Integrand size = 25, antiderivative size = 515 \[ \int \frac {\sqrt {d \sec (e+f x)}}{(a+b \tan (e+f x))^3} \, dx=-\frac {3 \sqrt {b} \left (5 a^2-2 b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sqrt {d \sec (e+f x)}}{8 \left (a^2+b^2\right )^{11/4} f \sqrt [4]{\sec ^2(e+f x)}}-\frac {3 \sqrt {b} \left (5 a^2-2 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sqrt {d \sec (e+f x)}}{8 \left (a^2+b^2\right )^{11/4} f \sqrt [4]{\sec ^2(e+f x)}}-\frac {7 a \operatorname {EllipticF}\left (\frac {1}{2} \arctan (\tan (e+f x)),2\right ) \sqrt {d \sec (e+f x)}}{4 \left (a^2+b^2\right )^2 f \sqrt [4]{\sec ^2(e+f x)}}+\frac {3 a \left (5 a^2-2 b^2\right ) \cot (e+f x) \operatorname {EllipticPi}\left (-\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right ) \sqrt {d \sec (e+f x)} \sqrt {-\tan ^2(e+f x)}}{8 \left (a^2+b^2\right )^3 f \sqrt [4]{\sec ^2(e+f x)}}+\frac {3 a \left (5 a^2-2 b^2\right ) \cot (e+f x) \operatorname {EllipticPi}\left (\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right ) \sqrt {d \sec (e+f x)} \sqrt {-\tan ^2(e+f x)}}{8 \left (a^2+b^2\right )^3 f \sqrt [4]{\sec ^2(e+f x)}}-\frac {b \sqrt {d \sec (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {7 a b \sqrt {d \sec (e+f x)}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))} \]
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Time = 0.62 (sec) , antiderivative size = 515, 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, 759, 849, 858, 237, 761, 410, 109, 418, 1227, 551, 455, 65, 218, 214, 211} \[ \int \frac {\sqrt {d \sec (e+f x)}}{(a+b \tan (e+f x))^3} \, dx=\frac {3 a \left (5 a^2-2 b^2\right ) \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \sqrt {d \sec (e+f x)} \operatorname {EllipticPi}\left (-\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right )}{8 f \left (a^2+b^2\right )^3 \sqrt [4]{\sec ^2(e+f x)}}+\frac {3 a \left (5 a^2-2 b^2\right ) \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \sqrt {d \sec (e+f x)} \operatorname {EllipticPi}\left (\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right )}{8 f \left (a^2+b^2\right )^3 \sqrt [4]{\sec ^2(e+f x)}}-\frac {3 \sqrt {b} \left (5 a^2-2 b^2\right ) \sqrt {d \sec (e+f x)} \arctan \left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{8 f \left (a^2+b^2\right )^{11/4} \sqrt [4]{\sec ^2(e+f x)}}-\frac {7 a \sqrt {d \sec (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} \arctan (\tan (e+f x)),2\right )}{4 f \left (a^2+b^2\right )^2 \sqrt [4]{\sec ^2(e+f x)}}-\frac {3 \sqrt {b} \left (5 a^2-2 b^2\right ) \sqrt {d \sec (e+f x)} \text {arctanh}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{8 f \left (a^2+b^2\right )^{11/4} \sqrt [4]{\sec ^2(e+f x)}}-\frac {7 a b \sqrt {d \sec (e+f x)}}{4 f \left (a^2+b^2\right )^2 (a+b \tan (e+f x))}-\frac {b \sqrt {d \sec (e+f x)}}{2 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2} \]
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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 759
Rule 761
Rule 849
Rule 858
Rule 1227
Rule 3593
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {d \sec (e+f x)} \text {Subst}\left (\int \frac {1}{(a+x)^3 \left (1+\frac {x^2}{b^2}\right )^{3/4}} \, dx,x,b \tan (e+f x)\right )}{b f \sqrt [4]{\sec ^2(e+f x)}} \\ & = -\frac {b \sqrt {d \sec (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {\sqrt {d \sec (e+f x)} \text {Subst}\left (\int \frac {-2 a+\frac {3 x}{2}}{(a+x)^2 \left (1+\frac {x^2}{b^2}\right )^{3/4}} \, dx,x,b \tan (e+f x)\right )}{2 b \left (a^2+b^2\right ) f \sqrt [4]{\sec ^2(e+f x)}} \\ & = -\frac {b \sqrt {d \sec (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {7 a b \sqrt {d \sec (e+f x)}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}+\frac {\left (b \sqrt {d \sec (e+f x)}\right ) \text {Subst}\left (\int \frac {\frac {1}{2} \left (-3+\frac {4 a^2}{b^2}\right )-\frac {7 a x}{4 b^2}}{(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 \sqrt [4]{\sec ^2(e+f x)}} \\ & = -\frac {b \sqrt {d \sec (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {7 a b \sqrt {d \sec (e+f x)}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac {\left (7 a \sqrt {d \sec (e+f x)}\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 )}{8 b \left (a^2+b^2\right )^2 f \sqrt [4]{\sec ^2(e+f x)}}-\frac {\left (3 \left (2-\frac {5 a^2}{b^2}\right ) b \sqrt {d \sec (e+f x)}\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 )}{8 \left (a^2+b^2\right )^2 f \sqrt [4]{\sec ^2(e+f x)}} \\ & = -\frac {7 a \operatorname {EllipticF}\left (\frac {1}{2} \arctan (\tan (e+f x)),2\right ) \sqrt {d \sec (e+f x)}}{4 \left (a^2+b^2\right )^2 f \sqrt [4]{\sec ^2(e+f x)}}-\frac {b \sqrt {d \sec (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {7 a b \sqrt {d \sec (e+f x)}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}+\frac {\left (3 \left (2-\frac {5 a^2}{b^2}\right ) b \sqrt {d \sec (e+f x)}\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 )}{8 \left (a^2+b^2\right )^2 f \sqrt [4]{\sec ^2(e+f x)}}-\frac {\left (3 a \left (2-\frac {5 a^2}{b^2}\right ) b \sqrt {d \sec (e+f x)}\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 )}{8 \left (a^2+b^2\right )^2 f \sqrt [4]{\sec ^2(e+f x)}} \\ & = -\frac {7 a \operatorname {EllipticF}\left (\frac {1}{2} \arctan (\tan (e+f x)),2\right ) \sqrt {d \sec (e+f x)}}{4 \left (a^2+b^2\right )^2 f \sqrt [4]{\sec ^2(e+f x)}}-\frac {b \sqrt {d \sec (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {7 a b \sqrt {d \sec (e+f x)}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}+\frac {\left (3 \left (2-\frac {5 a^2}{b^2}\right ) b \sqrt {d \sec (e+f x)}\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 )}{16 \left (a^2+b^2\right )^2 f \sqrt [4]{\sec ^2(e+f x)}}-\frac {\left (3 a \left (2-\frac {5 a^2}{b^2}\right ) \cot (e+f x) \sqrt {d \sec (e+f x)} \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 )}{16 \left (a^2+b^2\right )^2 f \sqrt [4]{\sec ^2(e+f x)}} \\ & = -\frac {7 a \operatorname {EllipticF}\left (\frac {1}{2} \arctan (\tan (e+f x)),2\right ) \sqrt {d \sec (e+f x)}}{4 \left (a^2+b^2\right )^2 f \sqrt [4]{\sec ^2(e+f x)}}-\frac {b \sqrt {d \sec (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {7 a b \sqrt {d \sec (e+f x)}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}+\frac {\left (3 \left (2-\frac {5 a^2}{b^2}\right ) b^3 \sqrt {d \sec (e+f x)}\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 )}{4 \left (a^2+b^2\right )^2 f \sqrt [4]{\sec ^2(e+f x)}}+\frac {\left (3 a \left (2-\frac {5 a^2}{b^2}\right ) \cot (e+f x) \sqrt {d \sec (e+f x)} \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 )}{4 \left (a^2+b^2\right )^2 f \sqrt [4]{\sec ^2(e+f x)}} \\ & = -\frac {7 a \operatorname {EllipticF}\left (\frac {1}{2} \arctan (\tan (e+f x)),2\right ) \sqrt {d \sec (e+f x)}}{4 \left (a^2+b^2\right )^2 f \sqrt [4]{\sec ^2(e+f x)}}-\frac {b \sqrt {d \sec (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {7 a b \sqrt {d \sec (e+f x)}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}+\frac {\left (3 \left (2-\frac {5 a^2}{b^2}\right ) b^3 \sqrt {d \sec (e+f x)}\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 )}{8 \left (a^2+b^2\right )^{5/2} f \sqrt [4]{\sec ^2(e+f x)}}+\frac {\left (3 \left (2-\frac {5 a^2}{b^2}\right ) b^3 \sqrt {d \sec (e+f x)}\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 )}{8 \left (a^2+b^2\right )^{5/2} f \sqrt [4]{\sec ^2(e+f x)}}-\frac {\left (3 a \left (2-\frac {5 a^2}{b^2}\right ) b^2 \cot (e+f x) \sqrt {d \sec (e+f x)} \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 )}{8 \left (a^2+b^2\right )^3 f \sqrt [4]{\sec ^2(e+f x)}}-\frac {\left (3 a \left (2-\frac {5 a^2}{b^2}\right ) b^2 \cot (e+f x) \sqrt {d \sec (e+f x)} \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 )}{8 \left (a^2+b^2\right )^3 f \sqrt [4]{\sec ^2(e+f x)}} \\ & = \frac {3 \left (2-\frac {5 a^2}{b^2}\right ) b^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sqrt {d \sec (e+f x)}}{8 \left (a^2+b^2\right )^{11/4} f \sqrt [4]{\sec ^2(e+f x)}}+\frac {3 \left (2-\frac {5 a^2}{b^2}\right ) b^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sqrt {d \sec (e+f