3.15.61 \(\int \frac {x^8}{a+b x^8} \, dx\) [1461]

Optimal. Leaf size=272 \[ \frac {x}{b}-\frac {\sqrt [8]{-a} \tan ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 b^{9/8}}+\frac {\sqrt [8]{-a} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} b^{9/8}}-\frac {\sqrt [8]{-a} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} b^{9/8}}-\frac {\sqrt [8]{-a} \tanh ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 b^{9/8}}+\frac {\sqrt [8]{-a} \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} b^{9/8}}-\frac {\sqrt [8]{-a} \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} b^{9/8}} \]

[Out]

x/b-1/4*(-a)^(1/8)*arctan(b^(1/8)*x/(-a)^(1/8))/b^(9/8)-1/4*(-a)^(1/8)*arctanh(b^(1/8)*x/(-a)^(1/8))/b^(9/8)-1
/8*(-a)^(1/8)*arctan(-1+b^(1/8)*x*2^(1/2)/(-a)^(1/8))/b^(9/8)*2^(1/2)-1/8*(-a)^(1/8)*arctan(1+b^(1/8)*x*2^(1/2
)/(-a)^(1/8))/b^(9/8)*2^(1/2)+1/16*(-a)^(1/8)*ln((-a)^(1/4)+b^(1/4)*x^2-(-a)^(1/8)*b^(1/8)*x*2^(1/2))/b^(9/8)*
2^(1/2)-1/16*(-a)^(1/8)*ln((-a)^(1/4)+b^(1/4)*x^2+(-a)^(1/8)*b^(1/8)*x*2^(1/2))/b^(9/8)*2^(1/2)

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Rubi [A]
time = 0.20, antiderivative size = 272, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.846, Rules used = {327, 220, 218, 214, 211, 217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {\sqrt [8]{-a} \text {ArcTan}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 b^{9/8}}+\frac {\sqrt [8]{-a} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} b^{9/8}}-\frac {\sqrt [8]{-a} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{4 \sqrt {2} b^{9/8}}+\frac {\sqrt [8]{-a} \log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} b^{9/8}}-\frac {\sqrt [8]{-a} \log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} b^{9/8}}-\frac {\sqrt [8]{-a} \tanh ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 b^{9/8}}+\frac {x}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^8/(a + b*x^8),x]

[Out]

x/b - ((-a)^(1/8)*ArcTan[(b^(1/8)*x)/(-a)^(1/8)])/(4*b^(9/8)) + ((-a)^(1/8)*ArcTan[1 - (Sqrt[2]*b^(1/8)*x)/(-a
)^(1/8)])/(4*Sqrt[2]*b^(9/8)) - ((-a)^(1/8)*ArcTan[1 + (Sqrt[2]*b^(1/8)*x)/(-a)^(1/8)])/(4*Sqrt[2]*b^(9/8)) -
((-a)^(1/8)*ArcTanh[(b^(1/8)*x)/(-a)^(1/8)])/(4*b^(9/8)) + ((-a)^(1/8)*Log[(-a)^(1/4) - Sqrt[2]*(-a)^(1/8)*b^(
1/8)*x + b^(1/4)*x^2])/(8*Sqrt[2]*b^(9/8)) - ((-a)^(1/8)*Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*b^(1/8)*x + b^(1/
4)*x^2])/(8*Sqrt[2]*b^(9/8))

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 211

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

Rule 214

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

Rule 217

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/b, 0]

Rule 220

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]
}, Dist[r/(2*a), Int[1/(r - s*x^(n/2)), x], x] + Dist[r/(2*a), Int[1/(r + s*x^(n/2)), x], x]] /; FreeQ[{a, b},
x] && IGtQ[n/4, 1] &&  !GtQ[a/b, 0]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
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 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rubi steps

