Integrand size = 25, antiderivative size = 456 \[ \int \frac {(d \sec (e+f x))^{7/2}}{a+b \tan (e+f x)} \, dx=\frac {2 d^2 (d \sec (e+f x))^{3/2}}{3 b f}+\frac {\left (a^2+b^2\right )^{3/4} d^2 \arctan \left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) (d \sec (e+f x))^{3/2}}{b^{5/2} f \sec ^2(e+f x)^{3/4}}-\frac {\left (a^2+b^2\right )^{3/4} d^2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) (d \sec (e+f x))^{3/2}}{b^{5/2} f \sec ^2(e+f x)^{3/4}}+\frac {2 a d^2 E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right ) (d \sec (e+f x))^{3/2}}{b^2 f \sec ^2(e+f x)^{3/4}}-\frac {2 a d^2 \cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{b^2 f}-\frac {a \sqrt {a^2+b^2} d^2 \cot (e+f x) \operatorname {EllipticPi}\left (-\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right ) (d \sec (e+f x))^{3/2} \sqrt {-\tan ^2(e+f x)}}{b^3 f \sec ^2(e+f x)^{3/4}}+\frac {a \sqrt {a^2+b^2} d^2 \cot (e+f x) \operatorname {EllipticPi}\left (\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right ) (d \sec (e+f x))^{3/2} \sqrt {-\tan ^2(e+f x)}}{b^3 f \sec ^2(e+f x)^{3/4}} \]
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Time = 0.46 (sec) , antiderivative size = 456, 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, 749, 858, 233, 202, 760, 408, 504, 1227, 551, 455, 65, 304, 211, 214} \[ \int \frac {(d \sec (e+f x))^{7/2}}{a+b \tan (e+f x)} \, dx=-\frac {a d^2 \sqrt {a^2+b^2} \sqrt {-\tan ^2(e+f x)} \cot (e+f x) (d \sec (e+f x))^{3/2} \operatorname {EllipticPi}\left (-\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right )}{b^3 f \sec ^2(e+f x)^{3/4}}+\frac {a d^2 \sqrt {a^2+b^2} \sqrt {-\tan ^2(e+f x)} \cot (e+f x) (d \sec (e+f x))^{3/2} \operatorname {EllipticPi}\left (\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right )}{b^3 f \sec ^2(e+f x)^{3/4}}+\frac {d^2 \left (a^2+b^2\right )^{3/4} (d \sec (e+f x))^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{b^{5/2} f \sec ^2(e+f x)^{3/4}}-\frac {d^2 \left (a^2+b^2\right )^{3/4} (d \sec (e+f x))^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{b^{5/2} f \sec ^2(e+f x)^{3/4}}+\frac {2 a d^2 (d \sec (e+f x))^{3/2} E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right )}{b^2 f \sec ^2(e+f x)^{3/4}}-\frac {2 a d^2 \sin (e+f x) \cos (e+f x) (d \sec (e+f x))^{3/2}}{b^2 f}+\frac {2 d^2 (d \sec (e+f x))^{3/2}}{3 b f} \]
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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 749
Rule 760
Rule 858
Rule 1227
Rule 3593
Rubi steps \begin{align*} \text {integral}& = \frac {\left (d^2 (d \sec (e+f x))^{3/2}\right ) \text {Subst}\left (\int \frac {\left (1+\frac {x^2}{b^2}\right )^{3/4}}{a+x} \, dx,x,b \tan (e+f x)\right )}{b f \sec ^2(e+f x)^{3/4}} \\ & = \frac {2 d^2 (d \sec (e+f x))^{3/2}}{3 b f}+\frac {\left (d^2 (d \sec (e+f x))^{3/2}\right ) \text {Subst}\left (\int \frac {1-\frac {a x}{b^2}}{(a+x) \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 {2 d^2 (d \sec (e+f x))^{3/2}}{3 b f}-\frac {\left (a d^2 (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 )}{b^3 f \sec ^2(e+f x)^{3/4}}+\frac {\left (\left (1+\frac {a^2}{b^2}\right ) d^2 (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 )}{b f \sec ^2(e+f x)^{3/4}} \\ & = \frac {2 d^2 (d \sec (e+f x))^{3/2}}{3 b f}-\frac {2 a d^2 \cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{b^2 f}+\frac {\left (a d^2 (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 )}{b^3 f \sec ^2(e+f x)^{3/4}}-\frac {\left (\left (1+\frac {a^2}{b^2}\right ) d^2 (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 )}{b f \sec ^2(e+f x)^{3/4}}+\frac {\left (a \left (1+\frac {a^2}{b^2}\right ) d^2 (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 )}{b f \sec ^2(e+f x)^{3/4}} \\ & = \frac {2 d^2 (d \sec (e+f x))^{3/2}}{3 b f}+\frac {2 a d^2 E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right ) (d \sec (e+f x))^{3/2}}{b^2 f \sec ^2(e+f x)^{3/4}}-\frac {2 a d^2 \cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{b^2 