x)}}{8 \left (a^2+b^2\right )^{11/4} f \sqrt [4]{\sec ^2(e+f x)}}-\frac {7 a \operatorname {EllipticF}\left (\frac {1}{2} \arctan (\tan (e+f x)),2\right ) \sqrt {d \sec (e+f x)}}{4 \left (a^2+b^2\right )^2 f \sqrt [4]{\sec ^2(e+f x)}}-\frac {b \sqrt {d \sec (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {7 a b \sqrt {d \sec (e+f x)}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac {\left (3 a \left (2-\frac {5 a^2}{b^2}\right ) b^2 \cot (e+f x) \sqrt {d \sec (e+f x)} \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 )}{8 \left (a^2+b^2\right )^3 f \sqrt [4]{\sec ^2(e+f x)}}-\frac {\left (3 a \left (2-\frac {5 a^2}{b^2}\right ) b^2 \cot (e+f x) \sqrt {d \sec (e+f x)} \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 )}{8 \left (a^2+b^2\right )^3 f \sqrt [4]{\sec ^2(e+f x)}} \\ & = \frac {3 \left (2-\frac {5 a^2}{b^2}\right ) b^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sqrt {d \sec (e+f x)}}{8 \left (a^2+b^2\right )^{11/4} f \sqrt [4]{\sec ^2(e+f x)}}+\frac {3 \left (2-\frac {5 a^2}{b^2}\right ) b^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sqrt {d \sec (e+f x)}}{8 \left (a^2+b^2\right )^{11/4} f \sqrt [4]{\sec ^2(e+f x)}}-\frac {7 a \operatorname {EllipticF}\left (\frac {1}{2} \arctan (\tan (e+f x)),2\right ) \sqrt {d \sec (e+f x)}}{4 \left (a^2+b^2\right )^2 f \sqrt [4]{\sec ^2(e+f x)}}+\frac {3 a \left (5 a^2-2 b^2\right ) \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 {d \sec (e+f x)} \sqrt {-\tan ^2(e+f x)}}{8 \left (a^2+b^2\right )^3 f \sqrt [4]{\sec ^2(e+f x)}}+\frac {3 a \left (5 a^2-2 b^2\right ) \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 {d \sec (e+f x)} \sqrt {-\tan ^2(e+f x)}}{8 \left (a^2+b^2\right )^3 f \sqrt [4]{\sec ^2(e+f x)}}-\frac {b \sqrt {d \sec (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {7 a b \sqrt {d \sec (e+f x)}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 9.70 (sec) , antiderivative size = 369, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt {d \sec (e+f x)}}{(a+b \tan (e+f x))^3} \, dx=\frac {(d \sec (e+f x))^{3/2} (a \cos (e+f x)+b \sin (e+f x))^3 \left (-\frac {2 b \sec ^2(e+f x) \left (9 a^2+2 b^2+7 a b \tan (e+f x)\right )}{(a-i b)^2 (a+i b)^2 (a+b \tan (e+f x))^2}+\frac {\sec ^2(e+f x)^{3/4} \left (-7 a \left (a^2+b^2\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},-\tan ^2(e+f x)\right ) \tan (e+f x)+3 \left (5 a^2-2 b^2\right ) \left (-\sqrt {b} \sqrt [4]{a^2+b^2} \left (\arctan \left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )+\text {arctanh}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )\right )+a \cot (e+f x) \operatorname {EllipticPi}\left (-\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right ) \sqrt {-\tan ^2(e+f x)}+a \cot (e+f x) \operatorname {EllipticPi}\left (\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right ) \sqrt {-\tan ^2(e+f x)}\right )\right )}{\left (a^2+b^2\right )^3}\right )}{8 d f (a+b \tan (e+f x))^3} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 75606 vs. \(2 (472 ) = 944\).
Time = 14.53 (sec) , antiderivative size = 75607, normalized size of antiderivative = 146.81
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Timed out. \[ \int \frac {\sqrt {d \sec (e+f x)}}{(a+b \tan (e+f x))^3} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {d \sec (e+f x)}}{(a+b \tan (e+f x))^3} \, dx=\int \frac {\sqrt {d \sec {\left (e + f x \right )}}}{\left (a + b \tan {\left (e + f x \right )}\right )^{3}}\, dx \]
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Timed out. \[ \int \frac {\sqrt {d \sec (e+f x)}}{(a+b \tan (e+f x))^3} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {d \sec (e+f x)}}{(a+b \tan (e+f x))^3} \, dx=\int { \frac {\sqrt {d \sec \left (f x + e\right )}}{{\left (b \tan \left (f x + e\right ) + a\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {d \sec (e+f x)}}{(a+b \tan (e+f x))^3} \, dx=\int \frac {\sqrt {\frac {d}{\cos \left (e+f\,x\right )}}}{{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^3} \,d x \]
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