2(x)+c \coth ^4(x)}}\right ) \\ & = -\frac {\text {arctanh}\left (\frac {2 a+b \coth ^2(x)}{2 \sqrt {a} \sqrt {a+b \coth ^2(x)+c \coth ^4(x)}}\right )}{2 \sqrt {a}}-\frac {\text {arctanh}\left (\frac {-2 a-b+(-b-2 c) \coth ^2(x)}{2 \sqrt {a+b+c} \sqrt {a+b \coth ^2(x)+c \coth ^4(x)}}\right )}{2 \sqrt {a+b+c}} \\ \end{align*}
Time = 0.43 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.50 \[ \int \frac {\tanh (x)}{\sqrt {a+b \coth ^2(x)+c \coth ^4(x)}} \, dx=-\frac {\left ((a+b+c) \text {arctanh}\left (\frac {b+2 a \tanh ^2(x)}{2 \sqrt {a} \sqrt {c+b \tanh ^2(x)+a \tanh ^4(x)}}\right )-\sqrt {a} \sqrt {a+b+c} \text {arctanh}\left (\frac {b+2 c+(2 a+b) \tanh ^2(x)}{2 \sqrt {a+b+c} \sqrt {c+b \tanh ^2(x)+a \tanh ^4(x)}}\right )\right ) \coth ^2(x) \sqrt {c+b \tanh ^2(x)+a \tanh ^4(x)}}{2 \sqrt {a} (a+b+c) \sqrt {a+b \coth ^2(x)+c \coth ^4(x)}} \]
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\[\int \frac {\tanh \left (x \right )}{\sqrt {a +b \coth \left (x \right )^{2}+c \coth \left (x \right )^{4}}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 1524 vs. \(2 (86) = 172\).
Time = 0.89 (sec) , antiderivative size = 6705, normalized size of antiderivative = 63.25 \[ \int \frac {\tanh (x)}{\sqrt {a+b \coth ^2(x)+c \coth ^4(x)}} \, dx=\text {Too large to display} \]
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\[ \int \frac {\tanh (x)}{\sqrt {a+b \coth ^2(x)+c \coth ^4(x)}} \, dx=\int \frac {\tanh {\left (x \right )}}{\sqrt {a + b \coth ^{2}{\left (x \right )} + c \coth ^{4}{\left (x \right )}}}\, dx \]
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\[ \int \frac {\tanh (x)}{\sqrt {a+b \coth ^2(x)+c \coth ^4(x)}} \, dx=\int { \frac {\tanh \left (x\right )}{\sqrt {c \coth \left (x\right )^{4} + b \coth \left (x\right )^{2} + a}} \,d x } \]
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Timed out. \[ \int \frac {\tanh (x)}{\sqrt {a+b \coth ^2(x)+c \coth ^4(x)}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\tanh (x)}{\sqrt {a+b \coth ^2(x)+c \coth ^4(x)}} \, dx=\int \frac {\mathrm {tanh}\left (x\right )}{\sqrt {c\,{\mathrm {coth}\left (x\right )}^4+b\,{\mathrm {coth}\left (x\right )}^2+a}} \,d x \]
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