Optimal. Leaf size=285 \[ -\frac {\sqrt {3} b e \text {ArcTan}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{4 c^{2/3}}+\frac {\sqrt {3} b e \text {ArcTan}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{4 c^{2/3}}+\frac {\sqrt {3} b d \text {ArcTan}\left (\frac {1+2 c^{2/3} x^2}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}-\frac {b e \tanh ^{-1}\left (\sqrt [3]{c} x\right )}{2 c^{2/3}}-\frac {b d^2 \tanh ^{-1}\left (c x^3\right )}{2 e}+\frac {(d+e x)^2 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{2 e}+\frac {b d \log \left (1-c^{2/3} x^2\right )}{2 \sqrt [3]{c}}+\frac {b e \log \left (1-\sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}-\frac {b e \log \left (1+\sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}-\frac {b d \log \left (1+c^{2/3} x^2+c^{4/3} x^4\right )}{4 \sqrt [3]{c}} \]
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Rubi [A]
time = 0.31, antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps
used = 22, number of rules used = 12, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6071, 1845,
281, 212, 298, 31, 648, 631, 210, 642, 302, 632} \begin {gather*} \frac {(d+e x)^2 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{2 e}+\frac {\sqrt {3} b d \text {ArcTan}\left (\frac {2 c^{2/3} x^2+1}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}-\frac {\sqrt {3} b e \text {ArcTan}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{4 c^{2/3}}+\frac {\sqrt {3} b e \text {ArcTan}\left (\frac {2 \sqrt [3]{c} x}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{4 c^{2/3}}+\frac {b d \log \left (1-c^{2/3} x^2\right )}{2 \sqrt [3]{c}}-\frac {b d \log \left (c^{4/3} x^4+c^{2/3} x^2+1\right )}{4 \sqrt [3]{c}}+\frac {b e \log \left (c^{2/3} x^2-\sqrt [3]{c} x+1\right )}{8 c^{2/3}}-\frac {b e \log \left (c^{2/3} x^2+\sqrt [3]{c} x+1\right )}{8 c^{2/3}}-\frac {b e \tanh ^{-1}\left (\sqrt [3]{c} x\right )}{2 c^{2/3}}-\frac {b d^2 \tanh ^{-1}\left (c x^3\right )}{2 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 210
Rule 212
Rule 281
Rule 298
Rule 302
Rule 631
Rule 632
Rule 642
Rule 648
Rule 1845
Rule 6071
Rubi steps
\begin {align*} \int (d+e x) \left (a+b \tanh ^{-1}\left (c x^3\right )\right ) \, dx &=\frac {(d+e x)^2 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{2 e}-\frac {b \int \frac {3 c x^2 (d+e x)^2}{1-c^2 x^6} \, dx}{2 e}\\ &=\frac {(d+e x)^2 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{2 e}-\frac {(3 b c) \int \frac {x^2 (d+e x)^2}{1-c^2 x^6} \, dx}{2 e}\\ &=\frac {(d+e x)^2 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{2 e}-\frac {(3 b c) \int \left (\frac {d^2 x^2}{1-c^2 x^6}+\frac {2 d e x^3}{1-c^2 x^6}+\frac {e^2 x^4}{1-c^2 x^6}\right ) \, dx}{2 e}\\ &=\frac {(d+e x)^2 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{2 e}-(3 b c d) \int \frac {x^3}{1-c^2 x^6} \, dx-\frac {\left (3 b c d^2\right ) \int \frac {x^2}{1-c^2 x^6} \, dx}{2 e}-\frac {1}{2} (3 b c e) \int \frac {x^4}{1-c^2 x^6} \, dx\\ &=\frac {(d+e x)^2 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{2 e}-\frac {1}{2} (3 b c d) \text {Subst}\left (\int \frac {x}{1-c^2 x^3} \, dx,x,x^2\right )-\frac {\left (b c d^2\right ) \text {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,x^3\right )}{2 e}-\frac {(b e) \int \frac {1}{1-c^{2/3} x^2} \, dx}{2 \sqrt [3]{c}}-\frac {(b e) \int \frac {-\frac {1}{2}-\frac {\sqrt [3]{c} x}{2}}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx}{2 \sqrt [3]{c}}-\frac {(b e) \int \frac {-\frac {1}{2}+\frac {\sqrt [3]{c} x}{2}}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx}{2 \sqrt [3]{c}}\\ &=-\frac {b e \tanh ^{-1}\left (\sqrt [3]{c} x\right )}{2 c^{2/3}}-\frac {b d^2 \tanh ^{-1}\left (c x^3\right )}{2 e}+\frac {(d+e x)^2 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{2 e}-\frac {1}{2} \left (b \sqrt [3]{c} d\right ) \text {Subst}\left (\int \frac {1}{1-c^{2/3} x} \, dx,x,x^2\right )+\frac {1}{2} \left (b \sqrt [3]{c} d\right ) \text {Subst}\left (\int \frac {1-c^{2/3} x}{1+c^{2/3} x+c^{4/3} x^2} \, dx,x,x^2\right )+\frac {(b e) \int \frac {-\sqrt [3]{c}+2 c^{2/3} x}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx}{8 c^{2/3}}-\frac {(b e) \int \frac {\sqrt [3]{c}+2 c^{2/3} x}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx}{8 c^{2/3}}+\frac {(3 b e) \int \frac {1}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx}{8 \sqrt [3]{c}}+\frac {(3 