Integrand size = 21, antiderivative size = 144 \[ \int \frac {\sec (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\frac {\left (8 a^2+8 a b+3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )}{8 a^{5/2} (a+b)^{5/2} f}-\frac {b \cos ^2(e+f x) \sin (e+f x)}{4 a (a+b) f \left (a+b-a \sin ^2(e+f x)\right )^2}-\frac {3 b (2 a+b) \sin (e+f x)}{8 a^2 (a+b)^2 f \left (a+b-a \sin ^2(e+f x)\right )} \]
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Time = 0.14 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {4232, 424, 393, 214} \[ \int \frac {\sec (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=-\frac {3 b (2 a+b) \sin (e+f x)}{8 a^2 f (a+b)^2 \left (-a \sin ^2(e+f x)+a+b\right )}+\frac {\left (8 a^2+8 a b+3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )}{8 a^{5/2} f (a+b)^{5/2}}-\frac {b \sin (e+f x) \cos ^2(e+f x)}{4 a f (a+b) \left (-a \sin ^2(e+f x)+a+b\right )^2} \]
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Rule 214
Rule 393
Rule 424
Rule 4232
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^2}{\left (a+b-a x^2\right )^3} \, dx,x,\sin (e+f x)\right )}{f} \\ & = -\frac {b \cos ^2(e+f x) \sin (e+f x)}{4 a (a+b) f \left (a+b-a \sin ^2(e+f x)\right )^2}-\frac {\text {Subst}\left (\int \frac {-4 a-b+(4 a+3 b) x^2}{\left (a+b-a x^2\right )^2} \, dx,x,\sin (e+f x)\right )}{4 a (a+b) f} \\ & = -\frac {b \cos ^2(e+f x) \sin (e+f x)}{4 a (a+b) f \left (a+b-a \sin ^2(e+f x)\right )^2}-\frac {3 b (2 a+b) \sin (e+f x)}{8 a^2 (a+b)^2 f \left (a+b-a \sin ^2(e+f x)\right )}+\frac {\left (8 a^2+8 a b+3 b^2\right ) \text {Subst}\left (\int \frac {1}{a+b-a x^2} \, dx,x,\sin (e+f x)\right )}{8 a^2 (a+b)^2 f} \\ & = \frac {\left (8 a^2+8 a b+3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )}{8 a^{5/2} (a+b)^{5/2} f}-\frac {b \cos ^2(e+f x) \sin (e+f x)}{4 a (a+b) f \left (a+b-a \sin ^2(e+f x)\right )^2}-\frac {3 b (2 a+b) \sin (e+f x)}{8 a^2 (a+b)^2 f \left (a+b-a \sin ^2(e+f x)\right )} \\ \end{align*}
Result contains complex when optimal does not.
Time = 5.45 (sec) , antiderivative size = 927, normalized size of antiderivative = 6.44 \[ \int \frac {\sec (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\frac {(a+2 b+a \cos (2 (e+f x))) \sec ^5(e+f x) \left (-2 i \left (8 a^2+8 a b+3 b^2\right ) \arctan \left (\frac {(a+b) \sin (e)}{(a+b) \cos (e)-\sqrt {a} \sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2} (\cos (2 e)+i \sin (2 e)) \sin (e+f x)}\right ) (a+2 b+a \cos (2 (e+f x)))^2 \sec (e+f x) (\cos (e)-i \sin (e))+\left (8 a^2+8 a b+3 b^2\right ) (a+2 b+a \cos (2 (e+f x)))^2 \log \left (a+2 (a+b) \cos (2 e)-a \cos (2 (e+f x))-2 i a \sin (2 e)-2 i b \sin (2 e)+2 \sqrt {a} \sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2} \sin (f x)+2 \sqrt {a} \sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2} \sin (2 e+f x)\right ) \sec (e+f x) (\cos (e)-i \sin (e))-\left (8 a^2+8 a b+3 b^2\right ) (a+2 b+a \cos (2 (e+f x)))^2 \log \left (-a-2 (a+b) \cos (2 e)+a \cos (2 (e+f x))+2 i a \sin (2 e)+2 i b \sin (2 e)+2 \sqrt {a} \sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2} \sin (f x)+2 \sqrt {a} \sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2} \sin (2 e+f x)\right ) \sec (e+f x) (\cos (e)-i \sin (e))+2 \left (8 a^2+8 a b+3 b^2\right ) \arctan \left (\frac {2 \sin (e) \left (i a+i b+i (a+b) \cos (2 e)+\sqrt {a} \sqrt {a+b} \cos (f x) \sqrt {(\cos (e)-i \sin (e))^2}-\sqrt {a} \sqrt {a+b} \cos (2 e+f x) \sqrt {(\cos (e)-i \sin (e))^2}+a \sin (2 e)+b \sin (2 e)-i \sqrt {a} \sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2} \sin (f x)-i \sqrt {a} \sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2} \sin (2 e+f x)\right )}{i (a+3 b) \cos (e)+i (a+b) \cos (3 e)+i a \cos (e+2 f x)+i a \cos (3 e+2 f x)+3 a \sin (e)+b \sin (e)+a \sin (3 e)+b \sin (3 e)+a \sin (e+2 f x)-a \sin (3 e+2 f x)}\right ) (a+2 b+a \cos (2 (e+f x)))^2 \sec (e+f x) (i \cos (e)+\sin (e))+32 \sqrt {a} b^2 (a+b)^{3/2} \sqrt {(\cos (e)-i \sin (e))^2} \tan (e+f x)-8 \sqrt {a} b \sqrt {a+b} (8 a+5 b) (a+2 b+a \cos (2 (e+f x))) \sqrt {(\cos (e)-i \sin (e))^2} \tan (e+f x)\right )}{256 a^{5/2} (a+b)^{5/2} f \left (a+b \sec ^2(e+f x)\right )^3 \sqrt {(\cos (e)-i \sin (e))^2}} \]