\begin {align*} \int \frac {x^8}{a+b x^8} \, dx &=\frac {x}{b}-\frac {a \int \frac {1}{a+b x^8} \, dx}{b}\\ &=\frac {x}{b}-\frac {\sqrt {-a} \int \frac {1}{\sqrt {-a}-\sqrt {b} x^4} \, dx}{2 b}-\frac {\sqrt {-a} \int \frac {1}{\sqrt {-a}+\sqrt {b} x^4} \, dx}{2 b}\\ &=\frac {x}{b}-\frac {\sqrt [4]{-a} \int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{b} x^2} \, dx}{4 b}-\frac {\sqrt [4]{-a} \int \frac {1}{\sqrt [4]{-a}+\sqrt [4]{b} x^2} \, dx}{4 b}-\frac {\sqrt [4]{-a} \int \frac {\sqrt [4]{-a}-\sqrt [4]{b} x^2}{\sqrt {-a}+\sqrt {b} x^4} \, dx}{4 b}-\frac {\sqrt [4]{-a} \int \frac {\sqrt [4]{-a}+\sqrt [4]{b} x^2}{\sqrt {-a}+\sqrt {b} x^4} \, dx}{4 b}\\ &=\frac {x}{b}-\frac {\sqrt [8]{-a} \tan ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 b^{9/8}}-\frac {\sqrt [8]{-a} \tanh ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 b^{9/8}}-\frac {\sqrt [4]{-a} \int \frac {1}{\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+x^2} \, dx}{8 b^{5/4}}-\frac {\sqrt [4]{-a} \int \frac {1}{\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+x^2} \, dx}{8 b^{5/4}}+\frac {\sqrt [8]{-a} \int \frac {\frac {\sqrt {2} \sqrt [8]{-a}}{\sqrt [8]{b}}+2 x}{-\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}-x^2} \, dx}{8 \sqrt {2} b^{9/8}}+\frac {\sqrt [8]{-a} \int \frac {\frac {\sqrt {2} \sqrt [8]{-a}}{\sqrt [8]{b}}-2 x}{-\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}-x^2} \, dx}{8 \sqrt {2} b^{9/8}}\\ &=\frac {x}{b}-\frac {\sqrt [8]{-a} \tan ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 b^{9/8}}-\frac {\sqrt [8]{-a} \tanh ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 b^{9/8}}+\frac {\sqrt [8]{-a} \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} b^{9/8}}-\frac {\sqrt [8]{-a} \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} b^{9/8}}-\frac {\sqrt [8]{-a} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} b^{9/8}}+\frac {\sqrt [8]{-a} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} b^{9/8}}\\ &=\frac {x}{b}-\frac {\sqrt [8]{-a} \tan ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 b^{9/8}}+\frac {\sqrt [8]{-a} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} b^{9/8}}-\frac {\sqrt [8]{-a} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} b^{9/8}}-\frac {\sqrt [8]{-a} \tanh ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 b^{9/8}}+\frac {\sqrt [8]{-a} \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} b^{9/8}}-\frac {\sqrt [8]{-a} \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} b^{9/8}}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 367, normalized size = 1.35 \begin {gather*} \frac {8 \sqrt [8]{b} x-2 \sqrt [8]{a} \tan ^{-1}\left (\frac {\sqrt [8]{b} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac {\pi }{8}\right )\right ) \cos \left (\frac {\pi }{8}\right )-2 \sqrt [8]{a} \tan ^{-1}\left (\frac {\sqrt [8]{b} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac {\pi }{8}\right )\right ) \cos \left (\frac {\pi }{8}\right )+\sqrt [8]{a} \cos \left (\frac {\pi }{8}\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right )\right )-\sqrt [8]{a} \cos \left (\frac {\pi }{8}\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2+2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right )\right )+2 \sqrt [8]{a} \tan ^{-1}\left (\cot \left (\frac {\pi }{8}\right )-\frac {\sqrt [8]{b} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}\right ) \sin \left (\frac {\pi }{8}\right )-2 \sqrt [8]{a} \tan ^{-1}\left (\cot \left (\frac {\pi }{8}\right )+\frac {\sqrt [8]{b} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}\right ) \sin \left (\frac {\pi }{8}\right )+\sqrt [8]{a} \log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )-\sqrt [8]{a} \log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2+2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )}{8 b^{9/8}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^8/(a + b*x^8),x]

[Out]

