Integrand size = 25, antiderivative size = 204 \[ \int \frac {\cos ^6(e+f x)}{\sqrt {a+b \sec ^2(e+f x)}} \, dx=\frac {(a-b) \left (5 a^2+2 a b+5 b^2\right ) \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+b+b \tan ^2(e+f x)}}\right )}{16 a^{7/2} f}+\frac {\left (15 a^2-14 a b+15 b^2\right ) \cos (e+f x) \sin (e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{48 a^3 f}+\frac {5 (a-b) \cos ^3(e+f x) \sin (e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{24 a^2 f}+\frac {\cos ^5(e+f x) \sin (e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{6 a f} \]
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Time = 0.24 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {4231, 425, 541, 12, 385, 209} \[ \int \frac {\cos ^6(e+f x)}{\sqrt {a+b \sec ^2(e+f x)}} \, dx=\frac {5 (a-b) \sin (e+f x) \cos ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)+b}}{24 a^2 f}+\frac {(a-b) \left (5 a^2+2 a b+5 b^2\right ) \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)+b}}\right )}{16 a^{7/2} f}+\frac {\left (15 a^2-14 a b+15 b^2\right ) \sin (e+f x) \cos (e+f x) \sqrt {a+b \tan ^2(e+f x)+b}}{48 a^3 f}+\frac {\sin (e+f x) \cos ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)+b}}{6 a f} \]
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Rule 12
Rule 209
Rule 385
Rule 425
Rule 541
Rule 4231
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right )^4 \sqrt {a+b+b x^2}} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {\cos ^5(e+f x) \sin (e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{6 a f}-\frac {\text {Subst}\left (\int \frac {-5 a+b-4 b x^2}{\left (1+x^2\right )^3 \sqrt {a+b+b x^2}} \, dx,x,\tan (e+f x)\right )}{6 a f} \\ & = \frac {5 (a-b) \cos ^3(e+f x) \sin (e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{24 a^2 f}+\frac {\cos ^5(e+f x) \sin (e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{6 a f}+\frac {\text {Subst}\left (\int \frac {15 a^2-4 a b+5 b^2+10 (a-b) b x^2}{\left (1+x^2\right )^2 \sqrt {a+b+b x^2}} \, dx,x,\tan (e+f x)\right )}{24 a^2 f} \\ & = \frac {\left (15 a^2-14 a b+15 b^2\right ) \cos (e+f x) \sin (e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{48 a^3 f}+\frac {5 (a-b) \cos ^3(e+f x) \sin (e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{24 a^2 f}+\frac {\cos ^5(e+f x) \sin (e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{6 a f}-\frac {\text {Subst}\left (\int -\frac {3 (a-b) \left (5 a^2+2 a b+5 b^2\right )}{\left (1+x^2\right ) \sqrt {a+b+b x^2}} \, dx,x,\tan (e+f x)\right )}{48 a^3 f} \\ & = \frac {\left (15 a^2-14 a b+15 b^2\right ) \cos (e+f x) \sin (e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{48 a^3 f}+\frac {5 (a-b) \cos ^3(e+f x) \sin (e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{24 a^2 f}+\frac {\cos ^5(e+f x) \sin (e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{6 a f}+\frac {\left ((a-b) \left (5 a^2+2 a b+5 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {a+b+b x^2}} \, dx,x,\tan (e+f x)\right )}{16 a^3 f} \\ & = \frac {\left (15 a^2-14 a b+15 b^2\right ) \cos (e+f x) \sin (e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{48 a^3 f}+\frac {5 (a-b) \cos ^3(e+f x) \sin (e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{24 a^2 f}+\frac {\cos ^5(e+f x) \sin (e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{6 a f}+\frac {\left ((a-b) \left (5 a^2+2 a b+5 b^2\right )\right ) \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 )}{16 a^3 f} \\ & = \frac {(a-b) \left (5 a^2+2 a b+5 b^2\right ) \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+b+b \tan ^2(e+f x)}}\right )}{16 a^{7/2} f}+\frac {\left (15 a^2-14 a b+15 b^2\right ) \cos (e+f x) \sin (e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{48 a^3 f}+\frac {5 (a-b) \cos ^3(e+f x) \sin (e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{24 a^2 f}+\frac {\cos ^5(e+f x) \sin (e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{6 a f} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 15.44 (sec) , antiderivative size = 1739, normalized size of antiderivative = 8.52 \[ \int \frac {\cos ^6(e+f x)}{\sqrt {a+b \sec ^2(e+f x)}} \, dx=\frac {3 (a+b) \operatorname {AppellF1}\left (\frac {1}{2},-3,\frac {1}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right ) \cos ^{12}(e+f x) \sin (e+f x)}{f \sqrt {a+2 b+a \cos (2 (e+f x))} \sqrt {a+b \sec ^2(e+f x)} \left (3 (a+b) \operatorname {AppellF1}\left (\frac {1}{2},-3,\frac {1}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )+\left (a \operatorname {AppellF1}\left (\frac {3}{2},-3,\frac {3}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )-6 (a+b) \operatorname {AppellF1}\left (\frac {3}{2},-2,\frac {1}{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 {3 (a+b) \operatorname {AppellF1}\left (\frac {1}{2},-3,\frac {1}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right ) \cos ^7(e+f x)}{\sqrt {a+2 b+a \cos (2 (e+f x))} \left (3 (a+b) \operatorname {AppellF1}\left (\frac {1}{2},-3,\frac {1}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )+\left (a \operatorname {AppellF1}\left (\frac {3}{2},-3,\frac {3}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )-6 (a+b) \operatorname {AppellF1}\left (\frac {3}{2},-2,\frac {1}{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 {18 (a+b) \operatorname {AppellF1}\left (\frac {1}{2},-3,\frac {1}{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)}{\sqrt {a+2 b+a \cos (2 (e+f x))} \left (3 (a+b) \operatorname {AppellF1}\left (\frac {1}{2},-3,\frac {1}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )+\left (a \operatorname {AppellF1}\left (\frac {3}{2},-3,\frac {3}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )-6 (a+b) \operatorname {AppellF1}\left (\frac {3}{2},-2,\frac {1}{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 ^6(e+f x) \sin (e+f x) \left (\frac {a f \operatorname {AppellF1}\left (\frac {3}{2},-3,\frac {3}{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)}-2 f \operatorname {AppellF1}\left (\frac {3}{2},-2,\frac {1}{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 )}{f \sqrt {a+2 b+a \cos (2 (e+f x))} \left (3 (a+b) \operatorname {AppellF1}\left (\frac {1}{2},-3,\frac {1}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )+\left (a \operatorname {AppellF1}\left (\frac {3}{2},-3,\frac {3}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )-6 (a+b) \operatorname {AppellF1}\left (\frac {3}{2},-2,\frac {1}{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},-3,\frac {1}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right ) \cos ^6(e+f x) \sin (e+f x) \left (2 f \left (a \operatorname {AppellF1}\left (\frac {3}{2},-3,\frac {3}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )-6 (a+b) \operatorname {AppellF1}\left (\frac {3}{2},-2,\frac {1}{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 {a f \operatorname {AppellF1}\left (\frac {3}{2},-3,\frac {3}{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)}-2 f \operatorname {AppellF1}\left (\frac {3}{2},-2,\frac {1}{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 (a \left (\frac {9 a f \operatorname {AppellF1}\left (\frac {5}{2},-3,\frac {5}{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 {18}{5} f \operatorname {AppellF1}\left (\frac {5}{2},-2,\frac {3}{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 )-6 (a+b) \left (\frac {3 a f \operatorname {AppellF1}\left (\frac {5}{2},-2,\frac {3}{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 {1}{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 )\right )\right )}{f \sqrt {a+2 b+a \cos (2 (e+f x))} \left (3 (a+b) \operatorname {AppellF1}\left (\frac {1}{2},-3,\frac {1}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )+\left (a \operatorname {AppellF1}\left (\frac {3}{2},-3,\frac {3}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )-6 (a+b) \operatorname {AppellF1}\left (\frac {3}{2},-2,\frac {1}{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}+\frac {3 a (a+b) \operatorname {AppellF1}\left (\frac {1}{2},-3,\frac {1}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right ) \cos ^6(e+f x) \sin (e+f x) \sin (2 (e+f x))}{(a+2 b+a \cos (2 (e+f x)))^{3/2} \left (3 (a+b) \operatorname {AppellF1}\left (\frac {1}{2},-3,\frac {1}{2},\frac {3}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )+\left (a \operatorname {AppellF1}\left (\frac {3}{2},-3,\frac {3}{2},\frac {5}{2},\sin ^2(e+f x),\frac {a \sin ^2(e+f x)}{a+b}\right )-6 (a+b) \operatorname {AppellF1}\left (\frac {3}{2},-2,\frac {1}{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 )}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1135\) vs. \(2(184)=368\).
Time = 9.68 (sec) , antiderivative size = 1136, normalized size of antiderivative = 5.57
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Time = 1.14 (sec) , antiderivative size = 643, normalized size of antiderivative = 3.15 \[ \int \frac {\cos ^6(e+f x)}{\sqrt {a+b \sec ^2(e+f x)}} \, dx=\left [\frac {3 \, {\left (5 \, a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - 5 \, b^{3}\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 (8 \, a^{3} \cos \left (f x + e\right )^{5} + 10 \, {\left (a^{3} - a^{2} b\right )} \cos \left (f x + e\right )^{3} + {\left (15 \, a^{3} - 14 \, a^{2} b + 15 \, a b^{2}\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 )}{384 \, a^{4} f}, -\frac {3 \, {\left (5 \, a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - 5 \, b^{3}\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 (8 \, a^{3} \cos \left (f x + e\right )^{5} + 10 \, {\left (a^{3} - a^{2} b\right )} \cos \left (f x + e\right )^{3} + {\left (15 \, a^{3} - 14 \, a^{2} b + 15 \, a b^{2}\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 )}{192 \, a^{4} f}\right ] \]
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\[ \int \frac {\cos ^6(e+f x)}{\sqrt {a+b \sec ^2(e+f x)}} \, dx=\int \frac {\cos ^{6}{\left (e + f x \right )}}{\sqrt {a + b \sec ^{2}{\left (e + f x \right )}}}\, dx \]
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\[ \int \frac {\cos ^6(e+f x)}{\sqrt {a+b \sec ^2(e+f x)}} \, dx=\int { \frac {\cos \left (f x + e\right )^{6}}{\sqrt {b \sec \left (f x + e\right )^{2} + a}} \,d x } \]
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\[ \int \frac {\cos ^6(e+f x)}{\sqrt {a+b \sec ^2(e+f x)}} \, dx=\int { \frac {\cos \left (f x + e\right )^{6}}{\sqrt {b \sec \left (f x + e\right )^{2} + a}} \,d x } \]
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Timed out. \[ \int \frac {\cos ^6(e+f x)}{\sqrt {a+b \sec ^2(e+f x)}} \, dx=\int \frac {{\cos \left (e+f\,x\right )}^6}{\sqrt {a+\frac {b}{{\cos \left (e+f\,x\right )}^2}}} \,d x \]
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