3.1443 \(\int \frac {1}{a+b x^7} \, dx\)

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

[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 = 0.54, antiderivative size = 335, normalized size of antiderivative = 1.00, 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.

[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 201

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] && PosQ[a/b]

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 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 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 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]

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*}

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Mathematica [A]  time = 0.39, 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.

[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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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]

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

[Out]

Exception raised: RuntimeError >> no explicit roots found

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giac [A]  time = 0.25, size = 310, normalized size = 0.93 \[ \frac {\left (-\frac {a}{b}\right )^{\frac {1}{7}} \cos \left (\frac {3}{7} \, \pi \right ) \log \left (2 \, x \left (-\frac {a}{b}\right )^{\frac {1}{7}} \cos \left (\frac {3}{7} \, \pi \right ) + x^{2} + \left (-\frac {a}{b}\right )^{\frac {2}{7}}\right )}{7 \, a} - \frac {\left (-\frac {a}{b}\right )^{\frac {1}{7}} \cos \left (\frac {2}{7} \, \pi \right ) \log \left (-2 \, x \left (-\frac {a}{b}\right )^{\frac {1}{7}} \cos \left (\frac {2}{7} \, \pi \right ) + x^{2} + \left (-\frac {a}{b}\right )^{\frac {2}{7}}\right )}{7 \, a} + \frac {\left (-\frac {a}{b}\right )^{\frac {1}{7}} \cos \left (\frac {1}{7} \, \pi \right ) \log \left (2 \, x \left (-\frac {a}{b}\right )^{\frac {1}{7}} \cos \left (\frac {1}{7} \, \pi \right ) + x^{2} + \left (-\frac {a}{b}\right )^{\frac {2}{7}}\right )}{7 \, a} + \frac {2 \, \left (-\frac {a}{b}\right )^{\frac {1}{7}} \arctan \left (\frac {\left (-\frac {a}{b}\right )^{\frac {1}{7}} \cos \left (\frac {3}{7} \, \pi \right ) + x}{\left (-\frac {a}{b}\right )^{\frac {1}{7}} \sin \left (\frac {3}{7} \, \pi \right )}\right ) \sin \left (\frac {3}{7} \, \pi \right )}{7 \, a} + \frac {2 \, \left (-\frac {a}{b}\right )^{\frac {1}{7}} \arctan \left (-\frac {\left (-\frac {a}{b}\right )^{\frac {1}{7}} \cos \left (\frac {2}{7} \, \pi \right ) - x}{\left (-\frac {a}{b}\right )^{\frac {1}{7}} \sin \left (\frac {2}{7} \, \pi \right )}\right ) \sin \left (\frac {2}{7} \, \pi \right )}{7 \, a} + \frac {2 \, \left (-\frac {a}{b}\right )^{\frac {1}{7}} \arctan \left (\frac {\left (-\frac {a}{b}\right )^{\frac {1}{7}} \cos \left (\frac {1}{7} \, \pi \right ) + x}{\left (-\frac {a}{b}\right )^{\frac {1}{7}} \sin \left (\frac {1}{7} \, \pi \right )}\right ) \sin \left (\frac {1}{7} \, \pi \right )}{7 \, a} - \frac {\left (-\frac {a}{b}\right )^{\frac {1}{7}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{7}} \right |}\right )}{7 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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*pi)*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)*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)*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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maple [C]  time = 0.44, size = 27, normalized size = 0.08 \[ \frac {\ln \left (-\RootOf \left (b \,\textit {\_Z}^{7}+a \right )+x \right )}{7 b \RootOf \left (b \,\textit {\_Z}^{7}+a \right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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 = 0.00, 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]

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

[Out]

integrate(1/(b*x^7 + a), x)

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mupad [B]  time = 1.85, size = 196, normalized size = 0.59 \[ \frac {\ln \left (b^{1/7}\,x+a^{1/7}\right )}{7\,a^{6/7}\,b^{1/7}}-\frac {{\mathrm {e}}^{\frac {\pi \,1{}\mathrm {i}}{7}}\,\ln \left (b^{1/7}\,x-a^{1/7}\,{\mathrm {e}}^{\frac {\pi \,1{}\mathrm {i}}{7}}\right )}{7\,a^{6/7}\,b^{1/7}}+\frac {{\mathrm {e}}^{\frac {\pi \,2{}\mathrm {i}}{7}}\,\ln \left (a^{1/7}\,{\mathrm {e}}^{\frac {\pi \,2{}\mathrm {i}}{7}}+b^{1/7}\,x\right )}{7\,a^{6/7}\,b^{1/7}}-\frac {{\mathrm {e}}^{\frac {\pi \,3{}\mathrm {i}}{7}}\,\ln \left (b^{1/7}\,x-a^{1/7}\,{\mathrm {e}}^{\frac {\pi \,3{}\mathrm {i}}{7}}\right )}{7\,a^{6/7}\,b^{1/7}}+\frac {{\mathrm {e}}^{\frac {\pi \,4{}\mathrm {i}}{7}}\,\ln \left (a^{1/7}\,{\mathrm {e}}^{\frac {\pi \,4{}\mathrm {i}}{7}}+b^{1/7}\,x\right )}{7\,a^{6/7}\,b^{1/7}}-\frac {{\mathrm {e}}^{\frac {\pi \,5{}\mathrm {i}}{7}}\,\ln \left (b^{1/7}\,x-a^{1/7}\,{\mathrm {e}}^{\frac {\pi \,5{}\mathrm {i}}{7}}\right )}{7\,a^{6/7}\,b^{1/7}}+\frac {{\mathrm {e}}^{\frac {\pi \,6{}\mathrm {i}}{7}}\,\ln \left (a^{1/7}\,{\mathrm {e}}^{\frac {\pi \,6{}\mathrm {i}}{7}}+b^{1/7}\,x\right )}{7\,a^{6/7}\,b^{1/7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*x^7),x)

[Out]

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

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sympy [A]  time = 0.27, size = 20, normalized size = 0.06 \[ \operatorname {RootSum} {\left (823543 t^{7} a^{6} b - 1, \left (t \mapsto t \log {\left (7 t a + x \right )} \right )\right )} \]

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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