3.302 \(\int \frac{(1+x^7)^{2/3}}{x^8} \, dx\)

Optimal. Leaf size=70 \[ -\frac{\left (x^7+1\right )^{2/3}}{7 x^7}+\frac{1}{7} \log \left (1-\sqrt [3]{x^7+1}\right )+\frac{2 \tan ^{-1}\left (\frac{2 \sqrt [3]{x^7+1}+1}{\sqrt{3}}\right )}{7 \sqrt{3}}-\frac{\log (x)}{3} \]

[Out]

-(1 + x^7)^(2/3)/(7*x^7) + (2*ArcTan[(1 + 2*(1 + x^7)^(1/3))/Sqrt[3]])/(7*Sqrt[3]) - Log[x]/3 + Log[1 - (1 + x
^7)^(1/3)]/7

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Rubi [A]  time = 0.0378973, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462, Rules used = {266, 47, 55, 618, 204, 31} \[ -\frac{\left (x^7+1\right )^{2/3}}{7 x^7}+\frac{1}{7} \log \left (1-\sqrt [3]{x^7+1}\right )+\frac{2 \tan ^{-1}\left (\frac{2 \sqrt [3]{x^7+1}+1}{\sqrt{3}}\right )}{7 \sqrt{3}}-\frac{\log (x)}{3} \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^7)^(2/3)/x^8,x]

[Out]

-(1 + x^7)^(2/3)/(7*x^7) + (2*ArcTan[(1 + 2*(1 + x^7)^(1/3))/Sqrt[3]])/(7*Sqrt[3]) - Log[x]/3 + Log[1 - (1 + x
^7)^(1/3)]/7

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 55

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, -Simp[L
og[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/
3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && PosQ
[(b*c - a*d)/b]

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 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{\left (1+x^7\right )^{2/3}}{x^8} \, dx &=\frac{1}{7} \operatorname{Subst}\left (\int \frac{(1+x)^{2/3}}{x^2} \, dx,x,x^7\right )\\ &=-\frac{\left (1+x^7\right )^{2/3}}{7 x^7}+\frac{2}{21} \operatorname{Subst}\left (\int \frac{1}{x \sqrt [3]{1+x}} \, dx,x,x^7\right )\\ &=-\frac{\left (1+x^7\right )^{2/3}}{7 x^7}-\frac{\log (x)}{3}-\frac{1}{7} \operatorname{Subst}\left (\int \frac{1}{1-x} \, dx,x,\sqrt [3]{1+x^7}\right )+\frac{1}{7} \operatorname{Subst}\left (\int \frac{1}{1+x+x^2} \, dx,x,\sqrt [3]{1+x^7}\right )\\ &=-\frac{\left (1+x^7\right )^{2/3}}{7 x^7}-\frac{\log (x)}{3}+\frac{1}{7} \log \left (1-\sqrt [3]{1+x^7}\right )-\frac{2}{7} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1+x^7}\right )\\ &=-\frac{\left (1+x^7\right )^{2/3}}{7 x^7}+\frac{2 \tan ^{-1}\left (\frac{1+2 \sqrt [3]{1+x^7}}{\sqrt{3}}\right )}{7 \sqrt{3}}-\frac{\log (x)}{3}+\frac{1}{7} \log \left (1-\sqrt [3]{1+x^7}\right )\\ \end{align*}

Mathematica [C]  time = 0.0057785, size = 26, normalized size = 0.37 \[ \frac{3}{35} \left (x^7+1\right )^{5/3} \, _2F_1\left (\frac{5}{3},2;\frac{8}{3};x^7+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^7)^(2/3)/x^8,x]

[Out]

(3*(1 + x^7)^(5/3)*Hypergeometric2F1[5/3, 2, 8/3, 1 + x^7])/35

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Maple [C]  time = 0.039, size = 76, normalized size = 1.1 \begin{align*} -{\frac{1}{7\,{x}^{7}} \left ({x}^{7}+1 \right ) ^{{\frac{2}{3}}}}+{\frac{\sqrt{3}\Gamma \left ({\frac{2}{3}} \right ) }{21\,\pi } \left ({\frac{2\,\pi \,\sqrt{3}}{3\,\Gamma \left ( 2/3 \right ) } \left ( -{\frac{\pi \,\sqrt{3}}{6}}-{\frac{3\,\ln \left ( 3 \right ) }{2}}+7\,\ln \left ( x \right ) \right ) }-{\frac{2\,\pi \,\sqrt{3}{x}^{7}}{9\,\Gamma \left ( 2/3 \right ) }{\mbox{$_3$F$_2$}(1,1,{\frac{4}{3}};\,2,2;\,-{x}^{7})}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^7+1)^(2/3)/x^8,x)

