Optimal. Leaf size=335 \[ \frac{\sin \left (\frac{3 \pi }{14}\right ) \log \left (a^{2/7}+2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac{3 \pi }{14}\right )+b^{2/7} x^2\right )}{7 a^{6/7} \sqrt [7]{b}}-\frac{\sin \left (\frac{\pi }{14}\right ) \log \left (a^{2/7}-2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac{\pi }{14}\right )+b^{2/7} x^2\right )}{7 a^{6/7} \sqrt [7]{b}}-\frac{\cos \left (\frac{\pi }{7}\right ) \log \left (a^{2/7}-2 \sqrt [7]{a} \sqrt [7]{b} x \cos \left (\frac{\pi }{7}\right )+b^{2/7} x^2\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac{\log \left (\sqrt [7]{a}+\sqrt [7]{b} x\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac{2 \cos \left (\frac{3 \pi }{14}\right ) \tan ^{-1}\left (\frac{\sqrt [7]{b} x \sec \left (\frac{3 \pi }{14}\right )}{\sqrt [7]{a}}+\tan \left (\frac{3 \pi }{14}\right )\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac{2 \cos \left (\frac{\pi }{14}\right ) \tan ^{-1}\left (\frac{\sqrt [7]{b} x \sec \left (\frac{\pi }{14}\right )}{\sqrt [7]{a}}-\tan \left (\frac{\pi }{14}\right )\right )}{7 a^{6/7} \sqrt [7]{b}}-\frac{2 \sin \left (\frac{\pi }{7}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{7}\right )-\frac{\sqrt [7]{b} x \csc \left (\frac{\pi }{7}\right )}{\sqrt [7]{a}}\right )}{7 a^{6/7} \sqrt [7]{b}} \]
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Rubi [A] time = 0.53599, 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, Rules used = {201, 634, 618, 204, 628, 31} \[ \frac{\sin \left (\frac{3 \pi }{14}\right ) \log \left (a^{2/7}+2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac{3 \pi }{14}\right )+b^{2/7} x^2\right )}{7 a^{6/7} \sqrt [7]{b}}-\frac{\sin \left (\frac{\pi }{14}\right ) \log \left (a^{2/7}-2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac{\pi }{14}\right )+b^{2/7} x^2\right )}{7 a^{6/7} \sqrt [7]{b}}-\frac{\cos \left (\frac{\pi }{7}\right ) \log \left (a^{2/7}-2 \sqrt [7]{a} \sqrt [7]{b} x \cos \left (\frac{\pi }{7}\right )+b^{2/7} x^2\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac{\log \left (\sqrt [7]{a}+\sqrt [7]{b} x\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac{2 \cos \left (\frac{3 \pi }{14}\right ) \tan ^{-1}\left (\frac{\sqrt [7]{b} x \sec \left (\frac{3 \pi }{14}\right )}{\sqrt [7]{a}}+\tan \left (\frac{3 \pi }{14}\right )\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac{2 \cos \left (\frac{\pi }{14}\right ) \tan ^{-1}\left (\frac{\sqrt [7]{b} x \sec \left (\frac{\pi }{14}\right )}{\sqrt [7]{a}}-\tan \left (\frac{\pi }{14}\right )\right )}{7 a^{6/7} \sqrt [7]{b}}-\frac{2 \sin \left (\frac{\pi }{7}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{7}\right )-\frac{\sqrt [7]{b} x \csc \left (\frac{\pi }{7}\right )}{\sqrt [7]{a}}\right )}{7 a^{6/7} \sqrt [7]{b}} \]
Antiderivative was successfully verified.