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Time = 1.66 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.99
method | result | size |
derivativedivides | \(\frac {-\frac {-\frac {b \left (8 a +5 b \right ) \sin \left (f x +e \right )^{3}}{8 a \left (a^{2}+2 a b +b^{2}\right )}+\frac {\left (8 a +3 b \right ) b \sin \left (f x +e \right )}{8 a^{2} \left (a +b \right )}}{\left (a \sin \left (f x +e \right )^{2}-a -b \right )^{2}}+\frac {\left (8 a^{2}+8 a b +3 b^{2}\right ) \operatorname {arctanh}\left (\frac {a \sin \left (f x +e \right )}{\sqrt {a \left (a +b \right )}}\right )}{8 \left (a^{2}+2 a b +b^{2}\right ) a^{2} \sqrt {a \left (a +b \right )}}}{f}\) | \(142\) |
default | \(\frac {-\frac {-\frac {b \left (8 a +5 b \right ) \sin \left (f x +e \right )^{3}}{8 a \left (a^{2}+2 a b +b^{2}\right )}+\frac {\left (8 a +3 b \right ) b \sin \left (f x +e \right )}{8 a^{2} \left (a +b \right )}}{\left (a \sin \left (f x +e \right )^{2}-a -b \right )^{2}}+\frac {\left (8 a^{2}+8 a b +3 b^{2}\right ) \operatorname {arctanh}\left (\frac {a \sin \left (f x +e \right )}{\sqrt {a \left (a +b \right )}}\right )}{8 \left (a^{2}+2 a b +b^{2}\right ) a^{2} \sqrt {a \left (a +b \right )}}}{f}\) | \(142\) |
risch | \(\frac {i b \left (8 a^{2} {\mathrm e}^{7 i \left (f x +e \right )}+5 a b \,{\mathrm e}^{7 i \left (f x +e \right )}+8 a^{2} {\mathrm e}^{5 i \left (f x +e \right )}+29 a b \,{\mathrm e}^{5 i \left (f x +e \right )}+12 b^{2} {\mathrm e}^{5 i \left (f x +e \right )}-8 a^{2} {\mathrm e}^{3 i \left (f x +e \right )}-29 a b \,{\mathrm e}^{3 i \left (f x +e \right )}-12 b^{2} {\mathrm e}^{3 i \left (f x +e \right )}-8 a^{2} {\mathrm e}^{i \left (f x +e \right )}-5 a b \,{\mathrm e}^{i \left (f x +e \right )}\right )}{4 a^{2} \left (a +b \right )^{2} f \left (a \,{\mathrm e}^{4 i \left (f x +e \right )}+2 a \,{\mathrm e}^{2 i \left (f x +e \right )}+4 b \,{\mathrm e}^{2 i \left (f x +e \right )}+a \right )^{2}}+\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \left (a +b \right ) {\mathrm e}^{i \left (f x +e \right )}}{\sqrt {a^{2}+a b}}-1\right )}{2 \sqrt {a^{2}+a b}\, \left (a +b \right )^{2} f}+\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \left (a +b \right ) {\mathrm e}^{i \left (f x +e \right )}}{\sqrt {a^{2}+a b}}-1\right ) b}{2 \sqrt {a^{2}+a b}\, \left (a +b \right )^{2} f a}+\frac {3 \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \left (a +b \right ) {\mathrm e}^{i \left (f x +e \right )}}{\sqrt {a^{2}+a b}}-1\right ) b^{2}}{16 \sqrt {a^{2}+a b}\, \left (a +b \right )^{2} f \,a^{2}}-\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \left (a +b \right ) {\mathrm e}^{i \left (f x +e \right )}}{\sqrt {a^{2}+a b}}-1\right )}{2 \sqrt {a^{2}+a b}\, \left (a +b \right )^{2} f}-\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \left (a +b \right ) {\mathrm e}^{i \left (f x +e \right )}}{\sqrt {a^{2}+a b}}-1\right ) b}{2 \sqrt {a^{2}+a b}\, \left (a +b \right )^{2} f a}-\frac {3 \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \left (a +b \right ) {\mathrm e}^{i \left (f x +e \right )}}{\sqrt {a^{2}+a b}}-1\right ) b^{2}}{16 \sqrt {a^{2}+a b}\, \left (a +b \right )^{2} f \,a^{2}}\) | \(543\) |
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Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (136) = 272\).
