Integrand size = 45, antiderivative size = 255 \[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=-\frac {4 \sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {3} \sqrt [6]{c}}\right )}{a \sqrt [6]{c}}+\frac {4 \sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {3} \sqrt [6]{c}}\right )}{a \sqrt [6]{c}}-\frac {8 \text {arctanh}\left (\frac {\sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [6]{c}}\right )}{a \sqrt [6]{c}}-\frac {4 \text {arctanh}\left (\frac {\sqrt [6]{c}+\frac {\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [6]{c}}}{\sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}\right )}{a \sqrt [6]{c}} \]
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\[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx \\ \end{align*}
Time = 1.09 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=-\frac {4 \left (\sqrt {3} \left (\arctan \left (\frac {1-\frac {2 \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [6]{c}}}{\sqrt {3}}\right )-\arctan \left (\frac {1+\frac {2 \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [6]{c}}}{\sqrt {3}}\right )\right )+2 \text {arctanh}\left (\frac {\sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [6]{c}}\right )+\text {arctanh}\left (\frac {\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [6]{c} \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}\right )\right )}{a \sqrt [6]{c}} \]
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Time = 0.17 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.13
method | result | size |
derivativedivides | \(\frac {-\frac {2 \ln \left ({\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{3}}+c^{\frac {1}{6}} {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{6}}+c^{\frac {1}{3}}\right )}{c^{\frac {1}{6}}}+\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}}{3}+\frac {2 {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{6}} \sqrt {3}}{3 c^{\frac {1}{6}}}\right )}{c^{\frac {1}{6}}}-\frac {4 \ln \left ({\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{6}}+c^{\frac {1}{6}}\right )}{c^{\frac {1}{6}}}+\frac {4 \ln \left (-{\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{6}}+c^{\frac {1}{6}}\right )}{c^{\frac {1}{6}}}+\frac {2 \ln \left (c^{\frac {1}{6}} {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{6}}-{\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{3}}-c^{\frac {1}{3}}\right )}{c^{\frac {1}{6}}}+\frac {4 \sqrt {3}\, \arctan \left (-\frac {\sqrt {3}}{3}+\frac {2 {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{6}} \sqrt {3}}{3 c^{\frac {1}{6}}}\right )}{c^{\frac {1}{6}}}}{a}\) | \(288\) |
default | \(\frac {-\frac {2 \ln \left ({\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{3}}+c^{\frac {1}{6}} {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{6}}+c^{\frac {1}{3}}\right )}{c^{\frac {1}{6}}}+\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}}{3}+\frac {2 {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{6}} \sqrt {3}}{3 c^{\frac {1}{6}}}\right )}{c^{\frac {1}{6}}}-\frac {4 \ln \left ({\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{6}}+c^{\frac {1}{6}}\right )}{c^{\frac {1}{6}}}+\frac {4 \ln \left (-{\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{6}}+c^{\frac {1}{6}}\right )}{c^{\frac {1}{6}}}+\frac {2 \ln \left (c^{\frac {1}{6}} {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{6}}-{\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{3}}-c^{\frac {1}{3}}\right )}{c^{\frac {1}{6}}}+\frac {4 \sqrt {3}\, \arctan \left (-\frac {\sqrt {3}}{3}+\frac {2 {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{6}} \sqrt {3}}{3 c^{\frac {1}{6}}}\right )}{c^{\frac {1}{6}}}}{a}\) | \(288\) |
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Time = 0.37 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.47 \[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=2 \, {\left (\sqrt {-3} - 1\right )} \left (\frac {1}{a^{6} c}\right )^{\frac {1}{6}} \log \left ({\left (\sqrt {-3} a^{5} c + a^{5} c\right )} \left (\frac {1}{a^{6} c}\right )^{\frac {5}{6}} + 2 \, {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}}\right ) - 2 \, {\left (\sqrt {-3} - 1\right )} \left (\frac {1}{a^{6} c}\right )^{\frac {1}{6}} \log \left (-{\left (\sqrt {-3} a^{5} c + a^{5} c\right )} \left (\frac {1}{a^{6} c}\right )^{\frac {5}{6}} + 2 \, {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}}\right ) + 2 \, {\left (\sqrt {-3} + 1\right )} \left (\frac {1}{a^{6} c}\right )^{\frac {1}{6}} \log \left ({\left (\sqrt {-3} a^{5} c - a^{5} c\right )} \left (\frac {1}{a^{6} c}\right )^{\frac {5}{6}} + 2 \, {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}}\right ) - 2 \, {\left (\sqrt {-3} + 1\right )} \left (\frac {1}{a^{6} c}\right )^{\frac {1}{6}} \log \left (-{\left (\sqrt {-3} a^{5} c - a^{5} c\right )} \left (\frac {1}{a^{6} c}\right )^{\frac {5}{6}} + 2 \, {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}}\right ) - 4 \, \left (\frac {1}{a^{6} c}\right )^{\frac {1}{6}} \log \left (a^{5} c \left (\frac {1}{a^{6} c}\right )^{\frac {5}{6}} + {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}}\right ) + 4 \, \left (\frac {1}{a^{6} c}\right )^{\frac {1}{6}} \log \left (-a^{5} c \left (\frac {1}{a^{6} c}\right )^{\frac {5}{6}} + {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}}\right ) \]
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\[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int \frac {1}{\sqrt [6]{c + \sqrt [4]{a x + \sqrt {a^{2} x^{2} - b}}} \sqrt {a^{2} x^{2} - b}}\, dx \]
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\[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int { \frac {1}{\sqrt {a^{2} x^{2} - b} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int \frac {1}{{\left (c+{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}\right )}^{1/6}\,\sqrt {a^2\,x^2-b}} \,d x \]
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