Integrand size = 15, antiderivative size = 332 \[ \int \frac {x^{13/2}}{\left (a+c x^4\right )^3} \, dx=-\frac {x^{7/2}}{8 c \left (a+c x^4\right )^2}+\frac {7 x^{7/2}}{64 a c \left (a+c x^4\right )}+\frac {7 \arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{9/8} c^{15/8}}-\frac {7 \arctan \left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{9/8} c^{15/8}}-\frac {7 \arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{9/8} c^{15/8}}+\frac {7 \text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{9/8} c^{15/8}}-\frac {7 \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{9/8} c^{15/8}}+\frac {7 \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{9/8} c^{15/8}} \]
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Time = 0.22 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.867, Rules used = {294, 296, 335, 307, 303, 1176, 631, 210, 1179, 642, 304, 211, 214} \[ \int \frac {x^{13/2}}{\left (a+c x^4\right )^3} \, dx=\frac {7 \arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{9/8} c^{15/8}}-\frac {7 \arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{256 \sqrt {2} (-a)^{9/8} c^{15/8}}-\frac {7 \arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{9/8} c^{15/8}}+\frac {7 \text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{9/8} c^{15/8}}-\frac {7 \log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{9/8} c^{15/8}}+\frac {7 \log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{9/8} c^{15/8}}+\frac {7 x^{7/2}}{64 a c \left (a+c x^4\right )}-\frac {x^{7/2}}{8 c \left (a+c x^4\right )^2} \]
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Rule 210
Rule 211
Rule 214
Rule 294
Rule 296
Rule 303
Rule 304
Rule 307
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = -\frac {x^{7/2}}{8 c \left (a+c x^4\right )^2}+\frac {7 \int \frac {x^{5/2}}{\left (a+c x^4\right )^2} \, dx}{16 c} \\ & = -\frac {x^{7/2}}{8 c \left (a+c x^4\right )^2}+\frac {7 x^{7/2}}{64 a c \left (a+c x^4\right )}+\frac {7 \int \frac {x^{5/2}}{a+c x^4} \, dx}{128 a c} \\ & = -\frac {x^{7/2}}{8 c \left (a+c x^4\right )^2}+\frac {7 x^{7/2}}{64 a c \left (a+c x^4\right )}+\frac {7 \text {Subst}\left (\int \frac {x^6}{a+c x^8} \, dx,x,\sqrt {x}\right )}{64 a c} \\ & = -\frac {x^{7/2}}{8 c \left (a+c x^4\right )^2}+\frac {7 x^{7/2}}{64 a c \left (a+c x^4\right )}-\frac {7 \text {Subst}\left (\int \frac {x^2}{\sqrt {-a}-\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{128 a c^{3/2}}+\frac {7 \text {Subst}\left (\int \frac {x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{128 a c^{3/2}} \\ & = -\frac {x^{7/2}}{8 c \left (a+c x^4\right )^2}+\frac {7 x^{7/2}}{64 a c \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 )}{256 a c^{7/4}}+\frac {7 \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a}+\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{256 a c^{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 )}{256 a c^{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 )}{256 a c^{7/4}} \\ & = -\frac {x^{7/2}}{8 c \left (a+c x^4\right )^2}+\frac {7 x^{7/2}}{64 a c \left (a+c x^4\right )}-\frac {7 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{9/8} c^{15/8}}+\frac {7 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{9/8} c^{15/8}}+\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 )}{512 a c^2}+\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 )}{512 a c^2}-\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 )}{512 \sqrt {2} (-a)^{9/8} c^{15/8}}-\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 )}{512 \sqrt {2} (-a)^{9/8} c^{15/8}} \\ & = -\frac {x^{7/2}}{8 c \left (a+c x^4\right )^2}+\frac {7 x^{7/2}}{64 a c \left (a+c x^4\right )}-\frac {7 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{9/8} c^{15/8}}+\frac {7 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{9/8} c^{15/8}}-\frac {7 \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{9/8} c^{15/8}}+\frac {7 \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{9/8} c^{15/8}}-\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 )}{256 \sqrt {2} (-a)^{9/8} c^{15/8}}+\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 )}{256 \sqrt {2} (-a)^{9/8} c^{15/8}} \\ & = -\frac {x^{7/2}}{8 c \left (a+c x^4\right )^2}+\frac {7 x^{7/2}}{64 a c \left (a+c x^4\right )}+\frac {7 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{9/8} c^{15/8}}-\frac {7 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{9/8} c^{15/8}}-\frac {7 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{9/8} c^{15/8}}+\frac {7 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{9/8} c^{15/8}}-\frac {7 \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{9/8} c^{15/8}}+\frac {7 \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{9/8} c^{15/8}} \\ \end{align*}
Time = 1.43 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.83 \[ \int \frac {x^{13/2}}{\left (a+c x^4\right )^3} \, dx=\frac {-\frac {8 \sqrt [8]{a} c^{7/8} x^{7/2} \left (a-7 c x^4\right )}{\left (a+c x^4\right )^2}-7 \sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )-7 \sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )-7 \sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )-7 \sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [8]{c} \sqrt {-\left (\left (-2+\sqrt {2}\right ) x\right )}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )}{512 a^{9/8} c^{15/8}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.38 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.18
method | result | size |
