Integrand size = 16, antiderivative size = 125 \[ \int \frac {1}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\frac {\arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+b+b \tan ^2(e+f x)}}\right )}{a^{5/2} f}-\frac {b \tan (e+f x)}{3 a (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}-\frac {b (5 a+3 b) \tan (e+f x)}{3 a^2 (a+b)^2 f \sqrt {a+b+b \tan ^2(e+f x)}} \]
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Time = 0.12 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4213, 425, 541, 12, 385, 209} \[ \int \frac {1}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\frac {\arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)+b}}\right )}{a^{5/2} f}-\frac {b (5 a+3 b) \tan (e+f x)}{3 a^2 f (a+b)^2 \sqrt {a+b \tan ^2(e+f x)+b}}-\frac {b \tan (e+f x)}{3 a f (a+b) \left (a+b \tan ^2(e+f x)+b\right )^{3/2}} \]
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Rule 12
Rule 209
Rule 385
Rule 425
Rule 541
Rule 4213
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \left (a+b+b x^2\right )^{5/2}} \, dx,x,\tan (e+f x)\right )}{f} \\ & = -\frac {b \tan (e+f x)}{3 a (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {\text {Subst}\left (\int \frac {3 a+b-2 b x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{3 a (a+b) f} \\ & = -\frac {b \tan (e+f x)}{3 a (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}-\frac {b (5 a+3 b) \tan (e+f x)}{3 a^2 (a+b)^2 f \sqrt {a+b+b \tan ^2(e+f x)}}+\frac {\text {Subst}\left (\int \frac {3 (a+b)^2}{\left (1+x^2\right ) \sqrt {a+b+b x^2}} \, dx,x,\tan (e+f x)\right )}{3 a^2 (a+b)^2 f} \\ & = -\frac {b \tan (e+f x)}{3 a (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}-\frac {b (5 a+3 b) \tan (e+f x)}{3 a^2 (a+b)^2 f \sqrt {a+b+b \tan ^2(e+f x)}}+\frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {a+b+b x^2}} \, dx,x,\tan (e+f x)\right )}{a^2 f} \\ & = -\frac {b \tan (e+f x)}{3 a (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}-\frac {b (5 a+3 b) \tan (e+f x)}{3 a^2 (a+b)^2 f \sqrt {a+b+b \tan ^2(e+f x)}}+\frac {\text {Subst}\left (\int \frac {1}{1+a x^2} \, dx,x,\frac {\tan (e+f x)}{\sqrt {a+b+b \tan ^2(e+f x)}}\right )}{a^2 f} \\ & = \frac {\arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+b+b \tan ^2(e+f x)}}\right )}{a^{5/2} f}-\frac {b \tan (e+f x)}{3 a (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}-\frac {b (5 a+3 b) \tan (e+f x)}{3 a^2 (a+b)^2 f \sqrt {a+b+b \tan ^2(e+f x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 6.48 (sec) , antiderivative size = 1927, normalized size of antiderivative = 15.42 \[ \int \frac {1}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\frac {3 (a+b) \operatorname {AppellF1}\left (\frac {1}{2},-2,\frac {5}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right ) \cos ^4(e+f x) \sin (e+f x)}{4 \sqrt {2} f \left (a+b \sec ^2(e+f x)\right )^{5/2} \left (a+b-a \sin ^2(e+f x)\right )^{5/2} \left (3 (a+b) \operatorname {AppellF1}\left (\frac {1}{2},-2,\frac {5}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )+\left (5 a \operatorname {AppellF1}\left (\frac {3}{2},-2,\frac {7}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )-4 (a+b) \operatorname {AppellF1}\left (\frac {3}{2},-1,\frac {5}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )\right ) \sin ^2(e+f x)\right ) \left (\frac {15 a (a+b) \operatorname {AppellF1}\left (\frac {1}{2},-2,\frac {5}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right ) \cos ^5(e+f x) \sin ^2(e+f x)}{4 \sqrt {2} \left (a+b-a \sin ^2(e+f x)\right )^{7/2} \left (3 (a+b) \operatorname {AppellF1}\left (\frac {1}{2},-2,\frac {5}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )+\left (5 a \operatorname {AppellF1}\left (\frac {3}{2},-2,\frac {7}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )-4 (a+b) \operatorname {AppellF1}\left (\frac {3}{2},-1,\frac {5}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )\right ) \sin ^2(e+f x)\right )}+\frac {3 (a+b) \operatorname {AppellF1}\left (\frac {1}{2},-2,\frac {5}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right ) \cos ^5(e+f x)}{4 \sqrt {2} \left (a+b-a \sin ^2(e+f x)\right )^{5/2} \left (3 (a+b) \operatorname {AppellF1}\left (\frac {1}{2},-2,\frac {5}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )+\left (5 a \operatorname {AppellF1}\left (\frac {3}{2},-2,\frac {7}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )-4 (a+b) \operatorname {AppellF1}\left (\frac {3}{2},-1,\frac {5}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )\right ) \sin ^2(e+f x)\right )}-\frac {3 (a+b) \operatorname {AppellF1}\left (\frac {1}{2},-2,\frac {5}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right ) \cos ^3(e+f x) \sin ^2(e+f x)}{\sqrt {2} \left (a+b-a \sin ^2(e+f x)\right )^{5/2} \left (3 (a+b) \operatorname {AppellF1}\left (\frac {1}{2},-2,\frac {5}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )+\left (5 a \operatorname {AppellF1}\left (\frac {3}{2},-2,\frac {7}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )-4 (a+b) \operatorname {AppellF1}\left (\frac {3}{2},-1,\frac {5}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )\right ) \sin ^2(e+f x)\right )}+\frac {3 (a+b) \cos ^4(e+f x) \sin (e+f x) \left (\frac {5 a f \operatorname {AppellF1}\left (\frac {3}{2},-2,\frac {7}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right ) \cos (e+f x) \sin (e+f x)}{3 (a+b)}-\frac {4}{3} f \operatorname {AppellF1}\left (\frac {3}{2},-1,\frac {5}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right ) \cos (e+f x) \sin (e+f x)\right )}{4 \sqrt {2} f \left (a+b-a \sin ^2(e+f x)\right )^{5/2} \left (3 (a+b) \operatorname {AppellF1}\left (\frac {1}{2},-2,\frac {5}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )+\left (5 a \operatorname {AppellF1}\left (\frac {3}{2},-2,\frac {7}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )-4 (a+b) \operatorname {AppellF1}\left (\frac {3}{2},-1,\frac {5}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )\right ) \sin ^2(e+f x)\right )}-\frac {3 (a+b) \operatorname {AppellF1}\left (\frac {1}{2},-2,\frac {5}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right ) \cos ^4(e+f x) \sin (e+f x) \left (2 f \left (5 a \operatorname {AppellF1}\left (\frac {3}{2},-2,\frac {7}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )-4 (a+b) \operatorname {AppellF1}\left (\frac {3}{2},-1,\frac {5}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )\right ) \cos (e+f x) \sin (e+f x)+3 (a+b) \left (\frac {5 a f \operatorname {AppellF1}\left (\frac {3}{2},-2,\frac {7}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right ) \cos (e+f x) \sin (e+f x)}{3 (a+b)}-\frac {4}{3} f \operatorname {AppellF1}\left (\frac {3}{2},-1,\frac {5}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right ) \cos (e+f x) \sin (e+f x)\right )+\sin ^2(e+f x) \left (5 a \left (\frac {21 a f \operatorname {AppellF1}\left (\frac {5}{2},-2,\frac {9}{2},\frac {7}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right ) \cos (e+f x) \sin (e+f x)}{5 (a+b)}-\frac {12}{5} f \operatorname {AppellF1}\left (\frac {5}{2},-1,\frac {7}{2},\frac {7}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right ) \cos (e+f x) \sin (e+f x)\right )-4 (a+b) \left (\frac {3 a f \operatorname {AppellF1}\left (\frac {5}{2},-1,\frac {7}{2},\frac {7}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right ) \cos (e+f x) \sin (e+f x)}{a+b}-\frac {6 (a+b)^3 f \cot (e+f x) \csc ^4(e+f x) \left (-1+\frac {a \sin ^2(e+f x)}{a+b}\right )^2 \left (\frac {\sqrt {a} \arcsin \left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right ) \sin (e+f x)}{\sqrt {a+b} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}+\frac {a^2 \sin ^4(e+f x)}{3 (a+b)^2 \left (-1+\frac {a \sin ^2(e+f x)}{a+b}\right )^2}+\frac {a \sin ^2(e+f x)}{(a+b) \left (-1+\frac {a \sin ^2(e+f x)}{a+b}\right )}\right )}{a^3 \left (1-\frac {a \sin ^2(e+f x)}{a+b}\right )^{3/2}}\right )\right )\right )}{4 \sqrt {2} f \left (a+b-a \sin ^2(e+f x)\right )^{5/2} \left (3 (a+b) \operatorname {AppellF1}\left (\frac {1}{2},-2,\frac {5}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )+\left (5 a \operatorname {AppellF1}\left (\frac {3}{2},-2,\frac {7}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )-4 (a+b) \operatorname {AppellF1}\left (\frac {3}{2},-1,\frac {5}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )\right ) \sin ^2(e+f x)\right )^2}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1262\) vs. \(2(111)=222\).
