Integrand size = 25, antiderivative size = 480 \[ \int \frac {(d \sec (e+f x))^{7/2}}{(a+b \tan (e+f x))^2} \, dx=-\frac {3 a d^2 \arctan \left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) (d \sec (e+f x))^{3/2}}{2 b^{5/2} \sqrt [4]{a^2+b^2} f \sec ^2(e+f x)^{3/4}}+\frac {3 a d^2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) (d \sec (e+f x))^{3/2}}{2 b^{5/2} \sqrt [4]{a^2+b^2} f \sec ^2(e+f x)^{3/4}}-\frac {3 d^2 E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right ) (d \sec (e+f x))^{3/2}}{b^2 f \sec ^2(e+f x)^{3/4}}+\frac {3 d^2 \cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{b^2 f}+\frac {3 a^2 d^2 \cot (e+f x) \operatorname {EllipticPi}\left (-\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right ) (d \sec (e+f x))^{3/2} \sqrt {-\tan ^2(e+f x)}}{2 b^3 \sqrt {a^2+b^2} f \sec ^2(e+f x)^{3/4}}-\frac {3 a^2 d^2 \cot (e+f x) \operatorname {EllipticPi}\left (\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right ) (d \sec (e+f x))^{3/2} \sqrt {-\tan ^2(e+f x)}}{2 b^3 \sqrt {a^2+b^2} f \sec ^2(e+f x)^{3/4}}-\frac {d^2 (d \sec (e+f x))^{3/2}}{b f (a+b \tan (e+f x))} \]
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Time = 0.46 (sec) , antiderivative size = 480, 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, 747, 858, 233, 202, 760, 408, 504, 1227, 551, 455, 65, 304, 211, 214} \[ \int \frac {(d \sec (e+f x))^{7/2}}{(a+b \tan (e+f x))^2} \, dx=\frac {3 a^2 d^2 \sqrt {-\tan ^2(e+f x)} \cot (e+f x) (d \sec (e+f x))^{3/2} \operatorname {EllipticPi}\left (-\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right )}{2 b^3 f \sqrt {a^2+b^2} \sec ^2(e+f x)^{3/4}}-\frac {3 a^2 d^2 \sqrt {-\tan ^2(e+f x)} \cot (e+f x) (d \sec (e+f x))^{3/2} \operatorname {EllipticPi}\left (\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right )}{2 b^3 f \sqrt {a^2+b^2} \sec ^2(e+f x)^{3/4}}-\frac {3 a d^2 (d \sec (e+f x))^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{2 b^{5/2} f \sqrt [4]{a^2+b^2} \sec ^2(e+f x)^{3/4}}+\frac {3 a d^2 (d \sec (e+f x))^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{2 b^{5/2} f \sqrt [4]{a^2+b^2} \sec ^2(e+f x)^{3/4}}-\frac {d^2 (d \sec (e+f x))^{3/2}}{b f (a+b \tan (e+f x))}-\frac {3 d^2 (d \sec (e+f x))^{3/2} E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right )}{b^2 f \sec ^2(e+f x)^{3/4}}+\frac {3 d^2 \sin (e+f x) \cos (e+f x) (d \sec (e+f x))^{3/2}}{b^2 f} \]
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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 747
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)^2} \, dx,x,b \tan (e+f x)\right )}{b f \sec ^2(e+f x)^{3/4}} \\ & = -\frac {d^2 (d \sec (e+f x))^{3/2}}{b f (a+b \tan (e+f x))}+\frac {\left (3 d^2 (d \sec (e+f x))^{3/2}\right ) \text {Subst}\left (\int \frac {x}{(a+x) \sqrt [4]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{2 b^3 f \sec ^2(e+f x)^{3/4}} \\ & = -\frac {d^2 (d \sec (e+f x))^{3/2}}{b f (a+b \tan (e+f x))}+\frac {\left (3 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 )}{2 b^3 f \sec ^2(e+f x)^{3/4}}-\frac {\left (3 a 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 )}{2 b^3 f \sec ^2(e+f x)^{3/4}} \\ & = \frac {3 d^2 \cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{b^2 f}-\frac {d^2 (d \sec (e+f x))^{3/2}}{b f (a+b \tan (e+f x))}-\frac {\left (3 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 )}{2 b^3 f \sec ^2(e+f x)^{3/4}}+\frac {\left (3 a 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 )}{2 b^3 f \sec ^2(e+f x)^{3/4}}-\frac {\left (3 a^2 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 )}{2 b^3 f \sec ^2(e+f x)^{3/4}} \\ & = -\frac {3 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 {3 d^2 \cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{b^2 f}-\frac {d^2 (d \sec (e+f x))^{3/2}}{b f (a+b \tan (e+f x))}+\frac {\left (3 a 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 )}{4 b^3 f \sec ^2(e+f x)^{3/4}}-\frac {\left (3 a^2 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^4 f \sec ^2(e+f x)^{3/4}} \\ & = -\frac {3 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 {3 d^2 \cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{b^2 f}-\frac {d^2 (d \sec (e+f x))^{3/2}}{b f (a+b \tan (e+f x))}+\frac {\left (3 a 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 )}{b f \sec ^2(e+f x)^{3/4}}-\frac {\left (3 a^2 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 )}{2 b^3 f \sec ^2(e+f x)^{3/4}}+\frac {\left (3 a^2 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 )}{2 b^3 f \sec ^2(e+f x)^{3/4}} \\ & = -\frac {3 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 {3 d^2 \cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{b^2 f}-\frac {d^2 (d \sec (e+f x))^{3/2}}{b f (a+b \tan (e+f x))}+\frac {\left (3 a 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 )}{2 b^2 f \sec ^2(e+f x)^{3/4}}-\frac {\left (3 a 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 )}{2 b^2 f \sec ^2(e+f x)^{3/4}}-\frac {\left (3 a^2 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 )}{2 b^3 f \sec ^2(e+f x)^{3/4}}+\frac {\left (3 a^2 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 )}{2 b^3 f \sec ^2(e+f x)^{3/4}} \\ & = -\frac {3 a 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}}{2 b^{5/2} \sqrt [4]{a^2+b^2} f \sec ^2(e+f x)^{3/4}}+\frac {3 a 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}}{2 b^{5/2} \sqrt [4]{a^2+b^2} f \sec ^2(e+f x)^{3/4}}-\frac {3 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 {3 d^2 \cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{b^2 f}+\frac {3 a^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)}}{2 b^3 \sqrt {a^2+b^2} f \sec ^2(e+f x)^{3/4}}-\frac {3 a^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)}}{2 b^3 \sqrt {a^2+b^2} f \sec ^2(e+f x)^{3/4}}-\frac {d^2 (d \sec (e+f x))^{3/2}}{b f (a+b \tan (e+f x))} \\ \end{align*}
Result contains complex when optimal does not.
Time = 29.03 (sec) , antiderivative size = 1129, normalized size of antiderivative = 2.35 \[ \int \frac {(d \sec (e+f x))^{7/2}}{(a+b \tan (e+f x))^2} \, dx=\frac {\cos (e+f x) (d \sec (e+f x))^{7/2} (a \cos (e+f x)+b \sin (e+f x))^2 \left (\frac {3 \cos (e+f x)}{a b}+\frac {3 \sin (e+f x)}{b^2}-\frac {1}{b (a \cos (e+f x)+b \sin (e+f x))}\right )}{f (a+b \tan (e+f x))^2}+\frac {3 (d \sec (e+f x))^{7/2} (a \cos (e+f x)+b \sin (e+f x))^2 \left (-\frac {a E\left (\left .\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right )\right |-1\right ) \sqrt {1+\tan ^2\left (\frac {1}{2} (e+f x)\right )}}{\sqrt {1-\tan ^2\left (\frac {1}{2} (e+f x)\right )}}+\frac {2 a \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),-1\right ) \sqrt {1+\tan ^2\left (\frac {1}{2} (e+f x)\right )}}{\sqrt {1-\tan ^2\left (\frac {1}{2} (e+f x)\right )}}+\frac {-2 \sqrt {2} a b \sqrt {a^2+b^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(1+i) \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )}{i+\tan \left (\frac {1}{2} (e+f x)\right )}}}{\sqrt {2}}\right ),2\right ) \sqrt {-\frac {1+i \tan \left (\frac {1}{2} (e+f x)\right )}{i+\tan \left (\frac {1}{2} (e+f x)\right )}}+\sqrt {2} a^2 \sqrt {a^2+b^2} \operatorname {EllipticPi}\left (\frac {(1+i) \left (a-i \left (b+\sqrt {a^2+b^2}\right )\right )}{a+b+\sqrt {a^2+b^2}},\arcsin \left (\frac {\sqrt {\frac {(1+i) \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )}{i+\tan \left (\frac {1}{2} (e+f x)\right )}}}{\sqrt {2}}\right ),2\right ) \sqrt {-\frac {1+i \tan \left (\frac {1}{2} (e+f x)\right )}{i+\tan \left (\frac {1}{2} (e+f x)\right )}}+a^2 \left (a+i b+\sqrt {a^2+b^2}\right ) \operatorname {EllipticPi}\left (\frac {(1+i) \left (a+i \left (-b+\sqrt {a^2+b^2}\right )\right )}{a+b-\sqrt {a^2+b^2}},\arcsin \left (\frac {\sqrt {\frac {(1+i) \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )}{i+\tan \left (\frac {1}{2} (e+f x)\right )}}}{\sqrt {2}}\right ),2\right ) \sqrt {-\frac {2+2 i \tan \left (\frac {1}{2} (e+f x)\right )}{i+\tan \left (\frac {1}{2} (e+f x)\right )}}-a^3 \operatorname {EllipticPi}\left (\frac {(1+i) \left (a-i \left (b+\sqrt {a^2+b^2}\right )\right )}{a+b+\sqrt {a^2+b^2}},\arcsin \left (\frac {\sqrt {\frac {(1+i) \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )}{i+\tan \left (\frac {1}{2} (e+f x)\right )}}}{\sqrt {2}}\right ),2\right ) \sqrt {-\frac {2+2 i \tan \left (\frac {1}{2} (e+f x)\right )}{i+\tan \left (\frac {1}{2} (e+f x)\right )}}-i a^2 b \operatorname {EllipticPi}\left (\frac {(1+i) \left (a-i \left (b+\sqrt {a^2+b^2}\right )\right )}{a+b+\sqrt {a^2+b^2}},\arcsin \left (\frac {\sqrt {\frac {(1+i) \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )}{i+\tan \left (\frac {1}{2} (e+f x)\right )}}}{\sqrt {2}}\right ),2\right ) \sqrt {-\frac {2+2 i \tan \left (\frac {1}{2} (e+f x)\right )}{i+\tan \left (\frac {1}{2} (e+f x)\right )}}-2 b^2 \sqrt {a^2+b^2} \sqrt {\frac {-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )}{\left (i+\tan \left (\frac {1}{2} (e+f x)\right )\right )^2}}-2 a b \sqrt {a^2+b^2} \tan \left (\frac {1}{2} (e+f x)\right ) \sqrt {\frac {-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )}{\left (i+\tan \left (\frac {1}{2} (e+f x)\right )\right )^2}}}{2 b \sqrt {a^2+b^2} \sqrt {\frac {-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )}{\left (i+\tan \left (\frac {1}{2} (e+f x)\right )\right )^2}}}\right )}{a b^2 f \sec ^{\frac {3}{2}}(e+f x) \sqrt {\frac {1+\tan ^2\left (\frac {1}{2} (e+f x)\right )}{1-\tan ^2\left (\frac {1}{2} (e+f x)\right )}} (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. 33210 vs. \(2 (439 ) = 878\).
Time = 132.90 (sec) , antiderivative size = 33211, normalized size of antiderivative = 69.19
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\[ \int \frac {(d \sec (e+f x))^{7/2}}{(a+b \tan (e+f x))^2} \, dx=\int { \frac {\left (d \sec \left (f x + e\right )\right )^{\frac {7}{2}}}{{\left (b \tan \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {(d \sec (e+f x))^{7/2}}{(a+b \tan (e+f x))^2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(d \sec (e+f x))^{7/2}}{(a+b \tan (e+f x))^2} \, dx=\text {Timed out} \]
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\[ \int \frac {(d \sec (e+f x))^{7/2}}{(a+b \tan (e+f x))^2} \, dx=\int { \frac {\left (d \sec \left (f x + e\right )\right )^{\frac {7}{2}}}{{\left (b \tan \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {(d \sec (e+f x))^{7/2}}{(a+b \tan (e+f x))^2} \, dx=\int \frac {{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{7/2}}{{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^2} \,d x \]
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