Integrand size = 25, antiderivative size = 520 \[ \int \frac {1}{(d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2} \, dx=-\frac {7 a b^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sec ^2(e+f x)^{3/4}}{2 \left (a^2+b^2\right )^{11/4} f (d \sec (e+f x))^{3/2}}-\frac {7 a b^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sec ^2(e+f x)^{3/4}}{2 \left (a^2+b^2\right )^{11/4} f (d \sec (e+f x))^{3/2}}+\frac {\left (2 a^2-5 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \arctan (\tan (e+f x)),2\right ) \sec ^2(e+f x)^{3/4}}{3 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}}+\frac {7 a^2 b^2 \cot (e+f x) \operatorname {EllipticPi}\left (-\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right ) \sec ^2(e+f x)^{3/4} \sqrt {-\tan ^2(e+f x)}}{2 \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2}}+\frac {7 a^2 b^2 \cot (e+f x) \operatorname {EllipticPi}\left (\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right ) \sec ^2(e+f x)^{3/4} \sqrt {-\tan ^2(e+f x)}}{2 \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2}}+\frac {b \left (2 a^2-5 b^2\right ) \sec ^2(e+f x)}{3 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}+\frac {2 (b+a \tan (e+f x))}{3 \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))} \]
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Time = 0.67 (sec) , antiderivative size = 520, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {3593, 755, 849, 858, 237, 761, 410, 109, 418, 1227, 551, 455, 65, 218, 214, 211} \[ \int \frac {1}{(d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2} \, dx=\frac {7 a^2 b^2 \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \sec ^2(e+f x)^{3/4} \operatorname {EllipticPi}\left (-\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right )}{2 f \left (a^2+b^2\right )^3 (d \sec (e+f x))^{3/2}}+\frac {7 a^2 b^2 \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \sec ^2(e+f x)^{3/4} \operatorname {EllipticPi}\left (\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right )}{2 f \left (a^2+b^2\right )^3 (d \sec (e+f x))^{3/2}}+\frac {\left (2 a^2-5 b^2\right ) \sec ^2(e+f x)^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan (\tan (e+f x)),2\right )}{3 f \left (a^2+b^2\right )^2 (d \sec (e+f x))^{3/2}}-\frac {7 a b^{5/2} \sec ^2(e+f x)^{3/4} \arctan \left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{2 f \left (a^2+b^2\right )^{11/4} (d \sec (e+f x))^{3/2}}-\frac {7 a b^{5/2} \sec ^2(e+f x)^{3/4} \text {arctanh}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{2 f \left (a^2+b^2\right )^{11/4} (d \sec (e+f x))^{3/2}}+\frac {b \left (2 a^2-5 b^2\right ) \sec ^2(e+f x)}{3 f \left (a^2+b^2\right )^2 (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}+\frac {2 (a \tan (e+f x)+b)}{3 f \left (a^2+b^2\right ) (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))} \]
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Rule 65
Rule 109
Rule 211
Rule 214
Rule 218
Rule 237
Rule 410
Rule 418
Rule 455
Rule 551
Rule 755
Rule 761
Rule 849
Rule 858
Rule 1227
Rule 3593
Rubi steps \begin{align*} \text {integral}& = \frac {\sec ^2(e+f x)^{3/4} \text {Subst}\left (\int \frac {1}{(a+x)^2 \left (1+\frac {x^2}{b^2}\right )^{7/4}} \, dx,x,b \tan (e+f x)\right )}{b f (d \sec (e+f x))^{3/2}} \\ & = \frac {2 (b+a \tan (e+f x))}{3 \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}-\frac {\left (2 b \sec ^2(e+f x)^{3/4}\right ) \text {Subst}\left (\int \frac {\frac {1}{2} \left (-5-\frac {a^2}{b^2}\right )-\frac {3 a x}{2 b^2}}{(a+x)^2 \left (1+\frac {x^2}{b^2}\right )^{3/4}} \, dx,x,b \tan (e+f x)\right )}{3 \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2}} \\ & = \frac {b \left (2 a^2-5 b^2\right ) \sec ^2(e+f x)}{3 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}+\frac {2 (b+a \tan (e+f x))}{3 \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}+\frac {\left (2 b^3 \sec ^2(e+f x)^{3/4}\right ) \text {Subst}\left (\int \frac {\frac {a \left (a^2+8 b^2\right )}{2 b^4}+\frac {\left (2 a^2-5 b^2\right ) x}{4 b^4}}{(a+x) \left (1+\frac {x^2}{b^2}\right )^{3/4}} \, dx,x,b \tan (e+f x)\right )}{3 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}} \\ & = \frac {b \left (2 a^2-5 b^2\right ) \sec ^2(e+f x)}{3 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}+\frac {2 (b+a \tan (e+f x))}{3 \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}+\frac {\left (7 a b \sec ^2(e+f x)^{3/4}\right ) \text {Subst}\left (\int \frac {1}{(a+x) \left (1+\frac {x^2}{b^2}\right )^{3/4}} \, dx,x,b \tan (e+f x)\right )}{2 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}}+\frac {\left (\left (2 a^2-5 b^2\right ) \sec ^2(e+f x)^{3/4}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {x^2}{b^2}\right )^{3/4}} \, dx,x,b \tan (e+f x)\right )}{6 b \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}} \\ & = \frac {\left (2 a^2-5 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \arctan (\tan (e+f x)),2\right ) \sec ^2(e+f x)^{3/4}}{3 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}}+\frac {b \left (2 a^2-5 b^2\right ) \sec ^2(e+f x)}{3 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}+\frac {2 (b+a \tan (e+f x))}{3 \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}-\frac {\left (7 a b \sec ^2(e+f x)^{3/4}\right ) \text {Subst}\left (\int \frac {x}{\left (a^2-x^2\right ) \left (1+\frac {x^2}{b^2}\right )^{3/4}} \, dx,x,b \tan (e+f x)\right )}{2 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}}+\frac {\left (7 a^2 b \sec ^2(e+f x)^{3/4}\right ) \text {Subst}\left (\int \frac {1}{\left (a^2-x^2\right ) \left (1+\frac {x^2}{b^2}\right )^{3/4}} \, dx,x,b \tan (e+f x)\right )}{2 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}} \\ & = \frac {\left (2 a^2-5 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \arctan (\tan (e+f x)),2\right ) \sec ^2(e+f x)^{3/4}}{3 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}}+\frac {b \left (2 a^2-5 b^2\right ) \sec ^2(e+f x)}{3 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}+\frac {2 (b+a \tan (e+f x))}{3 \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}-\frac {\left (7 a b \sec ^2(e+f x)^{3/4}\right ) \text {Subst}\left (\int \frac {1}{\left (a^2-x\right ) \left (1+\frac {x}{b^2}\right )^{3/4}} \, dx,x,b^2 \tan ^2(e+f x)\right )}{4 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}}+\frac {\left (7 a^2 \cot (e+f x) \sec ^2(e+f x)^{3/4} \sqrt {-\tan ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\left (a^2-x\right ) \sqrt {-\frac {x}{b^2}} \left (1+\frac {x}{b^2}\right )^{3/4}} \, dx,x,b^2 \tan ^2(e+f x)\right )}{4 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}} \\ & = \frac {\left (2 a^2-5 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \arctan (\tan (e+f x)),2\right ) \sec ^2(e+f x)^{3/4}}{3 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}}+\frac {b \left (2 a^2-5 b^2\right ) \sec ^2(e+f x)}{3 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}+\frac {2 (b+a \tan (e+f x))}{3 \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}-\frac {\left (7 a b^3 \sec ^2(e+f x)^{3/4}\right ) \text {Subst}\left (\int \frac {1}{a^2+b^2-b^2 x^4} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{\left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}}-\frac {\left (7 a^2 \cot (e+f x) \sec ^2(e+f x)^{3/4} \sqrt {-\tan ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^4} \left (-1-\frac {a^2}{b^2}+x^4\right )} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{\left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}} \\ & = \frac {\left (2 a^2-5 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \arctan (\tan (e+f x)),2\right ) \sec ^2(e+f x)^{3/4}}{3 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}}+\frac {b \left (2 a^2-5 b^2\right ) \sec ^2(e+f x)}{3 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}+\frac {2 (b+a \tan (e+f x))}{3 \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}-\frac {\left (7 a b^3 \sec ^2(e+f x)^{3/4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a^2+b^2}-b x^2} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{2 \left (a^2+b^2\right )^{5/2} f (d \sec (e+f x))^{3/2}}-\frac {\left (7 a b^3 \sec ^2(e+f x)^{3/4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a^2+b^2}+b x^2} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{2 \left (a^2+b^2\right )^{5/2} f (d \sec (e+f x))^{3/2}}+\frac {\left (7 a^2 b^2 \cot (e+f x) \sec ^2(e+f x)^{3/4} \sqrt {-\tan ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {b x^2}{\sqrt {a^2+b^2}}\right ) \sqrt {1-x^4}} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{2 \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2}}+\frac {\left (7 a^2 b^2 \cot (e+f x) \sec ^2(e+f x)^{3/4} \sqrt {-\tan ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {b x^2}{\sqrt {a^2+b^2}}\right ) \sqrt {1-x^4}} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{2 \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2}} \\ & = -\frac {7 a b^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sec ^2(e+f x)^{3/4}}{2 \left (a^2+b^2\right )^{11/4} f (d \sec (e+f x))^{3/2}}-\frac {7 a b^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sec ^2(e+f x)^{3/4}}{2 \left (a^2+b^2\right )^{11/4} f (d \sec (e+f x))^{3/2}}+\frac {\left (2 a^2-5 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \arctan (\tan (e+f x)),2\right ) \sec ^2(e+f x)^{3/4}}{3 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}}+\frac {b \left (2 a^2-5 b^2\right ) \sec ^2(e+f x)}{3 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}+\frac {2 (b+a \tan (e+f x))}{3 \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}+\frac {\left (7 a^2 b^2 \cot (e+f x) \sec ^2(e+f x)^{3/4} \sqrt {-\tan ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (1-\frac {b x^2}{\sqrt {a^2+b^2}}\right )} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{2 \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2}}+\frac {\left (7 a^2 b^2 \cot (e+f x) \sec ^2(e+f x)^{3/4} \sqrt {-\tan ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (1+\frac {b x^2}{\sqrt {a^2+b^2}}\right )} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{2 \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2}} \\ & = -\frac {7 a b^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sec ^2(e+f x)^{3/4}}{2 \left (a^2+b^2\right )^{11/4} f (d \sec (e+f x))^{3/2}}-\frac {7 a b^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sec ^2(e+f x)^{3/4}}{2 \left (a^2+b^2\right )^{11/4} f (d \sec (e+f x))^{3/2}}+\frac {\left (2 a^2-5 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \arctan (\tan (e+f x)),2\right ) \sec ^2(e+f x)^{3/4}}{3 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}}+\frac {7 a^2 b^2 \cot (e+f x) \operatorname {EllipticPi}\left (-\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right ) \sec ^2(e+f x)^{3/4} \sqrt {-\tan ^2(e+f x)}}{2 \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2}}+\frac {7 a^2 b^2 \cot (e+f x) \operatorname {EllipticPi}\left (\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right ) \sec ^2(e+f x)^{3/4} \sqrt {-\tan ^2(e+f x)}}{2 \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2}}+\frac {b \left (2 a^2-5 b^2\right ) \sec ^2(e+f x)}{3 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}+\frac {2 (b+a \tan (e+f x))}{3 \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 12.23 (sec) , antiderivative size = 528, normalized size of antiderivative = 1.02 \[ \int \frac {1}{(d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2} \, dx=\frac {\sec ^4(e+f x) (a \cos (e+f x)+b \sin (e+f x))^2 \left (\frac {b \left (2 a^2-3 b^2\right )}{3 a (a-i b)^2 (a+i b)^2}+\frac {2 a b \cos (2 (e+f x))}{3 (a-i b)^2 (a+i b)^2}+\frac {b^4 \sin (e+f x)}{a (a-i b)^2 (a+i b)^2 (a \cos (e+f x)+b \sin (e+f x))}+\frac {\left (a^2-b^2\right ) \sin (2 (e+f x))}{3 (a-i b)^2 (a+i b)^2}\right )}{f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2}+\frac {\sec ^2(e+f x) \sec ^2(e+f x)^{3/4} (a \cos (e+f x)+b \sin (e+f x))^2 \left (\left (2 a^4-3 a^2 b^2-5 b^4\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},-\tan ^2(e+f x)\right ) \tan (e+f x)+21 a b^2 \left (-\sqrt {b} \sqrt [4]{a^2+b^2} \left (\arctan \left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )+\text {arctanh}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )\right )+a \cot (e+f x) \operatorname {EllipticPi}\left (-\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right ) \sqrt {-\tan ^2(e+f x)}+a \cot (e+f x) \operatorname {EllipticPi}\left (\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right ) \sqrt {-\tan ^2(e+f x)}\right )\right )}{6 \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 14916 vs. \(2 (477 ) = 954\).
Time = 13.29 (sec) , antiderivative size = 14917, normalized size of antiderivative = 28.69
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Timed out. \[ \int \frac {1}{(d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2} \, dx=\int \frac {1}{\left (d \sec {\left (e + f x \right )}\right )^{\frac {3}{2}} \left (a + b \tan {\left (e + f x \right )}\right )^{2}}\, dx \]
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\[ \int \frac {1}{(d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2} \, dx=\int { \frac {1}{\left (d \sec \left (f x + e\right )\right )^{\frac {3}{2}} {\left (b \tan \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {1}{(d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2} \, dx=\int { \frac {1}{\left (d \sec \left (f x + e\right )\right )^{\frac {3}{2}} {\left (b \tan \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {1}{(d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2} \, dx=\int \frac {1}{{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{3/2}\,{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^2} \,d x \]
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