3.1445 \(\int \frac{1}{a-b x^7} \, dx\)

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]

(2*ArcTan[(b^(1/7)*x*Sec[Pi/14])/a^(1/7) + Tan[Pi/14]]*Cos[Pi/14])/(7*a^(6/7)*b^
(1/7)) + (2*ArcTan[(b^(1/7)*x*Sec[(3*Pi)/14])/a^(1/7) - Tan[(3*Pi)/14]]*Cos[(3*P
i)/14])/(7*a^(6/7)*b^(1/7)) - Log[a^(1/7) - b^(1/7)*x]/(7*a^(6/7)*b^(1/7)) + (Co
s[Pi/7]*Log[a^(2/7) + b^(2/7)*x^2 + 2*a^(1/7)*b^(1/7)*x*Cos[Pi/7]])/(7*a^(6/7)*b
^(1/7)) + (Log[a^(2/7) + b^(2/7)*x^2 + 2*a^(1/7)*b^(1/7)*x*Sin[Pi/14]]*Sin[Pi/14
])/(7*a^(6/7)*b^(1/7)) + (2*ArcTan[Cot[Pi/7] + (b^(1/7)*x*Csc[Pi/7])/a^(1/7)]*Si
n[Pi/7])/(7*a^(6/7)*b^(1/7)) - (Log[a^(2/7) + b^(2/7)*x^2 - 2*a^(1/7)*b^(1/7)*x*
Sin[(3*Pi)/14]]*Sin[(3*Pi)/14])/(7*a^(6/7)*b^(1/7))

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Rubi [A]  time = 0.80842, antiderivative size = 335, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6 \[ -\frac{\sin \left (\frac{3 \pi }{14}\right ) \log \left (a^{2/7}-2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac{3 \pi }{14}\right )+b^{2/7} x^2\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac{\sin \left (\frac{\pi }{14}\right ) \log \left (a^{2/7}+2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac{\pi }{14}\right )+b^{2/7} x^2\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac{\cos \left (\frac{\pi }{7}\right ) \log \left (a^{2/7}+2 \sqrt [7]{a} \sqrt [7]{b} x \cos \left (\frac{\pi }{7}\right )+b^{2/7} x^2\right )}{7 a^{6/7} \sqrt [7]{b}}-\frac{\log \left (\sqrt [7]{a}-\sqrt [7]{b} x\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac{2 \cos \left (\frac{3 \pi }{14}\right ) \tan ^{-1}\left (\frac{\sqrt [7]{b} x \sec \left (\frac{3 \pi }{14}\right )}{\sqrt [7]{a}}-\tan \left (\frac{3 \pi }{14}\right )\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac{2 \cos \left (\frac{\pi }{14}\right ) \tan ^{-1}\left (\frac{\sqrt [7]{b} x \sec \left (\frac{\pi }{14}\right )}{\sqrt [7]{a}}+\tan \left (\frac{\pi }{14}\right )\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac{2 \sin \left (\frac{\pi }{7}\right ) \tan ^{-1}\left (\frac{\sqrt [7]{b} x \csc \left (\frac{\pi }{7}\right )}{\sqrt [7]{a}}+\cot \left (\frac{\pi }{7}\right )\right )}{7 a^{6/7} \sqrt [7]{b}} \]

Warning: Unable to verify antiderivative.

[In]  Int[(a - b*x^7)^(-1),x]

[Out]

(2*ArcTan[(b^(1/7)*x*Sec[Pi/14])/a^(1/7) + Tan[Pi/14]]*Cos[Pi/14])/(7*a^(6/7)*b^
(1/7)) + (2*ArcTan[(b^(1/7)*x*Sec[(3*Pi)/14])/a^(1/7) - Tan[(3*Pi)/14]]*Cos[(3*P
i)/14])/(7*a^(6/7)*b^(1/7)) - Log[a^(1/7) - b^(1/7)*x]/(7*a^(6/7)*b^(1/7)) + (Co
s[Pi/7]*Log[a^(2/7) + b^(2/7)*x^2 + 2*a^(1/7)*b^(1/7)*x*Cos[Pi/7]])/(7*a^(6/7)*b
^(1/7)) + (Log[a^(2/7) + b^(2/7)*x^2 + 2*a^(1/7)*b^(1/7)*x*Sin[Pi/14]]*Sin[Pi/14
])/(7*a^(6/7)*b^(1/7)) + (2*ArcTan[Cot[Pi/7] + (b^(1/7)*x*Csc[Pi/7])/a^(1/7)]*Si
n[Pi/7])/(7*a^(6/7)*b^(1/7)) - (Log[a^(2/7) + b^(2/7)*x^2 - 2*a^(1/7)*b^(1/7)*x*
Sin[(3*Pi)/14]]*Sin[(3*Pi)/14])/(7*a^(6/7)*b^(1/7))

