\(\int \frac {-1+x^4+7 x^5+x^9}{-7+6 x^4+x^8} \, dx\) [367]

Optimal result

Integrand size = 26, antiderivative size = 148 \[ \int \frac {-1+x^4+7 x^5+x^9}{-7+6 x^4+x^8} \, dx=\frac {x^2}{2}-\frac {\arctan \left (1-\frac {\sqrt {2} x}{\sqrt [4]{7}}\right )}{2 \sqrt {2} 7^{3/4}}+\frac {\arctan \left (1+\frac {\sqrt {2} x}{\sqrt [4]{7}}\right )}{2 \sqrt {2} 7^{3/4}}-\frac {\text {arctanh}\left (x^2\right )}{2}-\frac {\log \left (\sqrt {7}-\sqrt {2} \sqrt [4]{7} x+x^2\right )}{4 \sqrt {2} 7^{3/4}}+\frac {\log \left (\sqrt {7}+\sqrt {2} \sqrt [4]{7} x+x^2\right )}{4 \sqrt {2} 7^{3/4}} \]

[Out]

1/2*x^2-1/2*arctanh(x^2)+1/28*arctan(-1+1/7*x*2^(1/2)*7^(3/4))*7^(1/4)*2^(1/2)+1/28*arctan(1+1/7*x*2^(1/2)*7^(
3/4))*7^(1/4)*2^(1/2)-1/56*ln(x^2-7^(1/4)*x*2^(1/2)+7^(1/2))*7^(1/4)*2^(1/2)+1/56*ln(x^2+7^(1/4)*x*2^(1/2)+7^(
1/2))*7^(1/4)*2^(1/2)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1804, 1417, 217, 1179, 642, 1176, 631, 210, 1598, 1492, 281, 327, 213} \[ \int \frac {-1+x^4+7 x^5+x^9}{-7+6 x^4+x^8} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} x}{\sqrt [4]{7}}\right )}{2 \sqrt {2} 7^{3/4}}+\frac {\arctan \left (\frac {\sqrt {2} x}{\sqrt [4]{7}}+1\right )}{2 \sqrt {2} 7^{3/4}}-\frac {\text {arctanh}\left (x^2\right )}{2}+\frac {x^2}{2}-\frac {\log \left (x^2-\sqrt {2} \sqrt [4]{7} x+\sqrt {7}\right )}{4 \sqrt {2} 7^{3/4}}+\frac {\log \left (x^2+\sqrt {2} \sqrt [4]{7} x+\sqrt {7}\right )}{4 \sqrt {2} 7^{3/4}} \]

[In]

Int[(-1 + x^4 + 7*x^5 + x^9)/(-7 + 6*x^4 + x^8),x]

[Out]

x^2/2 - ArcTan[1 - (Sqrt[2]*x)/7^(1/4)]/(2*Sqrt[2]*7^(3/4)) + ArcTan[1 + (Sqrt[2]*x)/7^(1/4)]/(2*Sqrt[2]*7^(3/
4)) - ArcTanh[x^2]/2 - Log[Sqrt[7] - Sqrt[2]*7^(1/4)*x + x^2]/(4*Sqrt[2]*7^(3/4)) + Log[Sqrt[7] + Sqrt[2]*7^(1
/4)*x + x^2]/(4*Sqrt[2]*7^(3/4))

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 213

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

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 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

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]

Rule 1417

Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_.), x_Symbol] :> Int[(d + e*
x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
&& EqQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]

Rule 1492

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_.)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_.), x_Sym
bol] :> Int[(f*x)^m*(d + e*x^n)^(q + p)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, q}, x] && Eq
Q[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]

