Integrand size = 25, antiderivative size = 566 \[ \int \frac {(d \sec (e+f x))^{3/2}}{(a+b \tan (e+f x))^3} \, dx=\frac {\left (3 a^2-2 b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) (d \sec (e+f x))^{3/2}}{8 \sqrt {b} \left (a^2+b^2\right )^{9/4} f \sec ^2(e+f x)^{3/4}}-\frac {\left (3 a^2-2 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) (d \sec (e+f x))^{3/2}}{8 \sqrt {b} \left (a^2+b^2\right )^{9/4} f \sec ^2(e+f x)^{3/4}}-\frac {5 a E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right ) (d \sec (e+f x))^{3/2}}{4 \left (a^2+b^2\right )^2 f \sec ^2(e+f x)^{3/4}}+\frac {5 a \cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{4 \left (a^2+b^2\right )^2 f}-\frac {a \left (3 a^2-2 b^2\right ) \cot (e+f x) \operatorname {EllipticPi}\left (-\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right ) (d \sec (e+f x))^{3/2} \sqrt {-\tan ^2(e+f x)}}{8 b \left (a^2+b^2\right )^{5/2} f \sec ^2(e+f x)^{3/4}}+\frac {a \left (3 a^2-2 b^2\right ) \cot (e+f x) \operatorname {EllipticPi}\left (\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right ) (d \sec (e+f x))^{3/2} \sqrt {-\tan ^2(e+f x)}}{8 b \left (a^2+b^2\right )^{5/2} f \sec ^2(e+f x)^{3/4}}-\frac {b (d \sec (e+f x))^{3/2}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {5 a b (d \sec (e+f x))^{3/2}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))} \]
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Time = 0.67 (sec) , antiderivative size = 566, 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, 233, 202, 760, 408, 504, 1227, 551, 455, 65, 304, 211, 214} \[ \int \frac {(d \sec (e+f x))^{3/2}}{(a+b \tan (e+f x))^3} \, dx=-\frac {a \left (3 a^2-2 b^2\right ) \sqrt {-\tan ^2(e+f x)} \cot (e+f x) (d \sec (e+f x))^{3/2} \operatorname {EllipticPi}\left (-\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right )}{8 b f \left (a^2+b^2\right )^{5/2} \sec ^2(e+f x)^{3/4}}+\frac {a \left (3 a^2-2 b^2\right ) \sqrt {-\tan ^2(e+f x)} \cot (e+f x) (d \sec (e+f x))^{3/2} \operatorname {EllipticPi}\left (\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right )}{8 b f \left (a^2+b^2\right )^{5/2} \sec ^2(e+f x)^{3/4}}+\frac {\left (3 a^2-2 b^2\right ) (d \sec (e+f x))^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{8 \sqrt {b} f \left (a^2+b^2\right )^{9/4} \sec ^2(e+f x)^{3/4}}-\frac {5 a (d \sec (e+f x))^{3/2} E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right )}{4 f \left (a^2+b^2\right )^2 \sec ^2(e+f x)^{3/4}}-\frac {\left (3 a^2-2 b^2\right ) (d \sec (e+f x))^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{8 \sqrt {b} f \left (a^2+b^2\right )^{9/4} \sec ^2(e+f x)^{3/4}}-\frac {5 a b (d \sec (e+f x))^{3/2}}{4 f \left (a^2+b^2\right )^2 (a+b \tan (e+f x))}-\frac {b (d \sec (e+f x))^{3/2}}{2 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {5 a \sin (e+f x) \cos (e+f x) (d \sec (e+f x))^{3/2}}{4 f \left (a^2+b^2\right )^2} \]
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Rule 65
Rule 202
Rule 211
Rule 214
Rule 233
Rule 304
Rule 408
Rule 455
Rule 504
Rule 551
Rule 759
Rule 760
Rule 849
Rule 858
Rule 1227
Rule 3593
Rubi steps \begin{align*} \text {integral}& = \frac {(d \sec (e+f x))^{3/2} \text {Subst}\left (\int \frac {1}{(a+x)^3 \sqrt [4]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{b f \sec ^2(e+f x)^{3/4}} \\ & = -\frac {b (d \sec (e+f x))^{3/2}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {(d \sec (e+f x))^{3/2} \text {Subst}\left (\int \frac {-2 a+\frac {x}{2}}{(a+x)^2 \sqrt [4]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{2 b \left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}} \\ & = -\frac {b (d \sec (e+f x))^{3/2}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {5 a b (d \sec (e+f x))^{3/2}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}+\frac {\left (b (d \sec (e+f x))^{3/2}\right ) \text {Subst}\left (\int \frac {\frac {1}{2} \left (-1+\frac {4 a^2}{b^2}\right )+\frac {5 a x}{4 