3.21.99 \(\int \frac {(1+x^8)^{3/4}}{-1+x^8} \, dx\)

Optimal. Leaf size=152 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^8+1}}\right )}{4 \sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^8+1}}\right )}{4 \sqrt [4]{2}}+\frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{x^8+1}}{\sqrt {2} x^2-\sqrt {x^8+1}}\right )}{4\ 2^{3/4}}-\frac {\tanh ^{-1}\left (\frac {2 \sqrt [4]{2} x \sqrt [4]{x^8+1}}{\sqrt {2} \sqrt {x^8+1}+2 x^2}\right )}{4\ 2^{3/4}} \]

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Rubi [C]  time = 0.01, antiderivative size = 22, normalized size of antiderivative = 0.14, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {429} \begin {gather*} -x F_1\left (\frac {1}{8};1,-\frac {3}{4};\frac {9}{8};x^8,-x^8\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[(1 + x^8)^(3/4)/(-1 + x^8),x]

[Out]

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

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])

Rubi steps

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

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Mathematica [C]  time = 0.12, size = 110, normalized size = 0.72 \begin {gather*} \frac {9 x \left (x^8+1\right )^{3/4} F_1\left (\frac {1}{8};-\frac {3}{4},1;\frac {9}{8};-x^8,x^8\right )}{\left (x^8-1\right ) \left (2 x^8 \left (4 F_1\left (\frac {9}{8};-\frac {3}{4},2;\frac {17}{8};-x^8,x^8\right )+3 F_1\left (\frac {9}{8};\frac {1}{4},1;\frac {17}{8};-x^8,x^8\right )\right )+9 F_1\left (\frac {1}{8};-\frac {3}{4},1;\frac {9}{8};-x^8,x^8\right )\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(1 + x^8)^(3/4)/(-1 + x^8),x]

[Out]

(9*x*(1 + x^8)^(3/4)*AppellF1[1/8, -3/4, 1, 9/8, -x^8, x^8])/((-1 + x^8)*(9*AppellF1[1/8, -3/4, 1, 9/8, -x^8,
x^8] + 2*x^8*(4*AppellF1[9/8, -3/4, 2, 17/8, -x^8, x^8] + 3*AppellF1[9/8, 1/4, 1, 17/8, -x^8, x^8])))

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IntegrateAlgebraic [A]  time = 0.52, size = 152, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^8}}\right )}{4 \sqrt [4]{2}}+\frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{1+x^8}}{\sqrt {2} x^2-\sqrt {1+x^8}}\right )}{4\ 2^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^8}}\right )}{4 \sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {2 \sqrt [4]{2} x \sqrt [4]{1+x^8}}{2 x^2+\sqrt {2} \sqrt {1+x^8}}\right )}{4\ 2^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 + x^8)^(3/4)/(-1 + x^8),x]

[Out]

-1/4*ArcTan[(2^(1/4)*x)/(1 + x^8)^(1/4)]/2^(1/4) + ArcTan[(2^(3/4)*x*(1 + x^8)^(1/4))/(Sqrt[2]*x^2 - Sqrt[1 +
x^8])]/(4*2^(3/4)) - ArcTanh[(2^(1/4)*x)/(1 + x^8)^(1/4)]/(4*2^(1/4)) - ArcTanh[(2*2^(1/4)*x*(1 + x^8)^(1/4))/
(2*x^2 + Sqrt[2]*Sqrt[1 + x^8])]/(4*2^(3/4))

