3.31.28 \(\int \frac {(-1+a x^8) (1+a x^8)^{3/4}}{1+x^8+a^2 x^{16}} \, dx\)

Optimal. Leaf size=432 \[ \frac {\left (1+\sqrt [4]{-1}\right ) \tan ^{-1}\left (\frac {(-1)^{7/8} \sqrt {2+\sqrt {2}} \sqrt [8]{2 a-1} x \sqrt [4]{a x^8+1}}{\sqrt {a x^8+1}+(-1)^{3/4} \sqrt [4]{2 a-1} x^2}\right )}{8 \sqrt [8]{2 a-1}}-\frac {i \left (\sqrt {2 \left (3-2 \sqrt {2}\right )}-i \sqrt {2}\right ) \tan ^{-1}\left (\frac {(-1)^{7/8} \left (\sqrt {2}-2\right ) \sqrt [8]{2 a-1} x \sqrt [4]{a x^8+1}}{\sqrt {2-\sqrt {2}} \sqrt {a x^8+1}+(-1)^{3/4} \sqrt {2-\sqrt {2}} \sqrt [4]{2 a-1} x^2}\right )}{16 \sqrt [8]{2 a-1}}+\frac {\left (\sqrt {2}+i \sqrt {2 \left (3-2 \sqrt {2}\right )}\right ) \tanh ^{-1}\left (\frac {(-1)^{7/8} \sqrt [4]{2 a-1} x^2-\sqrt [8]{-1} \sqrt {a x^8+1}}{\sqrt {2-\sqrt {2}} \sqrt [8]{2 a-1} x \sqrt [4]{a x^8+1}}\right )}{16 \sqrt [8]{2 a-1}}+\frac {\left (1+\sqrt [4]{-1}\right ) \tanh ^{-1}\left (\frac {(-1)^{7/8} \sqrt [4]{2 a-1} x^2-\sqrt [8]{-1} \sqrt {a x^8+1}}{\sqrt {2+\sqrt {2}} \sqrt [8]{2 a-1} x \sqrt [4]{a x^8+1}}\right )}{8 \sqrt [8]{2 a-1}} \]

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Rubi [C]  time = 0.38, antiderivative size = 160, normalized size of antiderivative = 0.37, number of steps used = 4, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {6728, 429} \begin {gather*} \frac {a \left (1-\frac {2 a+1}{\sqrt {1-4 a^2}}\right ) x F_1\left (\frac {1}{8};-\frac {3}{4},1;\frac {9}{8};-a x^8,-\frac {2 a^2 x^8}{1-\sqrt {1-4 a^2}}\right )}{1-\sqrt {1-4 a^2}}+\frac {a \left (\frac {2 a+1}{\sqrt {1-4 a^2}}+1\right ) x F_1\left (\frac {1}{8};-\frac {3}{4},1;\frac {9}{8};-a x^8,-\frac {2 a^2 x^8}{\sqrt {1-4 a^2}+1}\right )}{\sqrt {1-4 a^2}+1} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[((-1 + a*x^8)*(1 + a*x^8)^(3/4))/(1 + x^8 + a^2*x^16),x]

[Out]

(a*(1 - (1 + 2*a)/Sqrt[1 - 4*a^2])*x*AppellF1[1/8, -3/4, 1, 9/8, -(a*x^8), (-2*a^2*x^8)/(1 - Sqrt[1 - 4*a^2])]
)/(1 - Sqrt[1 - 4*a^2]) + (a*(1 + (1 + 2*a)/Sqrt[1 - 4*a^2])*x*AppellF1[1/8, -3/4, 1, 9/8, -(a*x^8), (-2*a^2*x
^8)/(1 + Sqrt[1 - 4*a^2])])/(1 + Sqrt[1 - 4*a^2])

