Optimal. Leaf size=335 \[ -\frac {\sin \left (\frac {3 \pi }{14}\right ) \log \left (a^{2/7}-2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac {3 \pi }{14}\right )+b^{2/7} x^2\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac {\sin \left (\frac {\pi }{14}\right ) \log \left (a^{2/7}+2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac {\pi }{14}\right )+b^{2/7} x^2\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac {\cos \left (\frac {\pi }{7}\right ) \log \left (a^{2/7}+2 \sqrt [7]{a} \sqrt [7]{b} x \cos \left (\frac {\pi }{7}\right )+b^{2/7} x^2\right )}{7 a^{6/7} \sqrt [7]{b}}-\frac {\log \left (\sqrt [7]{a}-\sqrt [7]{b} x\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac {2 \cos \left (\frac {3 \pi }{14}\right ) \tan ^{-1}\left (\frac {\sqrt [7]{b} x \sec \left (\frac {3 \pi }{14}\right )}{\sqrt [7]{a}}-\tan \left (\frac {3 \pi }{14}\right )\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac {2 \cos \left (\frac {\pi }{14}\right ) \tan ^{-1}\left (\frac {\sqrt [7]{b} x \sec \left (\frac {\pi }{14}\right )}{\sqrt [7]{a}}+\tan \left (\frac {\pi }{14}\right )\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac {2 \sin \left (\frac {\pi }{7}\right ) \tan ^{-1}\left (\frac {\sqrt [7]{b} x \csc \left (\frac {\pi }{7}\right )}{\sqrt [7]{a}}+\cot \left (\frac {\pi }{7}\right )\right )}{7 a^{6/7} \sqrt [7]{b}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.44, antiderivative size = 335, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {202, 634, 618, 204, 628, 31} \[ -\frac {\sin \left (\frac {3 \pi }{14}\right ) \log \left (a^{2/7}-2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac {3 \pi }{14}\right )+b^{2/7} x^2\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac {\sin \left (\frac {\pi }{14}\right ) \log \left (a^{2/7}+2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac {\pi }{14}\right )+b^{2/7} x^2\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac {\cos \left (\frac {\pi }{7}\right ) \log \left (a^{2/7}+2 \sqrt [7]{a} \sqrt [7]{b} x \cos \left (\frac {\pi }{7}\right )+b^{2/7} x^2\right )}{7 a^{6/7} \sqrt [7]{b}}-\frac {\log \left (\sqrt [7]{a}-\sqrt [7]{b} x\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac {2 \cos \left (\frac {3 \pi }{14}\right ) \tan ^{-1}\left (\frac {\sqrt [7]{b} x \sec \left (\frac {3 \pi }{14}\right )}{\sqrt [7]{a}}-\tan \left (\frac {3 \pi }{14}\right )\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac {2 \cos \left (\frac {\pi }{14}\right ) \tan ^{-1}\left (\frac {\sqrt [7]{b} x \sec \left (\frac {\pi }{14}\right )}{\sqrt [7]{a}}+\tan \left (\frac {\pi }{14}\right )\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac {2 \sin \left (\frac {\pi }{7}\right ) \tan ^{-1}\left (\frac {\sqrt [7]{b} x \csc \left (\frac {\pi }{7}\right )}{\sqrt [7]{a}}+\cot \left (\frac {\pi }{7}\right )\right )}{7 a^{6/7} \sqrt [7]{b}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 202
Rule 204
Rule 618
Rule 628
Rule 634
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 = 0.30, size = 263, normalized size = 0.79 \[ \frac {-\sin \left (\frac {3 \pi }{14}\right ) \log \left (a^{2/7}-2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac {3 \pi }{14}\right )+b^{2/7} x^2\right )+\sin \left (\frac {\pi }{14}\right ) \log \left (a^{2/7}+2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac {\pi }{14}\right )+b^{2/7} x^2\right )+\cos \left (\frac {\pi }{7}\right ) \log \left (a^{2/7}+2 \sqrt [7]{a} \sqrt [7]{b} x \cos \left (\frac {\pi }{7}\right )+b^{2/7} x^2\right )-\log \left (\sqrt [7]{a}-\sqrt [7]{b} x\right )+2 \cos \left (\frac {3 \pi }{14}\right ) \tan ^{-1}\left (\frac {\sqrt [7]{b} x \sec \left (\frac {3 \pi }{14}\right )}{\sqrt [7]{a}}-\tan \left (\frac {3 \pi }{14}\right )\right )+2 \cos \left (\frac {\pi }{14}\right ) \tan ^{-1}\left (\frac {\sqrt [7]{b} x \sec \left (\frac {\pi }{14}\right )}{\sqrt [7]{a}}+\tan \left (\frac {\pi }{14}\right )\right )+2 \sin \left (\frac {\pi }{7}\right ) \tan ^{-1}\left (\frac {\sqrt [7]{b} x \csc \left (\frac {\pi }{7}\right )}{\sqrt [7]{a}}+\cot \left (\frac {\pi }{7}\right )\right )}{7 a^{6/7} \sqrt [7]{b}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 290, normalized size = 0.87 \[ \frac {\left (\frac {a}{b}\right )^{\frac {1}{7}} \cos \left (\frac {3}{7} \, \pi \right ) \log \left (2 \, x \left (\frac {a}{b}\right )^{\frac {1}{7}} \cos \left (\frac {3}{7} \, \pi \right ) + x^{2} + \left (\frac {a}{b}\right )^{\frac {2}{7}}\right )}{7 \, a} - \frac {\left (\frac {a}{b}\right )^{\frac {1}{7}} \cos \left (\frac {2}{7} \, \pi \right ) \log \left (-2 \, x \left (\frac {a}{b}\right )^{\frac {1}{7}} \cos \left (\frac {2}{7} \, \pi \right ) + x^{2} + \left (\frac {a}{b}\right )^{\frac {2}{7}}\right )}{7 \, a} + \frac {\left (\frac {a}{b}\right )^{\frac {1}{7}} \cos \left (\frac {1}{7} \, \pi \right ) \log \left (2 \, x \left (\frac {a}{b}\right )^{\frac {1}{7}} \cos \left (\frac {1}{7} \, \pi \right ) + x^{2} + \left (\frac {a}{b}\right )^{\frac {2}{7}}\right )}{7 \, a} + \frac {2 \, \left (\frac {a}{b}\right )^{\frac {1}{7}} \arctan \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{7}} \cos \left (\frac {3}{7} \, \pi \right ) + x}{\left (\frac {a}{b}\right )^{\frac {1}{7}} \sin \left (\frac {3}{7} \, \pi \right )}\right ) \sin \left (\frac {3}{7} \, \pi \right )}{7 \, a} + \frac {2 \, \left (\frac {a}{b}\right )^{\frac {1}{7}} \arctan \left (-\frac {\left (\frac {a}{b}\right )^{\frac {1}{7}} \cos \left (\frac {2}{7} \, \pi \right ) - x}{\left (\frac {a}{b}\right )^{\frac {1}{7}} \sin \left (\frac {2}{7} \, \pi \right )}\right ) \sin \left (\frac {2}{7} \, \pi \right )}{7 \, a} + \frac {2 \, \left (\frac {a}{b}\right )^{\frac {1}{7}} \arctan \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{7}} \cos \left (\frac {1}{7} \, \pi \right ) + x}{\left (\frac {a}{b}\right )^{\frac {1}{7}} \sin \left (\frac {1}{7} \, \pi \right )}\right ) \sin \left (\frac {1}{7} \, \pi \right )}{7 \, a} - \frac {\left (\frac {a}{b}\right )^{\frac {1}{7}} \log \left ({\left | x - \left (\frac {a}{b}\right )^{\frac {1}{7}} \right |}\right )}{7 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.06, size = 29, normalized size = 0.09 \[ -\frac {\ln \left (-\RootOf \left (b \,\textit {\_Z}^{7}-a \right )+x \right )}{7 b \RootOf \left (b \,\textit {\_Z}^{7}-a \right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {1}{b x^{7} - a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.89, size = 246, normalized size = 0.73 \[ \frac {\ln \left (a^{1/7}\,{\left (-b\right )}^{41/7}+b^6\,x\right )}{7\,a^{6/7}\,{\left (-b\right )}^{1/7}}-\frac {{\mathrm {e}}^{\frac {\pi \,1{}\mathrm {i}}{7}}\,\ln \left (b^6\,x-a^{1/7}\,{\left (-b\right )}^{41/7}\,{\mathrm {e}}^{\frac {\pi \,1{}\mathrm {i}}{7}}\right )}{7\,a^{6/7}\,{\left (-b\right )}^{1/7}}+\frac {{\mathrm {e}}^{\frac {\pi \,2{}\mathrm {i}}{7}}\,\ln \left (b^6\,x+a^{1/7}\,{\left (-b\right )}^{41/7}\,{\mathrm {e}}^{\frac {\pi \,2{}\mathrm {i}}{7}}\right )}{7\,a^{6/7}\,{\left (-b\right )}^{1/7}}-\frac {{\mathrm {e}}^{\frac {\pi \,3{}\mathrm {i}}{7}}\,\ln \left (b^6\,x-a^{1/7}\,{\left (-b\right )}^{41/7}\,{\mathrm {e}}^{\frac {\pi \,3{}\mathrm {i}}{7}}\right )}{7\,a^{6/7}\,{\left (-b\right )}^{1/7}}+\frac {{\mathrm {e}}^{\frac {\pi \,4{}\mathrm {i}}{7}}\,\ln \left (b^6\,x+a^{1/7}\,{\left (-b\right )}^{41/7}\,{\mathrm {e}}^{\frac {\pi \,4{}\mathrm {i}}{7}}\right )}{7\,a^{6/7}\,{\left (-b\right )}^{1/7}}-\frac {{\mathrm {e}}^{\frac {\pi \,5{}\mathrm {i}}{7}}\,\ln \left (b^6\,x-a^{1/7}\,{\left (-b\right )}^{41/7}\,{\mathrm {e}}^{\frac {\pi \,5{}\mathrm {i}}{7}}\right )}{7\,a^{6/7}\,{\left (-b\right )}^{1/7}}+\frac {{\mathrm {e}}^{\frac {\pi \,6{}\mathrm {i}}{7}}\,\ln \left (b^6\,x+a^{1/7}\,{\left (-b\right )}^{41/7}\,{\mathrm {e}}^{\frac {\pi \,6{}\mathrm {i}}{7}}\right )}{7\,a^{6/7}\,{\left (-b\right )}^{1/7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.22, size = 22, normalized size = 0.07 \[ - \operatorname {RootSum} {\left (823543 t^{7} a^{6} b - 1, \left (t \mapsto t \log {\left (- 7 t a + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________