Optimal. Leaf size=308 \[ \frac {\sqrt {x}}{4 a \left (a+c x^4\right )}-\frac {7 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{15/8} \sqrt [8]{c}}+\frac {7 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{15/8} \sqrt [8]{c}}+\frac {7 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{15/8} \sqrt [8]{c}}+\frac {7 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{15/8} \sqrt [8]{c}}-\frac {7 \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{15/8} \sqrt [8]{c}}+\frac {7 \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{15/8} \sqrt [8]{c}} \]
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Rubi [A]
time = 0.20, antiderivative size = 308, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 12, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {296, 335,
220, 218, 214, 211, 217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {7 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{15/8} \sqrt [8]{c}}+\frac {7 \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{16 \sqrt {2} (-a)^{15/8} \sqrt [8]{c}}+\frac {7 \text {ArcTan}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{15/8} \sqrt [8]{c}}+\frac {\sqrt {x}}{4 a \left (a+c x^4\right )}-\frac {7 \log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{15/8} \sqrt [8]{c}}+\frac {7 \log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{15/8} \sqrt [8]{c}}+\frac {7 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{15/8} \sqrt [8]{c}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 211
Rule 214
Rule 217
Rule 218
Rule 220
Rule 296
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x} \left (a+c x^4\right )^2} \, dx &=\frac {\sqrt {x}}{4 a \left (a+c x^4\right )}+\frac {7 \int \frac {1}{\sqrt {x} \left (a+c x^4\right )} \, dx}{8 a}\\ &=\frac {\sqrt {x}}{4 a \left (a+c x^4\right )}+\frac {7 \text {Subst}\left (\int \frac {1}{a+c x^8} \, dx,x,\sqrt {x}\right )}{4 a}\\ &=\frac {\sqrt {x}}{4 a \left (a+c x^4\right )}+\frac {7 \text {Subst}\left (\int \frac {1}{\sqrt {-a}-\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{8 (-a)^{3/2}}+\frac {7 \text {Subst}\left (\int \frac {1}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{8 (-a)^{3/2}}\\ &=\frac {\sqrt {x}}{4 a \left (a+c x^4\right )}+\frac {7 \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{16 (-a)^{7/4}}+\frac {7 \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a}+\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{16 (-a)^{7/4}}+\frac {7 \text {Subst}\left (\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{16 (-a)^{7/4}}+\frac {7 \text {Subst}\left (\int \frac {\sqrt [4]{-a}+\sqrt [4]{c} x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{16 (-a)^{7/4}}\\ &=\frac {\sqrt {x}}{4 a \left (a+c x^4\right )}+\frac {7 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{15/8} \sqrt [8]{c}}+\frac {7 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{15/8} \sqrt [8]{c}}+\frac {7 \text {Subst}\left (\int \frac {1}{\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{c}}+x^2} \, dx,x,\sqrt {x}\right )}{32 (-a)^{7/4} \sqrt [4]{c}}+\frac {7 \text {Subst}\left (\int \frac {1}{\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{c}}+x^2} \, dx,x,\sqrt {x}\right )}{32 (-a)^{7/4} \sqrt [4]{c}}-\frac {7 \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [8]{-a}}{\sqrt [8]{c}}+2 x}{-\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{c}}-x^2} \, dx,x,\sqrt {x}\right )}{32 \sqrt {2} (-a)^{15/8} \sqrt [8]{c}}-\frac {7 \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [8]{-a}}{\sqrt [8]{c}}-2 x}{-\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{c}}-x^2} \, dx,x,\sqrt {x}\right )}{32 \sqrt {2} (-a)^{15/8} \sqrt [8]{c}}\\ &=\frac {\sqrt {x}}{4 a \left (a+c x^4\right )}+\frac {7 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{15/8} \sqrt [8]{c}}+\frac {7 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{15/8} \sqrt [8]{c}}-\frac {7 \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{15/8} \sqrt [8]{c}}+\frac {7 \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{15/8} \sqrt [8]{c}}+\frac {7 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{15/8} \sqrt [8]{c}}-\frac {7 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{15/8} \sqrt [8]{c}}\\ &=\frac {\sqrt {x}}{4 a \left (a+c x^4\right )}-\frac {7 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{15/8} \sqrt [8]{c}}+\frac {7 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{15/8} \sqrt [8]{c}}+\frac {7 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{15/8} \sqrt [8]{c}}+\frac {7 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{15/8} \sqrt [8]{c}}-\frac {7 \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{15/8} \sqrt [8]{c}}+\frac {7 \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{15/8} \sqrt [8]{c}}\\ \end {align*}
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Mathematica [A]
time = 1.19, size = 277, normalized size = 0.90 \begin {gather*} \frac {\frac {8 a^{7/8} \sqrt {x}}{a+c x^4}-\frac {7 \sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )}{\sqrt [8]{c}}-\frac {7 \sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )}{\sqrt [8]{c}}+\frac {7 \sqrt {2+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )}{\sqrt [8]{c}}+\frac {7 \sqrt {2-\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [8]{c} \sqrt {-\left (\left (-2+\sqrt {2}\right ) x\right )}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )}{\sqrt [8]{c}}}{32 a^{15/8}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.15, size = 50, normalized size = 0.16
method | result | size |
derivativedivides | \(\frac {\sqrt {x}}{4 a \left (x^{4} c +a \right )}+\frac {7 \left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}^{7}}\right )}{32 a c}\) | \(50\) |
default | \(\frac {\sqrt {x}}{4 a \left (x^{4} c +a \right )}+\frac {7 \left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}^{7}}\right )}{32 a c}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 517 vs.
\(2 (207) = 414\).
time = 0.40, size = 517, normalized size = 1.68 \begin {gather*} \frac {28 \, \sqrt {2} {\left (a c x^{4} + a^{2}\right )} \left (-\frac {1}{a^{15} c}\right )^{\frac {1}{8}} \arctan \left (\sqrt {2} \sqrt {a^{4} \left (-\frac {1}{a^{15} c}\right )^{\frac {1}{4}} + \sqrt {2} a^{2} \sqrt {x} \left (-\frac {1}{a^{15} c}\right )^{\frac {1}{8}} + x} a^{13} c \left (-\frac {1}{a^{15} c}\right )^{\frac {7}{8}} - \sqrt {2} a^{13} c \sqrt {x} \left (-\frac {1}{a^{15} c}\right )^{\frac {7}{8}} + 1\right ) + 28 \, \sqrt {2} {\left (a c x^{4} + a^{2}\right )} \left (-\frac {1}{a^{15} c}\right )^{\frac {1}{8}} \arctan \left (\sqrt {2} \sqrt {a^{4} \left (-\frac {1}{a^{15} c}\right )^{\frac {1}{4}} - \sqrt {2} a^{2} \sqrt {x} \left (-\frac {1}{a^{15} c}\right )^{\frac {1}{8}} + x} a^{13} c \left (-\frac {1}{a^{15} c}\right )^{\frac {7}{8}} - \sqrt {2} a^{13} c \sqrt {x} \left (-\frac {1}{a^{15} c}\right )^{\frac {7}{8}} - 1\right ) + 7 \, \sqrt {2} {\left (a c x^{4} + a^{2}\right )} \left (-\frac {1}{a^{15} c}\right )^{\frac {1}{8}} \log \left (a^{4} \left (-\frac {1}{a^{15} c}\right )^{\frac {1}{4}} + \sqrt {2} a^{2} \sqrt {x} \left (-\frac {1}{a^{15} c}\right )^{\frac {1}{8}} + x\right ) - 7 \, \sqrt {2} {\left (a c x^{4} + a^{2}\right )} \left (-\frac {1}{a^{15} c}\right )^{\frac {1}{8}} \log \left (a^{4} \left (-\frac {1}{a^{15} c}\right )^{\frac {1}{4}} - \sqrt {2} a^{2} \sqrt {x} \left (-\frac {1}{a^{15} c}\right )^{\frac {1}{8}} + x\right ) + 56 \, {\left (a c x^{4} + a^{2}\right )} \left (-\frac {1}{a^{15} c}\right )^{\frac {1}{8}} \arctan \left (\sqrt {a^{4} \left (-\frac {1}{a^{15} c}\right )^{\frac {1}{4}} + x} a^{13} c \left (-\frac {1}{a^{15} c}\right )^{\frac {7}{8}} - a^{13} c \sqrt {x} \left (-\frac {1}{a^{15} c}\right )^{\frac {7}{8}}\right ) + 14 \, {\left (a c x^{4} + a^{2}\right )} \left (-\frac {1}{a^{15} c}\right )^{\frac {1}{8}} \log \left (a^{2} \left (-\frac {1}{a^{15} c}\right )^{\frac {1}{8}} + \sqrt {x}\right ) - 14 \, {\left (a c x^{4} + a^{2}\right )} \left (-\frac {1}{a^{15} c}\right )^{\frac {1}{8}} \log \left (-a^{2} \left (-\frac {1}{a^{15} c}\right )^{\frac {1}{8}} + \sqrt {x}\right ) + 16 \, \sqrt {x}}{64 \, {\left (a c x^{4} + a^{2}\right )}} \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 [B] Leaf count of result is larger than twice the leaf count of optimal. 462 vs.
\(2 (207) = 414\).
time = 0.83, size = 462, normalized size = 1.50 \begin {gather*} \frac {7 \, \left (\frac {a}{c}\right )^{\frac {1}{8}} \arctan \left (\frac {\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + 2 \, \sqrt {x}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{16 \, a^{2} \sqrt {-2 \, \sqrt {2} + 4}} + \frac {7 \, \left (\frac {a}{c}\right )^{\frac {1}{8}} \arctan \left (-\frac {\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} - 2 \, \sqrt {x}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{16 \, a^{2} \sqrt {-2 \, \sqrt {2} + 4}} + \frac {7 \, \left (\frac {a}{c}\right )^{\frac {1}{8}} \arctan \left (\frac {\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + 2 \, \sqrt {x}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{16 \, a^{2} \sqrt {2 \, \sqrt {2} + 4}} + \frac {7 \, \left (\frac {a}{c}\right )^{\frac {1}{8}} \arctan \left (-\frac {\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} - 2 \, \sqrt {x}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{16 \, a^{2} \sqrt {2 \, \sqrt {2} + 4}} + \frac {7 \, \left (\frac {a}{c}\right )^{\frac {1}{8}} \log \left (\sqrt {x} \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{32 \, a^{2} \sqrt {-2 \, \sqrt {2} + 4}} - \frac {7 \, \left (\frac {a}{c}\right )^{\frac {1}{8}} \log \left (-\sqrt {x} \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{32 \, a^{2} \sqrt {-2 \, \sqrt {2} + 4}} + \frac {7 \, \left (\frac {a}{c}\right )^{\frac {1}{8}} \log \left (\sqrt {x} \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{32 \, a^{2} \sqrt {2 \, \sqrt {2} + 4}} - \frac {7 \, \left (\frac {a}{c}\right )^{\frac {1}{8}} \log \left (-\sqrt {x} \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{32 \, a^{2} \sqrt {2 \, \sqrt {2} + 4}} + \frac {\sqrt {x}}{4 \, {\left (c x^{4} + a\right )} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.13, size = 135, normalized size = 0.44 \begin {gather*} \frac {\sqrt {x}}{4\,a\,\left (c\,x^4+a\right )}+\frac {7\,\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}}{{\left (-a\right )}^{1/8}}\right )}{16\,{\left (-a\right )}^{15/8}\,c^{1/8}}-\frac {\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/8}}\right )\,7{}\mathrm {i}}{16\,{\left (-a\right )}^{15/8}\,c^{1/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (\frac {7}{32}+\frac {7}{32}{}\mathrm {i}\right )}{{\left (-a\right )}^{15/8}\,c^{1/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (\frac {7}{32}-\frac {7}{32}{}\mathrm {i}\right )}{{\left (-a\right )}^{15/8}\,c^{1/8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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