Rule 1598

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 1804

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], j, k}, Int[
Sum[x^j*Sum[Coeff[Pq, x, j + k*n]*x^(k*n), {k, 0, (q - j)/n + 1}]*(a + b*x^n + c*x^(2*n))^p, {j, 0, n - 1}], x
]] /; FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] &&  !PolyQ[P
q, x^n]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {-1+x^4}{-7+6 x^4+x^8}+\frac {x \left (7 x^4+x^8\right )}{-7+6 x^4+x^8}\right ) \, dx \\ & = \int \frac {-1+x^4}{-7+6 x^4+x^8} \, dx+\int \frac {x \left (7 x^4+x^8\right )}{-7+6 x^4+x^8} \, dx \\ & = \int \frac {1}{7+x^4} \, dx+\int \frac {x^5 \left (7+x^4\right )}{-7+6 x^4+x^8} \, dx \\ & = \frac {\int \frac {\sqrt {7}-x^2}{7+x^4} \, dx}{2 \sqrt {7}}+\frac {\int \frac {\sqrt {7}+x^2}{7+x^4} \, dx}{2 \sqrt {7}}+\int \frac {x^5}{-1+x^4} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,x^2\right )-\frac {\int \frac {\sqrt {2} \sqrt [4]{7}+2 x}{-\sqrt {7}-\sqrt {2} \sqrt [4]{7} x-x^2} \, dx}{4 \sqrt {2} 7^{3/4}}-\frac {\int \frac {\sqrt {2} \sqrt [4]{7}-2 x}{-\sqrt {7}+\sqrt {2} \sqrt [4]{7} x-x^2} \, dx}{4 \sqrt {2} 7^{3/4}}+\frac {\int \frac {1}{\sqrt {7}-\sqrt {2} \sqrt [4]{7} x+x^2} \, dx}{4 \sqrt {7}}+\frac {\int \frac {1}{\sqrt {7}+\sqrt {2} \sqrt [4]{7} x+x^2} \, dx}{4 \sqrt {7}} \\ & = \frac {x^2}{2}-\frac {\log \left (\sqrt {7}-\sqrt {2} \sqrt [4]{7} x+x^2\right )}{4 \sqrt {2} 7^{3/4}}+\frac {\log \left (\sqrt {7}+\sqrt {2} \sqrt [4]{7} x+x^2\right )}{4 \sqrt {2} 7^{3/4}}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,x^2\right )+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} x}{\sqrt [4]{7}}\right )}{2 \sqrt {2} 7^{3/4}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} x}{\sqrt [4]{7}}\right )}{2 \sqrt {2} 7^{3/4}} \\ & = \frac {x^2}{2}-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} x}{\sqrt [4]{7}}\right )}{2 \sqrt {2} 7^{3/4}}+\frac {\tan ^{-1}\left (1+\frac {\sqrt {2} x}{\sqrt [4]{7}}\right )}{2 \sqrt {2} 7^{3/4}}-\frac {1}{2} \tanh ^{-1}\left (x^2\right )-\frac {\log \left (\sqrt {7}-\sqrt {2} \sqrt [4]{7} x+x^2\right )}{4 \sqrt {2} 7^{3/4}}+\frac {\log \left (\sqrt {7}+\sqrt {2} \sqrt [4]{7} x+x^2\right )}{4 \sqrt {2} 7^{3/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.07 \[ \int \frac {-1+x^4+7 x^5+x^9}{-7+6 x^4+x^8} \, dx=\frac {1}{56} \left (28 x^2-2 \sqrt {2} \sqrt [4]{7} \arctan \left (1-\frac {\sqrt {2} x}{\sqrt [4]{7}}\right )+2 \sqrt {2} \sqrt [4]{7} \arctan \left (1+\frac {\sqrt {2} x}{\sqrt [4]{7}}\right )+14 \log (1-x)+14 \log (1+x)-14 \log \left (1+x^2\right )-\sqrt {2} \sqrt [4]{7} \log \left (7-\sqrt {2} 7^{3/4} x+\sqrt {7} x^2\right )+\sqrt {2} \sqrt [4]{7} \log \left (7+\sqrt {2} 7^{3/4} x+\sqrt {7} x^2\right )\right ) \]

[In]

Integrate[(-1 + x^4 + 7*x^5 + x^9)/(-7 + 6*x^4 + x^8),x]

[Out]

(28*x^2 - 2*Sqrt[2]*7^(1/4)*ArcTan[1 - (Sqrt[2]*x)/7^(1/4)] + 2*Sqrt[2]*7^(1/4)*ArcTan[1 + (Sqrt[2]*x)/7^(1/4)
] + 14*Log[1 - x] + 14*Log[1 + x] - 14*Log[1 + x^2] - Sqrt[2]*7^(1/4)*Log[7 - Sqrt[2]*7^(3/4)*x + Sqrt[7]*x^2]
+ Sqrt[2]*7^(1/4)*Log[7 + Sqrt[2]*7^(3/4)*x + Sqrt[7]*x^2])/56