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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]

Timed out

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Mathematica [A]  time = 0.549048, size = 263, normalized size = 0.79 \[ \frac{-\sin \left (\frac{3 \pi }{14}\right ) \log \left (a^{2/7}-2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac{3 \pi }{14}\right )+b^{2/7} x^2\right )+\sin \left (\frac{\pi }{14}\right ) \log \left (a^{2/7}+2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac{\pi }{14}\right )+b^{2/7} x^2\right )+\cos \left (\frac{\pi }{7}\right ) \log \left (a^{2/7}+2 \sqrt [7]{a} \sqrt [7]{b} x \cos \left (\frac{\pi }{7}\right )+b^{2/7} x^2\right )-\log \left (\sqrt [7]{a}-\sqrt [7]{b} x\right )+2 \cos \left (\frac{3 \pi }{14}\right ) \tan ^{-1}\left (\frac{\sqrt [7]{b} x \sec \left (\frac{3 \pi }{14}\right )}{\sqrt [7]{a}}-\tan \left (\frac{3 \pi }{14}\right )\right )+2 \cos \left (\frac{\pi }{14}\right ) \tan ^{-1}\left (\frac{\sqrt [7]{b} x \sec \left (\frac{\pi }{14}\right )}{\sqrt [7]{a}}+\tan \left (\frac{\pi }{14}\right )\right )+2 \sin \left (\frac{\pi }{7}\right ) \tan ^{-1}\left (\frac{\sqrt [7]{b} x \csc \left (\frac{\pi }{7}\right )}{\sqrt [7]{a}}+\cot \left (\frac{\pi }{7}\right )\right )}{7 a^{6/7} \sqrt [7]{b}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a - b*x^7)^(-1),x]

[Out]

(2*ArcTan[(b^(1/7)*x*Sec[Pi/14])/a^(1/7) + Tan[Pi/14]]*Cos[Pi/14] + 2*ArcTan[(b^
(1/7)*x*Sec[(3*Pi)/14])/a^(1/7) - Tan[(3*Pi)/14]]*Cos[(3*Pi)/14] - Log[a^(1/7) -
b^(1/7)*x] + Cos[Pi/7]*Log[a^(2/7) + b^(2/7)*x^2 + 2*a^(1/7)*b^(1/7)*x*Cos[Pi/7
]] + Log[a^(2/7) + b^(2/7)*x^2 + 2*a^(1/7)*b^(1/7)*x*Sin[Pi/14]]*Sin[Pi/14] + 2*
ArcTan[Cot[Pi/7] + (b^(1/7)*x*Csc[Pi/7])/a^(1/7)]*Sin[Pi/7] - Log[a^(2/7) + b^(2
/7)*x^2 - 2*a^(1/7)*b^(1/7)*x*Sin[(3*Pi)/14]]*Sin[(3*Pi)/14])/(7*a^(6/7)*b^(1/7)
)

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Maple [C]  time = 0.156, size = 29, 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]

-1/7/b*sum(1/_R^6*ln(x-_R),_R=RootOf(_Z^7*b-a))

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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]

-integrate(1/(b*x^7 - a), x)

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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]

Exception raised: NotImplementedError

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Sympy [A]  time = 0.476669, 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]  integrate(1/(-b*x**7+a),x)

[Out]

-RootSum(823543*_t**7*a**6*b - 1, Lambda(_t, _t*log(-7*_t*a + x)))