Rule 429

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, -((b*x^n)/a), -((d*x^n)/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\left (-1+a x^8\right ) \left (1+a x^8\right )^{3/4}}{1+x^8+a^2 x^{16}} \, dx &=\int \left (\frac {\left (a-\frac {a (1+2 a)}{\sqrt {1-4 a^2}}\right ) \left (1+a x^8\right )^{3/4}}{1-\sqrt {1-4 a^2}+2 a^2 x^8}+\frac {\left (a+\frac {a (1+2 a)}{\sqrt {1-4 a^2}}\right ) \left (1+a x^8\right )^{3/4}}{1+\sqrt {1-4 a^2}+2 a^2 x^8}\right ) \, dx\\ &=\left (a \left (1-\frac {1+2 a}{\sqrt {1-4 a^2}}\right )\right ) \int \frac {\left (1+a x^8\right )^{3/4}}{1-\sqrt {1-4 a^2}+2 a^2 x^8} \, dx+\left (a \left (1+\frac {1+2 a}{\sqrt {1-4 a^2}}\right )\right ) \int \frac {\left (1+a x^8\right )^{3/4}}{1+\sqrt {1-4 a^2}+2 a^2 x^8} \, dx\\ &=\frac {a \left (1-\frac {1+2 a}{\sqrt {1-4 a^2}}\right ) x F_1\left (\frac {1}{8};-\frac {3}{4},1;\frac {9}{8};-a x^8,-\frac {2 a^2 x^8}{1-\sqrt {1-4 a^2}}\right )}{1-\sqrt {1-4 a^2}}+\frac {a \left (1+\frac {1+2 a}{\sqrt {1-4 a^2}}\right ) x F_1\left (\frac {1}{8};-\frac {3}{4},1;\frac {9}{8};-a x^8,-\frac {2 a^2 x^8}{1+\sqrt {1-4 a^2}}\right )}{1+\sqrt {1-4 a^2}}\\ \end {align*}

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Mathematica [F]  time = 0.24, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+a x^8\right ) \left (1+a x^8\right )^{3/4}}{1+x^8+a^2 x^{16}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((-1 + a*x^8)*(1 + a*x^8)^(3/4))/(1 + x^8 + a^2*x^16),x]

[Out]

Integrate[((-1 + a*x^8)*(1 + a*x^8)^(3/4))/(1 + x^8 + a^2*x^16), x]

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IntegrateAlgebraic [A]  time = 14.48, size = 486, normalized size = 1.12 \begin {gather*} \frac {i \left (i \sqrt {2}+\sqrt {2 \left (3+2 \sqrt {2}\right )}\right ) \tan ^{-1}\left (\frac {\left ((-1+i)-(1+i) (-1)^{3/4}\right ) \sqrt [4]{-1+2 a} x^2+(1+i) \sqrt {1+a x^8}+(1+i) (-1)^{3/4} \sqrt {1+a x^8}}{2 \sqrt [8]{-1+2 a} x \sqrt [4]{1+a x^8}}\right )}{16 \sqrt [8]{-1+2 a}}+\frac {\left (\sqrt {2}+i \sqrt {2 \left (3-2 \sqrt {2}\right )}\right ) \tan ^{-1}\left (\frac {2 \sqrt [8]{-1+2 a} x \sqrt [4]{1+a x^8}}{\left ((1-i)+\sqrt {2}\right ) \sqrt [4]{-1+2 a} x^2-(1+i) \sqrt {1+a x^8}-\sqrt {2} \sqrt {1+a x^8}}\right )}{16 \sqrt [8]{-1+2 a}}+\frac {\left (i+(-1)^{3/4}\right ) \tanh ^{-1}\left (\frac {\left ((-2+2 i)-(2+2 i) (-1)^{3/4}\right ) \sqrt [4]{-1+2 a} x^2-(2+2 i) \sqrt {1+a x^8}-(2+2 i) (-1)^{3/4} \sqrt {1+a x^8}}{4 \sqrt [8]{-1+2 a} x \sqrt [4]{1+a x^8}}\right )}{8 \sqrt [8]{-1+2 a}}-\frac {i \left (-i \sqrt {2}+\sqrt {2 \left (3-2 \sqrt {2}\right )}\right ) \tanh ^{-1}\left (\frac {\left ((1-i)+\sqrt {2}\right ) \sqrt [4]{-1+2 a} x^2+(1+i) \sqrt {1+a x^8}+\sqrt {2} \sqrt {1+a x^8}}{2 \sqrt [8]{-1+2 a} x \sqrt [4]{1+a x^8}}\right )}{16 \sqrt [8]{-1+2 a}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 + a*x^8)*(1 + a*x^8)^(3/4))/(1 + x^8 + a^2*x^16),x]