(8*b^(1/8)*x - 2*a^(1/8)*ArcTan[(b^(1/8)*x*Sec[Pi/8])/a^(1/8) - Tan[Pi/8]]*Cos[Pi/8] - 2*a^(1/8)*ArcTan[(b^(1/
8)*x*Sec[Pi/8])/a^(1/8) + Tan[Pi/8]]*Cos[Pi/8] + a^(1/8)*Cos[Pi/8]*Log[a^(1/4) + b^(1/4)*x^2 - 2*a^(1/8)*b^(1/
8)*x*Cos[Pi/8]] - a^(1/8)*Cos[Pi/8]*Log[a^(1/4) + b^(1/4)*x^2 + 2*a^(1/8)*b^(1/8)*x*Cos[Pi/8]] + 2*a^(1/8)*Arc
Tan[Cot[Pi/8] - (b^(1/8)*x*Csc[Pi/8])/a^(1/8)]*Sin[Pi/8] - 2*a^(1/8)*ArcTan[Cot[Pi/8] + (b^(1/8)*x*Csc[Pi/8])/
a^(1/8)]*Sin[Pi/8] + a^(1/8)*Log[a^(1/4) + b^(1/4)*x^2 - 2*a^(1/8)*b^(1/8)*x*Sin[Pi/8]]*Sin[Pi/8] - a^(1/8)*Lo
g[a^(1/4) + b^(1/4)*x^2 + 2*a^(1/8)*b^(1/8)*x*Sin[Pi/8]]*Sin[Pi/8])/(8*b^(9/8))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.18, size = 34, normalized size = 0.12

method	result	size
default	\(\frac {x}{b}-\frac {a \left (\munderset {\textit {\_R} =\RootOf \left (b \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}}\right )}{8 b^{2}}\)	\(34\)
risch	\(\frac {x}{b}-\frac {a \left (\munderset {\textit {\_R} =\RootOf \left (b \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}}\right )}{8 b^{2}}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(b*x^8+a),x,method=_RETURNVERBOSE)

[Out]

x/b-1/8*a/b^2*sum(1/_R^7*ln(x-_R),_R=RootOf(_Z^8*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(x^8/(b*x^8+a),x, algorithm="maxima")

[Out]

-a*integrate(1/(b*x^8 + a), x)/b + x/b

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 385 vs. \(2 (187) = 374\).
time = 0.40, size = 385, normalized size = 1.42 \begin {gather*} -\frac {4 \, \sqrt {2} b \left (-\frac {a}{b^{9}}\right )^{\frac {1}{8}} \arctan \left (-\frac {\sqrt {2} b^{8} x \left (-\frac {a}{b^{9}}\right )^{\frac {7}{8}} - \sqrt {2} \sqrt {\sqrt {2} b x \left (-\frac {a}{b^{9}}\right )^{\frac {1}{8}} + b^{2} \left (-\frac {a}{b^{9}}\right )^{\frac {1}{4}} + x^{2}} b^{8} \left (-\frac {a}{b^{9}}\right )^{\frac {7}{8}} - a}{a}\right ) + 4 \, \sqrt {2} b \left (-\frac {a}{b^{9}}\right )^{\frac {1}{8}} \arctan \left (-\frac {\sqrt {2} b^{8} x \left (-\frac {a}{b^{9}}\right )^{\frac {7}{8}} - \sqrt {2} \sqrt {-\sqrt {2} b x \left (-\frac {a}{b^{9}}\right )^{\frac {1}{8}} + b^{2} \left (-\frac {a}{b^{9}}\right )^{\frac {1}{4}} + x^{2}} b^{8} \left (-\frac {a}{b^{9}}\right )^{\frac {7}{8}} + a}{a}\right ) + \sqrt {2} b \left (-\frac {a}{b^{9}}\right )^{\frac {1}{8}} \log \left (\sqrt {2} b x \left (-\frac {a}{b^{9}}\right )^{\frac {1}{8}} + b^{2} \left (-\frac {a}{b^{9}}\right )^{\frac {1}{4}} + x^{2}\right ) - \sqrt {2} b \left (-\frac {a}{b^{9}}\right )^{\frac {1}{8}} \log \left (-\sqrt {2} b x \left (-\frac {a}{b^{9}}\right )^{\frac {1}{8}} + b^{2} \left (-\frac {a}{b^{9}}\right )^{\frac {1}{4}} + x^{2}\right ) + 8 \, b \left (-\frac {a}{b^{9}}\right )^{\frac {1}{8}} \arctan \left (-\frac {b^{8} x \left (-\frac {a}{b^{9}}\right )^{\frac {7}{8}} - \sqrt {b^{2} \left (-\frac {a}{b^{9}}\right )^{\frac {1}{4}} + x^{2}} b^{8} \left (-\frac {a}{b^{9}}\right )^{\frac {7}{8}}}{a}\right ) + 2 \, b \left (-\frac {a}{b^{9}}\right )^{\frac {1}{8}} \log \left (b \left (-\frac {a}{b^{9}}\right )^{\frac {1}{8}} + x\right ) - 2 \, b \left (-\frac {a}{b^{9}}\right )^{\frac {1}{8}} \log \left (-b \left (-\frac {a}{b^{9}}\right )^{\frac {1}{8}} + x\right ) - 16 \, x}{16 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(b*x^8+a),x, algorithm="fricas")

