Integrand size = 25, antiderivative size = 717 \[ \int \frac {1}{(d \sec (e+f x))^{5/3} (a+b \tan (e+f x))^2} \, dx=\frac {11 a b}{5 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{5/3}}+\frac {11 a b^{8/3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt {3} \sqrt [6]{a^2+b^2}}\right ) \sec ^2(e+f x)^{5/6}}{2 \sqrt {3} \left (a^2+b^2\right )^{17/6} f (d \sec (e+f x))^{5/3}}-\frac {11 a b^{8/3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt {3} \sqrt [6]{a^2+b^2}}\right ) \sec ^2(e+f x)^{5/6}}{2 \sqrt {3} \left (a^2+b^2\right )^{17/6} f (d \sec (e+f x))^{5/3}}-\frac {11 a b^{8/3} \text {arctanh}\left (\frac {\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right ) \sec ^2(e+f x)^{5/6}}{3 \left (a^2+b^2\right )^{17/6} f (d \sec (e+f x))^{5/3}}+\frac {11 a b^{8/3} \log \left (\sqrt [3]{a^2+b^2}-\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right ) \sec ^2(e+f x)^{5/6}}{12 \left (a^2+b^2\right )^{17/6} f (d \sec (e+f x))^{5/3}}-\frac {11 a b^{8/3} \log \left (\sqrt [3]{a^2+b^2}+\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right ) \sec ^2(e+f x)^{5/6}}{12 \left (a^2+b^2\right )^{17/6} f (d \sec (e+f x))^{5/3}}+\frac {\operatorname {AppellF1}\left (\frac {1}{2},2,\frac {11}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{5/6} \tan (e+f x)}{a^2 f (d \sec (e+f x))^{5/3}}+\frac {b^2 \operatorname {AppellF1}\left (\frac {3}{2},2,\frac {11}{6},\frac {5}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{5/6} \tan ^3(e+f x)}{3 a^4 f (d \sec (e+f x))^{5/3}}-\frac {a b}{\left (a^2+b^2\right ) f (d \sec (e+f x))^{5/3} \left (a^2-b^2 \tan ^2(e+f x)\right )} \]
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Time = 1.24 (sec) , antiderivative size = 717, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {3593, 771, 440, 455, 44, 53, 65, 216, 648, 632, 210, 642, 214, 524} \[ \int \frac {1}{(d \sec (e+f x))^{5/3} (a+b \tan (e+f x))^2} \, dx=\frac {\tan (e+f x) \sec ^2(e+f x)^{5/6} \operatorname {AppellF1}\left (\frac {1}{2},2,\frac {11}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right )}{a^2 f (d \sec (e+f x))^{5/3}}+\frac {11 a b^{8/3} \sec ^2(e+f x)^{5/6} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt {3} \sqrt [6]{a^2+b^2}}\right )}{2 \sqrt {3} f \left (a^2+b^2\right )^{17/6} (d \sec (e+f x))^{5/3}}-\frac {11 a b^{8/3} \sec ^2(e+f x)^{5/6} \arctan \left (\frac {2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt {3} \sqrt [6]{a^2+b^2}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3} f \left (a^2+b^2\right )^{17/6} (d \sec (e+f x))^{5/3}}-\frac {11 a b^{8/3} \sec ^2(e+f x)^{5/6} \text {arctanh}\left (\frac {\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right )}{3 f \left (a^2+b^2\right )^{17/6} (d \sec (e+f x))^{5/3}}+\frac {11 a b}{5 f \left (a^2+b^2\right )^2 (d \sec (e+f x))^{5/3}}-\frac {a b}{f \left (a^2+b^2\right ) (d \sec (e+f x))^{5/3} \left (a^2-b^2 \tan ^2(e+f x)\right )}+\frac {11 a b^{8/3} \sec ^2(e+f x)^{5/6} \log \left (-\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+\sqrt [3]{a^2+b^2}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right )}{12 f \left (a^2+b^2\right )^{17/6} (d \sec (e+f x))^{5/3}}-\frac {11 a b^{8/3} \sec ^2(e+f x)^{5/6} \log \left (\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+\sqrt [3]{a^2+b^2}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right )}{12 f \left (a^2+b^2\right )^{17/6} (d \sec (e+f x))^{5/3}}+\frac {b^2 \tan ^3(e+f x) \sec ^2(e+f x)^{5/6} \operatorname {AppellF1}\left (\frac {3}{2},2,\frac {11}{6},\frac {5}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right )}{3 a^4 f (d \sec (e+f x))^{5/3}} \]
