Optimal. Leaf size=77 \[ x F_1\left (\frac {1}{4};2,-p;\frac {5}{4};x^4,-b x^4\right )+\frac {2}{3} x^3 F_1\left (\frac {3}{4};2,-p;\frac {7}{4};x^4,-b x^4\right )+\frac {1}{5} x^5 F_1\left (\frac {5}{4};2,-p;\frac {9}{4};x^4,-b x^4\right ) \]
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Rubi [A]
time = 0.05, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1254, 440, 524}
\begin {gather*} x F_1\left (\frac {1}{4};2,-p;\frac {5}{4};x^4,-b x^4\right )+\frac {1}{5} x^5 F_1\left (\frac {5}{4};2,-p;\frac {9}{4};x^4,-b x^4\right )+\frac {2}{3} x^3 F_1\left (\frac {3}{4};2,-p;\frac {7}{4};x^4,-b x^4\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 440
Rule 524
Rule 1254
Rubi steps
\begin {align*} \int \frac {\left (1+b x^4\right )^p}{\left (1-x^2\right )^2} \, dx &=\int \left (\frac {\left (1+b x^4\right )^p}{\left (-1+x^4\right )^2}+\frac {2 x^2 \left (1+b x^4\right )^p}{\left (-1+x^4\right )^2}+\frac {x^4 \left (1+b x^4\right )^p}{\left (-1+x^4\right )^2}\right ) \, dx\\ &=2 \int \frac {x^2 \left (1+b x^4\right )^p}{\left (-1+x^4\right )^2} \, dx+\int \frac {\left (1+b x^4\right )^p}{\left (-1+x^4\right )^2} \, dx+\int \frac {x^4 \left (1+b x^4\right )^p}{\left (-1+x^4\right )^2} \, dx\\ &=x F_1\left (\frac {1}{4};2,-p;\frac {5}{4};x^4,-b x^4\right )+\frac {2}{3} x^3 F_1\left (\frac {3}{4};2,-p;\frac {7}{4};x^4,-b x^4\right )+\frac {1}{5} x^5 F_1\left (\frac {5}{4};2,-p;\frac {9}{4};x^4,-b x^4\right )\\ \end {align*}
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Mathematica [F]
time = 0.99, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1+b x^4\right )^p}{\left (1-x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{4}+1\right )^{p}}{\left (-x^{2}+1\right )^{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (b\,x^4+1\right )}^p}{{\left (x^2-1\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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