[Out]

-1/16*(4*sqrt(2)*b*(-a/b^9)^(1/8)*arctan(-(sqrt(2)*b^8*x*(-a/b^9)^(7/8) - sqrt(2)*sqrt(sqrt(2)*b*x*(-a/b^9)^(1
/8) + b^2*(-a/b^9)^(1/4) + x^2)*b^8*(-a/b^9)^(7/8) - a)/a) + 4*sqrt(2)*b*(-a/b^9)^(1/8)*arctan(-(sqrt(2)*b^8*x
*(-a/b^9)^(7/8) - sqrt(2)*sqrt(-sqrt(2)*b*x*(-a/b^9)^(1/8) + b^2*(-a/b^9)^(1/4) + x^2)*b^8*(-a/b^9)^(7/8) + a)
/a) + sqrt(2)*b*(-a/b^9)^(1/8)*log(sqrt(2)*b*x*(-a/b^9)^(1/8) + b^2*(-a/b^9)^(1/4) + x^2) - sqrt(2)*b*(-a/b^9)
^(1/8)*log(-sqrt(2)*b*x*(-a/b^9)^(1/8) + b^2*(-a/b^9)^(1/4) + x^2) + 8*b*(-a/b^9)^(1/8)*arctan(-(b^8*x*(-a/b^9
)^(7/8) - sqrt(b^2*(-a/b^9)^(1/4) + x^2)*b^8*(-a/b^9)^(7/8))/a) + 2*b*(-a/b^9)^(1/8)*log(b*(-a/b^9)^(1/8) + x)
- 2*b*(-a/b^9)^(1/8)*log(-b*(-a/b^9)^(1/8) + x) - 16*x)/b

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Sympy [A]
time = 0.08, size = 22, normalized size = 0.08 \begin {gather*} \operatorname {RootSum} {\left (16777216 t^{8} b^{9} + a, \left ( t \mapsto t \log {\left (- 8 t b + x \right )} \right )\right )} + \frac {x}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8/(b*x**8+a),x)

[Out]

RootSum(16777216*_t**8*b**9 + a, Lambda(_t, _t*log(-8*_t*b + x))) + x/b

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 442 vs. \(2 (187) = 374\).
time = 1.56, size = 442, normalized size = 1.62 \begin {gather*} \frac {x}{b} - \frac {\left (\frac {a}{b}\right )^{\frac {1}{8}} \arctan \left (\frac {2 \, x + \sqrt {-\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}\right )}{4 \, b \sqrt {-2 \, \sqrt {2} + 4}} - \frac {\left (\frac {a}{b}\right )^{\frac {1}{8}} \arctan \left (\frac {2 \, x - \sqrt {-\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}\right )}{4 \, b \sqrt {-2 \, \sqrt {2} + 4}} - \frac {\left (\frac {a}{b}\right )^{\frac {1}{8}} \arctan \left (\frac {2 \, x + \sqrt {\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}\right )}{4 \, b \sqrt {2 \, \sqrt {2} + 4}} - \frac {\left (\frac {a}{b}\right )^{\frac {1}{8}} \arctan \left (\frac {2 \, x - \sqrt {\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}\right )}{4 \, b \sqrt {2 \, \sqrt {2} + 4}} - \frac {\left (\frac {a}{b}\right )^{\frac {1}{8}} \log \left (x^{2} + x \sqrt {\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}} + \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}{8 \, b \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{b}\right )^{\frac {1}{8}} \log \left (x^{2} - x \sqrt {\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}} + \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}{8 \, b \sqrt {-2 \, \sqrt {2} + 4}} - \frac {\left (\frac {a}{b}\right )^{\frac {1}{8}} \log \left (x^{2} + x \sqrt {-\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}} + \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}{8 \, b \sqrt {2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{b}\right )^{\frac {1}{8}} \log \left (x^{2} - x \sqrt {-\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}} + \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}{8 \, b \sqrt {2 \, \sqrt {2} + 4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(b*x^8+a),x, algorithm="giac")

