Optimal. Leaf size=335 \[ \frac{\sin \left (\frac{3 \pi }{14}\right ) \log \left (a^{2/7}+2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac{3 \pi }{14}\right )+b^{2/7} x^2\right )}{7 a^{6/7} \sqrt [7]{b}}-\frac{\sin \left (\frac{\pi }{14}\right ) \log \left (a^{2/7}-2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac{\pi }{14}\right )+b^{2/7} x^2\right )}{7 a^{6/7} \sqrt [7]{b}}-\frac{\cos \left (\frac{\pi }{7}\right ) \log \left (a^{2/7}-2 \sqrt [7]{a} \sqrt [7]{b} x \cos \left (\frac{\pi }{7}\right )+b^{2/7} x^2\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac{\log \left (\sqrt [7]{a}+\sqrt [7]{b} x\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac{2 \cos \left (\frac{3 \pi }{14}\right ) \tan ^{-1}\left (\frac{\sqrt [7]{b} x \sec \left (\frac{3 \pi }{14}\right )}{\sqrt [7]{a}}+\tan \left (\frac{3 \pi }{14}\right )\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac{2 \cos \left (\frac{\pi }{14}\right ) \tan ^{-1}\left (\frac{\sqrt [7]{b} x \sec \left (\frac{\pi }{14}\right )}{\sqrt [7]{a}}-\tan \left (\frac{\pi }{14}\right )\right )}{7 a^{6/7} \sqrt [7]{b}}-\frac{2 \sin \left (\frac{\pi }{7}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{7}\right )-\frac{\sqrt [7]{b} x \csc \left (\frac{\pi }{7}\right )}{\sqrt [7]{a}}\right )}{7 a^{6/7} \sqrt [7]{b}} \]
[Out]
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Rubi [A] time = 0.90296, antiderivative size = 335, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667 \[ \frac{\sin \left (\frac{3 \pi }{14}\right ) \log \left (a^{2/7}+2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac{3 \pi }{14}\right )+b^{2/7} x^2\right )}{7 a^{6/7} \sqrt [7]{b}}-\frac{\sin \left (\frac{\pi }{14}\right ) \log \left (a^{2/7}-2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac{\pi }{14}\right )+b^{2/7} x^2\right )}{7 a^{6/7} \sqrt [7]{b}}-\frac{\cos \left (\frac{\pi }{7}\right ) \log \left (a^{2/7}-2 \sqrt [7]{a} \sqrt [7]{b} x \cos \left (\frac{\pi }{7}\right )+b^{2/7} x^2\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac{\log \left (\sqrt [7]{a}+\sqrt [7]{b} x\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac{2 \cos \left (\frac{3 \pi }{14}\right ) \tan ^{-1}\left (\frac{\sqrt [7]{b} x \sec \left (\frac{3 \pi }{14}\right )}{\sqrt [7]{a}}+\tan \left (\frac{3 \pi }{14}\right )\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac{2 \cos \left (\frac{\pi }{14}\right ) \tan ^{-1}\left (\frac{\sqrt [7]{b} x \sec \left (\frac{\pi }{14}\right )}{\sqrt [7]{a}}-\tan \left (\frac{\pi }{14}\right )\right )}{7 a^{6/7} \sqrt [7]{b}}-\frac{2 \sin \left (\frac{\pi }{7}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{7}\right )-\frac{\sqrt [7]{b} x \csc \left (\frac{\pi }{7}\right )}{\sqrt [7]{a}}\right )}{7 a^{6/7} \sqrt [7]{b}} \]
Warning: Unable to verify antiderivative.
