Optimal. Leaf size=687 \[ \frac {\tan (e+f x) (d \sec (e+f x))^{5/3} F_1\left (\frac {1}{2};2,\frac {1}{6};\frac {3}{2};\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right )}{a^2 f \sec ^2(e+f x)^{5/6}}-\frac {a b (d \sec (e+f x))^{5/3}}{f \left (a^2+b^2\right ) \left (a^2-b^2 \tan ^2(e+f x)\right )}+\frac {a (d \sec (e+f x))^{5/3} \log \left (-\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+\sqrt [3]{a^2+b^2}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right )}{12 b^{2/3} f \left (a^2+b^2\right )^{7/6} \sec ^2(e+f x)^{5/6}}-\frac {a (d \sec (e+f x))^{5/3} \log \left (\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+\sqrt [3]{a^2+b^2}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right )}{12 b^{2/3} f \left (a^2+b^2\right )^{7/6} \sec ^2(e+f x)^{5/6}}-\frac {a (d \sec (e+f x))^{5/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt {3} \sqrt [6]{a^2+b^2}}\right )}{2 \sqrt {3} b^{2/3} f \left (a^2+b^2\right )^{7/6} \sec ^2(e+f x)^{5/6}}+\frac {a (d \sec (e+f x))^{5/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt {3} \sqrt [6]{a^2+b^2}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3} b^{2/3} f \left (a^2+b^2\right )^{7/6} \sec ^2(e+f x)^{5/6}}-\frac {a (d \sec (e+f x))^{5/3} \tanh ^{-1}\left (\frac {\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right )}{3 b^{2/3} f \left (a^2+b^2\right )^{7/6} \sec ^2(e+f x)^{5/6}}+\frac {b^2 \tan ^3(e+f x) (d \sec (e+f x))^{5/3} F_1\left (\frac {3}{2};2,\frac {1}{6};\frac {5}{2};\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right )}{3 a^4 f \sec ^2(e+f x)^{5/6}} \]
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Rubi [A] time = 0.94, antiderivative size = 687, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 13, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {3512, 757, 429, 444, 51, 63, 296, 634, 618, 204, 628, 208, 510} \[ \frac {b^2 \tan ^3(e+f x) (d \sec (e+f x))^{5/3} F_1\left (\frac {3}{2};2,\frac {1}{6};\frac {5}{2};\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right )}{3 a^4 f \sec ^2(e+f x)^{5/6}}+\frac {\tan (e+f x) (d \sec (e+f x))^{5/3} F_1\left (\frac {1}{2};2,\frac {1}{6};\frac {3}{2};\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right )}{a^2 f \sec ^2(e+f x)^{5/6}}+\frac {a (d \sec (e+f x))^{5/3} \log \left (-\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+\sqrt [3]{a^2+b^2}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right )}{12 b^{2/3} f \left (a^2+b^2\right )^{7/6} \sec ^2(e+f x)^{5/6}}-\frac {a (d \sec (e+f x))^{5/3} \log \left (\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+\sqrt [3]{a^2+b^2}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right )}{12 b^{2/3} f \left (a^2+b^2\right )^{7/6} \sec ^2(e+f x)^{5/6}}-\frac {a (d \sec (e+f x))^{5/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt {3} \sqrt [6]{a^2+b^2}}\right )}{2 \sqrt {3} b^{2/3} f \left (a^2+b^2\right )^{7/6} \sec ^2(e+f x)^{5/6}}+\frac {a (d \sec (e+f x))^{5/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt {3} \sqrt [6]{a^2+b^2}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3} b^{2/3} f \left (a^2+b^2\right )^{7/6} \sec ^2(e+f x)^{5/6}}-\frac {a (d \sec (e+f x))^{5/3} \tanh ^{-1}\left (\frac {\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right )}{3 b^{2/3} f \left (a^2+b^2\right )^{7/6} \sec ^2(e+f x)^{5/6}}-\frac {a b (d \sec (e+f x))^{5/3}}{f \left (a^2+b^2\right ) \left (a^2-b^2 \tan ^2(e+f x)\right )} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 204