derivativedivides | \(\frac {-\frac {x^{\frac {7}{2}}}{64 c}+\frac {7 x^{\frac {15}{2}}}{64 a}}{\left (x^{4} c +a \right )^{2}}+\frac {7 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}}\right )}{512 a \,c^{2}}\) | \(61\) |
default | \(\frac {-\frac {x^{\frac {7}{2}}}{64 c}+\frac {7 x^{\frac {15}{2}}}{64 a}}{\left (x^{4} c +a \right )^{2}}+\frac {7 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}}\right )}{512 a \,c^{2}}\) | \(61\) |
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Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 563, normalized size of antiderivative = 1.70 \[ \int \frac {x^{13/2}}{\left (a+c x^4\right )^3} \, dx=-\frac {7 \, \sqrt {2} {\left (\left (i - 1\right ) \, a c^{3} x^{8} + \left (2 i - 2\right ) \, a^{2} c^{2} x^{4} + \left (i - 1\right ) \, a^{3} c\right )} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a^{8} c^{13} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {7}{8}} + \sqrt {x}\right ) + 7 \, \sqrt {2} {\left (-\left (i + 1\right ) \, a c^{3} x^{8} - \left (2 i + 2\right ) \, a^{2} c^{2} x^{4} - \left (i + 1\right ) \, a^{3} c\right )} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a^{8} c^{13} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {7}{8}} + \sqrt {x}\right ) + 7 \, \sqrt {2} {\left (\left (i + 1\right ) \, a c^{3} x^{8} + \left (2 i + 2\right ) \, a^{2} c^{2} x^{4} + \left (i + 1\right ) \, a^{3} c\right )} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a^{8} c^{13} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {7}{8}} + \sqrt {x}\right ) + 7 \, \sqrt {2} {\left (-\left (i - 1\right ) \, a c^{3} x^{8} - \left (2 i - 2\right ) \, a^{2} c^{2} x^{4} - \left (i - 1\right ) \, a^{3} c\right )} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a^{8} c^{13} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {7}{8}} + \sqrt {x}\right ) - 14 \, {\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {1}{8}} \log \left (a^{8} c^{13} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {7}{8}} + \sqrt {x}\right ) + 14 \, {\left (i \, a c^{3} x^{8} + 2 i \, a^{2} c^{2} x^{4} + i \, a^{3} c\right )} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {1}{8}} \log \left (i \, a^{8} c^{13} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {7}{8}} + \sqrt {x}\right ) + 14 \, {\left (-i \, a c^{3} x^{8} - 2 i \, a^{2} c^{2} x^{4} - i \, a^{3} c\right )} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {1}{8}} \log \left (-i \, a^{8} c^{13} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {7}{8}} + \sqrt {x}\right ) + 14 \, {\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {1}{8}} \log \left (-a^{8} c^{13} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {7}{8}} + \sqrt {x}\right ) - 16 \, {\left (7 \, c x^{7} - a x^{3}\right )} \sqrt {x}}{1024 \, {\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )}} \]
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Timed out. \[ \int \frac {x^{13/2}}{\left (a+c x^4\right )^3} \, dx=\text {Timed out} \]
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\[ \int \frac {x^{13/2}}{\left (a+c x^4\right )^3} \, dx=\int { \frac {x^{\frac {13}{2}}}{{\left (c x^{4} + a\right )}^{3}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 499 vs. \(2 (227) = 454\).
Time = 0.49 (sec) , antiderivative size = 499, normalized size of antiderivative = 1.50 \[ \int \frac {x^{13/2}}{\left (a+c x^4\right )^3} \, dx=\frac {7 \, \left (\frac {a}{c}\right )^{\frac {7}{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 )}{256 \, a^{2} c \sqrt {-2 \, \sqrt {2} + 4}} + \frac {7 \, \left (\frac {a}{c}\right )^{\frac {7}{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 )}{256 \, a^{2} c \sqrt {-2 \, \sqrt {2} + 4}} + \frac {7 \, \left (\frac {a}{c}\right )^{\frac {7}{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 )}{256 \, a^{2} c \sqrt {2 \, \sqrt {2} + 4}} + \frac {7 \, \left (\frac {a}{c}\right )^{\frac {7}{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 )}{256 \, a^{2} c \sqrt {2 \, \sqrt {2} + 4}} - \frac {7 \, \left (\frac {a}{c}\right )^{\frac {7}{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 )}{512 \, a^{2} c \sqrt {-2 \, \sqrt {2} + 4}} + \frac {7 \, \left (\frac {a}{c}\right )^{\frac {7}{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 )}{512 \, a^{2} c \sqrt {-2 \, \sqrt {2} + 4}} - \frac {7 \, \left (\frac {a}{c}\right )^{\frac {7}{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 )}{512 \, a^{2} c \sqrt {2 \, \sqrt {2} + 4}} + \frac {7 \, \left (\frac {a}{c}\right )^{\frac {7}{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 )}{512 \, a^{2} c \sqrt {2 \, \sqrt {2} + 4}} + \frac {7 \, c x^{\frac {15}{2}} - a x^{\frac {7}{2}}}{64 \, {\left (c x^{4} + a\right )}^{2} a c} \]
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Time = 5.55 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.47 \[ \int \frac {x^{13/2}}{\left (a+c x^4\right )^3} \, dx=\frac {\frac {7\,x^{15/2}}{64\,a}-\frac {x^{7/2}}{64\,c}}{a^2+2\,a\,c\,x^4+c^2\,x^8}-\frac {7\,\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}}{{\left (-a\right )}^{1/8}}\right )}{256\,{\left (-a\right )}^{9/8}\,c^{15/8}}-\frac {\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/8}}\right )\,7{}\mathrm {i}}{256\,{\left (-a\right )}^{9/8}\,c^{15/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (-\frac {7}{512}+\frac {7}{512}{}\mathrm {i}\right )}{{\left (-a\right )}^{9/8}\,c^{15/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (-\frac {7}{512}-\frac {7}{512}{}\mathrm {i}\right )}{{\left (-a\right )}^{9/8}\,c^{15/8}} \]
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