[In] Int[(a + b*x^7)^(-1),x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x**7+a),x)
[Out]
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Mathematica [A] time = 0.528852, size = 262, normalized size = 0.78 \[ \frac{\sin \left (\frac{3 \pi }{14}\right ) \log \left (a^{2/7}+2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac{3 \pi }{14}\right )+b^{2/7} x^2\right )-\sin \left (\frac{\pi }{14}\right ) \log \left (a^{2/7}-2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac{\pi }{14}\right )+b^{2/7} x^2\right )-\cos \left (\frac{\pi }{7}\right ) \log \left (a^{2/7}-2 \sqrt [7]{a} \sqrt [7]{b} x \cos \left (\frac{\pi }{7}\right )+b^{2/7} x^2\right )+\log \left (\sqrt [7]{a}+\sqrt [7]{b} x\right )+2 \cos \left (\frac{3 \pi }{14}\right ) \tan ^{-1}\left (\frac{\sqrt [7]{b} x \sec \left (\frac{3 \pi }{14}\right )}{\sqrt [7]{a}}+\tan \left (\frac{3 \pi }{14}\right )\right )+2 \cos \left (\frac{\pi }{14}\right ) \tan ^{-1}\left (\frac{\sqrt [7]{b} x \sec \left (\frac{\pi }{14}\right )}{\sqrt [7]{a}}-\tan \left (\frac{\pi }{14}\right )\right )-2 \sin \left (\frac{\pi }{7}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{7}\right )-\frac{\sqrt [7]{b} x \csc \left (\frac{\pi }{7}\right )}{\sqrt [7]{a}}\right )}{7 a^{6/7} \sqrt [7]{b}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(a + b*x^7)^(-1),x]
[Out]
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Maple [C] time = 0.648, size = 27, normalized size = 0.1 \[{\frac{1}{7\,b}\sum _{{\it \_R}={\it RootOf} \left ( b{{\it \_Z}}^{7}+a \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{6}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x^7+a),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{b x^{7} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x^7 + a),x, algorithm="maxima")
[Out]
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x^7 + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.448503, size = 20, normalized size = 0.06 \[ \operatorname{RootSum}{\left (823543 t^{7} a^{6} b - 1, \left ( t \mapsto t \log{\left (7 t a + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x**7+a),x)
[Out]
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GIAC/XCAS [A] time = 0.228784, size = 419, normalized size = 1.25 \[ \frac{\left (-\frac{a}{b}\right )^{\frac{1}{7}} \cos \left (\frac{3}{7} \, \pi \right ){\rm ln}\left (2 \, x \left (-\frac{a}{b}\right )^{\frac{1}{7}} \cos \left (\frac{3}{7} \, \pi \right ) + x^{2} + \left (-\frac{a}{b}\right )^{\frac{2}{7}}\right )}{7 \, a} - \frac{\left (-\frac{a}{b}\right )^{\frac{1}{7}} \cos \left (\frac{2}{7} \, \pi \right ){\rm ln}\left (-2 \, x \left (-\frac{a}{b}\right )^{\frac{1}{7}} \cos \left (\frac{2}{7} \, \pi \right ) + x^{2} + \left (-\frac{a}{b}\right )^{\frac{2}{7}}\right )}{7 \, a} + \frac{\left (-\frac{a}{b}\right )^{\frac{1}{7}} \cos \left (\frac{1}{7} \, \pi \right ){\rm ln}\left (2 \, x \left (-\frac{a}{b}\right )^{\frac{1}{7}} \cos \left (\frac{1}{7} \, \pi \right ) + x^{2} + \left (-\frac{a}{b}\right )^{\frac{2}{7}}\right )}{7 \, a} + \frac{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{7}} \arctan \left (\frac{\left (-\frac{a}{b}\right )^{\frac{1}{7}} \cos \left (\frac{3}{7} \, \pi \right ) + x}{\left (-\frac{a}{b}\right )^{\frac{1}{7}} \sin \left (\frac{3}{7} \, \pi \right )}\right ) \sin \left (\frac{3}{7} \, \pi \right )}{7 \, a} + \frac{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{7}} \arctan \left (-\frac{\left (-\frac{a}{b}\right )^{\frac{1}{7}} \cos \left (\frac{2}{7} \, \pi \right ) - x}{\left (-\frac{a}{b}\right )^{\frac{1}{7}} \sin \left (\frac{2}{7} \, \pi \right )}\right ) \sin \left (\frac{2}{7} \, \pi \right )}{7 \, a} + \frac{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{7}} \arctan \left (\frac{\left (-\frac{a}{b}\right )^{\frac{1}{7}} \cos \left (\frac{1}{7} \, \pi \right ) + x}{\left (-\frac{a}{b}\right )^{\frac{1}{7}} \sin \left (\frac{1}{7} \, \pi \right )}\right ) \sin \left (\frac{1}{7} \, \pi \right )}{7 \, a} - \frac{\left (-\frac{a}{b}\right )^{\frac{1}{7}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{7}} \right |}\right )}{7 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x^7 + a),x, algorithm="giac")
[Out]