Optimal. Leaf size=70 \[ \frac{a^2 \log \left (a+b x^4\right )}{4 b^2 (b c-a d)}-\frac{c^2 \log \left (c+d x^4\right )}{4 d^2 (b c-a d)}+\frac{x^4}{4 b d} \]
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Rubi [A] time = 0.0637166, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {446, 72} \[ \frac{a^2 \log \left (a+b x^4\right )}{4 b^2 (b c-a d)}-\frac{c^2 \log \left (c+d x^4\right )}{4 d^2 (b c-a d)}+\frac{x^4}{4 b d} \]
Antiderivative was successfully verified.
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Rule 446
Rule 72
Rubi steps
\begin{align*} \int \frac{x^{11}}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x^2}{(a+b x) (c+d x)} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{1}{b d}+\frac{a^2}{b (b c-a d) (a+b x)}+\frac{c^2}{d (-b c+a d) (c+d x)}\right ) \, dx,x,x^4\right )\\ &=\frac{x^4}{4 b d}+\frac{a^2 \log \left (a+b x^4\right )}{4 b^2 (b c-a d)}-\frac{c^2 \log \left (c+d x^4\right )}{4 d^2 (b c-a d)}\\ \end{align*}
Mathematica [A] time = 0.0331131, size = 66, normalized size = 0.94 \[ \frac{a^2 d^2 \log \left (a+b x^4\right )-b \left (d x^4 (a d-b c)+b c^2 \log \left (c+d x^4\right )\right )}{4 b^2 d^2 (b c-a d)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 65, normalized size = 0.9 \begin{align*}{\frac{{x}^{4}}{4\,bd}}+{\frac{{c}^{2}\ln \left ( d{x}^{4}+c \right ) }{4\,{d}^{2} \left ( ad-bc \right ) }}-{\frac{{a}^{2}\ln \left ( b{x}^{4}+a \right ) }{4\,{b}^{2} \left ( ad-bc \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.928949, size = 92, normalized size = 1.31 \begin{align*} \frac{x^{4}}{4 \, b d} + \frac{a^{2} \log \left (b x^{4} + a\right )}{4 \,{\left (b^{3} c - a b^{2} d\right )}} - \frac{c^{2} \log \left (d x^{4} + c\right )}{4 \,{\left (b c d^{2} - a d^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 7.86221, size = 142, normalized size = 2.03 \begin{align*} \frac{{\left (b^{2} c d - a b d^{2}\right )} x^{4} + a^{2} d^{2} \log \left (b x^{4} + a\right ) - b^{2} c^{2} \log \left (d x^{4} + c\right )}{4 \,{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 7.35492, size = 201, normalized size = 2.87 \begin{align*} - \frac{a^{2} \log{\left (x^{4} + \frac{\frac{a^{4} d^{3}}{b \left (a d - b c\right )} - \frac{2 a^{3} c d^{2}}{a d - b c} + \frac{a^{2} b c^{2} d}{a d - b c} + a^{2} c d + a b c^{2}}{a^{2} d^{2} + b^{2} c^{2}} \right )}}{4 b^{2} \left (a d - b c\right )} + \frac{c^{2} \log{\left (x^{4} + \frac{- \frac{a^{2} b c^{2} d}{a d - b c} + a^{2} c d + \frac{2 a b^{2} c^{3}}{a d - b c} + a b c^{2} - \frac{b^{3} c^{4}}{d \left (a d - b c\right )}}{a^{2} d^{2} + b^{2} c^{2}} \right )}}{4 d^{2} \left (a d - b c\right )} + \frac{x^{4}}{4 b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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