Integrand size = 21, antiderivative size = 75 \[ \int \frac {\cos (c+d x)}{\sqrt [3]{a+a \sec (c+d x)}} \, dx=-\frac {3 \sqrt {2} \operatorname {AppellF1}\left (\frac {1}{6},\frac {1}{2},2,\frac {7}{6},\frac {1}{2} (1+\sec (c+d x)),1+\sec (c+d x)\right ) \tan (c+d x)}{d \sqrt {1-\sec (c+d x)} \sqrt [3]{a+a \sec (c+d x)}} \]
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Time = 0.13 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3913, 3912, 141} \[ \int \frac {\cos (c+d x)}{\sqrt [3]{a+a \sec (c+d x)}} \, dx=-\frac {3 \sqrt {2} \tan (c+d x) \operatorname {AppellF1}\left (\frac {1}{6},\frac {1}{2},2,\frac {7}{6},\frac {1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right )}{d \sqrt {1-\sec (c+d x)} \sqrt [3]{a \sec (c+d x)+a}} \]
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Rule 141
Rule 3912
Rule 3913
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [3]{1+\sec (c+d x)} \int \frac {\cos (c+d x)}{\sqrt [3]{1+\sec (c+d x)}} \, dx}{\sqrt [3]{a+a \sec (c+d x)}} \\ & = -\frac {\tan (c+d x) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} x^2 (1+x)^{5/6}} \, dx,x,\sec (c+d x)\right )}{d \sqrt {1-\sec (c+d x)} \sqrt [6]{1+\sec (c+d x)} \sqrt [3]{a+a \sec (c+d x)}} \\ & = -\frac {3 \sqrt {2} \operatorname {AppellF1}\left (\frac {1}{6},\frac {1}{2},2,\frac {7}{6},\frac {1}{2} (1+\sec (c+d x)),1+\sec (c+d x)\right ) \tan (c+d x)}{d \sqrt {1-\sec (c+d x)} \sqrt [3]{a+a \sec (c+d x)}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(240\) vs. \(2(75)=150\).
Time = 1.90 (sec) , antiderivative size = 240, normalized size of antiderivative = 3.20 \[ \int \frac {\cos (c+d x)}{\sqrt [3]{a+a \sec (c+d x)}} \, dx=\frac {(a (1+\sec (c+d x)))^{2/3} \left (\frac {20 \operatorname {AppellF1}\left (\frac {3}{2},\frac {2}{3},1,\frac {5}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \cos \left (\frac {1}{2} (c+d x)\right ) \sin ^3\left (\frac {1}{2} (c+d x)\right )}{6 \left (3 \operatorname {AppellF1}\left (\frac {5}{2},\frac {2}{3},2,\frac {7}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )-2 \operatorname {AppellF1}\left (\frac {5}{2},\frac {5}{3},1,\frac {7}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )\right ) (-1+\cos (c+d x))+45 \operatorname {AppellF1}\left (\frac {3}{2},\frac {2}{3},1,\frac {5}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) (1+\cos (c+d x))}+\sin (c+d x)-\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a d} \]
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\[\int \frac {\cos \left (d x +c \right )}{\left (a +a \sec \left (d x +c \right )\right )^{\frac {1}{3}}}d x\]
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Timed out. \[ \int \frac {\cos (c+d x)}{\sqrt [3]{a+a \sec (c+d x)}} \, dx=\text {Timed out} \]
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\[ \int \frac {\cos (c+d x)}{\sqrt [3]{a+a \sec (c+d x)}} \, dx=\int \frac {\cos {\left (c + d x \right )}}{\sqrt [3]{a \left (\sec {\left (c + d x \right )} + 1\right )}}\, dx \]
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\[ \int \frac {\cos (c+d x)}{\sqrt [3]{a+a \sec (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right )}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {\cos (c+d x)}{\sqrt [3]{a+a \sec (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right )}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {\cos (c+d x)}{\sqrt [3]{a+a \sec (c+d x)}} \, dx=\int \frac {\cos \left (c+d\,x\right )}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{1/3}} \,d x \]
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