3.590 \(\int \frac{1+x^3}{(1-x^4) \sqrt [4]{1+x^4}} \, dx\)

Optimal. Leaf size=103 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{2}}-\frac{\tan ^{-1}\left (\frac{\sqrt [4]{x^4+1}}{\sqrt [4]{2}}\right )}{2 \sqrt [4]{2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{x^4+1}}{\sqrt [4]{2}}\right )}{2 \sqrt [4]{2}} \]

[Out]

ArcTan[(2^(1/4)*x)/(1 + x^4)^(1/4)]/(2*2^(1/4)) - ArcTan[(1 + x^4)^(1/4)/2^(1/4)]/(2*2^(1/4)) + ArcTanh[(2^(1/
4)*x)/(1 + x^4)^(1/4)]/(2*2^(1/4)) + ArcTanh[(1 + x^4)^(1/4)/2^(1/4)]/(2*2^(1/4))

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Rubi [A]  time = 0.10257, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1899, 377, 212, 206, 203, 444, 63, 298} \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{2}}-\frac{\tan ^{-1}\left (\frac{\sqrt [4]{x^4+1}}{\sqrt [4]{2}}\right )}{2 \sqrt [4]{2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{x^4+1}}{\sqrt [4]{2}}\right )}{2 \sqrt [4]{2}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^3)/((1 - x^4)*(1 + x^4)^(1/4)),x]

[Out]

ArcTan[(2^(1/4)*x)/(1 + x^4)^(1/4)]/(2*2^(1/4)) - ArcTan[(1 + x^4)^(1/4)/2^(1/4)]/(2*2^(1/4)) + ArcTanh[(2^(1/
4)*x)/(1 + x^4)^(1/4)]/(2*2^(1/4)) + ArcTanh[(1 + x^4)^(1/4)/2^(1/4)]/(2*2^(1/4))

Rule 1899

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

Rule 377

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

Rule 212

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] &&
!GtQ[a/b, 0]

Rule 206

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

Rule 203

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

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(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] && EqQ[m
- n + 1, 0]

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 298

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

Rubi steps

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

Mathematica [C]  time = 0.168213, size = 93, normalized size = 0.9 \[ \frac{1}{4} x^4 F_1\left (1;\frac{1}{4},1;2;-x^4,x^4\right )+\frac{-\log \left (1-\frac{\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )+\log \left (\frac{\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}+1\right )+2 \tan ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{4 \sqrt [4]{2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(1 + x^3)/((1 - x^4)*(1 + x^4)^(1/4)),x]

[Out]

(x^4*AppellF1[1, 1/4, 1, 2, -x^4, x^4])/4 + (2*ArcTan[(2^(1/4)*x)/(1 + x^4)^(1/4)] - Log[1 - (2^(1/4)*x)/(1 +
x^4)^(1/4)] + Log[1 + (2^(1/4)*x)/(1 + x^4)^(1/4)])/(4*2^(1/4))

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Maple [F]  time = 0.387, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}+1}{-{x}^{4}+1}{\frac{1}{\sqrt [4]{{x}^{4}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3+1)/(-x^4+1)/(x^4+1)^(1/4),x)

[Out]

int((x^3+1)/(-x^4+1)/(x^4+1)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{x^{3} + 1}{{\left (x^{4} + 1\right )}^{\frac{1}{4}}{\left (x^{4} - 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)/(-x^4+1)/(x^4+1)^(1/4),x, algorithm="maxima")

[Out]

-integrate((x^3 + 1)/((x^4 + 1)^(1/4)*(x^4 - 1)), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)/(-x^4+1)/(x^4+1)^(1/4),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \frac{x}{x^{3} \sqrt [4]{x^{4} + 1} - x^{2} \sqrt [4]{x^{4} + 1} + x \sqrt [4]{x^{4} + 1} - \sqrt [4]{x^{4} + 1}}\, dx - \int \frac{x^{2}}{x^{3} \sqrt [4]{x^{4} + 1} - x^{2} \sqrt [4]{x^{4} + 1} + x \sqrt [4]{x^{4} + 1} - \sqrt [4]{x^{4} + 1}}\, dx - \int \frac{1}{x^{3} \sqrt [4]{x^{4} + 1} - x^{2} \sqrt [4]{x^{4} + 1} + x \sqrt [4]{x^{4} + 1} - \sqrt [4]{x^{4} + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3+1)/(-x**4+1)/(x**4+1)**(1/4),x)

[Out]

-Integral(-x/(x**3*(x**4 + 1)**(1/4) - x**2*(x**4 + 1)**(1/4) + x*(x**4 + 1)**(1/4) - (x**4 + 1)**(1/4)), x) -
Integral(x**2/(x**3*(x**4 + 1)**(1/4) - x**2*(x**4 + 1)**(1/4) + x*(x**4 + 1)**(1/4) - (x**4 + 1)**(1/4)), x)
- Integral(1/(x**3*(x**4 + 1)**(1/4) - x**2*(x**4 + 1)**(1/4) + x*(x**4 + 1)**(1/4) - (x**4 + 1)**(1/4)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{x^{3} + 1}{{\left (x^{4} + 1\right )}^{\frac{1}{4}}{\left (x^{4} - 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)/(-x^4+1)/(x^4+1)^(1/4),x, algorithm="giac")

[Out]

integrate(-(x^3 + 1)/((x^4 + 1)^(1/4)*(x^4 - 1)), x)