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GIAC/XCAS [A]  time = 0.227717, size = 392, normalized size = 1.17 \[ \frac{\left (\frac{a}{b}\right )^{\frac{1}{7}} \cos \left (\frac{3}{7} \, \pi \right ){\rm ln}\left (2 \, x \left (\frac{a}{b}\right )^{\frac{1}{7}} \cos \left (\frac{3}{7} \, \pi \right ) + x^{2} + \left (\frac{a}{b}\right )^{\frac{2}{7}}\right )}{7 \, a} - \frac{\left (\frac{a}{b}\right )^{\frac{1}{7}} \cos \left (\frac{2}{7} \, \pi \right ){\rm ln}\left (-2 \, x \left (\frac{a}{b}\right )^{\frac{1}{7}} \cos \left (\frac{2}{7} \, \pi \right ) + x^{2} + \left (\frac{a}{b}\right )^{\frac{2}{7}}\right )}{7 \, a} + \frac{\left (\frac{a}{b}\right )^{\frac{1}{7}} \cos \left (\frac{1}{7} \, \pi \right ){\rm ln}\left (2 \, x \left (\frac{a}{b}\right )^{\frac{1}{7}} \cos \left (\frac{1}{7} \, \pi \right ) + x^{2} + \left (\frac{a}{b}\right )^{\frac{2}{7}}\right )}{7 \, a} + \frac{2 \, \left (\frac{a}{b}\right )^{\frac{1}{7}} \arctan \left (\frac{\left (\frac{a}{b}\right )^{\frac{1}{7}} \cos \left (\frac{3}{7} \, \pi \right ) + x}{\left (\frac{a}{b}\right )^{\frac{1}{7}} \sin \left (\frac{3}{7} \, \pi \right )}\right ) \sin \left (\frac{3}{7} \, \pi \right )}{7 \, a} + \frac{2 \, \left (\frac{a}{b}\right )^{\frac{1}{7}} \arctan \left (-\frac{\left (\frac{a}{b}\right )^{\frac{1}{7}} \cos \left (\frac{2}{7} \, \pi \right ) - x}{\left (\frac{a}{b}\right )^{\frac{1}{7}} \sin \left (\frac{2}{7} \, \pi \right )}\right ) \sin \left (\frac{2}{7} \, \pi \right )}{7 \, a} + \frac{2 \, \left (\frac{a}{b}\right )^{\frac{1}{7}} \arctan \left (\frac{\left (\frac{a}{b}\right )^{\frac{1}{7}} \cos \left (\frac{1}{7} \, \pi \right ) + x}{\left (\frac{a}{b}\right )^{\frac{1}{7}} \sin \left (\frac{1}{7} \, \pi \right )}\right ) \sin \left (\frac{1}{7} \, \pi \right )}{7 \, a} - \frac{\left (\frac{a}{b}\right )^{\frac{1}{7}}{\rm ln}\left ({\left | x - \left (\frac{a}{b}\right )^{\frac{1}{7}} \right |}\right )}{7 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(b*x^7 - a),x, algorithm="giac")

[Out]

1/7*(a/b)^(1/7)*cos(3/7*pi)*ln(2*x*(a/b)^(1/7)*cos(3/7*pi) + x^2 + (a/b)^(2/7))/
a - 1/7*(a/b)^(1/7)*cos(2/7*pi)*ln(-2*x*(a/b)^(1/7)*cos(2/7*pi) + x^2 + (a/b)^(2
/7))/a + 1/7*(a/b)^(1/7)*cos(1/7*pi)*ln(2*x*(a/b)^(1/7)*cos(1/7*pi) + x^2 + (a/b
)^(2/7))/a + 2/7*(a/b)^(1/7)*arctan(((a/b)^(1/7)*cos(3/7*pi) + x)/((a/b)^(1/7)*s
in(3/7*pi)))*sin(3/7*pi)/a + 2/7*(a/b)^(1/7)*arctan(-((a/b)^(1/7)*cos(2/7*pi) -
x)/((a/b)^(1/7)*sin(2/7*pi)))*sin(2/7*pi)/a + 2/7*(a/b)^(1/7)*arctan(((a/b)^(1/7
)*cos(1/7*pi) + x)/((a/b)^(1/7)*sin(1/7*pi)))*sin(1/7*pi)/a - 1/7*(a/b)^(1/7)*ln
(abs(x - (a/b)^(1/7)))/a