Integrand size = 19, antiderivative size = 50 \[ \int \frac {\left (1+b x^4\right )^p}{1-x^2} \, dx=x \operatorname {AppellF1}\left (\frac {1}{4},1,-p,\frac {5}{4},x^4,-b x^4\right )+\frac {1}{3} x^3 \operatorname {AppellF1}\left (\frac {3}{4},1,-p,\frac {7}{4},x^4,-b x^4\right ) \]
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Time = 0.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1254, 440, 524} \[ \int \frac {\left (1+b x^4\right )^p}{1-x^2} \, dx=x \operatorname {AppellF1}\left (\frac {1}{4},1,-p,\frac {5}{4},x^4,-b x^4\right )+\frac {1}{3} x^3 \operatorname {AppellF1}\left (\frac {3}{4},1,-p,\frac {7}{4},x^4,-b x^4\right ) \]
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Rule 440
Rule 524
Rule 1254
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (1+b x^4\right )^p}{1-x^4}-\frac {x^2 \left (1+b x^4\right )^p}{-1+x^4}\right ) \, dx \\ & = \int \frac {\left (1+b x^4\right )^p}{1-x^4} \, dx-\int \frac {x^2 \left (1+b x^4\right )^p}{-1+x^4} \, dx \\ & = x F_1\left (\frac {1}{4};1,-p;\frac {5}{4};x^4,-b x^4\right )+\frac {1}{3} x^3 F_1\left (\frac {3}{4};1,-p;\frac {7}{4};x^4,-b x^4\right ) \\ \end{align*}
\[ \int \frac {\left (1+b x^4\right )^p}{1-x^2} \, dx=\int \frac {\left (1+b x^4\right )^p}{1-x^2} \, dx \]
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\[\int \frac {\left (b \,x^{4}+1\right )^{p}}{-x^{2}+1}d x\]
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\[ \int \frac {\left (1+b x^4\right )^p}{1-x^2} \, dx=\int { -\frac {{\left (b x^{4} + 1\right )}^{p}}{x^{2} - 1} \,d x } \]
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Timed out. \[ \int \frac {\left (1+b x^4\right )^p}{1-x^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (1+b x^4\right )^p}{1-x^2} \, dx=\int { -\frac {{\left (b x^{4} + 1\right )}^{p}}{x^{2} - 1} \,d x } \]
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\[ \int \frac {\left (1+b x^4\right )^p}{1-x^2} \, dx=\int { -\frac {{\left (b x^{4} + 1\right )}^{p}}{x^{2} - 1} \,d x } \]
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Timed out. \[ \int \frac {\left (1+b x^4\right )^p}{1-x^2} \, dx=-\int \frac {{\left (b\,x^4+1\right )}^p}{x^2-1} \,d x \]
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