Integrand size = 25, antiderivative size = 242 \[ \int \frac {\sin ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\frac {5 (a+b)^2 (a+7 b) \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+b+b \tan ^2(e+f x)}}\right )}{16 a^{9/2} f}-\frac {(a+b) (33 a+35 b) \cos (e+f x) \sin (e+f x)}{48 a^3 f \sqrt {a+b+b \tan ^2(e+f x)}}+\frac {(9 a+7 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \sqrt {a+b+b \tan ^2(e+f x)}}+\frac {\cos ^3(e+f x) \sin ^3(e+f x)}{6 a f \sqrt {a+b+b \tan ^2(e+f x)}}-\frac {b \left (81 a^2+190 a b+105 b^2\right ) \tan (e+f x)}{48 a^4 f \sqrt {a+b+b \tan ^2(e+f x)}} \]
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Time = 0.41 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {4217, 481, 592, 541, 12, 385, 209} \[ \int \frac {\sin ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\frac {5 (a+b)^2 (a+7 b) \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)+b}}\right )}{16 a^{9/2} f}-\frac {(a+b) (33 a+35 b) \sin (e+f x) \cos (e+f x)}{48 a^3 f \sqrt {a+b \tan ^2(e+f x)+b}}+\frac {(9 a+7 b) \sin (e+f x) \cos ^3(e+f x)}{24 a^2 f \sqrt {a+b \tan ^2(e+f x)+b}}-\frac {b \left (81 a^2+190 a b+105 b^2\right ) \tan (e+f x)}{48 a^4 f \sqrt {a+b \tan ^2(e+f x)+b}}+\frac {\sin ^3(e+f x) \cos ^3(e+f x)}{6 a f \sqrt {a+b \tan ^2(e+f x)+b}} \]
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Rule 12
Rule 209
Rule 385
Rule 481
Rule 541
Rule 592
Rule 4217
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^6}{\left (1+x^2\right )^4 \left (a+b+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {\cos ^3(e+f x) \sin ^3(e+f x)}{6 a f \sqrt {a+b+b \tan ^2(e+f x)}}-\frac {\text {Subst}\left (\int \frac {x^2 \left (3 (a+b)+2 (b-3 (a+b)) x^2\right )}{\left (1+x^2\right )^3 \left (a+b+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{6 a f} \\ & = \frac {(9 a+7 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \sqrt {a+b+b \tan ^2(e+f x)}}+\frac {\cos ^3(e+f x) \sin ^3(e+f x)}{6 a f \sqrt {a+b+b \tan ^2(e+f x)}}-\frac {\text {Subst}\left (\int \frac {(a+b) (9 a+7 b)-4 (a+b) (6 a+7 b) x^2}{\left (1+x^2\right )^2 \left (a+b+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{24 a^2 f} \\ & = -\frac {(a+b) (33 a+35 b) \cos (e+f x) \sin (e+f x)}{48 a^3 f \sqrt {a+b+b \tan ^2(e+f x)}}+\frac {(9 a+7 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \sqrt {a+b+b \tan ^2(e+f x)}}+\frac {\cos ^3(e+f x) \sin ^3(e+f x)}{6 a f \sqrt {a+b+b \tan ^2(e+f x)}}+\frac {\text {Subst}\left (\int \frac {(a+b) \left (15 a^2+54 a b+35 b^2\right )-2 b (a+b) (33 a+35 b) x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{48 a^3 f} \\ & = -\frac {(a+b) (33 a+35 b) \cos (e+f x) \sin (e+f x)}{48 a^3 f \sqrt {a+b+b \tan ^2(e+f x)}}+\frac {(9 a+7 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \sqrt {a+b+b \tan ^2(e+f x)}}+\frac {\cos ^3(e+f x) \sin ^3(e+f x)}{6 a f \sqrt {a+b+b \tan ^2(e+f x)}}-\frac {b \left (81 a^2+190 a b+105 b^2\right ) \tan (e+f x)}{48 a^4 f \sqrt {a+b+b \tan ^2(e+f x)}}+\frac {\text {Subst}\left (\int \frac {15 (a+b)^3 (a+7 b)}{\left (1+x^2\right ) \sqrt {a+b+b x^2}} \, dx,x,\tan (e+f x)\right )}{48 a^4 (a+b) f} \\ & = -\frac {(a+b) (33 a+35 b) \cos (e+f x) \sin (e+f x)}{48 a^3 f \sqrt {a+b+b \tan ^2(e+f x)}}+\frac {(9 a+7 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \sqrt {a+b+b \tan ^2(e+f x)}}+\frac {\cos ^3(e+f x) \sin ^3(e+f x)}{6 a f \sqrt {a+b+b \tan ^2(e+f x)}}-\frac {b \left (81 a^2+190 a b+105 b^2\right ) \tan (e+f x)}{48 a^4 f \sqrt {a+b+b \tan ^2(e+f x)}}+\frac {\left (5 (a+b)^2 (a+7 b)\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^4 f} \\ & = -\frac {(a+b) (33 a+35 b) \cos (e+f x) \sin (e+f x)}{48 a^3 f \sqrt {a+b+b \tan ^2(e+f x)}}+\frac {(9 a+7 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \sqrt {a+b+b \tan ^2(e+f x)}}+\frac {\cos ^3(e+f x) \sin ^3(e+f