[Out]

((I/16)*(I*Sqrt[2] + Sqrt[2*(3 + 2*Sqrt[2])])*ArcTan[(((-1 + I) - (1 + I)*(-1)^(3/4))*(-1 + 2*a)^(1/4)*x^2 + (
1 + I)*Sqrt[1 + a*x^8] + (1 + I)*(-1)^(3/4)*Sqrt[1 + a*x^8])/(2*(-1 + 2*a)^(1/8)*x*(1 + a*x^8)^(1/4))])/(-1 +
2*a)^(1/8) + ((Sqrt[2] + I*Sqrt[2*(3 - 2*Sqrt[2])])*ArcTan[(2*(-1 + 2*a)^(1/8)*x*(1 + a*x^8)^(1/4))/(((1 - I)
+ Sqrt[2])*(-1 + 2*a)^(1/4)*x^2 - (1 + I)*Sqrt[1 + a*x^8] - Sqrt[2]*Sqrt[1 + a*x^8])])/(16*(-1 + 2*a)^(1/8)) +
((I + (-1)^(3/4))*ArcTanh[(((-2 + 2*I) - (2 + 2*I)*(-1)^(3/4))*(-1 + 2*a)^(1/4)*x^2 - (2 + 2*I)*Sqrt[1 + a*x^
8] - (2 + 2*I)*(-1)^(3/4)*Sqrt[1 + a*x^8])/(4*(-1 + 2*a)^(1/8)*x*(1 + a*x^8)^(1/4))])/(8*(-1 + 2*a)^(1/8)) - (
(I/16)*((-I)*Sqrt[2] + Sqrt[2*(3 - 2*Sqrt[2])])*ArcTanh[(((1 - I) + Sqrt[2])*(-1 + 2*a)^(1/4)*x^2 + (1 + I)*Sq
rt[1 + a*x^8] + Sqrt[2]*Sqrt[1 + a*x^8])/(2*(-1 + 2*a)^(1/8)*x*(1 + a*x^8)^(1/4))])/(-1 + 2*a)^(1/8)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^8-1)*(a*x^8+1)^(3/4)/(a^2*x^16+x^8+1),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{8} + 1\right )}^{\frac {3}{4}} {\left (a x^{8} - 1\right )}}{a^{2} x^{16} + x^{8} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^8-1)*(a*x^8+1)^(3/4)/(a^2*x^16+x^8+1),x, algorithm="giac")

[Out]

integrate((a*x^8 + 1)^(3/4)*(a*x^8 - 1)/(a^2*x^16 + x^8 + 1), x)

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{8}-1\right ) \left (a \,x^{8}+1\right )^{\frac {3}{4}}}{a^{2} x^{16}+x^{8}+1}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^8-1)*(a*x^8+1)^(3/4)/(a^2*x^16+x^8+1),x)

[Out]

int((a*x^8-1)*(a*x^8+1)^(3/4)/(a^2*x^16+x^8+1),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{8} + 1\right )}^{\frac {3}{4}} {\left (a x^{8} - 1\right )}}{a^{2} x^{16} + x^{8} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^8-1)*(a*x^8+1)^(3/4)/(a^2*x^16+x^8+1),x, algorithm="maxima")

[Out]

integrate((a*x^8 + 1)^(3/4)*(a*x^8 - 1)/(a^2*x^16 + x^8 + 1), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (a\,x^8-1\right )\,{\left (a\,x^8+1\right )}^{3/4}}{a^2\,x^{16}+x^8+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*x^8 - 1)*(a*x^8 + 1)^(3/4))/(x^8 + a^2*x^16 + 1),x)

[Out]

int(((a*x^8 - 1)*(a*x^8 + 1)^(3/4))/(x^8 + a^2*x^16 + 1), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x**8-1)*(a*x**8+1)**(3/4)/(a**2*x**16+x**8+1),x)

[Out]

Timed out

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