Rule 208
Rule 296
Rule 429
Rule 444
Rule 510
Rule 618
Rule 628
Rule 634
Rule 757
Rule 3512
Rubi steps
\begin {align*} \int \frac {(d \sec (e+f x))^{5/3}}{(a+b \tan (e+f x))^2} \, dx &=\frac {(d \sec (e+f x))^{5/3} \operatorname {Subst}\left (\int \frac {1}{(a+x)^2 \sqrt [6]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{b f \sec ^2(e+f x)^{5/6}}\\ &=\frac {(d \sec (e+f x))^{5/3} \operatorname {Subst}\left (\int \left (\frac {a^2}{\left (a^2-x^2\right )^2 \sqrt [6]{1+\frac {x^2}{b^2}}}-\frac {2 a x}{\left (a^2-x^2\right )^2 \sqrt [6]{1+\frac {x^2}{b^2}}}+\frac {x^2}{\left (-a^2+x^2\right )^2 \sqrt [6]{1+\frac {x^2}{b^2}}}\right ) \, dx,x,b \tan (e+f x)\right )}{b f \sec ^2(e+f x)^{5/6}}\\ &=\frac {(d \sec (e+f x))^{5/3} \operatorname {Subst}\left (\int \frac {x^2}{\left (-a^2+x^2\right )^2 \sqrt [6]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{b f \sec ^2(e+f x)^{5/6}}-\frac {\left (2 a (d \sec (e+f x))^{5/3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (a^2-x^2\right )^2 \sqrt [6]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{b f \sec ^2(e+f x)^{5/6}}+\frac {\left (a^2 (d \sec (e+f x))^{5/3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (a^2-x^2\right )^2 \sqrt [6]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{b f \sec ^2(e+f x)^{5/6}}\\ &=\frac {F_1\left (\frac {1}{2};2,\frac {1}{6};\frac {3}{2};\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) (d \sec (e+f x))^{5/3} \tan (e+f x)}{a^2 f \sec ^2(e+f x)^{5/6}}+\frac {b^2 F_1\left (\frac {3}{2};2,\frac {1}{6};\frac {5}{2};\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) (d \sec (e+f x))^{5/3} \tan ^3(e+f x)}{3 a^4 f \sec ^2(e+f x)^{5/6}}-\frac {\left (a (d \sec (e+f x))^{5/3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (a^2-x\right )^2 \sqrt [6]{1+\frac {x}{b^2}}} \, dx,x,b^2 \tan ^2(e+f x)\right )}{b f \sec ^2(e+f x)^{5/6}}\\ &=\frac {F_1\left (\frac {1}{2};2,\frac {1}{6};\frac {3}{2};\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) (d \sec (e+f x))^{5/3} \tan (e+f x)}{a^2 f \sec ^2(e+f x)^{5/6}}+\frac {b^2 F_1\left (\frac {3}{2};2,\frac {1}{6};\frac {5}{2};\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) (d \sec (e+f x))^{5/3} \tan ^3(e+f x)}{3 a^4 f \sec ^2(e+f x)^{5/6}}-\frac {a b (d \sec (e+f x))^{5/3}}{\left (a^2+b^2\right ) f \left (a^2-b^2 \tan ^2(e+f x)\right )}-\frac {\left (a (d \sec (e+f x))^{5/3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (a^2-x\right ) \sqrt [6]{1+\frac {x}{b^2}}} \, dx,x,b^2 \tan ^2(e+f x)\right )}{6 b \left (a^2+b^2\right ) f \sec ^2(e+f x)^{5/6}}\\ &=\frac {F_1\left (\frac {1}{2};2,\frac {1}{6};\frac {3}{2};\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) (d \sec (e+f x))^{5/3} \tan (e+f x)}{a^2 f \sec ^2(e+f x)^{5/6}}+\frac {b^2 F_1\left (\frac {3}{2};2,\frac {1}{6};\frac {5}{2};\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) (d \sec (e+f x))^{5/3} \tan ^3(e+f x)}{3 a^4 f \sec ^2(e+f x)^{5/6}}-\frac {a b (d \sec (e+f x))^{5/3}}{\left (a^2+b^2\right ) f \left (a^2-b^2 \tan ^2(e+f x)\right )}-\frac {\left (a b (d \sec (e+f x))^{5/3}\right ) \operatorname {Subst}\left (\int \frac {x^4}{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 \sec ^2(e+f x)^{5/6}}\\ &=\frac {F_1\left (\frac {1}{2};2,\frac {1}{6};\frac {3}{2};\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) (d \sec (e+f x))^{5/3} \tan (e+f x)}{a^2 f \sec ^2(e+f x)^{5/6}}+\frac {b^2 F_1\left (\frac {3}{2};2,\frac {1}{6};\frac {5}{2};\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) (d \sec (e+f x))^{5/3} \tan ^3(e+f x)}{3 a^4 f \sec ^2(e+f x)^{5/6}}-\frac {a b (d \sec (e+f x))^{5/3}}{\left (a^2+b^2\right ) f \left (a^2-b^2 \tan ^2(e+f x)\right )}-\frac {\left (a (d \sec (e+f x))^{5/3}\right ) \operatorname {Subst}\left (\int \frac {-\frac {1}{2} \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 \sqrt [3]{b} \left (a^2+b^2\right )^{7/6} f \sec ^2(e+f x)^{5/6}}-\frac {\left (a (d \sec (e+f x))^{5/3}\right ) \operatorname {Subst}\left (\int \frac {-\frac {1}{2} \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 \sqrt [3]{b} \left (a^2+b^2\right )^{7/6} f \sec ^2(e+f x)^{5/6}}-\frac {\left (a (d \sec (e+f x))^{5/3}\right ) \operatorname {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 \sqrt [3]{b} \left (a^2+b^2\right ) f \sec ^2(e+f x)^{5/6}}\\ &=-\frac {a \tanh ^{-1}\left (\frac {\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right ) (d \sec (e+f x))^{5/3}}{3 b^{2/3} \left (a^2+b^2\right )^{7/6} f \sec ^2(e+f x)^{5/6}}+\frac {F_1\left (\frac {1}{2};2,\frac {1}{6};\frac {3}{2};\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) (d \sec (e+f x))^{5/3} \tan (e+f x)}{a^2 f \sec ^2(e+f x)^{5/6}}+\frac {b^2 F_1\left (\frac {3}{2};2,\frac {1}{6};\frac {5}{2};\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) (d \sec (e+f x))^{5/3} \tan ^3(e+f x)}{3 a^4 f \sec ^2(e+f x)^{5/6}}-\frac {a b (d \sec (e+f x))^{5/3}}{\left (a^2+b^2\right ) f \left (a^2-b^2 \tan ^2(e+f x)\right )}+\frac {\left (a (d \sec (e+f x))^{5/3}\right ) \operatorname {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 b^{2/3} \left (a^2+b^2\right )^{7/6} f \sec ^2(e+f x)^{5/6}}-\frac {\left (a (d \sec (e+f x))^{5/3}\right ) \operatorname {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 b^{2/3} \left (a^2+b^2\right )^{7/6} f \sec ^2(e+f x)^{5/6}}+\frac {\left (a (d \sec (e+f x))^{5/3}\right ) \operatorname {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 \sqrt [3]{b} \left (a^2+b^2\right ) f \sec ^2(e+f x)^{5/6}}+\frac {\left (a (d \sec (e+f x))^{5/3}\right ) \operatorname {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 \sqrt [3]{b} \left (a^2+b^2\right ) f \sec ^2(e+f x)^{5/6}}\\ &=-\frac {a \tanh ^{-1}\left (\frac {\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right ) (d \sec (e+f x))^{5/3}}{3 b^{2/3} \left (a^2+b^2\right )^{7/6} f \sec ^2(e+f x)^{5/6}}+\frac {a \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 ) (d \sec (e+f x))^{5/3}}{12 b^{2/3} \left (a^2+b^2\right )^{7/6} f \sec ^2(e+f x)^{5/6}}-\frac {a \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 ) (d \sec (e+f x))^{5/3}}{12 b^{2/3} \left (a^2+b^2\right )^{7/6} f \sec ^2(e+f x)^{5/6}}+\frac {F_1\left (\frac {1}{2};2,\frac {1}{6};\frac {3}{2};\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) (d \sec (e+f x))^{5/3} \tan (e+f x)}{a^2 f \sec ^2(e+f x)^{5/6}}+\frac {b^2 F_1\left (\frac {3}{2};2,\frac {1}{6};\frac {5}{2};\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) (d \sec (e+f x))^{5/3} \tan ^3(e+f x)}{3 a^4 f \sec ^2(e+f x)^{5/6}}-\frac {a b (d \sec (e+f x))^{5/3}}{\left (a^2+b^2\right ) f \left (a^2-b^2 \tan ^2(e+f x)\right )}+\frac {\left (a (d \sec (e+f x))^{5/3}\right ) \operatorname {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 b^{2/3} \left (a^2+b^2\right )^{7/6} f \sec ^2(e+f x)^{5/6}}-\frac {\left (a (d \sec (e+f x))^{5/3}\right ) \operatorname {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 b^{2/3} \left (a^2+b^2\right )^{7/6} f \sec ^2(e+f x)^{5/6}}\\ &=-\frac {a \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}}{\sqrt {3}}\right ) (d \sec (e+f x))^{5/3}}{2 \sqrt {3} b^{2/3} \left (a^2+b^2\right )^{7/6} f \sec ^2(e+f x)^{5/6}}+\frac {a \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}}{\sqrt {3}}\right ) (d \sec (e+f x))^{5/3}}{2 \sqrt {3} b^{2/3} \left (a^2+b^2\right )^{7/6} f \sec ^2(e+f x)^{5/6}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right ) (d \sec (e+f x))^{5/3}}{3 b^{2/3} \left (a^2+b^2\right )^{7/6} f \sec ^2(e+f x)^{5/6}}+\frac {a \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 ) (d \sec (e+f x))^{5/3}}{12 b^{2/3} \left (a^2+b^2\right )^{7/6} f \sec ^2(e+f x)^{5/6}}-\frac {a \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 ) (d \sec (e+f x))^{5/3}}{12 b^{2/3} \left (a^2+b^2\right )^{7/6} f \sec ^2(e+f x)^{5/6}}+\frac {F_1\left (\frac {1}{2};2,\frac {1}{6};\frac {3}{2};\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) (d \sec (e+f x))^{5/3} \tan (e+f x)}{a^2 f \sec ^2(e+f x)^{5/6}}+\frac {b^2 F_1\left (\frac {3}{2};2,\frac {1}{6};\frac {5}{2};\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) (d \sec (e+f x))^{5/3} \tan ^3(e+f x)}{3 a^4 f \sec ^2(e+f x)^{5/6}}-\frac {a b (d \sec (e+f x))^{5/3}}{\left (a^2+b^2\right ) f \left (a^2-b^2 \tan ^2(e+f x)\right )}\\ \end {align*}
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Mathematica [C] time = 39.52, size = 3398, normalized size = 4.95 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \sec \left (f x + e\right )\right )^{\frac {5}{3}}}{{\left (b \tan \left (f x + e\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.38, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \sec \left (f x +e \right )\right )^{\frac {5}{3}}}{\left (a +b \tan \left (f x +e \right )\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{5/3}}{{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \sec {\left (e + f x \right )}\right )^{\frac {5}{3}}}{\left (a + b \tan {\left (e + f x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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