b e) \int \frac {1}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx}{8 \sqrt [3]{c}}\\ &=-\frac {b e \tanh ^{-1}\left (\sqrt [3]{c} x\right )}{2 c^{2/3}}-\frac {b d^2 \tanh ^{-1}\left (c x^3\right )}{2 e}+\frac {(d+e x)^2 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{2 e}+\frac {b d \log \left (1-c^{2/3} x^2\right )}{2 \sqrt [3]{c}}+\frac {b e \log \left (1-\sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}-\frac {b e \log \left (1+\sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}-\frac {(b d) \text {Subst}\left (\int \frac {c^{2/3}+2 c^{4/3} x}{1+c^{2/3} x+c^{4/3} x^2} \, dx,x,x^2\right )}{4 \sqrt [3]{c}}+\frac {1}{4} \left (3 b \sqrt [3]{c} d\right ) \text {Subst}\left (\int \frac {1}{1+c^{2/3} x+c^{4/3} x^2} \, dx,x,x^2\right )+\frac {(3 b e) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-2 \sqrt [3]{c} x\right )}{4 c^{2/3}}-\frac {(3 b e) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{c} x\right )}{4 c^{2/3}}\\ &=-\frac {\sqrt {3} b e \tan ^{-1}\left (\frac {1-2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{4 c^{2/3}}+\frac {\sqrt {3} b e \tan ^{-1}\left (\frac {1+2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{4 c^{2/3}}-\frac {b e \tanh ^{-1}\left (\sqrt [3]{c} x\right )}{2 c^{2/3}}-\frac {b d^2 \tanh ^{-1}\left (c x^3\right )}{2 e}+\frac {(d+e x)^2 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{2 e}+\frac {b d \log \left (1-c^{2/3} x^2\right )}{2 \sqrt [3]{c}}+\frac {b e \log \left (1-\sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}-\frac {b e \log \left (1+\sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}-\frac {b d \log \left (1+c^{2/3} x^2+c^{4/3} x^4\right )}{4 \sqrt [3]{c}}-\frac {(3 b d) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 c^{2/3} x^2\right )}{2 \sqrt [3]{c}}\\ &=-\frac {\sqrt {3} b e \tan ^{-1}\left (\frac {1-2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{4 c^{2/3}}+\frac {\sqrt {3} b e \tan ^{-1}\left (\frac {1+2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{4 c^{2/3}}+\frac {\sqrt {3} b d \tan ^{-1}\left (\frac {1+2 c^{2/3} x^2}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}-\frac {b e \tanh ^{-1}\left (\sqrt [3]{c} x\right )}{2 c^{2/3}}-\frac {b d^2 \tanh ^{-1}\left (c x^3\right )}{2 e}+\frac {(d+e x)^2 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{2 e}+\frac {b d \log \left (1-c^{2/3} x^2\right )}{2 \sqrt [3]{c}}+\frac {b e \log \left (1-\sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}-\frac {b e \log \left (1+\sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}-\frac {b d \log \left (1+c^{2/3} x^2+c^{4/3} x^4\right )}{4 \sqrt [3]{c}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 333, normalized size = 1.17 \begin {gather*} a d x+\frac {1}{2} a e x^2+\frac {\sqrt {3} b e \text {ArcTan}\left (\frac {-1+2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{4 c^{2/3}}+\frac {\sqrt {3} b e \text {ArcTan}\left (\frac {1+2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{4 c^{2/3}}+b d x \tanh ^{-1}\left (c x^3\right )+\frac {1}{2} b e x^2 \tanh ^{-1}\left (c x^3\right )+\frac {b e \log \left (1-\sqrt [3]{c} x\right )}{4 c^{2/3}}-\frac {b e \log \left (1+\sqrt [3]{c} x\right )}{4 c^{2/3}}+\frac {b e \log \left (1-\sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}-\frac {b e \log \left (1+\sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}-\frac {b d \left (-2 \sqrt {3} \text {ArcTan}\left (\frac {-1+2 \sqrt [3]{c} x}{\sqrt {3}}\right )+2 \sqrt {3} \text {ArcTan}\left (\frac {1+2 \sqrt [3]{c} x}{\sqrt {3}}\right )-2 \log \left (1-\sqrt [3]{c} x\right )-2 \log \left (1+\sqrt [3]{c} x\right )+\log \left (1-\sqrt [3]{c} x+c^{2/3} x^2\right )+\log \left (1+\sqrt [3]{c} x+c^{2/3} x^2\right )\right )}{4 \sqrt [3]{c}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 363, normalized size = 1.27
method | result | size |
