3.1447 \(\int \frac{1}{a-b x^7} \, dx\)

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*Pi)/14])/(7*a^(6/7)*b^(1/7)) - Log[a^(1/7) - b^(1/7)*x]/(7*a^
(6/7)*b^(1/7)) + (Cos[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[P
i/7] + (b^(1/7)*x*Csc[Pi/7])/a^(1/7)]*Sin[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 = 0.437613, 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, 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]

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*Pi)/14])/(7*a^(6/7)*b^(1/7)) - Log[a^(1/7) - b^(1/7)*x]/(7*a^
(6/7)*b^(1/7)) + (Cos[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[P
i/7] + (b^(1/7)*x*Csc[Pi/7])/a^(1/7)]*Sin[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))

Rule 202

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[-(a/b), n]], s = Denominator[Rt[-(a/b
), n]], k, u}, Simp[u = Int[(r + s*Cos[((2*k - 1)*Pi)/n]*x)/(r^2 + 2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x
]; (r*Int[1/(r - s*x), x])/(a*n) + Dist[(2*r)/(a*n), Sum[u, {k, 1, (n - 1)/2}], x], x]] /; FreeQ[{a, b}, x] &&
IGtQ[(n - 3)/2, 0] && NegQ[a/b]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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 = 0.211067, 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]

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 = 0.024, size = 29, 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.

[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. \begin{align*} -\int \frac{1}{b x^{7} - a}\,{d x} \end{align*}

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. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-b*x^7+a),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [A]  time = 0.167195, size = 22, normalized size = 0.07 \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.

[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 [A]  time = 1.50956, size = 392, normalized size = 1.17 \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.

[In]

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

[Out]

1/7*(a/b)^(1/7)*cos(3/7*pi)*log(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*p
i)*log(-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)*log(2*x*(a/b)^(1/7)*c
os(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)*sin(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)*log(abs(x - (a/b)^(1/7)))/a