Integrand size = 25, antiderivative size = 193 \[ \int \cos ^6(e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx=\frac {(5 a-b) (a+b)^2 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+b+b \tan ^2(e+f x)}}\right )}{16 a^{3/2} f}+\frac {(5 a-b) (a+b) \cos (e+f x) \sin (e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{16 a f}+\frac {(5 a-b) \cos ^3(e+f x) \sin (e+f x) \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}{24 a f}+\frac {\cos ^5(e+f x) \sin (e+f x) \left (a+b+b \tan ^2(e+f x)\right )^{5/2}}{6 a f} \]
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Time = 0.20 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4231, 390, 386, 385, 209} \[ \int \cos ^6(e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx=\frac {(5 a-b) (a+b)^2 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)+b}}\right )}{16 a^{3/2} f}+\frac {\sin (e+f x) \cos ^5(e+f x) \left (a+b \tan ^2(e+f x)+b\right )^{5/2}}{6 a f}+\frac {(5 a-b) \sin (e+f x) \cos ^3(e+f x) \left (a+b \tan ^2(e+f x)+b\right )^{3/2}}{24 a f}+\frac {(5 a-b) (a+b) \sin (e+f x) \cos (e+f x) \sqrt {a+b \tan ^2(e+f x)+b}}{16 a f} \]
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Rule 209
Rule 385
Rule 386
Rule 390
Rule 4231
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (a+b+b x^2\right )^{3/2}}{\left (1+x^2\right )^4} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {\cos ^5(e+f x) \sin (e+f x) \left (a+b+b \tan ^2(e+f x)\right )^{5/2}}{6 a f}+\frac {(5 a-b) \text {Subst}\left (\int \frac {\left (a+b+b x^2\right )^{3/2}}{\left (1+x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{6 a f} \\ & = \frac {(5 a-b) \cos ^3(e+f x) \sin (e+f x) \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}{24 a f}+\frac {\cos ^5(e+f x) \sin (e+f x) \left (a+b+b \tan ^2(e+f x)\right )^{5/2}}{6 a f}+\frac {((5 a-b) (a+b)) \text {Subst}\left (\int \frac {\sqrt {a+b+b x^2}}{\left (1+x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{8 a f} \\ & = \frac {(5 a-b) (a+b) \cos (e+f x) \sin (e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{16 a f}+\frac {(5 a-b) \cos ^3(e+f x) \sin (e+f x) \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}{24 a f}+\frac {\cos ^5(e+f x) \sin (e+f x) \left (a+b+b \tan ^2(e+f x)\right )^{5/2}}{6 a f}+\frac {\left ((5 a-b) (a+b)^2\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 f} \\ & = \frac {(5 a-b) (a+b) \cos (e+f x) \sin (e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{16 a f}+\frac {(5 a-b) \cos ^3(e+f x) \sin (e+f x) \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}{24 a f}+\frac {\cos ^5(e+f x) \sin (e+f x) \left (a+b+b \tan ^2(e+f x)\right )^{5/2}}{6 a f}+\frac {\left ((5 a-b) (a+b)^2\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 f} \\ & = \frac {(5 a-b) (a+b)^2 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+b+b \tan ^2(e+f x)}}\right )}{16 a^{3/2} f}+\frac {(5 a-b) (a+b) \cos (e+f x) \sin (e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{16 a f}+\frac {(5 a-b) \cos ^3(e+f x) \sin (e+f x) \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}{24 a f}+\frac {\cos ^5(e+f x) \sin (e+f x) \left (a+b+b \tan ^2(e+f x)\right )^{5/2}}{6 a f} \\ \end{align*}
Time = 2.22 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.85 \[ \int \cos ^6(e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx=\frac {\cos (e+f x) \sqrt {a+b \sec ^2(e+f x)} \left (\frac {3 \sqrt {2} \sqrt {a+b} \left (5 a^2+4 a b-b^2\right ) \arcsin \left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )}{\sqrt {\frac {a+2 b+a \cos (2 (e+f x))}{a+b}}}+\sqrt {a} \left (23 a^2+29 a b+3 b^2+a (9 a+7 b) \cos (2 (e+f x))+a^2 \cos (4 (e+f x))\right ) \sin (e+f x)\right )}{48 a^{3/2} f} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(985\) vs. \(2(173)=346\).
Time = 6.04 (sec) , antiderivative size = 986, normalized size of antiderivative = 5.11
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Time = 1.19 (sec) , antiderivative size = 647, normalized size of antiderivative = 3.35 \[ \int \cos ^6(e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx=\left [\frac {3 \, {\left (5 \, a^{3} + 9 \, a^{2} b + 3 \, a b^{2} - 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} + 2 \, {\left (5 \, a^{3} + 7 \, a^{2} b\right )} \cos \left (f x + e\right )^{3} + {\left (15 \, a^{3} + 22 \, a^{2} b + 3 \, 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^{2} f}, -\frac {3 \, {\left (5 \, a^{3} + 9 \, a^{2} b + 3 \, a b^{2} - 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} + 2 \, {\left (5 \, a^{3} + 7 \, a^{2} b\right )} \cos \left (f x + e\right )^{3} + {\left (15 \, a^{3} + 22 \, a^{2} b + 3 \, 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^{2} f}\right ] \]
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Timed out. \[ \int \cos ^6(e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx=\text {Timed out} \]
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\[ \int \cos ^6(e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \cos \left (f x + e\right )^{6} \,d x } \]
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\[ \int \cos ^6(e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \cos \left (f x + e\right )^{6} \,d x } \]
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Timed out. \[ \int \cos ^6(e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx=\int {\cos \left (e+f\,x\right )}^6\,{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^{3/2} \,d x \]
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