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fricas [B]  time = 17.23, size = 677, normalized size = 4.45 \begin {gather*} -\frac {1}{8} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {2^{\frac {1}{4}} {\left (x^{8} + 1\right )}^{\frac {3}{4}} x^{2}}{x^{9} + x}\right ) - \frac {1}{32} \cdot 2^{\frac {3}{4}} \log \left (-\frac {4 \cdot 2^{\frac {1}{4}} {\left (x^{8} + 1\right )}^{\frac {1}{4}} x^{3} + 2 \cdot 2^{\frac {3}{4}} {\left (x^{8} + 1\right )}^{\frac {3}{4}} x + 4 \, \sqrt {x^{8} + 1} x^{2} + \sqrt {2} {\left (x^{8} + 2 \, x^{4} + 1\right )}}{x^{8} - 2 \, x^{4} + 1}\right ) + \frac {1}{32} \cdot 2^{\frac {3}{4}} \log \left (\frac {4 \cdot 2^{\frac {1}{4}} {\left (x^{8} + 1\right )}^{\frac {1}{4}} x^{3} + 2 \cdot 2^{\frac {3}{4}} {\left (x^{8} + 1\right )}^{\frac {3}{4}} x - 4 \, \sqrt {x^{8} + 1} x^{2} - \sqrt {2} {\left (x^{8} + 2 \, x^{4} + 1\right )}}{x^{8} - 2 \, x^{4} + 1}\right ) - \frac {1}{8} \cdot 2^{\frac {1}{4}} \arctan \left (\frac {4 \cdot 2^{\frac {1}{4}} {\left (x^{8} + 1\right )}^{\frac {1}{4}} x^{3} + 2 \cdot 2^{\frac {3}{4}} {\left (x^{8} + 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (2 \cdot 2^{\frac {3}{4}} {\left (x^{8} + 1\right )}^{\frac {1}{4}} x^{3} - 4 \, \sqrt {x^{8} + 1} x^{2} + 2 \cdot 2^{\frac {1}{4}} {\left (x^{8} + 1\right )}^{\frac {3}{4}} x - \sqrt {2} {\left (x^{8} + 2 \, x^{4} + 1\right )}\right )} \sqrt {\frac {x^{8} + 2 \, x^{4} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{8} + 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {2} \sqrt {x^{8} + 1} x^{2} + 2 \cdot 2^{\frac {3}{4}} {\left (x^{8} + 1\right )}^{\frac {3}{4}} x + 1}{x^{8} + 2 \, x^{4} + 1}}}{2 \, {\left (x^{8} - 2 \, x^{4} + 1\right )}}\right ) - \frac {1}{8} \cdot 2^{\frac {1}{4}} \arctan \left (\frac {4 \cdot 2^{\frac {1}{4}} {\left (x^{8} + 1\right )}^{\frac {1}{4}} x^{3} + 2 \cdot 2^{\frac {3}{4}} {\left (x^{8} + 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (2 \cdot 2^{\frac {3}{4}} {\left (x^{8} + 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{8} + 1} x^{2} + 2 \cdot 2^{\frac {1}{4}} {\left (x^{8} + 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (x^{8} + 2 \, x^{4} + 1\right )}\right )} \sqrt {\frac {x^{8} + 2 \, x^{4} - 4 \cdot 2^{\frac {1}{4}} {\left (x^{8} + 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {2} \sqrt {x^{8} + 1} x^{2} - 2 \cdot 2^{\frac {3}{4}} {\left (x^{8} + 1\right )}^{\frac {3}{4}} x + 1}{x^{8} + 2 \, x^{4} + 1}}}{2 \, {\left (x^{8} - 2 \, x^{4} + 1\right )}}\right ) - \frac {1}{32} \cdot 2^{\frac {1}{4}} \log \left (\frac {2 \, {\left (x^{8} + 2 \, x^{4} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{8} + 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {2} \sqrt {x^{8} + 1} x^{2} + 2 \cdot 2^{\frac {3}{4}} {\left (x^{8} + 1\right )}^{\frac {3}{4}} x + 1\right )}}{x^{8} + 2 \, x^{4} + 1}\right ) + \frac {1}{32} \cdot 2^{\frac {1}{4}} \log \left (\frac {2 \, {\left (x^{8} + 2 \, x^{4} - 4 \cdot 2^{\frac {1}{4}} {\left (x^{8} + 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {2} \sqrt {x^{8} + 1} x^{2} - 2 \cdot 2^{\frac {3}{4}} {\left (x^{8} + 1\right )}^{\frac {3}{4}} x + 1\right )}}{x^{8} + 2 \, x^{4} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*2^(3/4)*arctan(2^(1/4)*(x^8 + 1)^(3/4)*x^2/(x^9 + x)) - 1/32*2^(3/4)*log(-(4*2^(1/4)*(x^8 + 1)^(1/4)*x^3
+ 2*2^(3/4)*(x^8 + 1)^(3/4)*x + 4*sqrt(x^8 + 1)*x^2 + sqrt(2)*(x^8 + 2*x^4 + 1))/(x^8 - 2*x^4 + 1)) + 1/32*2^(
3/4)*log((4*2^(1/4)*(x^8 + 1)^(1/4)*x^3 + 2*2^(3/4)*(x^8 + 1)^(3/4)*x - 4*sqrt(x^8 + 1)*x^2 - sqrt(2)*(x^8 + 2
*x^4 + 1))/(x^8 - 2*x^4 + 1)) - 1/8*2^(1/4)*arctan(1/2*(4*2^(1/4)*(x^8 + 1)^(1/4)*x^3 + 2*2^(3/4)*(x^8 + 1)^(3
/4)*x + sqrt(2)*(2*2^(3/4)*(x^8 + 1)^(1/4)*x^3 - 4*sqrt(x^8 + 1)*x^2 + 2*2^(1/4)*(x^8 + 1)^(3/4)*x - sqrt(2)*(
x^8 + 2*x^4 + 1))*sqrt((x^8 + 2*x^4 + 4*2^(1/4)*(x^8 + 1)^(1/4)*x^3 + 4*sqrt(2)*sqrt(x^8 + 1)*x^2 + 2*2^(3/4)*
(x^8 + 1)^(3/4)*x + 1)/(x^8 + 2*x^4 + 1)))/(x^8 - 2*x^4 + 1)) - 1/8*2^(1/4)*arctan(1/2*(4*2^(1/4)*(x^8 + 1)^(1
/4)*x^3 + 2*2^(3/4)*(x^8 + 1)^(3/4)*x + sqrt(2)*(2*2^(3/4)*(x^8 + 1)^(1/4)*x^3 + 4*sqrt(x^8 + 1)*x^2 + 2*2^(1/
4)*(x^8 + 1)^(3/4)*x + sqrt(2)*(x^8 + 2*x^4 + 1))*sqrt((x^8 + 2*x^4 - 4*2^(1/4)*(x^8 + 1)^(1/4)*x^3 + 4*sqrt(2
)*sqrt(x^8 + 1)*x^2 - 2*2^(3/4)*(x^8 + 1)^(3/4)*x + 1)/(x^8 + 2*x^4 + 1)))/(x^8 - 2*x^4 + 1)) - 1/32*2^(1/4)*l
og(2*(x^8 + 2*x^4 + 4*2^(1/4)*(x^8 + 1)^(1/4)*x^3 + 4*sqrt(2)*sqrt(x^8 + 1)*x^2 + 2*2^(3/4)*(x^8 + 1)^(3/4)*x
+ 1)/(x^8 + 2*x^4 + 1)) + 1/32*2^(1/4)*log(2*(x^8 + 2*x^4 - 4*2^(1/4)*(x^8 + 1)^(1/4)*x^3 + 4*sqrt(2)*sqrt(x^8
+ 1)*x^2 - 2*2^(3/4)*(x^8 + 1)^(3/4)*x + 1)/(x^8 + 2*x^4 + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x^8 + 1)^(3/4)/(x^8 - 1), x)

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maple [F]  time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{8}+1\right )^{\frac {3}{4}}}{x^{8}-1}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^8+1)^(3/4)/(x^8-1),x)

[Out]

int((x^8+1)^(3/4)/(x^8-1),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x^8 + 1)^(3/4)/(x^8 - 1), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^8+1\right )}^{3/4}}{x^8-1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^8 + 1)^(3/4)/(x^8 - 1),x)

[Out]

int((x^8 + 1)^(3/4)/(x^8 - 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((x**8 + 1)**(3/4)/((x - 1)*(x + 1)*(x**2 + 1)*(x**4 + 1)), x)

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