Optimal. Leaf size=493 \[ \frac {(a+b) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {(a+b) \tan ^{-1}\left (\frac {1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {c \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {(a-c) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {(b+c) \tan ^{-1}\left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {(a+b) \log \left ((1-x) (1+x)^2\right )}{12 \sqrt [3]{2}}-\frac {(a-c) \log \left (1+x^3\right )}{6 \sqrt [3]{2}}-\frac {(b+c) \log \left (1+x^3\right )}{6 \sqrt [3]{2}}+\frac {(a+b) \log \left (1+\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{6 \sqrt [3]{2}}-\frac {(a+b) \log \left (1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{3 \sqrt [3]{2}}+\frac {(b+c) \log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}+\frac {(a-c) \log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}+\frac {1}{2} c \log \left (x+\sqrt [3]{1-x^3}\right )-\frac {(a+b) \log \left (-1+x+2^{2/3} \sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.28, antiderivative size = 570, normalized size of antiderivative = 1.16, number of steps
used = 19, number of rules used = 14, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {2183, 384,
502, 2174, 206, 31, 648, 631, 210, 642, 455, 57, 494, 245} \begin {gather*} \frac {(a+b) \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {(a+b) \text {ArcTan}\left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {a \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {(a+b) \log \left (\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{6 \sqrt [3]{2}}-\frac {(a+b) \log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3 \sqrt [3]{2}}-\frac {(a+b) \log \left (2^{2/3} \sqrt [3]{1-x^3}+x-1\right )}{4 \sqrt [3]{2}}+\frac {(a+b) \log \left ((1-x) (x+1)^2\right )}{12 \sqrt [3]{2}}-\frac {a \log \left (x^3+1\right )}{6 \sqrt [3]{2}}+\frac {a \log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{2 \sqrt [3]{2}}+\frac {\text {ArcTan}\left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right ) (b+c)}{\sqrt [3]{2} \sqrt {3}}-\frac {c \text {ArcTan}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {c \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {(b+c) \log \left (x^3+1\right )}{6 \sqrt [3]{2}}+\frac {(b+c) \log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}+\frac {c \log \left (x^3+1\right )}{6 \sqrt [3]{2}}-\frac {c \log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{2 \sqrt [3]{2}}+\frac {1}{2} c \log \left (\sqrt [3]{1-x^3}+x\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 57
Rule 206
Rule 210
Rule 245
Rule 384
Rule 455
Rule 494
Rule 502
Rule 631
Rule 642
Rule 648
Rule 2174
Rule 2183
Rubi steps
\begin {align*} \int \frac {a+b x+c x^2}{\left (1-x+x^2\right ) \sqrt [3]{1-x^3}} \, dx &=\int \left (\frac {c}{\sqrt [3]{1-x^3}}+\frac {a-c+(b+c) x}{\left (1-x+x^2\right ) \sqrt [3]{1-x^3}}\right ) \, dx\\ &=c \int \frac {1}{\sqrt [3]{1-x^3}} \, dx+\int \frac {a-c+(b+c) x}{\left (1-x+x^2\right ) \sqrt [3]{1-x^3}} \, dx\\ &=-\frac {c \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} c \log \left (x+\sqrt [3]{1-x^3}\right )+\int \left (\frac {b-\frac {i (2 a+b-c)}{\sqrt {3}}+c}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt [3]{1-x^3}}+\frac {b+\frac {i (2 a+b-c)}{\sqrt {3}}+c}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt [3]{1-x^3}}\right ) \, dx\\ &=-\frac {c \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} c \log \left (x+\sqrt [3]{1-x^3}\right )+\frac {1}{3} \left (3 b-i \sqrt {3} (2 a+b-c)+3 c\right ) \int \frac {1}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt [3]{1-x^3}} \, dx+\frac {1}{3} \left (3 b+i \sqrt {3} (2 a+b-c)+3 c\right ) \int \frac {1}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt [3]{1-x^3}} \, dx\\ &=-\frac {c \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\left (2 a+b-i \sqrt {3} b-c-i \sqrt {3} c\right ) \tan ^{-1}\left (\frac {2-\frac {\sqrt [3]{2} \left (1-i \sqrt {3}+2 x\right )}{\sqrt [3]{1-x^3}}}{2 \sqrt {3}}\right )}{2 \sqrt [3]{2} \left (i+\sqrt {3}\right )}+\frac {\left (2 a+b+i \sqrt {3} b-c+i \sqrt {3} c\right ) \tan ^{-1}\left (\frac {2-\frac {\sqrt [3]{2} \left (1+i \sqrt {3}+2 x\right )}{\sqrt [3]{1-x^3}}}{2 \sqrt {3}}\right )}{2 \sqrt [3]{2} \left (i-\sqrt {3}\right )}+\frac {\left (3 i b-\sqrt {3} \left (2 a+b-c-i \sqrt {3} c\right )\right ) \log \left (-\left (1-i \sqrt {3}-2 x\right )^2 \left (1-i \sqrt {3}+2 x\right )\right )}{12 \sqrt [3]{2} \left (i+\sqrt {3}\right )}+\frac {\left (3 i b+\sqrt {3} \left (2 a+b-c+i \sqrt {3} c\right )\right ) \log \left (-\left (1+i \sqrt {3}-2 x\right )^2 \left (1+i \sqrt {3}+2 x\right )\right )}{12 \sqrt [3]{2} \left (i-\sqrt {3}\right )}+\frac {1}{2} c \log \left (x+\sqrt [3]{1-x^3}\right )-\frac {\left (3 i b-\sqrt {3} \left (2 a+b-c-i \sqrt {3} c\right )\right ) \log \left (1-i \sqrt {3}+2 x+2\ 2^{2/3} \sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2} \left (i+\sqrt {3}\right )}-\frac {\left (3 i b+\sqrt {3} \left (2 a+b-c+i \sqrt {3} c\right )\right ) \log \left (1+i \sqrt {3}+2 x+2\ 2^{2/3} \sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2} \left (i-\sqrt {3}\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F]
time = 10.25, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b x+c x^2}{\left (1-x+x^2\right ) \sqrt [3]{1-x^3}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {c \,x^{2}+b x +a}{\left (x^{2}-x +1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}}}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b x + c x^{2}}{\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} - x + 1\right )}\, dx \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*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {c\,x^2+b\,x+a}{{\left (1-x^3\right )}^{1/3}\,\left (x^2-x+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________