Optimal. Leaf size=209 \[ -\frac{\sqrt{c} e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \left (a e^2+c d^2\right )}-\frac{c d \left (2 a e^2+c d^2\right ) \log \left (a+c x^4\right )}{4 a^2 \left (a e^2+c d^2\right )^2}+\frac{\log (x)}{a^2 d}+\frac{c \left (d-e x^2\right )}{4 a \left (a+c x^4\right ) \left (a e^2+c d^2\right )}-\frac{e^4 \log \left (d+e x^2\right )}{2 d \left (a e^2+c d^2\right )^2}-\frac{\sqrt{c} e^3 \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \left (a e^2+c d^2\right )^2} \]
[Out]
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Rubi [A] time = 0.485347, 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 \[ -\frac{\sqrt{c} e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \left (a e^2+c d^2\right )}-\frac{c d \left (2 a e^2+c d^2\right ) \log \left (a+c x^4\right )}{4 a^2 \left (a e^2+c d^2\right )^2}+\frac{\log (x)}{a^2 d}+\frac{c \left (d-e x^2\right )}{4 a \left (a+c x^4\right ) \left (a e^2+c d^2\right )}-\frac{e^4 \log \left (d+e x^2\right )}{2 d \left (a e^2+c d^2\right )^2}-\frac{\sqrt{c} e^3 \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \left (a e^2+c d^2\right )^2} \]
Antiderivative was successfully verified.
[In] Int[1/(x*(d + e*x^2)*(a + c*x^4)^2),x]
[Out]
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Rubi in Sympy [A] time = 74.1124, size = 189, normalized size = 0.9 \[ - \frac{e^{4} \log{\left (d + e x^{2} \right )}}{2 d \left (a e^{2} + c d^{2}\right )^{2}} + \frac{c \left (d - e x^{2}\right )}{4 a \left (a + c x^{4}\right ) \left (a e^{2} + c d^{2}\right )} - \frac{c d \left (2 a e^{2} + c d^{2}\right ) \log{\left (a + c x^{4} \right )}}{4 a^{2} \left (a e^{2} + c d^{2}\right )^{2}} + \frac{\log{\left (x^{2} \right )}}{2 a^{2} d} - \frac{\sqrt{c} e^{3} \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{a} \left (a e^{2} + c d^{2}\right )^{2}} - \frac{\sqrt{c} e \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{4 a^{\frac{3}{2}} \left (a e^{2} + c d^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x/(e*x**2+d)/(c*x**4+a)**2,x)
[Out]
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Mathematica [A] time = 0.320713, size = 241, normalized size = 1.15 \[ \frac{-2 a^2 e^4 \left (a+c x^4\right ) \log \left (d+e x^2\right )+4 \log (x) \left (a+c x^4\right ) \left (a e^2+c d^2\right )^2-c d^2 \left (a+c x^4\right ) \left (2 a e^2+c d^2\right ) \log \left (a+c x^4\right )+\sqrt{a} \sqrt{c} d e \left (a+c x^4\right ) \left (3 a e^2+c d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+\sqrt{a} \sqrt{c} d e \left (a+c x^4\right ) \left (3 a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )+a c d \left (d-e x^2\right ) \left (a e^2+c d^2\right )}{4 a^2 d \left (a+c x^4\right ) \left (a e^2+c d^2\right )^2} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x*(d + e*x^2)*(a + c*x^4)^2),x]
[Out]
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Maple [A] time = 0.029, size = 309, normalized size = 1.5 \[{\frac{\ln \left ( x \right ) }{{a}^{2}d}}-{\frac{c{x}^{2}{e}^{3}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{4}+a \right ) }}-{\frac{{c}^{2}{x}^{2}{d}^{2}e}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}a \left ( c{x}^{4}+a \right ) }}+{\frac{cd{e}^{2}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{4}+a \right ) }}+{\frac{{c}^{2}{d}^{3}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}a \left ( c{x}^{4}+a \right ) }}-{\frac{c\ln \left ( c{x}^{4}+a \right ) d{e}^{2}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}a}}-{\frac{{c}^{2}\ln \left ( c{x}^{4}+a \right ){d}^{3}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}{a}^{2}}}-{\frac{3\,c{e}^{3}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-{\frac{{d}^{2}e{c}^{2}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}a}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-{\frac{{e}^{4}\ln \left ( e{x}^{2}+d \right ) }{2\,d \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x/(e*x^2+d)/(c*x^4+a)^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^4 + a)^2*(e*x^2 + d)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 130.493, size = 1, normalized size = 0. \[ \left [\frac{2 \, a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} - 2 \,{\left (a c^{2} d^{3} e + a^{2} c d e^{3}\right )} x^{2} +{\left (a^{2} c d^{3} e + 3 \, a^{3} d e^{3} +{\left (a c^{2} d^{3} e + 3 \, a^{2} c d e^{3}\right )} x^{4}\right )} \sqrt{-\frac{c}{a}} \log \left (\frac{c x^{4} - 2 \, a x^{2} \sqrt{-\frac{c}{a}} - a}{c x^{4} + a}\right ) - 2 \,{\left (a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} +{\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2}\right )} x^{4}\right )} \log \left (c x^{4} + a\right ) - 4 \,{\left (a^{2} c e^{4} x^{4} + a^{3} e^{4}\right )} \log \left (e x^{2} + d\right ) + 8 \,{\left (a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4} +{\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} x^{4}\right )} \log \left (x\right )}{8 \,{\left (a^{3} c^{2} d^{5} + 2 \, a^{4} c d^{3} e^{2} + a^{5} d e^{4} +{\left (a^{2} c^{3} d^{5} + 2 \, a^{3} c^{2} d^{3} e^{2} + a^{4} c d e^{4}\right )} x^{4}\right )}}, \frac{a c^{2} d^{4} + a^{2} c d^{2} e^{2} -{\left (a c^{2} d^{3} e + a^{2} c d e^{3}\right )} x^{2} +{\left (a^{2} c d^{3} e + 3 \, a^{3} d e^{3} +{\left (a c^{2} d^{3} e + 3 \, a^{2} c d e^{3}\right )} x^{4}\right )} \sqrt{\frac{c}{a}} \arctan \left (\frac{a \sqrt{\frac{c}{a}}}{c x^{2}}\right ) -{\left (a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} +{\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2}\right )} x^{4}\right )} \log \left (c x^{4} + a\right ) - 2 \,{\left (a^{2} c e^{4} x^{4} + a^{3} e^{4}\right )} \log \left (e x^{2} + d\right ) + 4 \,{\left (a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4} +{\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} x^{4}\right )} \log \left (x\right )}{4 \,{\left (a^{3} c^{2} d^{5} + 2 \, a^{4} c d^{3} e^{2} + a^{5} d e^{4} +{\left (a^{2} c^{3} d^{5} + 2 \, a^{3} c^{2} d^{3} e^{2} + a^{4} c d e^{4}\right )} x^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^4 + a)^2*(e*x^2 + d)*x),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x/(e*x**2+d)/(c*x**4+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.276621, size = 377, normalized size = 1.8 \[ -\frac{{\left (c^{2} d^{3} + 2 \, a c d e^{2}\right )}{\rm ln}\left (c x^{4} + a\right )}{4 \,{\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )}} - \frac{e^{5}{\rm ln}\left ({\left | x^{2} e + d \right |}\right )}{2 \,{\left (c^{2} d^{5} e + 2 \, a c d^{3} e^{3} + a^{2} d e^{5}\right )}} - \frac{{\left (c^{2} d^{2} e + 3 \, a c e^{3}\right )} \arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{4 \,{\left (a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} \sqrt{a c}} + \frac{c^{3} d^{3} x^{4} + 2 \, a c^{2} d x^{4} e^{2} - a c^{2} d^{2} x^{2} e + 2 \, a c^{2} d^{3} - a^{2} c x^{2} e^{3} + 3 \, a^{2} c d e^{2}}{4 \,{\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )}{\left (c x^{4} + a\right )}} + \frac{{\rm ln}\left (x^{2}\right )}{2 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^4 + a)^2*(e*x^2 + d)*x),x, algorithm="giac")
[Out]