Optimal. Leaf size=195 \[ \frac{c^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{7/4}}-\frac{c^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{7/4}}+\frac{c^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{7/4}}-\frac{c^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{7/4}}-\frac{1}{3 a x^3} \]
[Out]
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Rubi [A] time = 0.239959, antiderivative size = 195, 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 \[ \frac{c^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{7/4}}-\frac{c^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{7/4}}+\frac{c^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{7/4}}-\frac{c^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{7/4}}-\frac{1}{3 a x^3} \]
Antiderivative was successfully verified.
[In] Int[1/(x^4*(a + c*x^4)),x]
[Out]
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Rubi in Sympy [A] time = 52.7651, size = 180, normalized size = 0.92 \[ - \frac{1}{3 a x^{3}} + \frac{\sqrt{2} c^{\frac{3}{4}} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{8 a^{\frac{7}{4}}} - \frac{\sqrt{2} c^{\frac{3}{4}} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{8 a^{\frac{7}{4}}} + \frac{\sqrt{2} c^{\frac{3}{4}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{4 a^{\frac{7}{4}}} - \frac{\sqrt{2} c^{\frac{3}{4}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{4 a^{\frac{7}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**4/(c*x**4+a),x)
[Out]
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Mathematica [A] time = 0.0664477, size = 188, normalized size = 0.96 \[ \frac{-8 a^{3/4}+6 \sqrt{2} c^{3/4} x^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )-6 \sqrt{2} c^{3/4} x^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )+3 \sqrt{2} c^{3/4} x^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )-3 \sqrt{2} c^{3/4} x^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{24 a^{7/4} x^3} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^4*(a + c*x^4)),x]
[Out]
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Maple [A] time = 0.006, size = 139, normalized size = 0.7 \[ -{\frac{c\sqrt{2}}{8\,{a}^{2}}\sqrt [4]{{\frac{a}{c}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }-{\frac{c\sqrt{2}}{4\,{a}^{2}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }-{\frac{c\sqrt{2}}{4\,{a}^{2}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }-{\frac{1}{3\,a{x}^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^4/(c*x^4+a),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)*x^4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.249701, size = 200, normalized size = 1.03 \[ \frac{12 \, a x^{3} \left (-\frac{c^{3}}{a^{7}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{2} \left (-\frac{c^{3}}{a^{7}}\right )^{\frac{1}{4}}}{c x + c \sqrt{\frac{a^{4} \sqrt{-\frac{c^{3}}{a^{7}}} + c^{2} x^{2}}{c^{2}}}}\right ) - 3 \, a x^{3} \left (-\frac{c^{3}}{a^{7}}\right )^{\frac{1}{4}} \log \left (a^{2} \left (-\frac{c^{3}}{a^{7}}\right )^{\frac{1}{4}} + c x\right ) + 3 \, a x^{3} \left (-\frac{c^{3}}{a^{7}}\right )^{\frac{1}{4}} \log \left (-a^{2} \left (-\frac{c^{3}}{a^{7}}\right )^{\frac{1}{4}} + c x\right ) - 4}{12 \, a x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^4 + a)*x^4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.58711, size = 32, normalized size = 0.16 \[ \operatorname{RootSum}{\left (256 t^{4} a^{7} + c^{3}, \left ( t \mapsto t \log{\left (- \frac{4 t a^{2}}{c} + x \right )} \right )\right )} - \frac{1}{3 a x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**4/(c*x**4+a),x)
[Out]
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GIAC/XCAS [A] time = 0.22824, size = 236, normalized size = 1.21 \[ -\frac{\sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{4 \, a^{2}} - \frac{\sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{4 \, a^{2}} - \frac{\sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{8 \, a^{2}} + \frac{\sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{8 \, a^{2}} - \frac{1}{3 \, a x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^4 + a)*x^4),x, algorithm="giac")
[Out]