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.13 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.30

method	result	size
risch	\(\frac {x^{2}}{2}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (343 \textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left (x +7 \textit {\_R} \right )\right )}{4}+\frac {\ln \left (x^{2}-1\right )}{4}-\frac {\ln \left (x^{2}+1\right )}{4}\)	\(44\)
default	\(\frac {x^{2}}{2}+\frac {\ln \left (x +1\right )}{4}+\frac {7^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+7^{\frac {1}{4}} x \sqrt {2}+\sqrt {7}}{x^{2}-7^{\frac {1}{4}} x \sqrt {2}+\sqrt {7}}\right )+2 \arctan \left (1+\frac {x \sqrt {2}\, 7^{\frac {3}{4}}}{7}\right )+2 \arctan \left (-1+\frac {x \sqrt {2}\, 7^{\frac {3}{4}}}{7}\right )\right )}{56}-\frac {\ln \left (x^{2}+1\right )}{4}+\frac {\ln \left (x -1\right )}{4}\)	\(99\)

[In]

int((x^9+7*x^5+x^4-1)/(x^8+6*x^4-7),x,method=_RETURNVERBOSE)

[Out]

1/2*x^2+1/4*sum(_R*ln(x+7*_R),_R=RootOf(343*_Z^4+1))+1/4*ln(x^2-1)-1/4*ln(x^2+1)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.72 \[ \int \frac {-1+x^4+7 x^5+x^9}{-7+6 x^4+x^8} \, dx=\left (\frac {1}{2744} i + \frac {1}{2744}\right ) \cdot 343^{\frac {3}{4}} \sqrt {2} \log \left (\left (i + 1\right ) \cdot 343^{\frac {3}{4}} \sqrt {2} + 98 \, x\right ) - \left (\frac {1}{2744} i - \frac {1}{2744}\right ) \cdot 343^{\frac {3}{4}} \sqrt {2} \log \left (-\left (i - 1\right ) \cdot 343^{\frac {3}{4}} \sqrt {2} + 98 \, x\right ) + \left (\frac {1}{2744} i - \frac {1}{2744}\right ) \cdot 343^{\frac {3}{4}} \sqrt {2} \log \left (\left (i - 1\right ) \cdot 343^{\frac {3}{4}} \sqrt {2} + 98 \, x\right ) - \left (\frac {1}{2744} i + \frac {1}{2744}\right ) \cdot 343^{\frac {3}{4}} \sqrt {2} \log \left (-\left (i + 1\right ) \cdot 343^{\frac {3}{4}} \sqrt {2} + 98 \, x\right ) + \frac {1}{2} \, x^{2} - \frac {1}{4} \, \log \left (x^{2} + 1\right ) + \frac {1}{4} \, \log \left (x^{2} - 1\right ) \]

[In]

integrate((x^9+7*x^5+x^4-1)/(x^8+6*x^4-7),x, algorithm="fricas")

[Out]

(1/2744*I + 1/2744)*343^(3/4)*sqrt(2)*log((I + 1)*343^(3/4)*sqrt(2) + 98*x) - (1/2744*I - 1/2744)*343^(3/4)*sq
rt(2)*log(-(I - 1)*343^(3/4)*sqrt(2) + 98*x) + (1/2744*I - 1/2744)*343^(3/4)*sqrt(2)*log((I - 1)*343^(3/4)*sqr
t(2) + 98*x) - (1/2744*I + 1/2744)*343^(3/4)*sqrt(2)*log(-(I + 1)*343^(3/4)*sqrt(2) + 98*x) + 1/2*x^2 - 1/4*lo
g(x^2 + 1) + 1/4*log(x^2 - 1)

