Optimal. Leaf size=209 \[ -\frac{c d \left (2 a e^2+c d^2\right ) \log \left (a+c x^4\right )}{4 a^2 \left (a e^2+c d^2\right )^2}-\frac{\sqrt{c} e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \left (a e^2+c d^2\right )}+\frac{\log (x)}{a^2 d}+\frac{c \left (d-e x^2\right )}{4 a \left (a+c x^4\right ) \left (a e^2+c d^2\right )}-\frac{e^4 \log \left (d+e x^2\right )}{2 d \left (a e^2+c d^2\right )^2}-\frac{\sqrt{c} e^3 \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \left (a e^2+c d^2\right )^2} \]
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Rubi [A] time = 0.239187, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {1252, 894, 639, 205, 635, 260} \[ -\frac{c d \left (2 a e^2+c d^2\right ) \log \left (a+c x^4\right )}{4 a^2 \left (a e^2+c d^2\right )^2}-\frac{\sqrt{c} e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \left (a e^2+c d^2\right )}+\frac{\log (x)}{a^2 d}+\frac{c \left (d-e x^2\right )}{4 a \left (a+c x^4\right ) \left (a e^2+c d^2\right )}-\frac{e^4 \log \left (d+e x^2\right )}{2 d \left (a e^2+c d^2\right )^2}-\frac{\sqrt{c} e^3 \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \left (a e^2+c d^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 1252
Rule 894
Rule 639
Rule 205
Rule 635
Rule 260
Rubi steps
\begin{align*} \int \frac{1}{x \left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x (d+e x) \left (a+c x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{a^2 d x}-\frac{e^5}{d \left (c d^2+a e^2\right )^2 (d+e x)}-\frac{c (a e+c d x)}{a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}+\frac{c \left (-a^2 e^3-c d \left (c d^2+2 a e^2\right ) x\right )}{a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}\right ) \, dx,x,x^2\right )\\ &=\frac{\log (x)}{a^2 d}-\frac{e^4 \log \left (d+e x^2\right )}{2 d \left (c d^2+a e^2\right )^2}+\frac{c \operatorname{Subst}\left (\int \frac{-a^2 e^3-c d \left (c d^2+2 a e^2\right ) x}{a+c x^2} \, dx,x,x^2\right )}{2 a^2 \left (c d^2+a e^2\right )^2}-\frac{c \operatorname{Subst}\left (\int \frac{a e+c d x}{\left (a+c x^2\right )^2} \, dx,x,x^2\right )}{2 a \left (c d^2+a e^2\right )}\\ &=\frac{c \left (d-e x^2\right )}{4 a \left (c d^2+a e^2\right ) \left (a+c x^4\right )}+\frac{\log (x)}{a^2 d}-\frac{e^4 \log \left (d+e x^2\right )}{2 d \left (c d^2+a e^2\right )^2}-\frac{\left (c e^3\right ) \operatorname{Subst}\left (\int \frac{1}{a+c x^2} \, dx,x,x^2\right )}{2 \left (c d^2+a e^2\right )^2}-\frac{(c e) \operatorname{Subst}\left (\int \frac{1}{a+c x^2} \, dx,x,x^2\right )}{4 a \left (c d^2+a e^2\right )}-\frac{\left (c^2 d \left (c d^2+2 a e^2\right )\right ) \operatorname{Subst}\left (\int \frac{x}{a+c x^2} \, dx,x,x^2\right )}{2 a^2 \left (c d^2+a e^2\right )^2}\\ &=\frac{c \left (d-e x^2\right )}{4 a \left (c d^2+a e^2\right ) \left (a+c x^4\right )}-\frac{\sqrt{c} e^3 \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \left (c d^2+a e^2\right )^2}-\frac{\sqrt{c} e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \left (c d^2+a e^2\right )}+\frac{\log (x)}{a^2 d}-\frac{e^4 \log \left (d+e x^2\right )}{2 d \left (c d^2+a e^2\right )^2}-\frac{c d \left (c d^2+2 a e^2\right ) \log \left (a+c x^4\right )}{4 a^2 \left (c d^2+a e^2\right )^2}\\ \end{align*}
Mathematica [A] time = 0.195079, size = 241, normalized size = 1.15 \[ \frac{-2 a^2 e^4 \left (a+c x^4\right ) \log \left (d+e x^2\right )+a c d \left (d-e x^2\right ) \left (a e^2+c d^2\right )+4 \log (x) \left (a+c x^4\right ) \left (a e^2+c d^2\right )^2-c d^2 \left (a+c x^4\right ) \left (2 a e^2+c d^2\right ) \log \left (a+c x^4\right )+\sqrt{a} \sqrt{c} d e \left (a+c x^4\right ) \left (3 a e^2+c d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+\sqrt{a} \sqrt{c} d e \left (a+c x^4\right ) \left (3 a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{4 a^2 d \left (a+c x^4\right ) \left (a e^2+c d^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 309, normalized size = 1.5 \begin{align*} -{\frac{{e}^{3}c{x}^{2}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{4}+a \right ) }}-{\frac{{c}^{2}{x}^{2}e{d}^{2}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}a \left ( c{x}^{4}+a \right ) }}+{\frac{cd{e}^{2}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{4}+a \right ) }}+{\frac{{c}^{2}{d}^{3}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}a \left ( c{x}^{4}+a \right ) }}-{\frac{c\ln \left ( c{x}^{4}+a \right ){e}^{2}d}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}a}}-{\frac{{c}^{2}\ln \left ( c{x}^{4}+a \right ){d}^{3}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}{a}^{2}}}-{\frac{3\,{e}^{3}c}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-{\frac{e{c}^{2}{d}^{2}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}a}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{\ln \left ( x \right ) }{d{a}^{2}}}-{\frac{{e}^{4}\ln \left ( e{x}^{2}+d \right ) }{2\,d \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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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 [A] time = 1.07542, size = 377, normalized size = 1.8 \begin{align*} -\frac{{\left (c^{2} d^{3} + 2 \, a c d e^{2}\right )} \log \left (c x^{4} + a\right )}{4 \,{\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )}} - \frac{e^{5} \log \left ({\left | x^{2} e + d \right |}\right )}{2 \,{\left (c^{2} d^{5} e + 2 \, a c d^{3} e^{3} + a^{2} d e^{5}\right )}} - \frac{{\left (c^{2} d^{2} e + 3 \, a c e^{3}\right )} \arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{4 \,{\left (a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} \sqrt{a c}} + \frac{c^{3} d^{3} x^{4} + 2 \, a c^{2} d x^{4} e^{2} - a c^{2} d^{2} x^{2} e + 2 \, a c^{2} d^{3} - a^{2} c x^{2} e^{3} + 3 \, a^{2} c d e^{2}}{4 \,{\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )}{\left (c x^{4} + a\right )}} + \frac{\log \left (x^{2}\right )}{2 \, a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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