[Out]

x/b - 1/4*(a/b)^(1/8)*arctan((2*x + sqrt(-sqrt(2) + 2)*(a/b)^(1/8))/(sqrt(sqrt(2) + 2)*(a/b)^(1/8)))/(b*sqrt(-
2*sqrt(2) + 4)) - 1/4*(a/b)^(1/8)*arctan((2*x - sqrt(-sqrt(2) + 2)*(a/b)^(1/8))/(sqrt(sqrt(2) + 2)*(a/b)^(1/8)
))/(b*sqrt(-2*sqrt(2) + 4)) - 1/4*(a/b)^(1/8)*arctan((2*x + sqrt(sqrt(2) + 2)*(a/b)^(1/8))/(sqrt(-sqrt(2) + 2)
*(a/b)^(1/8)))/(b*sqrt(2*sqrt(2) + 4)) - 1/4*(a/b)^(1/8)*arctan((2*x - sqrt(sqrt(2) + 2)*(a/b)^(1/8))/(sqrt(-s
qrt(2) + 2)*(a/b)^(1/8)))/(b*sqrt(2*sqrt(2) + 4)) - 1/8*(a/b)^(1/8)*log(x^2 + x*sqrt(sqrt(2) + 2)*(a/b)^(1/8)
+ (a/b)^(1/4))/(b*sqrt(-2*sqrt(2) + 4)) + 1/8*(a/b)^(1/8)*log(x^2 - x*sqrt(sqrt(2) + 2)*(a/b)^(1/8) + (a/b)^(1
/4))/(b*sqrt(-2*sqrt(2) + 4)) - 1/8*(a/b)^(1/8)*log(x^2 + x*sqrt(-sqrt(2) + 2)*(a/b)^(1/8) + (a/b)^(1/4))/(b*s
qrt(2*sqrt(2) + 4)) + 1/8*(a/b)^(1/8)*log(x^2 - x*sqrt(-sqrt(2) + 2)*(a/b)^(1/8) + (a/b)^(1/4))/(b*sqrt(2*sqrt
(2) + 4))

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Mupad [B]
time = 0.15, size = 115, normalized size = 0.42 \begin {gather*} \frac {x}{b}-\frac {{\left (-a\right )}^{1/8}\,\mathrm {atan}\left (\frac {b^{1/8}\,x}{{\left (-a\right )}^{1/8}}\right )}{4\,b^{9/8}}+\frac {{\left (-a\right )}^{1/8}\,\mathrm {atan}\left (\frac {b^{1/8}\,x\,1{}\mathrm {i}}{{\left (-a\right )}^{1/8}}\right )\,1{}\mathrm {i}}{4\,b^{9/8}}+\frac {\sqrt {2}\,{\left (-a\right )}^{1/8}\,\mathrm {atan}\left (\frac {\sqrt {2}\,b^{1/8}\,x\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (-\frac {1}{8}-\frac {1}{8}{}\mathrm {i}\right )}{b^{9/8}}+\frac {\sqrt {2}\,{\left (-a\right )}^{1/8}\,\mathrm {atan}\left (\frac {\sqrt {2}\,b^{1/8}\,x\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (-\frac {1}{8}+\frac {1}{8}{}\mathrm {i}\right )}{b^{9/8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(a + b*x^8),x)

[Out]

x/b - ((-a)^(1/8)*atan((b^(1/8)*x)/(-a)^(1/8)))/(4*b^(9/8)) + ((-a)^(1/8)*atan((b^(1/8)*x*1i)/(-a)^(1/8))*1i)/
(4*b^(9/8)) - (2^(1/2)*(-a)^(1/8)*atan((2^(1/2)*b^(1/8)*x*(1/2 - 1i/2))/(-a)^(1/8))*(1/8 + 1i/8))/b^(9/8) - (2
^(1/2)*(-a)^(1/8)*atan((2^(1/2)*b^(1/8)*x*(1/2 + 1i/2))/(-a)^(1/8))*(1/8 - 1i/8))/b^(9/8)

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