Integrand size = 25, antiderivative size = 568 \[ \int \frac {1}{(d \sec (e+f x))^{5/2} (a+b \tan (e+f x))} \, dx=\frac {b^{7/2} \arctan \left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sqrt [4]{\sec ^2(e+f x)}}{\left (a^2+b^2\right )^{9/4} d^2 f \sqrt {d \sec (e+f x)}}-\frac {b^{7/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sqrt [4]{\sec ^2(e+f x)}}{\left (a^2+b^2\right )^{9/4} d^2 f \sqrt {d \sec (e+f x)}}+\frac {2 a \left (3 a^2+8 b^2\right ) E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right ) \sqrt [4]{\sec ^2(e+f x)}}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)}}-\frac {2 a \left (3 a^2+8 b^2\right ) \tan (e+f x)}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)}}-\frac {a b^3 \cot (e+f x) \operatorname {EllipticPi}\left (-\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right ) \sqrt [4]{\sec ^2(e+f x)} \sqrt {-\tan ^2(e+f x)}}{\left (a^2+b^2\right )^{5/2} d^2 f \sqrt {d \sec (e+f x)}}+\frac {a b^3 \cot (e+f x) \operatorname {EllipticPi}\left (\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right ) \sqrt [4]{\sec ^2(e+f x)} \sqrt {-\tan ^2(e+f x)}}{\left (a^2+b^2\right )^{5/2} d^2 f \sqrt {d \sec (e+f x)}}+\frac {2 \cos ^2(e+f x) (b+a \tan (e+f x))}{5 \left (a^2+b^2\right ) d^2 f \sqrt {d \sec (e+f x)}}+\frac {2 \left (5 b^3+a \left (3 a^2+8 b^2\right ) \tan (e+f x)\right )}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)}} \]
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Time = 0.74 (sec) , antiderivative size = 568, 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, 755, 837, 858, 233, 202, 760, 408, 504, 1227, 551, 455, 65, 304, 211, 214} \[ \int \frac {1}{(d \sec (e+f x))^{5/2} (a+b \tan (e+f x))} \, dx=-\frac {a b^3 \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \sqrt [4]{\sec ^2(e+f x)} \operatorname {EllipticPi}\left (-\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right )}{d^2 f \left (a^2+b^2\right )^{5/2} \sqrt {d \sec (e+f x)}}+\frac {a b^3 \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \sqrt [4]{\sec ^2(e+f x)} \operatorname {EllipticPi}\left (\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right )}{d^2 f \left (a^2+b^2\right )^{5/2} \sqrt {d \sec (e+f x)}}+\frac {2 a \left (3 a^2+8 b^2\right ) \sqrt [4]{\sec ^2(e+f x)} E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right )}{5 d^2 f \left (a^2+b^2\right )^2 \sqrt {d \sec (e+f x)}}+\frac {b^{7/2} \sqrt [4]{\sec ^2(e+f x)} \arctan \left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{d^2 f \left (a^2+b^2\right )^{9/4} \sqrt {d \sec (e+f x)}}-\frac {b^{7/2} \sqrt [4]{\sec ^2(e+f x)} \text {arctanh}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{d^2 f \left (a^2+b^2\right )^{9/4} \sqrt {d \sec (e+f x)}}-\frac {2 a \left (3 a^2+8 b^2\right ) \tan (e+f x)}{5 d^2 f \left (a^2+b^2\right )^2 \sqrt {d \sec (e+f x)}}+\frac {2 \cos ^2(e+f x) (a \tan (e+f x)+b)}{5 d^2 f \left (a^2+b^2\right ) \sqrt {d \sec (e+f x)}}+\frac {2 \left (a \left (3 a^2+8 b^2\right ) \tan (e+f x)+5 b^3\right )}{5 d^2 f \left (a^2+b^2\right )^2 \sqrt {d \sec (e+f x)}} \]
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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 755
Rule 760
Rule 837
Rule 858
Rule 1227
Rule 3593
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{\sec ^2(e+f x)} \text {Subst}\left (\int \frac {1}{(a+x) \left (1+\frac {x^2}{b^2}\right )^{9/4}} \, dx,x,b \tan (e+f x)\right )}{b d^2 f \sqrt {d \sec (e+f x)}} \\ & = \frac {2 \cos ^2(e+f x) (b+a \tan (e+f x))}{5 \left (a^2+b^2\right ) d^2 f \sqrt {d \sec (e+f x)}}-\frac {\left (2 b \sqrt [4]{\sec ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {\frac {1}{2} \left (-5-\frac {3 a^2}{b^2}\right )-\frac {3 a x}{2 b^2}}{(a+x) \left (1+\frac {x^2}{b^2}\right )^{5/4}} \, dx,x,b \tan (e+f x)\right )}{5 \left (a^2+b^2\right ) d^2 f \sqrt {d \sec (e+f x)}} \\ & = \frac {2 \cos ^2(e+f x) (b+a \tan (e+f x))}{5 \left (a^2+b^2\right ) d^2 f \sqrt {d \sec (e+f x)}}+\frac {2 \left (5 b^3+a \left (3 a^2+8 b^2\right ) \tan (e+f x)\right )}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)}}+\frac {\left (4 b^5 \sqrt [4]{\sec ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {-\frac {3 a^4+8 a^2 b^2-5 b^4}{4 b^6}-\frac {a \left (3 a^2+8 b^2\right ) x}{4 b^6}}{(a+x) \sqrt [4]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)}} \\ & = \frac {2 \cos ^2(e+f x) (b+a \tan (e+f x))}{5 \left (a^2+b^2\right ) d^2 f \sqrt {d \sec (e+f x)}}+\frac {2 \left (5 b^3+a \left (3 a^2+8 b^2\right ) \tan (e+f x)\right )}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)}}+\frac {\left (b^3 \sqrt [4]{\sec ^2(e+f x)}\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 )}{\left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)}}-\frac {\left (a \left (3 a^2+8 b^2\right ) \sqrt [4]{\sec ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{5 b \left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)}} \\ & = -\frac {2 a \left (3 a^2+8 b^2\right ) \tan (e+f x)}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)}}+\frac {2 \cos ^2(e+f x) (b+a \tan (e+f x))}{5 \left (a^2+b^2\right ) d^2 f \sqrt {d \sec (e+f x)}}+\frac {2 \left (5 b^3+a \left (3 a^2+8 b^2\right ) \tan (e+f x)\right )}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)}}-\frac {\left (b^3 \sqrt [4]{\sec ^2(e+f x)}\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 )}{\left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)}}+\frac {\left (a b^3 \sqrt [4]{\sec ^2(e+f x)}\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 )}{\left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)}}+\frac {\left (a \left (3 a^2+8 b^2\right ) \sqrt [4]{\sec ^2(e+f x)}\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 )}{5 b \left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)}} \\ & = \frac {2 a \left (3 a^2+8 b^2\right ) E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right ) \sqrt [4]{\sec ^2(e+f x)}}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)}}-\frac {2 a \left (3 a^2+8 b^2\right ) \tan (e+f x)}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)}}+\frac {2 \cos ^2(e+f x) (b+a \tan (e+f x))}{5 \left (a^2+b^2\right ) d^2 f \sqrt {d \sec (e+f x)}}+\frac {2 \left (5 b^3+a \left (3 a^2+8 b^2\right ) \tan (e+f x)\right )}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)}}-\frac {\left (b^3 \sqrt [4]{\sec ^2(e+f x)}\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 \left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)}}+\frac {\left (2 a b^2 \cot (e+f x) \sqrt [4]{\sec ^2(e+f x)} \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 )}{\left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)}} \\ & = \frac {2 a \left (3 a^2+8 b^2\right ) E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right ) \sqrt [4]{\sec ^2(e+f x)}}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)}}-\frac {2 a \left (3 a^2+8 b^2\right ) \tan (e+f x)}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)}}+\frac {2 \cos ^2(e+f x) (b+a \tan (e+f x))}{5 \left (a^2+b^2\right ) d^2 f \sqrt {d \sec (e+f x)}}+\frac {2 \left (5 b^3+a \left (3 a^2+8 b^2\right ) \tan (e+f x)\right )}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)}}-\frac {\left (2 b^5 \sqrt [4]{\sec ^2(e+f x)}\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 )}{\left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)}}+\frac {\left (a b^3 \cot (e+f x) \sqrt [4]{\sec ^2(e+f x)} \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 )}{\left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)}}-\frac {\left (a b^3 \cot (e+f x) \sqrt [4]{\sec ^2(e+f x)} \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 )}{\left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)}} \\ & = \frac {2 a \left (3 a^2+8 b^2\right ) E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right ) \sqrt [4]{\sec ^2(e+f x)}}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)}}-\frac {2 a \left (3 a^2+8 b^2\right ) \tan (e+f x)}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)}}+\frac {2 \cos ^2(e+f x) (b+a \tan (e+f x))}{5 \left (a^2+b^2\right ) d^2 f \sqrt {d \sec (e+f x)}}+\frac {2 \left (5 b^3+a \left (3 a^2+8 b^2\right ) \tan (e+f x)\right )}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)}}-\frac {\left (b^4 \sqrt [4]{\sec ^2(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 )}{\left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)}}+\frac {\left (b^4 \sqrt [4]{\sec ^2(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 )}{\left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)}}+\frac {\left (a b^3 \cot (e+f x) \sqrt [4]{\sec ^2(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 (\sqrt {a^2+b^2}-b x^2\right )} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{\left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)}}-\frac {\left (a b^3 \cot (e+f x) \sqrt [4]{\sec ^2(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 (\sqrt {a^2+b^2}+b x^2\right )} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{\left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)}} \\ & = \frac {b^{7/2} \arctan \left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sqrt [4]{\sec ^2(e+f x)}}{\left (a^2+b^2\right )^{9/4} d^2 f \sqrt {d \sec (e+f x)}}-\frac {b^{7/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sqrt [4]{\sec ^2(e+f x)}}{\left (a^2+b^2\right )^{9/4} d^2 f \sqrt {d \sec (e+f x)}}+\frac {2 a \left (3 a^2+8 b^2\right ) E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right ) \sqrt [4]{\sec ^2(e+f x)}}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)}}-\frac {2 a \left (3 a^2+8 b^2\right ) \tan (e+f x)}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)}}-\frac {a b^3 \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 [4]{\sec ^2(e+f x)} \sqrt {-\tan ^2(e+f x)}}{\left (a^2+b^2\right )^{5/2} d^2 f \sqrt {d \sec (e+f x)}}+\frac {a b^3 \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 [4]{\sec ^2(e+f x)} \sqrt {-\tan ^2(e+f x)}}{\left (a^2+b^2\right )^{5/2} d^2 f \sqrt {d \sec (e+f x)}}+\frac {2 \cos ^2(e+f x) (b+a \tan (e+f x))}{5 \left (a^2+b^2\right ) d^2 f \sqrt {d \sec (e+f x)}}+\frac {2 \left (5 b^3+a \left (3 a^2+8 b^2\right ) \tan (e+f x)\right )}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 77.21 (sec) , antiderivative size = 2596, normalized size of antiderivative = 4.57 \[ \int \frac {1}{(d \sec (e+f x))^{5/2} (a+b \tan (e+f x))} \, dx=\text {Result too large to show} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 12840 vs. \(2 (525 ) = 1050\).
Time = 14.01 (sec) , antiderivative size = 12841, normalized size of antiderivative = 22.61
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Timed out. \[ \int \frac {1}{(d \sec (e+f x))^{5/2} (a+b \tan (e+f x))} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(d \sec (e+f x))^{5/2} (a+b \tan (e+f x))} \, dx=\int \frac {1}{\left (d \sec {\left (e + f x \right )}\right )^{\frac {5}{2}} \left (a + b \tan {\left (e + f x \right )}\right )}\, dx \]
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Exception generated. \[ \int \frac {1}{(d \sec (e+f x))^{5/2} (a+b \tan (e+f x))} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {1}{(d \sec (e+f x))^{5/2} (a+b \tan (e+f x))} \, dx=\int { \frac {1}{\left (d \sec \left (f x + e\right )\right )^{\frac {5}{2}} {\left (b \tan \left (f x + e\right ) + a\right )}} \,d x } \]
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Timed out. \[ \int \frac {1}{(d \sec (e+f x))^{5/2} (a+b \tan (e+f x))} \, dx=\int \frac {1}{{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{5/2}\,\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )} \,d x \]
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