[Out]

-1/7*(x^7+1)^(2/3)/x^7+1/21/Pi*3^(1/2)*GAMMA(2/3)*(2/3*(-1/6*Pi*3^(1/2)-3/2*ln(3)+7*ln(x))*Pi*3^(1/2)/GAMMA(2/
3)-2/9*Pi*3^(1/2)/GAMMA(2/3)*x^7*hypergeom([1,1,4/3],[2,2],-x^7))

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Maxima [A]  time = 1.45476, size = 89, normalized size = 1.27 \begin{align*} \frac{2}{21} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \,{\left (x^{7} + 1\right )}^{\frac{1}{3}} + 1\right )}\right ) - \frac{{\left (x^{7} + 1\right )}^{\frac{2}{3}}}{7 \, x^{7}} - \frac{1}{21} \, \log \left ({\left (x^{7} + 1\right )}^{\frac{2}{3}} +{\left (x^{7} + 1\right )}^{\frac{1}{3}} + 1\right ) + \frac{2}{21} \, \log \left ({\left (x^{7} + 1\right )}^{\frac{1}{3}} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^7+1)^(2/3)/x^8,x, algorithm="maxima")

[Out]

2/21*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^7 + 1)^(1/3) + 1)) - 1/7*(x^7 + 1)^(2/3)/x^7 - 1/21*log((x^7 + 1)^(2/3)
+ (x^7 + 1)^(1/3) + 1) + 2/21*log((x^7 + 1)^(1/3) - 1)

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Fricas [A]  time = 1.81303, size = 240, normalized size = 3.43 \begin{align*} \frac{2 \, \sqrt{3} x^{7} \arctan \left (\frac{2}{3} \, \sqrt{3}{\left (x^{7} + 1\right )}^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right ) - x^{7} \log \left ({\left (x^{7} + 1\right )}^{\frac{2}{3}} +{\left (x^{7} + 1\right )}^{\frac{1}{3}} + 1\right ) + 2 \, x^{7} \log \left ({\left (x^{7} + 1\right )}^{\frac{1}{3}} - 1\right ) - 3 \,{\left (x^{7} + 1\right )}^{\frac{2}{3}}}{21 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^7+1)^(2/3)/x^8,x, algorithm="fricas")

[Out]

1/21*(2*sqrt(3)*x^7*arctan(2/3*sqrt(3)*(x^7 + 1)^(1/3) + 1/3*sqrt(3)) - x^7*log((x^7 + 1)^(2/3) + (x^7 + 1)^(1
/3) + 1) + 2*x^7*log((x^7 + 1)^(1/3) - 1) - 3*(x^7 + 1)^(2/3))/x^7

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Sympy [C]  time = 1.89561, size = 34, normalized size = 0.49 \begin{align*} - \frac{\Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{e^{i \pi }}{x^{7}}} \right )}}{7 x^{\frac{7}{3}} \Gamma \left (\frac{4}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**7+1)**(2/3)/x**8,x)

[Out]

-gamma(1/3)*hyper((-2/3, 1/3), (4/3,), exp_polar(I*pi)/x**7)/(7*x**(7/3)*gamma(4/3))

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Giac [A]  time = 1.09018, size = 90, normalized size = 1.29 \begin{align*} \frac{2}{21} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \,{\left (x^{7} + 1\right )}^{\frac{1}{3}} + 1\right )}\right ) - \frac{{\left (x^{7} + 1\right )}^{\frac{2}{3}}}{7 \, x^{7}} - \frac{1}{21} \, \log \left ({\left (x^{7} + 1\right )}^{\frac{2}{3}} +{\left (x^{7} + 1\right )}^{\frac{1}{3}} + 1\right ) + \frac{2}{21} \, \log \left ({\left |{\left (x^{7} + 1\right )}^{\frac{1}{3}} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^7+1)^(2/3)/x^8,x, algorithm="giac")

[Out]

2/21*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^7 + 1)^(1/3) + 1)) - 1/7*(x^7 + 1)^(2/3)/x^7 - 1/21*log((x^7 + 1)^(2/3)
+ (x^7 + 1)^(1/3) + 1) + 2/21*log(abs((x^7 + 1)^(1/3) - 1))