Integrand size = 25, antiderivative size = 687 \[ \int \frac {\sqrt [3]{d \sec (e+f x)}}{(a+b \tan (e+f x))^2} \, dx=\frac {5 a b^{2/3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt {3} \sqrt [6]{a^2+b^2}}\right ) \sqrt [3]{d \sec (e+f x)}}{2 \sqrt {3} \left (a^2+b^2\right )^{11/6} f \sqrt [6]{\sec ^2(e+f x)}}-\frac {5 a b^{2/3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt {3} \sqrt [6]{a^2+b^2}}\right ) \sqrt [3]{d \sec (e+f x)}}{2 \sqrt {3} \left (a^2+b^2\right )^{11/6} f \sqrt [6]{\sec ^2(e+f x)}}-\frac {5 a b^{2/3} \text {arctanh}\left (\frac {\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right ) \sqrt [3]{d \sec (e+f x)}}{3 \left (a^2+b^2\right )^{11/6} f \sqrt [6]{\sec ^2(e+f x)}}+\frac {5 a b^{2/3} \log \left (\sqrt [3]{a^2+b^2}-\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right ) \sqrt [3]{d \sec (e+f x)}}{12 \left (a^2+b^2\right )^{11/6} f \sqrt [6]{\sec ^2(e+f x)}}-\frac {5 a b^{2/3} \log \left (\sqrt [3]{a^2+b^2}+\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right ) \sqrt [3]{d \sec (e+f x)}}{12 \left (a^2+b^2\right )^{11/6} f \sqrt [6]{\sec ^2(e+f x)}}+\frac {\operatorname {AppellF1}\left (\frac {1}{2},2,\frac {5}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [3]{d \sec (e+f x)} \tan (e+f x)}{a^2 f \sqrt [6]{\sec ^2(e+f x)}}+\frac {b^2 \operatorname {AppellF1}\left (\frac {3}{2},2,\frac {5}{6},\frac {5}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [3]{d \sec (e+f x)} \tan ^3(e+f x)}{3 a^4 f \sqrt [6]{\sec ^2(e+f x)}}-\frac {a b \sqrt [3]{d \sec (e+f x)}}{\left (a^2+b^2\right ) f \left (a^2-b^2 \tan ^2(e+f x)\right )} \]
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Time = 1.16 (sec) , antiderivative size = 687, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {3593, 771, 440, 455, 44, 65, 216, 648, 632, 210, 642, 214, 524} \[ \int \frac {\sqrt [3]{d \sec (e+f x)}}{(a+b \tan (e+f x))^2} \, dx=\frac {\tan (e+f x) \sqrt [3]{d \sec (e+f x)} \operatorname {AppellF1}\left (\frac {1}{2},2,\frac {5}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right )}{a^2 f \sqrt [6]{\sec ^2(e+f x)}}+\frac {5 a b^{2/3} \sqrt [3]{d \sec (e+f x)} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt {3} \sqrt [6]{a^2+b^2}}\right )}{2 \sqrt {3} f \left (a^2+b^2\right )^{11/6} \sqrt [6]{\sec ^2(e+f x)}}-\frac {5 a b^{2/3} \sqrt [3]{d \sec (e+f x)} \arctan \left (\frac {2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt {3} \sqrt [6]{a^2+b^2}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3} f \left (a^2+b^2\right )^{11/6} \sqrt [6]{\sec ^2(e+f x)}}-\frac {5 a b^{2/3} \sqrt [3]{d \sec (e+f x)} \text {arctanh}\left (\frac {\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right )}{3 f \left (a^2+b^2\right )^{11/6} \sqrt [6]{\sec ^2(e+f x)}}-\frac {a b \sqrt [3]{d \sec (e+f x)}}{f \left (a^2+b^2\right ) \left (a^2-b^2 \tan ^2(e+f x)\right )}+\frac {5 a b^{2/3} \sqrt [3]{d \sec (e+f x)} \log \left (-\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+\sqrt [3]{a^2+b^2}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right )}{12 f \left (a^2+b^2\right )^{11/6} \sqrt [6]{\sec ^2(e+f x)}}-\frac {5 a b^{2/3} \sqrt [3]{d \sec (e+f x)} \log \left (\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+\sqrt [3]{a^2+b^2}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right )}{12 f \left (a^2+b^2\right )^{11/6} \sqrt [6]{\sec ^2(e+f x)}}+\frac {b^2 \tan ^3(e+f x) \sqrt [3]{d \sec (e+f x)} \operatorname {AppellF1}\left (\frac {3}{2},2,\frac {5}{6},\frac {5}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right )}{3 a^4 f \sqrt [6]{\sec ^2(e+f x)}} \]
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Rule 44
Rule 65
Rule 210
Rule 214
Rule 216
Rule 440
Rule 455
Rule 524
Rule 632
Rule 642
Rule 648
Rule 771
Rule 3593
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [3]{d \sec (e+f x)} \text {Subst}\left (\int \frac {1}{(a+x)^2 \left (1+\frac {x^2}{b^2}\right )^{5/6}} \, dx,x,b \tan (e+f x)\right )}{b f \sqrt [6]{\sec ^2(e+f x)}} \\ & = \frac {\sqrt [3]{d \sec (e+f x)} \text {Subst}\left (\int \left (\frac {a^2}{\left (a^2-x^2\right )^2 \left (1+\frac {x^2}{b^2}\right )^{5/6}}-\frac {2 a x}{\left (a^2-x^2\right )^2 \left (1+\frac {x^2}{b^2}\right )^{5/6}}+\frac {x^2}{\left (-a^2+x^2\right )^2 \left (1+\frac {x^2}{b^2}\right )^{5/6}}\right ) \, dx,x,b \tan (e+f x)\right )}{b f \sqrt [6]{\sec ^2(e+f x)}} \\ & = \frac {\sqrt [3]{d \sec (e+f x)} \text {Subst}\left (\int \frac {x^2}{\left (-a^2+x^2\right )^2 \left (1+\frac {x^2}{b^2}\right )^{5/6}} \, dx,x,b \tan (e+f x)\right )}{b f \sqrt [6]{\sec ^2(e+f x)}}-\frac {\left (2 a \sqrt [3]{d \sec (e+f x)}\right ) \text {Subst}\left (\int \frac {x}{\left (a^2-x^2\right )^2 \left (1+\frac {x^2}{b^2}\right )^{5/6}} \, dx,x,b \tan (e+f x)\right )}{b f \sqrt [6]{\sec ^2(e+f x)}}+\frac {\left (a^2 \sqrt [3]{d \sec (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\left (a^2-x^2\right )^2 \left (1+\frac {x^2}{b^2}\right )^{5/6}} \, dx,x,b \tan (e+f x)\right )}{b f \sqrt [6]{\sec ^2(e+f x)}} \\ & = \frac {\operatorname {AppellF1}\left (\frac {1}{2},2,\frac {5}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [3]{d \sec (e+f x)} \tan (e+f x)}{a^2 f \sqrt [6]{\sec ^2(e+f x)}}+\frac {b^2 \operatorname {AppellF1}\left (\frac {3}{2},2,\frac {5}{6},\frac {5}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [3]{d \sec (e+f x)} \tan ^3(e+f x)}{3 a^4 f \sqrt [6]{\sec ^2(e+f x)}}-\frac {\left (a \sqrt [3]{d \sec (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\left (a^2-x\right )^2 \left (1+\frac {x}{b^2}\right )^{5/6}} \, dx,x,b^2 \tan ^2(e+f x)\right )}{b f \sqrt [6]{\sec ^2(e+f x)}} \\ & = \frac {\operatorname {AppellF1}\left (\frac {1}{2},2,\frac {5}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [3]{d \sec (e+f x)} \tan (e+f x)}{a^2 f \sqrt [6]{\sec ^2(e+f x)}}+\frac {b^2 \operatorname {AppellF1}\left (\frac {3}{2},2,\frac {5}{6},\frac {5}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [3]{d \sec (e+f x)} \tan ^3(e+f x)}{3 a^4 f \sqrt [6]{\sec ^2(e+f x)}}-\frac {a b \sqrt [3]{d \sec (e+f x)}}{\left (a^2+b^2\right ) f \left (a^2-b^2 \tan ^2(e+f x)\right )}-\frac {\left (5 a \sqrt [3]{d \sec (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\left (a^2-x\right ) \left (1+\frac {x}{b^2}\right )^{5/6}} \, dx,x,b^2 \tan ^2(e+f x)\right )}{6 b \left (a^2+b^2\right ) f \sqrt [6]{\sec ^2(e+f x)}} \\ & = \frac {\operatorname {AppellF1}\left (\frac {1}{2},2,\frac {5}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [3]{d \sec (e+f x)} \tan (e+f x)}{a^2 f \sqrt [6]{\sec ^2(e+f x)}}+\frac {b^2 \operatorname {AppellF1}\left (\frac {3}{2},2,\frac {5}{6},\frac {5}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [3]{d \sec (e+f x)} \tan ^3(e+f x)}{3 a^4 f \sqrt [6]{\sec ^2(e+f x)}}-\frac {a b \sqrt [3]{d \sec (e+f x)}}{\left (a^2+b^2\right ) f \left (a^2-b^2 \tan ^2(e+f x)\right )}-\frac {\left (5 a b \sqrt [3]{d \sec (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{a^2+b^2-b^2 x^6} \, dx,x,\sqrt [6]{\sec ^2(e+f x)}\right )}{\left (a^2+b^2\right ) f \sqrt [6]{\sec ^2(e+f x)}} \\ & = \frac {\operatorname {AppellF1}\left (\frac {1}{2},2,\frac {5}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [3]{d \sec (e+f x)} \tan (e+f x)}{a^2 f \sqrt [6]{\sec ^2(e+f x)}}+\frac {b^2 \operatorname {AppellF1}\left (\frac {3}{2},2,\frac {5}{6},\frac {5}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [3]{d \sec (e+f x)} \tan ^3(e+f x)}{3 a^4 f \sqrt [6]{\sec ^2(e+f x)}}-\frac {a b \sqrt [3]{d \sec (e+f x)}}{\left (a^2+b^2\right ) f \left (a^2-b^2 \tan ^2(e+f x)\right )}-\frac {\left (5 a b \sqrt [3]{d \sec (e+f x)}\right ) \text {Subst}\left (\int \frac {\sqrt [6]{a^2+b^2}-\frac {\sqrt [3]{b} x}{2}}{\sqrt [3]{a^2+b^2}-\sqrt [3]{b} \sqrt [6]{a^2+b^2} x+b^{2/3} x^2} \, dx,x,\sqrt [6]{\sec ^2(e+f x)}\right )}{3 \left (a^2+b^2\right )^{11/6} f \sqrt [6]{\sec ^2(e+f x)}}-\frac {\left (5 a b \sqrt [3]{d \sec (e+f x)}\right ) \text {Subst}\left (\int \frac {\sqrt [6]{a^2+b^2}+\frac {\sqrt [3]{b} x}{2}}{\sqrt [3]{a^2+b^2}+\sqrt [3]{b} \sqrt [6]{a^2+b^2} x+b^{2/3} x^2} \, dx,x,\sqrt [6]{\sec ^2(e+f x)}\right )}{3 \left (a^2+b^2\right )^{11/6} f \sqrt [6]{\sec ^2(e+f x)}}-\frac {\left (5 a b \sqrt [3]{d \sec (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a^2+b^2}-b^{2/3} x^2} \, dx,x,\sqrt [6]{\sec ^2(e+f x)}\right )}{3 \left (a^2+b^2\right )^{5/3} f \sqrt [6]{\sec ^2(e+f x)}} \\ & = -\frac {5 a b^{2/3} \text {arctanh}\left (\frac {\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right ) \sqrt [3]{d \sec (e+f x)}}{3 \left (a^2+b^2\right )^{11/6} f \sqrt [6]{\sec ^2(e+f x)}}+\frac {\operatorname {AppellF1}\left (\frac {1}{2},2,\frac {5}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [3]{d \sec (e+f x)} \tan (e+f x)}{a^2 