x)}{6 a f \sqrt {a+b+b \tan ^2(e+f x)}}-\frac {b \left (81 a^2+190 a b+105 b^2\right ) \tan (e+f x)}{48 a^4 f \sqrt {a+b+b \tan ^2(e+f x)}}+\frac {\left (5 (a+b)^2 (a+7 b)\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^4 f} \\ & = \frac {5 (a+b)^2 (a+7 b) \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+b+b \tan ^2(e+f x)}}\right )}{16 a^{9/2} f}-\frac {(a+b) (33 a+35 b) \cos (e+f x) \sin (e+f x)}{48 a^3 f \sqrt {a+b+b \tan ^2(e+f x)}}+\frac {(9 a+7 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \sqrt {a+b+b \tan ^2(e+f x)}}+\frac {\cos ^3(e+f x) \sin ^3(e+f x)}{6 a f \sqrt {a+b+b \tan ^2(e+f x)}}-\frac {b \left (81 a^2+190 a b+105 b^2\right ) \tan (e+f x)}{48 a^4 f \sqrt {a+b+b \tan ^2(e+f x)}} \\ \end{align*}
Time = 6.30 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.06 \[ \int \frac {\sin ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\frac {(a+2 b+a \cos (2 (e+f x))) \sec ^3(e+f x) \left (120 (a+b)^2 (a+7 b) \arcsin \left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right ) (a+2 b+a \cos (2 (e+f x)))-2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\frac {a+2 b+a \cos (2 (e+f x))}{a+b}} \left (37 a^3+439 a^2 b+830 a b^2+420 b^3+a \left (29 a^2+108 a b+70 b^2\right ) \cos (2 (e+f x))-7 a^2 (a+b) \cos (4 (e+f x))+a^3 \cos (6 (e+f x))\right ) \sin (e+f x)\right )}{1536 a^{9/2} \sqrt {a+b} f \left (a+b \sec ^2(e+f x)\right )^{3/2} \sqrt {\frac {a+b-a \sin ^2(e+f x)}{a+b}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1161\) vs. \(2(218)=436\).
Time = 9.33 (sec) , antiderivative size = 1162, normalized size of antiderivative = 4.80
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Time = 5.91 (sec) , antiderivative size = 813, normalized size of antiderivative = 3.36 \[ \int \frac {\sin ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\left [-\frac {15 \, {\left (a^{3} b + 9 \, a^{2} b^{2} + 15 \, a b^{3} + 7 \, b^{4} + {\left (a^{4} + 9 \, a^{3} b + 15 \, a^{2} b^{2} + 7 \, 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 (8 \, a^{4} \cos \left (f x + e\right )^{7} - 2 \, {\left (13 \, a^{4} + 7 \, a^{3} b\right )} \cos \left (f x + e\right )^{5} + {\left (33 \, a^{4} + 68 \, a^{3} b + 35 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{3} + {\left (81 \, a^{3} b + 190 \, a^{2} b^{2} + 105 \, 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 )}{384 \, {\left (a^{6} f \cos \left (f x + e\right )^{2} + a^{5} b f\right )}}, -\frac {15 \, {\left (a^{3} b + 9 \, a^{2} b^{2} + 15 \, a b^{3} + 7 \, b^{4} + {\left (a^{4} + 9 \, a^{3} b + 15 \, a^{2} b^{2} + 7 \, 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 (8 \, a^{4} \cos \left (f x + e\right )^{7} - 2 \, {\left (13 \, a^{4} + 7 \, a^{3} b\right )} \cos \left (f x + e\right )^{5} + {\left (33 \, a^{4} + 68 \, a^{3} b + 35 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{3} + {\left (81 \, a^{3} b + 190 \, a^{2} b^{2} + 105 \, 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 )}{192 \, {\left (a^{6} f \cos \left (f x + e\right )^{2} + a^{5} b f\right )}}\right ] \]
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\[ \int \frac {\sin ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {\sin ^{6}{\left (e + f x \right )}}{\left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {\sin ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\int { \frac {\sin \left (f x + e\right )^{6}}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {\sin ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\int { \frac {\sin \left (f x + e\right )^{6}}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sin ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {{\sin \left (e+f\,x\right )}^6}{{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^{3/2}} \,d x \]
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