default | \(a \left (\frac {1}{2} e \,x^{2}+d x \right )+\frac {b \arctanh \left (c \,x^{3}\right ) x^{2} e}{2}+b \arctanh \left (c \,x^{3}\right ) d x +\frac {b d \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{2 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b d \ln \left (x^{2}-\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{4 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b d \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{2 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b e \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{4 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b e \ln \left (x^{2}-\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{8 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b e \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{4 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b d \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{2 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b d \ln \left (x^{2}+\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{4 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b d \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{2 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b e \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{4 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b e \ln \left (x^{2}+\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{8 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b e \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{4 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}\) | \(363\) |
risch | \(\frac {b \left (\frac {1}{2} e \,x^{2}+d x \right ) \ln \left (c \,x^{3}+1\right )}{2}+\frac {a e \,x^{2}}{2}+a d x -\frac {b d x \ln \left (-c \,x^{3}+1\right )}{2}+\frac {b d \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{2 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b d \ln \left (x^{2}+\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{4 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b d \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{2 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b e \ln \left (-c \,x^{3}+1\right ) x^{2}}{4}+\frac {b e \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{4 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b e \ln \left (x^{2}+\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{8 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b e \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{4 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b d \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{2 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b d \ln \left (x^{2}-\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{4 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b d \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{2 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b e \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{4 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b e \ln \left (x^{2}-\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{8 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b e \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{4 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}\) | \(390\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 250, normalized size = 0.88 \begin {gather*} \frac {1}{2} \, a x^{2} e + \frac {1}{4} \, {\left (c {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {4}{3}} x^{2} + c^{\frac {2}{3}}\right )}}{3 \, c^{\frac {2}{3}}}\right )}{c^{\frac {4}{3}}} - \frac {\log \left (c^{\frac {4}{3}} x^{4} + c^{\frac {2}{3}} x^{2} + 1\right )}{c^{\frac {4}{3}}} + \frac {2 \, \log \left (\frac {c^{\frac {2}{3}} x^{2} - 1}{c^{\frac {2}{3}}}\right )}{c^{\frac {4}{3}}}\right )} + 4 \, x \operatorname {artanh}\left (c x^{3}\right )\right )} b d + a d x + \frac {1}{8} \, {\left (4 \, x^{2} \operatorname {artanh}\left (c x^{3}\right ) + c {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} x + c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right )}{c^{\frac {5}{3}}} + \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} x - c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right )}{c^{\frac {5}{3}}} - \frac {\log \left (c^{\frac {2}{3}} x^{2} + c^{\frac {1}{3}} x + 1\right )}{c^{\frac {5}{3}}} + \frac {\log \left (c^{\frac {2}{3}} x^{2} - c^{\frac {1}{3}} x + 1\right )}{c^{\frac {5}{3}}} - \frac {2 \, \log \left (\frac {c^{\frac {1}{3}} x + 1}{c^{\frac {1}{3}}}\right )}{c^{\frac {5}{3}}} + \frac {2 \, \log \left (\frac {c^{\frac {1}{3}} x - 1}{c^{\frac {1}{3}}}\right )}{c^{\frac {5}{3}}}\right )}\right )} b e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 1.29, size = 14519, normalized size = 50.94 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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.
