Integrand size = 25, antiderivative size = 422 \[ \int \frac {1}{(d \sec (e+f x))^{3/2} (a+b \tan (e+f x))} \, dx=-\frac {b^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sec ^2(e+f x)^{3/4}}{\left (a^2+b^2\right )^{7/4} f (d \sec (e+f x))^{3/2}}-\frac {b^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sec ^2(e+f x)^{3/4}}{\left (a^2+b^2\right )^{7/4} f (d \sec (e+f x))^{3/2}}+\frac {2 a \operatorname {EllipticF}\left (\frac {1}{2} \arctan (\tan (e+f x)),2\right ) \sec ^2(e+f x)^{3/4}}{3 \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2}}+\frac {a b^2 \cot (e+f x) \operatorname {EllipticPi}\left (-\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right ) \sec ^2(e+f x)^{3/4} \sqrt {-\tan ^2(e+f x)}}{\left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}}+\frac {a b^2 \cot (e+f x) \operatorname {EllipticPi}\left (\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right ) \sec ^2(e+f x)^{3/4} \sqrt {-\tan ^2(e+f x)}}{\left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}}+\frac {2 (b+a \tan (e+f x))}{3 \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2}} \]
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Time = 0.53 (sec) , antiderivative size = 422, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3593, 755, 858, 237, 761, 410, 109, 418, 1227, 551, 455, 65, 218, 214, 211} \[ \int \frac {1}{(d \sec (e+f x))^{3/2} (a+b \tan (e+f x))} \, dx=\frac {a b^2 \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \sec ^2(e+f x)^{3/4} \operatorname {EllipticPi}\left (-\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right )}{f \left (a^2+b^2\right )^2 (d \sec (e+f x))^{3/2}}+\frac {a b^2 \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \sec ^2(e+f x)^{3/4} \operatorname {EllipticPi}\left (\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right )}{f \left (a^2+b^2\right )^2 (d \sec (e+f x))^{3/2}}+\frac {2 a \sec ^2(e+f x)^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan (\tan (e+f x)),2\right )}{3 f \left (a^2+b^2\right ) (d \sec (e+f x))^{3/2}}-\frac {b^{5/2} \sec ^2(e+f x)^{3/4} \arctan \left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{f \left (a^2+b^2\right )^{7/4} (d \sec (e+f x))^{3/2}}-\frac {b^{5/2} \sec ^2(e+f x)^{3/4} \text {arctanh}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{f \left (a^2+b^2\right )^{7/4} (d \sec (e+f x))^{3/2}}+\frac {2 (a \tan (e+f x)+b)}{3 f \left (a^2+b^2\right ) (d \sec (e+f x))^{3/2}} \]
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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 755
Rule 761
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) \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}}-\frac {\left (2 b \sec ^2(e+f x)^{3/4}\right ) \text {Subst}\left (\int \frac {\frac {1}{2} \left (-3-\frac {a^2}{b^2}\right )-\frac {a x}{2 b^2}}{(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 ) 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}}+\frac {\left (a \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 )}{3 b \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2}}+\frac {\left (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 )}{\left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2}} \\ & = \frac {2 a \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 ) 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}}-\frac {\left (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 )}{\left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2}}+\frac {\left (a 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 )}{\left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2}} \\ & = \frac {2 a \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 ) 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}}-\frac {\left (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 )}{2 \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2}}+\frac {\left (a \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 )}{2 \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2}} \\ & = \frac {2 a \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 ) 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}}-\frac {\left (2 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 ) f (d \sec (e+f x))^{3/2}}-\frac {\left (2 a \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 ) f (d \sec (e+f x))^{3/2}} \\ & = \frac {2 a \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 ) 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}}-\frac {\left (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 )}{\left (a^2+b^2\right )^{3/2} f (d \sec (e+f x))^{3/2}}-\frac {\left (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 )}{\left (a^2+b^2\right )^{3/2} f (d \sec (e+f x))^{3/2}}+\frac {\left (a 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 )}{\left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}}+\frac {\left (a 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 )}{\left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}} \\ & = -\frac {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}}{\left (a^2+b^2\right )^{7/4} f (d \sec (e+f x))^{3/2}}-\frac {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}}{\left (a^2+b^2\right )^{7/4} f (d \sec (e+f x))^{3/2}}+\frac {2 a \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 ) 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}}+\frac {\left (a 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 )}{\left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}}+\frac {\left (a 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 )}{\left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}} \\ & = -\frac {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}}{\left (a^2+b^2\right )^{7/4} f (d \sec (e+f x))^{3/2}}-\frac {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}}{\left (a^2+b^2\right )^{7/4} f (d \sec (e+f x))^{3/2}}+\frac {2 a \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 ) f (d \sec (e+f x))^{3/2}}+\frac {a 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)}}{\left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}}+\frac {a 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)}}{\left (a^2+b^2\right )^2 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}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 7.17 (sec) , antiderivative size = 418, normalized size of antiderivative = 0.99 \[ \int \frac {1}{(d \sec (e+f x))^{3/2} (a+b \tan (e+f x))} \, dx=\frac {a^2 b \sec ^2(e+f x)+b^3 \sec ^2(e+f x)+a^2 b \cos (2 (e+f x)) \sec ^2(e+f x)+b^3 \cos (2 (e+f x)) \sec ^2(e+f x)-3 b^{5/2} \sqrt [4]{a^2+b^2} \arctan \left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sec ^2(e+f x)^{3/4}-3 b^{5/2} \sqrt [4]{a^2+b^2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sec ^2(e+f x)^{3/4}+2 a^3 \tan (e+f x)+2 a b^2 \tan (e+f x)+a \left (a^2+b^2\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{3/4} \tan (e+f x)+3 a b^2 \cot (e+f x) \operatorname {EllipticPi}\left (-\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right ) \sec ^2(e+f x)^{3/4} \sqrt {-\tan ^2(e+f x)}+3 a b^2 \cot (e+f x) \operatorname {EllipticPi}\left (\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right ) \sec ^2(e+f x)^{3/4} \sqrt {-\tan ^2(e+f x)}}{3 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 6051 vs. \(2 (391 ) = 782\).
Time = 10.47 (sec) , antiderivative size = 6052, normalized size of antiderivative = 14.34
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Timed out. \[ \int \frac {1}{(d \sec (e+f x))^{3/2} (a+b \tan (e+f x))} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(d \sec (e+f x))^{3/2} (a+b \tan (e+f x))} \, dx=\int \frac {1}{\left (d \sec {\left (e + f x \right )}\right )^{\frac {3}{2}} \left (a + b \tan {\left (e + f x \right )}\right )}\, dx \]
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\[ \int \frac {1}{(d \sec (e+f x))^{3/2} (a+b \tan (e+f x))} \, dx=\int { \frac {1}{\left (d \sec \left (f x + e\right )\right )^{\frac {3}{2}} {\left (b \tan \left (f x + e\right ) + a\right )}} \,d x } \]
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\[ \int \frac {1}{(d \sec (e+f x))^{3/2} (a+b \tan (e+f x))} \, dx=\int { \frac {1}{\left (d \sec \left (f x + e\right )\right )^{\frac {3}{2}} {\left (b \tan \left (f x + e\right ) + a\right )}} \,d x } \]
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Timed out. \[ \int \frac {1}{(d \sec (e+f x))^{3/2} (a+b \tan (e+f x))} \, dx=\int \frac {1}{{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{3/2}\,\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )} \,d x \]
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