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Rule 44
Rule 53
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 {\sec ^2(e+f x)^{5/6} \text {Subst}\left (\int \frac {1}{(a+x)^2 \left (1+\frac {x^2}{b^2}\right )^{11/6}} \, dx,x,b \tan (e+f x)\right )}{b f (d \sec (e+f x))^{5/3}} \\ & = \frac {\sec ^2(e+f x)^{5/6} \text {Subst}\left (\int \left (\frac {a^2}{\left (a^2-x^2\right )^2 \left (1+\frac {x^2}{b^2}\right )^{11/6}}-\frac {2 a x}{\left (a^2-x^2\right )^2 \left (1+\frac {x^2}{b^2}\right )^{11/6}}+\frac {x^2}{\left (-a^2+x^2\right )^2 \left (1+\frac {x^2}{b^2}\right )^{11/6}}\right ) \, dx,x,b \tan (e+f x)\right )}{b f (d \sec (e+f x))^{5/3}} \\ & = \frac {\sec ^2(e+f x)^{5/6} \text {Subst}\left (\int \frac {x^2}{\left (-a^2+x^2\right )^2 \left (1+\frac {x^2}{b^2}\right )^{11/6}} \, dx,x,b \tan (e+f x)\right )}{b f (d \sec (e+f x))^{5/3}}-\frac {\left (2 a \sec ^2(e+f x)^{5/6}\right ) \text {Subst}\left (\int \frac {x}{\left (a^2-x^2\right )^2 \left (1+\frac {x^2}{b^2}\right )^{11/6}} \, dx,x,b \tan (e+f x)\right )}{b f (d \sec (e+f x))^{5/3}}+\frac {\left (a^2 \sec ^2(e+f x)^{5/6}\right ) \text {Subst}\left (\int \frac {1}{\left (a^2-x^2\right )^2 \left (1+\frac {x^2}{b^2}\right )^{11/6}} \, dx,x,b \tan (e+f x)\right )}{b f (d \sec (e+f x))^{5/3}} \\ & = \frac {\operatorname {AppellF1}\left (\frac {1}{2},2,\frac {11}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{5/6} \tan (e+f x)}{a^2 f (d \sec (e+f x))^{5/3}}+\frac {b^2 \operatorname {AppellF1}\left (\frac {3}{2},2,\frac {11}{6},\frac {5}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{5/6} \tan ^3(e+f x)}{3 a^4 f (d \sec (e+f x))^{5/3}}-\frac {\left (a \sec ^2(e+f x)^{5/6}\right ) \text {Subst}\left (\int \frac {1}{\left (a^2-x\right )^2 \left (1+\frac {x}{b^2}\right )^{11/6}} \, dx,x,b^2 \tan ^2(e+f x)\right )}{b f (d \sec (e+f x))^{5/3}} \\ & = \frac {\operatorname {AppellF1}\left (\frac {1}{2},2,\frac {11}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{5/6} \tan (e+f x)}{a^2 f (d \sec (e+f x))^{5/3}}+\frac {b^2 \operatorname {AppellF1}\left (\frac {3}{2},2,\frac {11}{6},\frac {5}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{5/6} \tan ^3(e+f x)}{3 a^4 f (d \sec (e+f x))^{5/3}}-\frac {a b}{\left (a^2+b^2\right ) f (d \sec (e+f x))^{5/3} \left (a^2-b^2 \tan ^2(e+f x)\right )}-\frac {\left (11 a \sec ^2(e+f x)^{5/6}\right ) \text {Subst}\left (\int \frac {1}{\left (a^2-x\right ) \left (1+\frac {x}{b^2}\right )^{11/6}} \, dx,x,b^2 \tan ^2(e+f x)\right )}{6 b \left (a^2+b^2\right ) f (d \sec (e+f x))^{5/3}} \\ & = \frac {11 a b}{5 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{5/3}}+\frac {\operatorname {AppellF1}\left (\frac {1}{2},2,\frac {11}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{5/6} \tan (e+f x)}{a^2 f (d \sec (e+f x))^{5/3}}+\frac {b^2 \operatorname {AppellF1}\left (\frac {3}{2},2,\frac {11}{6},\frac {5}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{5/6} \tan ^3(e+f x)}{3 a^4 f (d \sec (e+f x))^{5/3}}-\frac {a b}{\left (a^2+b^2\right ) f (d \sec (e+f x))^{5/3} \left (a^2-b^2 \tan ^2(e+f x)\right )}-\frac {\left (11 a b \sec ^2(e+f x)^{5/6}\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 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{5/3}} \\ & = \frac {11 a b}{5 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{5/3}}+\frac {\operatorname {AppellF1}\left (\frac {1}{2},2,\frac {11}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{5/6} \tan (e+f x)}{a^2 f (d \sec (e+f x))^{5/3}}+\frac {b^2 \operatorname {AppellF1}\left (\frac {3}{2},2,\frac {11}{6},\frac {5}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{5/6} \tan ^3(e+f x)}{3 a^4 f (d \sec (e+f x))^{5/3}}-\frac {a b}{\left (a^2+b^2\right ) f (d \sec (e+f x))^{5/3} \left (a^2-b^2 \tan ^2(e+f x)\right )}-\frac {\left (11 a b^3 \sec ^2(e+f x)^{5/6}\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 )^2 f (d \sec (e+f x))^{5/3}} \\ & = \frac {11 a b}{5 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{5/3}}+\frac {\operatorname {AppellF1}\left (\frac {1}{2},2,\frac {11}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{5/6} \tan (e+f x)}{a^2 f (d \sec (e+f x))^{5/3}}+\frac {b^2 \operatorname {AppellF1}\left (\frac {3}{2},2,\frac {11}{6},\frac {5}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{5/6} \tan ^3(e+f x)}{3 a^4 f (d \sec (e+f x))^{5/3}}-\frac {a b}{\left (a^2+b^2\right ) f (d \sec (e+f x))^{5/3} \left (a^2-b^2 \tan ^2(e+f x)\right )}-\frac {\left (11 a b^3 \sec ^2(e+f x)^{5/6}\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 )^{17/6} f (d \sec (e+f x))^{5/3}}-\frac {\left (11 a b^3 \sec ^2(e+f x)^{5/6}\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 )^{17/6} f (d \sec (e+f x))^{5/3}}-\frac {\left (11 a b^3 \sec ^2(e+f x)^{5/6}\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 )^{8/3} f (d \sec (e+f x))^{5/3}} \\ & = \frac {11 a b}{5 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{5/3}}-\frac {11 a b^{8/3} \text {arctanh}\left (\frac {\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right ) \sec ^2(e+f x)^{5/6}}{3 \left (a^2+b^2\right )^{17/6} f (d \sec (e+f x))^{5/3}}+\frac {\operatorname {AppellF1}\left (\frac {1}{2},2,\frac {11}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{5/6} \tan (e+f x)}{a^2 f (d \sec (e+f x))^{5/3}}+\frac {b^2 \operatorname {AppellF1}\left (\frac {3}{2},2,\frac {11}{6},\frac {5}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{5/6} \tan ^3(e+f x)}{3 a^4 f (d \sec (e+f x))^{5/3}}-\frac {a b}{\left (a^2+b^2\right ) f (d \sec (e+f x))^{5/3} \left (a^2-b^2 \tan ^2(e+f x)\right )}+\frac {\left (11 a b^{8/3} \sec ^2(e+f x)^{5/6}\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 )^{17/6} f (d \sec (e+f x))^{5/3}}-\frac {\left (11 a b^{8/3} \sec ^2(e+f x)^{5/6}\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 )^{17/6} f (d \sec (e+f x))^{5/3}}-\frac {\left (11 a b^3 \sec ^2(e+f x)^{5/6}\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 )^{8/3} f (d \sec (e+f x))^{5/3}}-\frac {\left (11 a b^3 \sec ^2(e+f x)^{5/6}\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 )^{8/3} f (d \sec (e+f x))^{5/3}} \\ & = \frac {11 a b}{5 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{5/3}}-\frac {11 a b^{8/3} \text {arctanh}\left (\frac {\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right ) \sec ^2(e+f x)^{5/6}}{3 \left (a^2+b^2\right )^{17/6} f (d \sec (e+f x))^{5/3}}+\frac {11 a b^{8/3} \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 ) \sec ^2(e+f x)^{5/6}}{12 \left (a^2+b^2\right )^{17/6} f (d \sec (e+f x))^{5/3}}-\frac {11 a b^{8/3} \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 ) \sec ^2(e+f x)^{5/6}}{12 \left (a^2+b^2\right )^{17/6} f (d \sec (e+f x))^{5/3}}+\frac {\operatorname {AppellF1}\left (\frac {1}{2},2,\frac {11}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{5/6} \tan (e+f x)}{a^2 f (d \sec (e+f x))^{5/3}}+\frac {b^2 \operatorname {AppellF1}\left (\frac {3}{2},2,\frac {11}{6},\frac {5}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{5/6} \tan ^3(e+f x)}{3 a^4 f (d \sec (e+f x))^{5/3}}-\frac {a b}{\left (a^2+b^2\right ) f (d \sec (e+f x))^{5/3} \left (a^2-b^2 \tan ^2(e+f x)\right )}-\frac {\left (11 a b^{8/3} \sec ^2(e+f x)^{5/6}\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 )^{17/6} f (d \sec (e+f x))^{5/3}}+\frac {\left (11 a b^{8/3} \sec ^2(e+f x)^{5/6}\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 )^{17/6} f (d \sec (e+f x))^{5/3}} \\ & = \frac {11 a b}{5 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{5/3}}+\frac {11 a b^{8/3} \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 ) \sec ^2(e+f x)^{5/6}}{2 \sqrt {3} \left (a^2+b^2\right )^{17/6} f (d \sec (e+f x))^{5/3}}-\frac {11 a b^{8/3} \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 ) \sec ^2(e+f x)^{5/6}}{2 \sqrt {3} \left (a^2+b^2\right )^{17/6} f (d \sec (e+f x))^{5/3}}-\frac {11 a b^{8/3} \text {arctanh}\left (\frac {\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right ) \sec ^2(e+f x)^{5/6}}{3 \left (a^2+b^2\right )^{17/6} f (d \sec (e+f x))^{5/3}}+\frac {11 a b^{8/3} \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 ) \sec ^2(e+f x)^{5/6}}{12 \left (a^2+b^2\right )^{17/6} f (d \sec (e+f x))^{5/3}}-\frac {11 a b^{8/3} \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 ) \sec ^2(e+f x)^{5/6}}{12 \left (a^2+b^2\right )^{17/6} f (d \sec (e+f x))^{5/3}}+\frac {\operatorname {AppellF1}\left (\frac {1}{2},2,\frac {11}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{5/6} \tan (e+f x)}{a^2 f (d \sec (e+f x))^{5/3}}+\frac {b^2 \operatorname {AppellF1}\left (\frac {3}{2},2,\frac {11}{6},\frac {5}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{5/6} \tan ^3(e+f x)}{3 a^4 f (d \sec (e+f x))^{5/3}}-\frac {a b}{\left (a^2+b^2\right ) f (d \sec (e+f x))^{5/3} \left (a^2-b^2 \tan ^2(e+f x)\right )} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(11783\) vs. \(2(717)=1434\).
Time = 81.40 (sec) , antiderivative size = 11783, normalized size of antiderivative = 16.43 \[ \int \frac {1}{(d \sec (e+f x))^{5/3} (a+b \tan (e+f x))^2} \, dx=\text {Result too large to show} \]
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\[\int \frac {1}{\left (d \sec \left (f x +e \right )\right )^{\frac {5}{3}} \left (a +b \tan \left (f x +e \right )\right )^{2}}d x\]
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Timed out. \[ \int \frac {1}{(d \sec (e+f x))^{5/3} (a+b \tan (e+f x))^2} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(d \sec (e+f x))^{5/3} (a+b \tan (e+f x))^2} \, dx=\int \frac {1}{\left (d \sec {\left (e + f x \right )}\right )^{\frac {5}{3}} \left (a + b \tan {\left (e + f x \right )}\right )^{2}}\, dx \]
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\[ \int \frac {1}{(d \sec (e+f x))^{5/3} (a+b \tan (e+f x))^2} \, dx=\int { \frac {1}{\left (d \sec \left (f x + e\right )\right )^{\frac {5}{3}} {\left (b \tan \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {1}{(d \sec (e+f x))^{5/3} (a+b \tan (e+f x))^2} \, dx=\int { \frac {1}{\left (d \sec \left (f x + e\right )\right )^{\frac {5}{3}} {\left (b \tan \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {1}{(d \sec (e+f x))^{5/3} (a+b \tan (e+f x))^2} \, dx=\int \frac {1}{{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{5/3}\,{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^2} \,d x \]
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