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Giac [A]
time = 1.09, size = 279, normalized size = 0.98 \begin {gather*} \frac {1}{2} \, a e x^{2} + a d x + \frac {1}{4} \, {\left (b e x^{2} + 2 \, b d x\right )} \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right ) - \frac {\sqrt {3} {\left (2 \, b c d {\left | c \right |}^{\frac {2}{3}} - b c e {\left | c \right |}^{\frac {1}{3}}\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + \frac {1}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{4 \, c^{2}} + \frac {\sqrt {3} {\left (2 \, b c d {\left | c \right |}^{\frac {2}{3}} + b c e {\left | c \right |}^{\frac {1}{3}}\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - \frac {1}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{4 \, c^{2}} - \frac {{\left (2 \, b c d {\left | c \right |}^{\frac {2}{3}} + b c e {\left | c \right |}^{\frac {1}{3}}\right )} \log \left (x^{2} + \frac {x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{8 \, c^{2}} - \frac {{\left (2 \, b c d {\left | c \right |}^{\frac {2}{3}} - b c e {\left | c \right |}^{\frac {1}{3}}\right )} \log \left (x^{2} - \frac {x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{8 \, c^{2}} + \frac {{\left (2 \, b c d {\left | c \right |}^{\frac {2}{3}} - b c e {\left | c \right |}^{\frac {1}{3}}\right )} \log \left ({\left | x + \frac {1}{{\left | c \right |}^{\frac {1}{3}}} \right |}\right )}{4 \, c^{2}} + \frac {{\left (2 \, b c d {\left | c \right |}^{\frac {2}{3}} + b c e {\left | c \right |}^{\frac {1}{3}}\right )} \log \left ({\left | x - \frac {1}{{\left | c \right |}^{\frac {1}{3}}} \right |}\right )}{4 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.03, size = 621, normalized size = 2.18 \begin {gather*} \left (\sum _{k=1}^3\ln \left (-\mathrm {root}\left (64\,c^2\,z^3+24\,b^2\,c\,d\,e\,z-8\,b^3\,c\,d^3+b^3\,e^3,z,k\right )\,\left (\mathrm {root}\left (64\,c^2\,z^3+24\,b^2\,c\,d\,e\,z-8\,b^3\,c\,d^3+b^3\,e^3,z,k\right )\,\left (\mathrm {root}\left (64\,c^2\,z^3+24\,b^2\,c\,d\,e\,z-8\,b^3\,c\,d^3+b^3\,e^3,z,k\right )\,\left (486\,b^2\,c^{10}\,e^2\,x-1944\,b^2\,c^{10}\,d\,e+\mathrm {root}\left (64\,c^2\,z^3+24\,b^2\,c\,d\,e\,z-8\,b^3\,c\,d^3+b^3\,e^3,z,k\right )\,b\,c^{11}\,d\,x\,3888\right )-\frac {243\,b^3\,c^9\,e^3}{2}\right )-486\,b^4\,c^{10}\,d^4\,x\right )+\frac {243\,b^5\,c^9\,d^4\,e}{2}+\frac {243\,b^5\,c^9\,d^3\,e^2\,x}{4}\right )\,\mathrm {root}\left (64\,c^2\,z^3+24\,b^2\,c\,d\,e\,z-8\,b^3\,c\,d^3+b^3\,e^3,z,k\right )\right )+\left (\sum _{k=1}^3\ln \left (-\mathrm {root}\left (64\,c^2\,z^3-24\,b^2\,c\,d\,e\,z-8\,b^3\,c\,d^3-b^3\,e^3,z,k\right )\,\left (\mathrm {root}\left (64\,c^2\,z^3-24\,b^2\,c\,d\,e\,z-8\,b^3\,c\,d^3-b^3\,e^3,z,k\right )\,\left (\mathrm {root}\left (64\,c^2\,z^3-24\,b^2\,c\,d\,e\,z-8\,b^3\,c\,d^3-b^3\,e^3,z,k\right )\,\left (486\,b^2\,c^{10}\,e^2\,x-1944\,b^2\,c^{10}\,d\,e+\mathrm {root}\left (64\,c^2\,z^3-24\,b^2\,c\,d\,e\,z-8\,b^3\,c\,d^3-b^3\,e^3,z,k\right )\,b\,c^{11}\,d\,x\,3888\right )-\frac {243\,b^3\,c^9\,e^3}{2}\right )-486\,b^4\,c^{10}\,d^4\,x\right )+\frac {243\,b^5\,c^9\,d^4\,e}{2}+\frac {243\,b^5\,c^9\,d^3\,e^2\,x}{4}\right )\,\mathrm {root}\left (64\,c^2\,z^3-24\,b^2\,c\,d\,e\,z-8\,b^3\,c\,d^3-b^3\,e^3,z,k\right )\right )+\ln \left (c\,x^3+1\right )\,\left (\frac {b\,e\,x^2}{4}+\frac {b\,d\,x}{2}\right )-\ln \left (1-c\,x^3\right )\,\left (\frac {b\,e\,x^2}{4}+\frac {b\,d\,x}{2}\right )+a\,d\,x+\frac {a\,e\,x^2}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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[Out]
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