f}-\frac {\left (\left (1+\frac {a^2}{b^2}\right ) d^2 (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 )}{2 b f \sec ^2(e+f x)^{3/4}}+\frac {\left (2 a \left (1+\frac {a^2}{b^2}\right ) d^2 \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 )}{b^2 f \sec ^2(e+f x)^{3/4}} \\ & = \frac {2 d^2 (d \sec (e+f x))^{3/2}}{3 b f}+\frac {2 a d^2 E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right ) (d \sec (e+f x))^{3/2}}{b^2 f \sec ^2(e+f x)^{3/4}}-\frac {2 a d^2 \cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{b^2 f}-\frac {\left (2 \left (1+\frac {a^2}{b^2}\right ) b d^2 (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 )}{f \sec ^2(e+f x)^{3/4}}+\frac {\left (a \left (1+\frac {a^2}{b^2}\right ) d^2 \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 )}{b f \sec ^2(e+f x)^{3/4}}-\frac {\left (a \left (1+\frac {a^2}{b^2}\right ) d^2 \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 )}{b f \sec ^2(e+f x)^{3/4}} \\ & = \frac {2 d^2 (d \sec (e+f x))^{3/2}}{3 b f}+\frac {2 a d^2 E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right ) (d \sec (e+f x))^{3/2}}{b^2 f \sec ^2(e+f x)^{3/4}}-\frac {2 a d^2 \cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{b^2 f}-\frac {\left (\left (1+\frac {a^2}{b^2}\right ) d^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 )}{f \sec ^2(e+f x)^{3/4}}+\frac {\left (\left (1+\frac {a^2}{b^2}\right ) d^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 )}{f \sec ^2(e+f x)^{3/4}}+\frac {\left (a \left (1+\frac {a^2}{b^2}\right ) d^2 \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 )}{b f \sec ^2(e+f x)^{3/4}}-\frac {\left (a \left (1+\frac {a^2}{b^2}\right ) d^2 \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 )}{b f \sec ^2(e+f x)^{3/4}} \\ & = \frac {2 d^2 (d \sec (e+f x))^{3/2}}{3 b f}+\frac {\left (a^2+b^2\right )^{3/4} d^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}}{b^{5/2} f \sec ^2(e+f x)^{3/4}}-\frac {\left (a^2+b^2\right )^{3/4} d^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}}{b^{5/2} f \sec ^2(e+f x)^{3/4}}+\frac {2 a d^2 E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right ) (d \sec (e+f x))^{3/2}}{b^2 f \sec ^2(e+f x)^{3/4}}-\frac {2 a d^2 \cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{b^2 f}-\frac {a \sqrt {a^2+b^2} d^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 ) (d \sec (e+f x))^{3/2} \sqrt {-\tan ^2(e+f x)}}{b^3 f \sec ^2(e+f x)^{3/4}}+\frac {a \sqrt {a^2+b^2} d^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 ) (d \sec (e+f x))^{3/2} \sqrt {-\tan ^2(e+f x)}}{b^3 f \sec ^2(e+f x)^{3/4}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(11117\) vs. \(2(456)=912\).
Time = 90.48 (sec) , antiderivative size = 11117, normalized size of antiderivative = 24.38 \[ \int \frac {(d \sec (e+f x))^{7/2}}{a+b \tan (e+f x)} \, dx=\text {Result too large to show} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 31612 vs. \(2 (421 ) = 842\).
Time = 33.06 (sec) , antiderivative size = 31613, normalized size of antiderivative = 69.33
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Timed out. \[ \int \frac {(d \sec (e+f x))^{7/2}}{a+b \tan (e+f x)} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(d \sec (e+f x))^{7/2}}{a+b \tan (e+f x)} \, dx=\text {Timed out} \]
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\[ \int \frac {(d \sec (e+f x))^{7/2}}{a+b \tan (e+f x)} \, dx=\int { \frac {\left (d \sec \left (f x + e\right )\right )^{\frac {7}{2}}}{b \tan \left (f x + e\right ) + a} \,d x } \]
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\[ \int \frac {(d \sec (e+f x))^{7/2}}{a+b \tan (e+f x)} \, dx=\int { \frac {\left (d \sec \left (f x + e\right )\right )^{\frac {7}{2}}}{b \tan \left (f x + e\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(d \sec (e+f x))^{7/2}}{a+b \tan (e+f x)} \, dx=\int \frac {{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{7/2}}{a+b\,\mathrm {tan}\left (e+f\,x\right )} \,d x \]
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