Optimal. Leaf size=144 \[ \frac {3 \sqrt [3]{1-x^7}}{2 x}-\frac {\log \left (2^{2/3} \sqrt [3]{1-x^7}-x\right )}{2\ 2^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2\ 2^{2/3} \sqrt [3]{1-x^7}+x}\right )}{2\ 2^{2/3}}+\frac {\log \left (2^{2/3} \sqrt [3]{1-x^7} x+2 \sqrt [3]{2} \left (1-x^7\right )^{2/3}+x^2\right )}{4\ 2^{2/3}} \]
________________________________________________________________________________________
Rubi [F] time = 1.60, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\sqrt [3]{1-x^7} \left (-2+x^3+2 x^7\right ) \left (3+4 x^7\right )}{x^2 \left (-1+x^7\right ) \left (-4+x^3+4 x^7\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{1-x^7} \left (-2+x^3+2 x^7\right ) \left (3+4 x^7\right )}{x^2 \left (-1+x^7\right ) \left (-4+x^3+4 x^7\right )} \, dx &=-\int \frac {\left (-2+x^3+2 x^7\right ) \left (3+4 x^7\right )}{x^2 \left (1-x^7\right )^{2/3} \left (-4+x^3+4 x^7\right )} \, dx\\ &=-\int \left (\frac {3}{2 x^2 \left (1-x^7\right )^{2/3}}+\frac {x}{2 \left (1-x^7\right )^{2/3}}+\frac {2 x^5}{\left (1-x^7\right )^{2/3}}-\frac {x \left (-7+x^3\right )}{2 \left (1-x^7\right )^{2/3} \left (-4+x^3+4 x^7\right )}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {x}{\left (1-x^7\right )^{2/3}} \, dx\right )+\frac {1}{2} \int \frac {x \left (-7+x^3\right )}{\left (1-x^7\right )^{2/3} \left (-4+x^3+4 x^7\right )} \, dx-\frac {3}{2} \int \frac {1}{x^2 \left (1-x^7\right )^{2/3}} \, dx-2 \int \frac {x^5}{\left (1-x^7\right )^{2/3}} \, dx\\ &=\frac {3 \, _2F_1\left (-\frac {1}{7},\frac {2}{3};\frac {6}{7};x^7\right )}{2 x}-\frac {1}{4} x^2 \, _2F_1\left (\frac {2}{7},\frac {2}{3};\frac {9}{7};x^7\right )-\frac {1}{3} x^6 \, _2F_1\left (\frac {2}{3},\frac {6}{7};\frac {13}{7};x^7\right )+\frac {1}{2} \int \left (-\frac {7 x}{\left (1-x^7\right )^{2/3} \left (-4+x^3+4 x^7\right )}+\frac {x^4}{\left (1-x^7\right )^{2/3} \left (-4+x^3+4 x^7\right )}\right ) \, dx\\ &=\frac {3 \, _2F_1\left (-\frac {1}{7},\frac {2}{3};\frac {6}{7};x^7\right )}{2 x}-\frac {1}{4} x^2 \, _2F_1\left (\frac {2}{7},\frac {2}{3};\frac {9}{7};x^7\right )-\frac {1}{3} x^6 \, _2F_1\left (\frac {2}{3},\frac {6}{7};\frac {13}{7};x^7\right )+\frac {1}{2} \int \frac {x^4}{\left (1-x^7\right )^{2/3} \left (-4+x^3+4 x^7\right )} \, dx-\frac {7}{2} \int \frac {x}{\left (1-x^7\right )^{2/3} \left (-4+x^3+4 x^7\right )} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 0.42, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{1-x^7} \left (-2+x^3+2 x^7\right ) \left (3+4 x^7\right )}{x^2 \left (-1+x^7\right ) \left (-4+x^3+4 x^7\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 18.09, size = 144, normalized size = 1.00 \begin {gather*} \frac {3 \sqrt [3]{1-x^7}}{2 x}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2\ 2^{2/3} \sqrt [3]{1-x^7}}\right )}{2\ 2^{2/3}}-\frac {\log \left (-x+2^{2/3} \sqrt [3]{1-x^7}\right )}{2\ 2^{2/3}}+\frac {\log \left (x^2+2^{2/3} x \sqrt [3]{1-x^7}+2 \sqrt [3]{2} \left (1-x^7\right )^{2/3}\right )}{4\ 2^{2/3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (4 \, x^{7} + 3\right )} {\left (2 \, x^{7} + x^{3} - 2\right )} {\left (-x^{7} + 1\right )}^{\frac {1}{3}}}{{\left (4 \, x^{7} + x^{3} - 4\right )} {\left (x^{7} - 1\right )} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 61.10, size = 1486, normalized size = 10.32 \[\text {Expression too large to display}\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (4 \, x^{7} + 3\right )} {\left (2 \, x^{7} + x^{3} - 2\right )} {\left (-x^{7} + 1\right )}^{\frac {1}{3}}}{{\left (4 \, x^{7} + x^{3} - 4\right )} {\left (x^{7} - 1\right )} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {\left (4\,x^7+3\right )\,\left (2\,x^7+x^3-2\right )}{x^2\,{\left (1-x^7\right )}^{2/3}\,\left (4\,x^7+x^3-4\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________