3.1 \(\int \frac {\sqrt {1+x^8} (1+2 x^8)}{x+2 x^9+x^{17}} \, dx\)

Optimal. Leaf size=28 \[ -\frac {1}{4 \sqrt {x^8+1}}-\frac {1}{4} \tanh ^{-1}\left (\sqrt {x^8+1}\right ) \]

[Out]

-1/4*arctanh((x^8+1)^(1/2))-1/4/(x^8+1)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1586, 1593, 446, 78, 63, 207} \[ -\frac {1}{4 \sqrt {x^8+1}}-\frac {1}{4} \tanh ^{-1}\left (\sqrt {x^8+1}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[1 + x^8]*(1 + 2*x^8))/(x + 2*x^9 + x^17),x]

[Out]

-1/(4*Sqrt[1 + x^8]) - ArcTanh[Sqrt[1 + x^8]]/4

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 207

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

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 1593

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

Rubi steps

\begin {align*} \int \frac {\sqrt {1+x^8} \left (1+2 x^8\right )}{x+2 x^9+x^{17}} \, dx &=\int \frac {1+2 x^8}{\sqrt {1+x^8} \left (x+x^9\right )} \, dx\\ &=\int \frac {1+2 x^8}{x \left (1+x^8\right )^{3/2}} \, dx\\ &=\frac {1}{8} \operatorname {Subst}\left (\int \frac {1+2 x}{x (1+x)^{3/2}} \, dx,x,x^8\right )\\ &=-\frac {1}{4 \sqrt {1+x^8}}+\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^8\right )\\ &=-\frac {1}{4 \sqrt {1+x^8}}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^8}\right )\\ &=-\frac {1}{4 \sqrt {1+x^8}}-\frac {1}{4} \tanh ^{-1}\left (\sqrt {1+x^8}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 28, normalized size = 1.00 \[ -\frac {1}{4 \sqrt {x^8+1}}-\frac {1}{4} \tanh ^{-1}\left (\sqrt {x^8+1}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[1 + x^8]*(1 + 2*x^8))/(x + 2*x^9 + x^17),x]

[Out]

-1/4*1/Sqrt[1 + x^8] - ArcTanh[Sqrt[1 + x^8]]/4

________________________________________________________________________________________

fricas [B]  time = 0.39, size = 52, normalized size = 1.86 \[ -\frac {{\left (x^{8} + 1\right )} \log \left (\sqrt {x^{8} + 1} + 1\right ) - {\left (x^{8} + 1\right )} \log \left (\sqrt {x^{8} + 1} - 1\right ) + 2 \, \sqrt {x^{8} + 1}}{8 \, {\left (x^{8} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^8+1)*(x^8+1)^(1/2)/(x^17+2*x^9+x),x, algorithm="fricas")

[Out]

-1/8*((x^8 + 1)*log(sqrt(x^8 + 1) + 1) - (x^8 + 1)*log(sqrt(x^8 + 1) - 1) + 2*sqrt(x^8 + 1))/(x^8 + 1)

________________________________________________________________________________________

giac [A]  time = 0.94, size = 34, normalized size = 1.21 \[ -\frac {1}{4 \, \sqrt {x^{8} + 1}} - \frac {1}{8} \, \log \left (\sqrt {x^{8} + 1} + 1\right ) + \frac {1}{8} \, \log \left (\sqrt {x^{8} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^8+1)*(x^8+1)^(1/2)/(x^17+2*x^9+x),x, algorithm="giac")

[Out]

-1/4/sqrt(x^8 + 1) - 1/8*log(sqrt(x^8 + 1) + 1) + 1/8*log(sqrt(x^8 + 1) - 1)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 29, normalized size = 1.04 \[ \frac {\ln \left (\frac {\sqrt {x^{8}+1}-1}{\sqrt {x^{8}}}\right )}{4}-\frac {1}{4 \sqrt {x^{8}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^8+1)*(x^8+1)^(1/2)/(x^17+2*x^9+x),x)

[Out]

-1/4/(x^8+1)^(1/2)+1/4*ln(((x^8+1)^(1/2)-1)/(x^8)^(1/2))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (2 \, x^{8} + 1\right )} \sqrt {x^{8} + 1}}{x^{17} + 2 \, x^{9} + x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^8+1)*(x^8+1)^(1/2)/(x^17+2*x^9+x),x, algorithm="maxima")

[Out]

integrate((2*x^8 + 1)*sqrt(x^8 + 1)/(x^17 + 2*x^9 + x), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\sqrt {x^8+1}\,\left (2\,x^8+1\right )}{x^{17}+2\,x^9+x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^8 + 1)^(1/2)*(2*x^8 + 1))/(x + 2*x^9 + x^17),x)

[Out]

int(((x^8 + 1)^(1/2)*(2*x^8 + 1))/(x + 2*x^9 + x^17), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {2 x^{8} + 1}{x \left (x^{8} + 1\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**8+1)*(x**8+1)**(1/2)/(x**17+2*x**9+x),x)

[Out]

Integral((2*x**8 + 1)/(x*(x**8 + 1)**(3/2)), x)

________________________________________________________________________________________