Sympy [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.99 \[ \int \frac {-1+x^4+7 x^5+x^9}{-7+6 x^4+x^8} \, dx=\frac {x^{2}}{2} + \frac {\log {\left (x^{2} - 1 \right )}}{4} - \frac {\log {\left (x^{2} + 1 \right )}}{4} - \frac {\sqrt {2} \cdot \sqrt [4]{7} \log {\left (x^{2} - \sqrt {2} \cdot \sqrt [4]{7} x + \sqrt {7} \right )}}{56} + \frac {\sqrt {2} \cdot \sqrt [4]{7} \log {\left (x^{2} + \sqrt {2} \cdot \sqrt [4]{7} x + \sqrt {7} \right )}}{56} + \frac {\sqrt {2} \cdot \sqrt [4]{7} \operatorname {atan}{\left (\frac {\sqrt {2} \cdot 7^{\frac {3}{4}} x}{7} - 1 \right )}}{28} + \frac {\sqrt {2} \cdot \sqrt [4]{7} \operatorname {atan}{\left (\frac {\sqrt {2} \cdot 7^{\frac {3}{4}} x}{7} + 1 \right )}}{28} \]

[In]

integrate((x**9+7*x**5+x**4-1)/(x**8+6*x**4-7),x)

[Out]

x**2/2 + log(x**2 - 1)/4 - log(x**2 + 1)/4 - sqrt(2)*7**(1/4)*log(x**2 - sqrt(2)*7**(1/4)*x + sqrt(7))/56 + sq
rt(2)*7**(1/4)*log(x**2 + sqrt(2)*7**(1/4)*x + sqrt(7))/56 + sqrt(2)*7**(1/4)*atan(sqrt(2)*7**(3/4)*x/7 - 1)/2
8 + sqrt(2)*7**(1/4)*atan(sqrt(2)*7**(3/4)*x/7 + 1)/28

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.89 \[ \int \frac {-1+x^4+7 x^5+x^9}{-7+6 x^4+x^8} \, dx=\frac {1}{2} \, x^{2} + \frac {1}{28} \cdot 7^{\frac {1}{4}} \sqrt {2} \arctan \left (\frac {1}{14} \cdot 7^{\frac {3}{4}} \sqrt {2} {\left (2 \, x + 7^{\frac {1}{4}} \sqrt {2}\right )}\right ) + \frac {1}{28} \cdot 7^{\frac {1}{4}} \sqrt {2} \arctan \left (\frac {1}{14} \cdot 7^{\frac {3}{4}} \sqrt {2} {\left (2 \, x - 7^{\frac {1}{4}} \sqrt {2}\right )}\right ) + \frac {1}{56} \cdot 7^{\frac {1}{4}} \sqrt {2} \log \left (x^{2} + 7^{\frac {1}{4}} \sqrt {2} x + \sqrt {7}\right ) - \frac {1}{56} \cdot 7^{\frac {1}{4}} \sqrt {2} \log \left (x^{2} - 7^{\frac {1}{4}} \sqrt {2} x + \sqrt {7}\right ) - \frac {1}{4} \, \log \left (x^{2} + 1\right ) + \frac {1}{4} \, \log \left (x + 1\right ) + \frac {1}{4} \, \log \left (x - 1\right ) \]

[In]

integrate((x^9+7*x^5+x^4-1)/(x^8+6*x^4-7),x, algorithm="maxima")

[Out]

1/2*x^2 + 1/28*7^(1/4)*sqrt(2)*arctan(1/14*7^(3/4)*sqrt(2)*(2*x + 7^(1/4)*sqrt(2))) + 1/28*7^(1/4)*sqrt(2)*arc
tan(1/14*7^(3/4)*sqrt(2)*(2*x - 7^(1/4)*sqrt(2))) + 1/56*7^(1/4)*sqrt(2)*log(x^2 + 7^(1/4)*sqrt(2)*x + sqrt(7)
) - 1/56*7^(1/4)*sqrt(2)*log(x^2 - 7^(1/4)*sqrt(2)*x + sqrt(7)) - 1/4*log(x^2 + 1) + 1/4*log(x + 1) + 1/4*log(
x - 1)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.82 \[ \int \frac {-1+x^4+7 x^5+x^9}{-7+6 x^4+x^8} \, dx=\frac {1}{2} \, x^{2} + \frac {1}{28} \cdot 28^{\frac {1}{4}} \arctan \left (\frac {1}{14} \cdot 7^{\frac {3}{4}} \sqrt {2} {\left (2 \, x + 7^{\frac {1}{4}} \sqrt {2}\right )}\right ) + \frac {1}{28} \cdot 28^{\frac {1}{4}} \arctan \left (\frac {1}{14} \cdot 7^{\frac {3}{4}} \sqrt {2} {\left (2 \, x - 7^{\frac {1}{4}} \sqrt {2}\right )}\right ) + \frac {1}{56} \cdot 28^{\frac {1}{4}} \log \left (x^{2} + 7^{\frac {1}{4}} \sqrt {2} x + \sqrt {7}\right ) - \frac {1}{56} \cdot 28^{\frac {1}{4}} \log \left (x^{2} - 7^{\frac {1}{4}} \sqrt {2} x + \sqrt {7}\right ) - \frac {1}{4} \, \log \left (x^{2} + 1\right ) + \frac {1}{4} \, \log \left ({\left | x + 1 \right |}\right ) + \frac {1}{4} \, \log \left ({\left | x - 1 \right |}\right ) \]

