3.15.47 \(\int \frac {1}{a-b x^7} \, dx\) [1447]

Optimal. Leaf size=335 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt [7]{b} x \sec \left (\frac {\pi }{14}\right )}{\sqrt [7]{a}}+\tan \left (\frac {\pi }{14}\right )\right ) \cos \left (\frac {\pi }{14}\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt [7]{b} x \sec \left (\frac {3 \pi }{14}\right )}{\sqrt [7]{a}}-\tan \left (\frac {3 \pi }{14}\right )\right ) \cos \left (\frac {3 \pi }{14}\right )}{7 a^{6/7} \sqrt [7]{b}}-\frac {\log \left (\sqrt [7]{a}-\sqrt [7]{b} x\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac {\cos \left (\frac {\pi }{7}\right ) \log \left (a^{2/7}+b^{2/7} x^2+2 \sqrt [7]{a} \sqrt [7]{b} x \cos \left (\frac {\pi }{7}\right )\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac {\log \left (a^{2/7}+b^{2/7} x^2+2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac {\pi }{14}\right )\right ) \sin \left (\frac {\pi }{14}\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac {2 \tan ^{-1}\left (\cot \left (\frac {\pi }{7}\right )+\frac {\sqrt [7]{b} x \csc \left (\frac {\pi }{7}\right )}{\sqrt [7]{a}}\right ) \sin \left (\frac {\pi }{7}\right )}{7 a^{6/7} \sqrt [7]{b}}-\frac {\log \left (a^{2/7}+b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac {3 \pi }{14}\right )\right ) \sin \left (\frac {3 \pi }{14}\right )}{7 a^{6/7} \sqrt [7]{b}} \]

[Out]

2/7*arctan(b^(1/7)*x*sec(1/14*Pi)/a^(1/7)+tan(1/14*Pi))*cos(1/14*Pi)/a^(6/7)/b^(1/7)+2/7*arctan(b^(1/7)*x*sec(
3/14*Pi)/a^(1/7)-tan(3/14*Pi))*cos(3/14*Pi)/a^(6/7)/b^(1/7)-1/7*ln(a^(1/7)-b^(1/7)*x)/a^(6/7)/b^(1/7)+1/7*cos(
1/7*Pi)*ln(a^(2/7)+b^(2/7)*x^2+2*a^(1/7)*b^(1/7)*x*cos(1/7*Pi))/a^(6/7)/b^(1/7)+1/7*ln(a^(2/7)+b^(2/7)*x^2+2*a
^(1/7)*b^(1/7)*x*sin(1/14*Pi))*sin(1/14*Pi)/a^(6/7)/b^(1/7)+2/7*arctan(cot(1/7*Pi)+b^(1/7)*x*csc(1/7*Pi)/a^(1/
7))*sin(1/7*Pi)/a^(6/7)/b^(1/7)-1/7*ln(a^(2/7)+b^(2/7)*x^2-2*a^(1/7)*b^(1/7)*x*sin(3/14*Pi))*sin(3/14*Pi)/a^(6
/7)/b^(1/7)

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Rubi [A]
time = 0.31, 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 = {208, 648, 632, 210, 642, 31} \begin {gather*} \frac {2 \cos \left (\frac {3 \pi }{14}\right ) \text {ArcTan}\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 ) \text {ArcTan}\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 ) \text {ArcTan}\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}}-\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}} \end {gather*}

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*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 31

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

Rule 208

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/(a*n))*Int[1/(r - s*x), x] + Dist[2*(r/(a*n)), Sum[u, {k, 1, (n - 1)/2}], x], x]] /; FreeQ[{a, b}, x] && IGt
Q[(n - 3)/2, 0] && NegQ[a/b]

Rule 210

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

Rule 632

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 642

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 648

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]

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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.16, size = 263, normalized size = 0.79 \begin {gather*} \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 )+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 )-\log \left (\sqrt [7]{a}-\sqrt [7]{b} x\right )+\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 )+\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 )+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 )-\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 {gather*}

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] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.17, size = 29, normalized size = 0.09

method	result	size
default	\(-\frac {\munderset {\textit {\_R} =\RootOf \left (b \,\textit {\_Z}^{7}-a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{6}}}{7 b}\)	\(29\)
risch	\(-\frac {\munderset {\textit {\_R} =\RootOf \left (b \,\textit {\_Z}^{7}-a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{6}}}{7 b}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-b*x^7+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> no explicit roots found

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Sympy [A]
time = 0.07, size = 22, normalized size = 0.07 \begin {gather*} - \operatorname {RootSum} {\left (823543 t^{7} a^{6} b - 1, \left ( t \mapsto t \log {\left (- 7 t a + x \right )} \right )\right )} \end {gather*}

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 = 1.42, size = 290, normalized size = 0.87 \begin {gather*} \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 {gather*}

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

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Mupad [B]
time = 1.89, size = 246, normalized size = 0.73 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a - b*x^7),x)

[Out]

log(a^(1/7)*(-b)^(41/7) + b^6*x)/(7*a^(6/7)*(-b)^(1/7)) - (exp((pi*1i)/7)*log(b^6*x - a^(1/7)*(-b)^(41/7)*exp(
(pi*1i)/7)))/(7*a^(6/7)*(-b)^(1/7)) + (exp((pi*2i)/7)*log(b^6*x + a^(1/7)*(-b)^(41/7)*exp((pi*2i)/7)))/(7*a^(6
/7)*(-b)^(1/7)) - (exp((pi*3i)/7)*log(b^6*x - a^(1/7)*(-b)^(41/7)*exp((pi*3i)/7)))/(7*a^(6/7)*(-b)^(1/7)) + (e
xp((pi*4i)/7)*log(b^6*x + a^(1/7)*(-b)^(41/7)*exp((pi*4i)/7)))/(7*a^(6/7)*(-b)^(1/7)) - (exp((pi*5i)/7)*log(b^
6*x - a^(1/7)*(-b)^(41/7)*exp((pi*5i)/7)))/(7*a^(6/7)*(-b)^(1/7)) + (exp((pi*6i)/7)*log(b^6*x + a^(1/7)*(-b)^(
41/7)*exp((pi*6i)/7)))/(7*a^(6/7)*(-b)^(1/7))

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