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Rule 201
Rule 634
Rule 618
Rule 204
Rule 628
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{a+b x^7} \, dx &=\frac{2 \int \frac{\sqrt [7]{a}-\sqrt [7]{b} x \cos \left (\frac{\pi }{7}\right )}{a^{2/7}+b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} x \cos \left (\frac{\pi }{7}\right )} \, dx}{7 a^{6/7}}+\frac{2 \int \frac{\sqrt [7]{a}-\sqrt [7]{b} x \sin \left (\frac{\pi }{14}\right )}{a^{2/7}+b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac{\pi }{14}\right )} \, dx}{7 a^{6/7}}+\frac{2 \int \frac{\sqrt [7]{a}+\sqrt [7]{b} x \sin \left (\frac{3 \pi }{14}\right )}{a^{2/7}+b^{2/7} x^2+2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac{3 \pi }{14}\right )} \, dx}{7 a^{6/7}}+\frac{\int \frac{1}{\sqrt [7]{a}+\sqrt [7]{b} x} \, dx}{7 a^{6/7}}\\ &=\frac{\log \left (\sqrt [7]{a}+\sqrt [7]{b} x\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac{\left (2 \cos ^2\left (\frac{\pi }{14}\right )\right ) \int \frac{1}{a^{2/7}+b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac{\pi }{14}\right )} \, dx}{7 a^{5/7}}-\frac{\cos \left (\frac{\pi }{7}\right ) \int \frac{2 b^{2/7} x-2 \sqrt [7]{a} \sqrt [7]{b} \cos \left (\frac{\pi }{7}\right )}{a^{2/7}+b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} x \cos \left (\frac{\pi }{7}\right )} \, dx}{7 a^{6/7} \sqrt [7]{b}}+\frac{\left (2 \cos ^2\left (\frac{3 \pi }{14}\right )\right ) \int \frac{1}{a^{2/7}+b^{2/7} x^2+2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac{3 \pi }{14}\right )} \, dx}{7 a^{5/7}}-\frac{\sin \left (\frac{\pi }{14}\right ) \int \frac{2 b^{2/7} x-2 \sqrt [7]{a} \sqrt [7]{b} \sin \left (\frac{\pi }{14}\right )}{a^{2/7}+b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac{\pi }{14}\right )} \, dx}{7 a^{6/7} \sqrt [7]{b}}+\frac{\left (2 \sin ^2\left (\frac{\pi }{7}\right )\right ) \int \frac{1}{a^{2/7}+b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} x \cos \left (\frac{\pi }{7}\right )} \, dx}{7 a^{5/7}}+\frac{\sin \left (\frac{3 \pi }{14}\right ) \int \frac{2 b^{2/7} x+2 \sqrt [7]{a} \sqrt [7]{b} \sin \left (\frac{3 \pi }{14}\right )}{a^{2/7}+b^{2/7} x^2+2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac{3 \pi }{14}\right )} \, dx}{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{\cos \left (\frac{\pi }{7}\right ) \log \left (a^{2/7}+b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} x \cos \left (\frac{\pi }{7}\right )\right )}{7 a^{6/7} \sqrt [7]{b}}-\frac{\log \left (a^{2/7}+b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac{\pi }{14}\right )\right ) \sin \left (\frac{\pi }{14}\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac{\log \left (a^{2/7}+b^{2/7} x^2+2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac{3 \pi }{14}\right )\right ) \sin \left (\frac{3 \pi }{14}\right )}{7 a^{6/7} \sqrt [7]{b}}-\frac{\left (4 \cos ^2\left (\frac{\pi }{14}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-x^2-4 a^{2/7} b^{2/7} \cos ^2\left (\frac{\pi }{14}\right )} \, dx,x,2 b^{2/7} x-2 \sqrt [7]{a} \sqrt [7]{b} \sin \left (\frac{\pi }{14}\right )\right )}{7 a^{5/7}}-\frac{\left (4 \cos ^2\left (\frac{3 \pi }{14}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-x^2-4 a^{2/7} b^{2/7} \cos ^2\left (\frac{3 \pi }{14}\right )} \, dx,x,2 b^{2/7} x+2 \sqrt [7]{a} \sqrt [7]{b} \sin \left (\frac{3 \pi }{14}\right )\right )}{7 a^{5/7}}-\frac{\left (4 \sin ^2\left (\frac{\pi }{7}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-x^2-4 a^{2/7} b^{2/7} \sin ^2\left (\frac{\pi }{7}\right )} \, dx,x,2 b^{2/7} x-2 \sqrt [7]{a} \sqrt [7]{b} \cos \left (\frac{\pi }{7}\right )\right )}{7 a^{5/7}}\\ &=\frac{2 \tan ^{-1}\left (\frac{\sqrt [7]{b} x \sec \left (\frac{\pi }{14}\right )}{\sqrt [7]{a}}-\tan \left (\frac{\pi }{14}\right )\right ) \cos \left (\frac{\pi }{14}\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac{2 \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 ) \cos \left (\frac{3 \pi }{14}\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{\cos \left (\frac{\pi }{7}\right ) \log \left (a^{2/7}+b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} x \cos \left (\frac{\pi }{7}\right )\right )}{7 a^{6/7} \sqrt [7]{b}}-\frac{\log \left (a^{2/7}+b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac{\pi }{14}\right )\right ) \sin \left (\frac{\pi }{14}\right )}{7 a^{6/7} \sqrt [7]{b}}-\frac{2 \tan ^{-1}\left (\cot \left (\frac{\pi }{7}\right )-\frac{\sqrt [7]{b} x \csc \left (\frac{\pi }{7}\right )}{\sqrt [7]{a}}\right ) \sin \left (\frac{\pi }{7}\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac{\log \left (a^{2/7}+b^{2/7} x^2+2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac{3 \pi }{14}\right )\right ) \sin \left (\frac{3 \pi }{14}\right )}{7 a^{6/7} \sqrt [7]{b}}\\ \end{align*}
Mathematica [A] time = 0.234218, size = 262, normalized size = 0.78 \[ \frac{\sin \left (\frac{3 \pi }{14}\right ) \log \left (a^{2/7}+2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac{3 \pi }{14}\right )+b^{2/7} x^2\right )-\sin \left (\frac{\pi }{14}\right ) \log \left (a^{2/7}-2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac{\pi }{14}\right )+b^{2/7} x^2\right )-\cos \left (\frac{\pi }{7}\right ) \log \left (a^{2/7}-2 \sqrt [7]{a} \sqrt [7]{b} x \cos \left (\frac{\pi }{7}\right )+b^{2/7} x^2\right )+\log \left (\sqrt [7]{a}+\sqrt [7]{b} x\right )+2 \cos \left (\frac{3 \pi }{14}\right ) \tan ^{-1}\left (\frac{\sqrt [7]{b} x \sec \left (\frac{3 \pi }{14}\right )}{\sqrt [7]{a}}+\tan \left (\frac{3 \pi }{14}\right )\right )+2 \cos \left (\frac{\pi }{14}\right ) \tan ^{-1}\left (\frac{\sqrt [7]{b} x \sec \left (\frac{\pi }{14}\right )}{\sqrt [7]{a}}-\tan \left (\frac{\pi }{14}\right )\right )-2 \sin \left (\frac{\pi }{7}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{7}\right )-\frac{\sqrt [7]{b} x \csc \left (\frac{\pi }{7}\right )}{\sqrt [7]{a}}\right )}{7 a^{6/7} \sqrt [7]{b}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.247, size = 27, normalized size = 0.1 \begin{align*}{\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}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{b x^{7} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.160876, size = 20, normalized size = 0.06 \begin{align*} \operatorname{RootSum}{\left (823543 t^{7} a^{6} b - 1, \left ( t \mapsto t \log{\left (7 t a + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22214, size = 419, normalized size = 1.25 \begin{align*} \frac{\left (-\frac{a}{b}\right )^{\frac{1}{7}} \cos \left (\frac{3}{7} \, \pi \right ) \log \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 ) \log \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 ) \log \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}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{7}} \right |}\right )}{7 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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