[In]

integrate((x^9+7*x^5+x^4-1)/(x^8+6*x^4-7),x, algorithm="giac")

[Out]

1/2*x^2 + 1/28*28^(1/4)*arctan(1/14*7^(3/4)*sqrt(2)*(2*x + 7^(1/4)*sqrt(2))) + 1/28*28^(1/4)*arctan(1/14*7^(3/
4)*sqrt(2)*(2*x - 7^(1/4)*sqrt(2))) + 1/56*28^(1/4)*log(x^2 + 7^(1/4)*sqrt(2)*x + sqrt(7)) - 1/56*28^(1/4)*log
(x^2 - 7^(1/4)*sqrt(2)*x + sqrt(7)) - 1/4*log(x^2 + 1) + 1/4*log(abs(x + 1)) + 1/4*log(abs(x - 1))

Mupad [B] (verification not implemented)

Time = 9.10 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.84 \[ \int \frac {-1+x^4+7 x^5+x^9}{-7+6 x^4+x^8} \, dx=\frac {\mathrm {atan}\left (x^2\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}+\frac {x^2}{2}+\sqrt {2}\,7^{1/4}\,\mathrm {atan}\left (\frac {\sqrt {2}\,7^{1/4}\,x\,\left (\frac {89653248}{2401}+\frac {89653248}{2401}{}\mathrm {i}\right )}{-\frac {1048576}{49}+\frac {\sqrt {7}\,179306496{}\mathrm {i}}{2401}}+\frac {\sqrt {2}\,7^{3/4}\,x\,\left (-\frac {524288}{343}+\frac {524288}{343}{}\mathrm {i}\right )}{-\frac {1048576}{49}+\frac {\sqrt {7}\,179306496{}\mathrm {i}}{2401}}\right )\,\left (\frac {1}{28}+\frac {1}{28}{}\mathrm {i}\right )+\sqrt {2}\,7^{1/4}\,\mathrm {atan}\left (\frac {\sqrt {2}\,7^{1/4}\,x\,\left (\frac {89653248}{2401}-\frac {89653248}{2401}{}\mathrm {i}\right )}{\frac {1048576}{49}+\frac {\sqrt {7}\,179306496{}\mathrm {i}}{2401}}+\frac {\sqrt {2}\,7^{3/4}\,x\,\left (-\frac {524288}{343}-\frac {524288}{343}{}\mathrm {i}\right )}{\frac {1048576}{49}+\frac {\sqrt {7}\,179306496{}\mathrm {i}}{2401}}\right )\,\left (-\frac {1}{28}+\frac {1}{28}{}\mathrm {i}\right ) \]

[In]

int((x^4 + 7*x^5 + x^9 - 1)/(6*x^4 + x^8 - 7),x)

[Out]

(atan(x^2*1i)*1i)/2 + x^2/2 + 2^(1/2)*7^(1/4)*atan((2^(1/2)*7^(1/4)*x*(89653248/2401 + 89653248i/2401))/((7^(1
/2)*179306496i)/2401 - 1048576/49) - (2^(1/2)*7^(3/4)*x*(524288/343 - 524288i/343))/((7^(1/2)*179306496i)/2401
- 1048576/49))*(1/28 + 1i/28) - 2^(1/2)*7^(1/4)*atan((2^(1/2)*7^(1/4)*x*(89653248/2401 - 89653248i/2401))/((7
^(1/2)*179306496i)/2401 + 1048576/49) - (2^(1/2)*7^(3/4)*x*(524288/343 + 524288i/343))/((7^(1/2)*179306496i)/2
401 + 1048576/49))*(1/28 - 1i/28)