b^2}}{(a+x) \sqrt [4]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{2 \left (a^2+b^2\right )^2 f \sec ^2(e+f x)^{3/4}} \\ & = -\frac {b (d \sec (e+f x))^{3/2}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {5 a b (d \sec (e+f x))^{3/2}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}+\frac {\left (5 a (d \sec (e+f x))^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{8 b \left (a^2+b^2\right )^2 f \sec ^2(e+f x)^{3/4}}+\frac {\left (\left (-2+\frac {3 a^2}{b^2}\right ) b (d \sec (e+f x))^{3/2}\right ) \text {Subst}\left (\int \frac {1}{(a+x) \sqrt [4]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{8 \left (a^2+b^2\right )^2 f \sec ^2(e+f x)^{3/4}} \\ & = \frac {5 a \cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{4 \left (a^2+b^2\right )^2 f}-\frac {b (d \sec (e+f x))^{3/2}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {5 a b (d \sec (e+f x))^{3/2}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac {\left (5 a (d \sec (e+f x))^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {x^2}{b^2}\right )^{5/4}} \, dx,x,b \tan (e+f x)\right )}{8 b \left (a^2+b^2\right )^2 f \sec ^2(e+f x)^{3/4}}-\frac {\left (\left (-2+\frac {3 a^2}{b^2}\right ) b (d \sec (e+f x))^{3/2}\right ) \text {Subst}\left (\int \frac {x}{\left (a^2-x^2\right ) \sqrt [4]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{8 \left (a^2+b^2\right )^2 f \sec ^2(e+f x)^{3/4}}+\frac {\left (a \left (-2+\frac {3 a^2}{b^2}\right ) b (d \sec (e+f x))^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\left (a^2-x^2\right ) \sqrt [4]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{8 \left (a^2+b^2\right )^2 f \sec ^2(e+f x)^{3/4}} \\ & = -\frac {5 a E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right ) (d \sec (e+f x))^{3/2}}{4 \left (a^2+b^2\right )^2 f \sec ^2(e+f x)^{3/4}}+\frac {5 a \cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{4 \left (a^2+b^2\right )^2 f}-\frac {b (d \sec (e+f x))^{3/2}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {5 a b (d \sec (e+f x))^{3/2}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac {\left (\left (-2+\frac {3 a^2}{b^2}\right ) b (d \sec (e+f x))^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\left (a^2-x\right ) \sqrt [4]{1+\frac {x}{b^2}}} \, dx,x,b^2 \tan ^2(e+f x)\right )}{16 \left (a^2+b^2\right )^2 f \sec ^2(e+f x)^{3/4}}+\frac {\left (a \left (-2+\frac {3 a^2}{b^2}\right ) \cot (e+f x) (d \sec (e+f x))^{3/2} \sqrt {-\tan ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {x^2}{\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 \sec ^2(e+f x)^{3/4}} \\ & = -\frac {5 a E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right ) (d \sec (e+f x))^{3/2}}{4 \left (a^2+b^2\right )^2 f \sec ^2(e+f x)^{3/4}}+\frac {5 a \cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{4 \left (a^2+b^2\right )^2 f}-\frac {b (d \sec (e+f x))^{3/2}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {5 a b (d \sec (e+f x))^{3/2}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac {\left (\left (-2+\frac {3 a^2}{b^2}\right ) b^3 (d \sec (e+f x))^{3/2}\right ) \text {Subst}\left (\int \frac {x^2}{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 \sec ^2(e+f x)^{3/4}}+\frac {\left (a \left (-2+\frac {3 a^2}{b^2}\right ) b \cot (e+f x) (d \sec (e+f x))^{3/2} \sqrt {-\tan ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a^2+b^2}-b x^2\right ) \sqrt {1-x^4}} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{8 \left (a^2+b^2\right )^2 f \sec ^2(e+f x)^{3/4}}-\frac {\left (a \left (-2+\frac {3 a^2}{b^2}\right ) b \cot (e+f x) (d \sec (e+f x))^{3/2} \sqrt {-\tan ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a^2+b^2}+b x^2\right ) \sqrt {1-x^4}} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{8 \left (a^2+b^2\right )^2 f \sec ^2(e+f x)^{3/4}} \\ & = -\frac {5 a E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right ) (d \sec (e+f x))^{3/2}}{4 \left (a^2+b^2\right )^2 f \sec ^2(e+f x)^{3/4}}+\frac {5 a \cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{4 \left (a^2+b^2\right )^2 f}-\frac {b (d \sec (e+f x))^{3/2}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {5 a b (d \sec (e+f x))^{3/2}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac {\left (\left (-2+\frac {3 a^2}{b^2}\right ) b^2 (d \sec (e+f x))^{3/2}\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 )^2 f \sec ^2(e+f x)^{3/4}}+\frac {\left (\left (-2+\frac {3 a^2}{b^2}\right ) b^2 (d \sec (e+f x))^{3/2}\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 )^2 f \sec ^2(e+f x)^{3/4}}+\frac {\left (a \left (-2+\frac {3 a^2}{b^2}\right ) b \cot (e+f x) (d \sec (e+f x))^{3/2} \sqrt {-\tan ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (\sqrt {a^2+b^2}-b x^2\right )} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{8 \left (a^2+b^2\right )^2 f \sec ^2(e+f x)^{3/4}}-\frac {\left (a \left (-2+\frac {3 a^2}{b^2}\right ) b \cot (e+f x) (d \sec (e+f x))^{3/2} \sqrt {-\tan ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (\sqrt {a^2+b^2}+b x^2\right )} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{8 \left (a^2+b^2\right )^2 f \sec ^2(e+f x)^{3/4}} \\ & = -\frac {\left (2-\frac {3 a^2}{b^2}\right ) b^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) (d \sec (e+f x))^{3/2}}{8 \left (a^2+b^2\right )^{9/4} f \sec ^2(e+f x)^{3/4}}+\frac {\left (2-\frac {3 a^2}{b^2}\right ) b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) (d \sec (e+f x))^{3/2}}{8 \left (a^2+b^2\right )^{9/4} f \sec ^2(e+f x)^{3/4}}-\frac {5 a E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right ) (d \sec (e+f x))^{3/2}}{4 \left (a^2+b^2\right )^2 f \sec ^2(e+f x)^{3/4}}+\frac {5 a \cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{4 \left (a^2+b^2\right )^2 f}+\frac {a \left (2-\frac {3 a^2}{b^2}\right ) b \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 ) (d \sec (e+f x))^{3/2} \sqrt {-\tan ^2(e+f x)}}{8 \left (a^2+b^2\right )^{5/2} f \sec ^2(e+f x)^{3/4}}-\frac {a \left (2-\frac {3 a^2}{b^2}\right ) b \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 ) (d \sec (e+f x))^{3/2} \sqrt {-\tan ^2(e+f x)}}{8 \left (a^2+b^2\right )^{5/2} f \sec ^2(e+f x)^{3/4}}-\frac {b (d \sec (e+f x))^{3/2}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {5 a b (d \sec (e+f x))^{3/2}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))} \\ \end{align*}
Result contains complex when optimal does not.
Time = 35.27 (sec) , antiderivative size = 14364, normalized size of antiderivative = 25.38 \[ \int \frac {(d \sec (e+f x))^{3/2}}{(a+b \tan (e+f x))^3} \, dx=\text {Result too large to show} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 60918 vs. \(2 (515 ) = 1030\).
Time = 13.11 (sec) , antiderivative size = 60919, normalized size of antiderivative = 107.63
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Timed out. \[ \int \frac {(d \sec (e+f x))^{3/2}}{(a+b \tan (e+f x))^3} \, dx=\text {Timed out} \]
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\[ \int \frac {(d \sec (e+f x))^{3/2}}{(a+b \tan (e+f x))^3} \, dx=\int \frac {\left (d \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}}{\left (a + b \tan {\left (e + f x \right )}\right )^{3}}\, dx \]
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Timed out. \[ \int \frac {(d \sec (e+f x))^{3/2}}{(a+b \tan (e+f x))^3} \, dx=\text {Timed out} \]
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\[ \int \frac {(d \sec (e+f x))^{3/2}}{(a+b \tan (e+f x))^3} \, dx=\int { \frac {\left (d \sec \left (f x + e\right )\right )^{\frac {3}{2}}}{{\left (b \tan \left (f x + e\right ) + a\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {(d \sec (e+f x))^{3/2}}{(a+b \tan (e+f x))^3} \, dx=\int \frac {{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{3/2}}{{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^3} \,d x \]
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