f \sqrt [6]{\sec ^2(e+f x)}}+\frac {b^2 \operatorname {AppellF1}\left (\frac {3}{2},2,\frac {5}{6},\frac {5}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [3]{d \sec (e+f x)} \tan ^3(e+f x)}{3 a^4 f \sqrt [6]{\sec ^2(e+f x)}}-\frac {a b \sqrt [3]{d \sec (e+f x)}}{\left (a^2+b^2\right ) f \left (a^2-b^2 \tan ^2(e+f x)\right )}+\frac {\left (5 a b^{2/3} \sqrt [3]{d \sec (e+f x)}\right ) \text {Subst}\left (\int \frac {-\sqrt [3]{b} \sqrt [6]{a^2+b^2}+2 b^{2/3} x}{\sqrt [3]{a^2+b^2}-\sqrt [3]{b} \sqrt [6]{a^2+b^2} x+b^{2/3} x^2} \, dx,x,\sqrt [6]{\sec ^2(e+f x)}\right )}{12 \left (a^2+b^2\right )^{11/6} f \sqrt [6]{\sec ^2(e+f x)}}-\frac {\left (5 a b^{2/3} \sqrt [3]{d \sec (e+f x)}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{b} \sqrt [6]{a^2+b^2}+2 b^{2/3} x}{\sqrt [3]{a^2+b^2}+\sqrt [3]{b} \sqrt [6]{a^2+b^2} x+b^{2/3} x^2} \, dx,x,\sqrt [6]{\sec ^2(e+f x)}\right )}{12 \left (a^2+b^2\right )^{11/6} f \sqrt [6]{\sec ^2(e+f x)}}-\frac {\left (5 a b \sqrt [3]{d \sec (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a^2+b^2}-\sqrt [3]{b} \sqrt [6]{a^2+b^2} x+b^{2/3} x^2} \, dx,x,\sqrt [6]{\sec ^2(e+f x)}\right )}{4 \left (a^2+b^2\right )^{5/3} f \sqrt [6]{\sec ^2(e+f x)}}-\frac {\left (5 a b \sqrt [3]{d \sec (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a^2+b^2}+\sqrt [3]{b} \sqrt [6]{a^2+b^2} x+b^{2/3} x^2} \, dx,x,\sqrt [6]{\sec ^2(e+f x)}\right )}{4 \left (a^2+b^2\right )^{5/3} f \sqrt [6]{\sec ^2(e+f x)}} \\ & = -\frac {5 a b^{2/3} \text {arctanh}\left (\frac {\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right ) \sqrt [3]{d \sec (e+f x)}}{3 \left (a^2+b^2\right )^{11/6} f \sqrt [6]{\sec ^2(e+f x)}}+\frac {5 a b^{2/3} \log \left (\sqrt [3]{a^2+b^2}-\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right ) \sqrt [3]{d \sec (e+f x)}}{12 \left (a^2+b^2\right )^{11/6} f \sqrt [6]{\sec ^2(e+f x)}}-\frac {5 a b^{2/3} \log \left (\sqrt [3]{a^2+b^2}+\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right ) \sqrt [3]{d \sec (e+f x)}}{12 \left (a^2+b^2\right )^{11/6} f \sqrt [6]{\sec ^2(e+f x)}}+\frac {\operatorname {AppellF1}\left (\frac {1}{2},2,\frac {5}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [3]{d \sec (e+f x)} \tan (e+f x)}{a^2 f \sqrt [6]{\sec ^2(e+f x)}}+\frac {b^2 \operatorname {AppellF1}\left (\frac {3}{2},2,\frac {5}{6},\frac {5}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [3]{d \sec (e+f x)} \tan ^3(e+f x)}{3 a^4 f \sqrt [6]{\sec ^2(e+f x)}}-\frac {a b \sqrt [3]{d \sec (e+f x)}}{\left (a^2+b^2\right ) f \left (a^2-b^2 \tan ^2(e+f x)\right )}-\frac {\left (5 a b^{2/3} \sqrt [3]{d \sec (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{11/6} f \sqrt [6]{\sec ^2(e+f x)}}+\frac {\left (5 a b^{2/3} \sqrt [3]{d \sec (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{11/6} f \sqrt [6]{\sec ^2(e+f x)}} \\ & = \frac {5 a b^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}}{\sqrt {3}}\right ) \sqrt [3]{d \sec (e+f x)}}{2 \sqrt {3} \left (a^2+b^2\right )^{11/6} f \sqrt [6]{\sec ^2(e+f x)}}-\frac {5 a b^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}}{\sqrt {3}}\right ) \sqrt [3]{d \sec (e+f x)}}{2 \sqrt {3} \left (a^2+b^2\right )^{11/6} f \sqrt [6]{\sec ^2(e+f x)}}-\frac {5 a b^{2/3} \text {arctanh}\left (\frac {\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right ) \sqrt [3]{d \sec (e+f x)}}{3 \left (a^2+b^2\right )^{11/6} f \sqrt [6]{\sec ^2(e+f x)}}+\frac {5 a b^{2/3} \log \left (\sqrt [3]{a^2+b^2}-\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right ) \sqrt [3]{d \sec (e+f x)}}{12 \left (a^2+b^2\right )^{11/6} f \sqrt [6]{\sec ^2(e+f x)}}-\frac {5 a b^{2/3} \log \left (\sqrt [3]{a^2+b^2}+\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right ) \sqrt [3]{d \sec (e+f x)}}{12 \left (a^2+b^2\right )^{11/6} f \sqrt [6]{\sec ^2(e+f x)}}+\frac {\operatorname {AppellF1}\left (\frac {1}{2},2,\frac {5}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [3]{d \sec (e+f x)} \tan (e+f x)}{a^2 f \sqrt [6]{\sec ^2(e+f x)}}+\frac {b^2 \operatorname {AppellF1}\left (\frac {3}{2},2,\frac {5}{6},\frac {5}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [3]{d \sec (e+f x)} \tan ^3(e+f x)}{3 a^4 f \sqrt [6]{\sec ^2(e+f x)}}-\frac {a b \sqrt [3]{d \sec (e+f x)}}{\left (a^2+b^2\right ) f \left (a^2-b^2 \tan ^2(e+f x)\right )} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(7801\) vs. \(2(687)=1374\).
Time = 78.55 (sec) , antiderivative size = 7801, normalized size of antiderivative = 11.36 \[ \int \frac {\sqrt [3]{d \sec (e+f x)}}{(a+b \tan (e+f x))^2} \, dx=\text {Result too large to show} \]
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\[\int \frac {\left (d \sec \left (f x +e \right )\right )^{\frac {1}{3}}}{\left (a +b \tan \left (f x +e \right )\right )^{2}}d x\]
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Timed out. \[ \int \frac {\sqrt [3]{d \sec (e+f x)}}{(a+b \tan (e+f x))^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt [3]{d \sec (e+f x)}}{(a+b \tan (e+f x))^2} \, dx=\int \frac {\sqrt [3]{d \sec {\left (e + f x \right )}}}{\left (a + b \tan {\left (e + f x \right )}\right )^{2}}\, dx \]
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\[ \int \frac {\sqrt [3]{d \sec (e+f x)}}{(a+b \tan (e+f x))^2} \, dx=\int { \frac {\left (d \sec \left (f x + e\right )\right )^{\frac {1}{3}}}{{\left (b \tan \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {\sqrt [3]{d \sec (e+f x)}}{(a+b \tan (e+f x))^2} \, dx=\int { \frac {\left (d \sec \left (f x + e\right )\right )^{\frac {1}{3}}}{{\left (b \tan \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt [3]{d \sec (e+f x)}}{(a+b \tan (e+f x))^2} \, dx=\int \frac {{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{1/3}}{{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^2} \,d x \]
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