Optimal. Leaf size=204 \[ -\frac {x^3}{4 c \left (a+c x^4\right )}-\frac {3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} \sqrt [4]{a} c^{7/4}}+\frac {3 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} \sqrt [4]{a} c^{7/4}}+\frac {3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} \sqrt [4]{a} c^{7/4}}-\frac {3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} \sqrt [4]{a} c^{7/4}} \]
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Rubi [A]
time = 0.09, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {294, 303,
1176, 631, 210, 1179, 642} \begin {gather*} -\frac {3 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} \sqrt [4]{a} c^{7/4}}+\frac {3 \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt {2} \sqrt [4]{a} c^{7/4}}+\frac {3 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{16 \sqrt {2} \sqrt [4]{a} c^{7/4}}-\frac {3 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{16 \sqrt {2} \sqrt [4]{a} c^{7/4}}-\frac {x^3}{4 c \left (a+c x^4\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 294
Rule 303
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {x^6}{\left (a+c x^4\right )^2} \, dx &=-\frac {x^3}{4 c \left (a+c x^4\right )}+\frac {3 \int \frac {x^2}{a+c x^4} \, dx}{4 c}\\ &=-\frac {x^3}{4 c \left (a+c x^4\right )}-\frac {3 \int \frac {\sqrt {a}-\sqrt {c} x^2}{a+c x^4} \, dx}{8 c^{3/2}}+\frac {3 \int \frac {\sqrt {a}+\sqrt {c} x^2}{a+c x^4} \, dx}{8 c^{3/2}}\\ &=-\frac {x^3}{4 c \left (a+c x^4\right )}+\frac {3 \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 c^2}+\frac {3 \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 c^2}+\frac {3 \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{16 \sqrt {2} \sqrt [4]{a} c^{7/4}}+\frac {3 \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{16 \sqrt {2} \sqrt [4]{a} c^{7/4}}\\ &=-\frac {x^3}{4 c \left (a+c x^4\right )}+\frac {3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} \sqrt [4]{a} c^{7/4}}-\frac {3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} \sqrt [4]{a} c^{7/4}}+\frac {3 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} \sqrt [4]{a} c^{7/4}}-\frac {3 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} \sqrt [4]{a} c^{7/4}}\\ &=-\frac {x^3}{4 c \left (a+c x^4\right )}-\frac {3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} \sqrt [4]{a} c^{7/4}}+\frac {3 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} \sqrt [4]{a} c^{7/4}}+\frac {3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} \sqrt [4]{a} c^{7/4}}-\frac {3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} \sqrt [4]{a} c^{7/4}}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 185, normalized size = 0.91 \begin {gather*} \frac {-\frac {8 c^{3/4} x^3}{a+c x^4}-\frac {6 \sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt [4]{a}}+\frac {6 \sqrt {2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt [4]{a}}+\frac {3 \sqrt {2} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{\sqrt [4]{a}}-\frac {3 \sqrt {2} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{\sqrt [4]{a}}}{32 c^{7/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 120, normalized size = 0.59
method | result | size |
risch | \(-\frac {x^{3}}{4 c \left (x^{4} c +a \right )}+\frac {3 \left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{4}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}}\right )}{16 c^{2}}\) | \(45\) |
default | \(-\frac {x^{3}}{4 c \left (x^{4} c +a \right )}+\frac {3 \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{32 c^{2} \left (\frac {a}{c}\right )^{\frac {1}{4}}}\) | \(120\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 192, normalized size = 0.94 \begin {gather*} -\frac {x^{3}}{4 \, {\left (c^{2} x^{4} + a c\right )}} + \frac {3 \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} - \frac {\sqrt {2} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {1}{4}} c^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {1}{4}} c^{\frac {3}{4}}}\right )}}{32 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 181, normalized size = 0.89 \begin {gather*} -\frac {4 \, x^{3} + 12 \, {\left (c^{2} x^{4} + a c\right )} \left (-\frac {1}{a c^{7}}\right )^{\frac {1}{4}} \arctan \left (-c^{2} x \left (-\frac {1}{a c^{7}}\right )^{\frac {1}{4}} + \sqrt {-a c^{3} \sqrt {-\frac {1}{a c^{7}}} + x^{2}} c^{2} \left (-\frac {1}{a c^{7}}\right )^{\frac {1}{4}}\right ) - 3 \, {\left (c^{2} x^{4} + a c\right )} \left (-\frac {1}{a c^{7}}\right )^{\frac {1}{4}} \log \left (a c^{5} \left (-\frac {1}{a c^{7}}\right )^{\frac {3}{4}} + x\right ) + 3 \, {\left (c^{2} x^{4} + a c\right )} \left (-\frac {1}{a c^{7}}\right )^{\frac {1}{4}} \log \left (-a c^{5} \left (-\frac {1}{a c^{7}}\right )^{\frac {3}{4}} + x\right )}{16 \, {\left (c^{2} x^{4} + a c\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.14, size = 44, normalized size = 0.22 \begin {gather*} - \frac {x^{3}}{4 a c + 4 c^{2} x^{4}} + \operatorname {RootSum} {\left (65536 t^{4} a c^{7} + 81, \left ( t \mapsto t \log {\left (\frac {4096 t^{3} a c^{5}}{27} + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.86, size = 196, normalized size = 0.96 \begin {gather*} -\frac {x^{3}}{4 \, {\left (c x^{4} + a\right )} c} + \frac {3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{16 \, a c^{4}} + \frac {3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{16 \, a c^{4}} - \frac {3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{32 \, a c^{4}} + \frac {3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{32 \, a c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.08, size = 60, normalized size = 0.29 \begin {gather*} \frac {3\,\mathrm {atan}\left (\frac {c^{1/4}\,x}{{\left (-a\right )}^{1/4}}\right )}{8\,{\left (-a\right )}^{1/4}\,c^{7/4}}-\frac {3\,\mathrm {atanh}\left (\frac {c^{1/4}\,x}{{\left (-a\right )}^{1/4}}\right )}{8\,{\left (-a\right )}^{1/4}\,c^{7/4}}-\frac {x^3}{4\,c\,\left (c\,x^4+a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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