Time = 6.40 (sec) , antiderivative size = 1263, normalized size of antiderivative = 10.10
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Leaf count of result is larger than twice the leaf count of optimal. 380 vs. \(2 (111) = 222\).
Time = 0.73 (sec) , antiderivative size = 881, normalized size of antiderivative = 7.05 \[ \int \frac {1}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\left [-\frac {3 \, {\left ({\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} \cos \left (f x + e\right )^{4} + a^{2} b^{2} + 2 \, a b^{3} + b^{4} + 2 \, {\left (a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-a} \log \left (128 \, a^{4} \cos \left (f x + e\right )^{8} - 256 \, {\left (a^{4} - a^{3} b\right )} \cos \left (f x + e\right )^{6} + 32 \, {\left (5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left (a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right )} \cos \left (f x + e\right )^{2} + 8 \, {\left (16 \, a^{3} \cos \left (f x + e\right )^{7} - 24 \, {\left (a^{3} - a^{2} b\right )} \cos \left (f x + e\right )^{5} + 2 \, {\left (5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right )} \cos \left (f x + e\right )^{3} - {\left (a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {-a} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \sin \left (f x + e\right )\right ) + 8 \, {\left (2 \, {\left (3 \, a^{3} b + 2 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{3} + {\left (5 \, a^{2} b^{2} + 3 \, a b^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \sin \left (f x + e\right )}{24 \, {\left ({\left (a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right )} f \cos \left (f x + e\right )^{4} + 2 \, {\left (a^{6} b + 2 \, a^{5} b^{2} + a^{4} b^{3}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{5} b^{2} + 2 \, a^{4} b^{3} + a^{3} b^{4}\right )} f\right )}}, -\frac {3 \, {\left ({\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} \cos \left (f x + e\right )^{4} + a^{2} b^{2} + 2 \, a b^{3} + b^{4} + 2 \, {\left (a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {a} \arctan \left (\frac {{\left (8 \, a^{2} \cos \left (f x + e\right )^{5} - 8 \, {\left (a^{2} - a b\right )} \cos \left (f x + e\right )^{3} + {\left (a^{2} - 6 \, a b + b^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {a} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{4 \, {\left (2 \, a^{3} \cos \left (f x + e\right )^{4} - a^{2} b + a b^{2} - {\left (a^{3} - 3 \, a^{2} b\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}\right ) + 4 \, {\left (2 \, {\left (3 \, a^{3} b + 2 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{3} + {\left (5 \, a^{2} b^{2} + 3 \, a b^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \sin \left (f x + e\right )}{12 \, {\left ({\left (a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right )} f \cos \left (f x + e\right )^{4} + 2 \, {\left (a^{6} b + 2 \, a^{5} b^{2} + a^{4} b^{3}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{5} b^{2} + 2 \, a^{4} b^{3} + a^{3} b^{4}\right )} f\right )}}\right ] \]
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\[ \int \frac {1}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {1}{\left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
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Timed out. \[ \int \frac {1}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {1}{{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^{5/2}} \,d x \]
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