Optimal. Leaf size=204 \[ \frac{3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} \sqrt [4]{a} c^{7/4}}-\frac{3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} \sqrt [4]{a} c^{7/4}}-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} \sqrt [4]{a} c^{7/4}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} \sqrt [4]{a} c^{7/4}}-\frac{x^3}{4 c \left (a+c x^4\right )} \]
[Out]
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Rubi [A] time = 0.263406, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538 \[ \frac{3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} \sqrt [4]{a} c^{7/4}}-\frac{3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} \sqrt [4]{a} c^{7/4}}-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} \sqrt [4]{a} c^{7/4}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} \sqrt [4]{a} c^{7/4}}-\frac{x^3}{4 c \left (a+c x^4\right )} \]
Antiderivative was successfully verified.
[In] Int[x^6/(a + c*x^4)^2,x]
[Out]
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Rubi in Sympy [A] time = 53.8217, size = 192, normalized size = 0.94 \[ - \frac{x^{3}}{4 c \left (a + c x^{4}\right )} + \frac{3 \sqrt{2} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{32 \sqrt [4]{a} c^{\frac{7}{4}}} - \frac{3 \sqrt{2} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{32 \sqrt [4]{a} c^{\frac{7}{4}}} - \frac{3 \sqrt{2} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{16 \sqrt [4]{a} c^{\frac{7}{4}}} + \frac{3 \sqrt{2} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{16 \sqrt [4]{a} c^{\frac{7}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**6/(c*x**4+a)**2,x)
[Out]
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Mathematica [A] time = 0.300103, size = 185, normalized size = 0.91 \[ \frac{-\frac{8 c^{3/4} x^3}{a+c x^4}+\frac{3 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{\sqrt [4]{a}}-\frac{3 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{\sqrt [4]{a}}-\frac{6 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt [4]{a}}+\frac{6 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{a}}}{32 c^{7/4}} \]
Antiderivative was successfully verified.
[In] Integrate[x^6/(a + c*x^4)^2,x]
[Out]
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Maple [A] time = 0.011, size = 145, normalized size = 0.7 \[ -{\frac{{x}^{3}}{4\,c \left ( c{x}^{4}+a \right ) }}+{\frac{3\,\sqrt{2}}{32\,{c}^{2}}\ln \left ({1 \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{3\,\sqrt{2}}{16\,{c}^{2}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{3\,\sqrt{2}}{16\,{c}^{2}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^6/(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(x^6/(c*x^4 + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.247078, size = 228, normalized size = 1.12 \[ -\frac{4 \, x^{3} - 12 \,{\left (c^{2} x^{4} + a c\right )} \left (-\frac{1}{a c^{7}}\right )^{\frac{1}{4}} \arctan \left (\frac{a c^{5} \left (-\frac{1}{a c^{7}}\right )^{\frac{3}{4}}}{x + \sqrt{-a c^{3} \sqrt{-\frac{1}{a c^{7}}} + x^{2}}}\right ) - 3 \,{\left (c^{2} x^{4} + a c\right )} \left (-\frac{1}{a c^{7}}\right )^{\frac{1}{4}} \log \left (a c^{5} \left (-\frac{1}{a c^{7}}\right )^{\frac{3}{4}} + x\right ) + 3 \,{\left (c^{2} x^{4} + a c\right )} \left (-\frac{1}{a c^{7}}\right )^{\frac{1}{4}} \log \left (-a c^{5} \left (-\frac{1}{a c^{7}}\right )^{\frac{3}{4}} + x\right )}{16 \,{\left (c^{2} x^{4} + a c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^6/(c*x^4 + a)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.88773, size = 44, normalized size = 0.22 \[ - \frac{x^{3}}{4 a c + 4 c^{2} x^{4}} + \operatorname{RootSum}{\left (65536 t^{4} a c^{7} + 81, \left ( t \mapsto t \log{\left (\frac{4096 t^{3} a c^{5}}{27} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**6/(c*x**4+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.22572, size = 265, normalized size = 1.3 \[ -\frac{x^{3}}{4 \,{\left (c x^{4} + a\right )} c} + \frac{3 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{16 \, a c^{4}} + \frac{3 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{16 \, a c^{4}} - \frac{3 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{32 \, a c^{4}} + \frac{3 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{32 \, a c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^6/(c*x^4 + a)^2,x, algorithm="giac")
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