Time = 0.29 (sec) , antiderivative size = 613, normalized size of antiderivative = 4.26 \[ \int \frac {\sec (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\left [\frac {{\left ({\left (8 \, a^{4} + 8 \, a^{3} b + 3 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{4} + 8 \, a^{2} b^{2} + 8 \, a b^{3} + 3 \, b^{4} + 2 \, {\left (8 \, a^{3} b + 8 \, a^{2} b^{2} + 3 \, a b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {a^{2} + a b} \log \left (-\frac {a \cos \left (f x + e\right )^{2} - 2 \, \sqrt {a^{2} + a b} \sin \left (f x + e\right ) - 2 \, a - b}{a \cos \left (f x + e\right )^{2} + b}\right ) - 2 \, {\left (6 \, a^{3} b^{2} + 9 \, a^{2} b^{3} + 3 \, a b^{4} + {\left (8 \, a^{4} b + 13 \, a^{3} b^{2} + 5 \, a^{2} b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}{16 \, {\left ({\left (a^{8} + 3 \, a^{7} b + 3 \, a^{6} b^{2} + a^{5} b^{3}\right )} f \cos \left (f x + e\right )^{4} + 2 \, {\left (a^{7} b + 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} + a^{4} b^{4}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{6} b^{2} + 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} + a^{3} b^{5}\right )} f\right )}}, -\frac {{\left ({\left (8 \, a^{4} + 8 \, a^{3} b + 3 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{4} + 8 \, a^{2} b^{2} + 8 \, a b^{3} + 3 \, b^{4} + 2 \, {\left (8 \, a^{3} b + 8 \, a^{2} b^{2} + 3 \, a b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-a^{2} - a b} \arctan \left (\frac {\sqrt {-a^{2} - a b} \sin \left (f x + e\right )}{a + b}\right ) + {\left (6 \, a^{3} b^{2} + 9 \, a^{2} b^{3} + 3 \, a b^{4} + {\left (8 \, a^{4} b + 13 \, a^{3} b^{2} + 5 \, a^{2} b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}{8 \, {\left ({\left (a^{8} + 3 \, a^{7} b + 3 \, a^{6} b^{2} + a^{5} b^{3}\right )} f \cos \left (f x + e\right )^{4} + 2 \, {\left (a^{7} b + 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} + a^{4} b^{4}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{6} b^{2} + 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} + a^{3} b^{5}\right )} f\right )}}\right ] \]
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\[ \int \frac {\sec (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\int \frac {\sec {\left (e + f x \right )}}{\left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{3}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.62 \[ \int \frac {\sec (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=-\frac {\frac {{\left (8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right )} \log \left (\frac {a \sin \left (f x + e\right ) - \sqrt {{\left (a + b\right )} a}}{a \sin \left (f x + e\right ) + \sqrt {{\left (a + b\right )} a}}\right )}{{\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} \sqrt {{\left (a + b\right )} a}} - \frac {2 \, {\left ({\left (8 \, a^{2} b + 5 \, a b^{2}\right )} \sin \left (f x + e\right )^{3} - {\left (8 \, a^{2} b + 11 \, a b^{2} + 3 \, b^{3}\right )} \sin \left (f x + e\right )\right )}}{a^{6} + 4 \, a^{5} b + 6 \, a^{4} b^{2} + 4 \, a^{3} b^{3} + a^{2} b^{4} + {\left (a^{6} + 2 \, a^{5} b + a^{4} b^{2}\right )} \sin \left (f x + e\right )^{4} - 2 \, {\left (a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right )} \sin \left (f x + e\right )^{2}}}{16 \, f} \]
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Time = 0.34 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.24 \[ \int \frac {\sec (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=-\frac {\frac {{\left (8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right )} \arctan \left (\frac {a \sin \left (f x + e\right )}{\sqrt {-a^{2} - a b}}\right )}{{\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} \sqrt {-a^{2} - a b}} - \frac {8 \, a^{2} b \sin \left (f x + e\right )^{3} + 5 \, a b^{2} \sin \left (f x + e\right )^{3} - 8 \, a^{2} b \sin \left (f x + e\right ) - 11 \, a b^{2} \sin \left (f x + e\right ) - 3 \, b^{3} \sin \left (f x + e\right )}{{\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} {\left (a \sin \left (f x + e\right )^{2} - a - b\right )}^{2}}}{8 \, f} \]
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Time = 0.33 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.03 \[ \int \frac {\sec (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\frac {\frac {{\sin \left (e+f\,x\right )}^3\,\left (5\,b^2+8\,a\,b\right )}{8\,a\,{\left (a+b\right )}^2}-\frac {\sin \left (e+f\,x\right )\,\left (3\,b^2+8\,a\,b\right )}{8\,a^2\,\left (a+b\right )}}{f\,\left (2\,a\,b+a^2+b^2-{\sin \left (e+f\,x\right )}^2\,\left (2\,a^2+2\,b\,a\right )+a^2\,{\sin \left (e+f\,x\right )}^4\right )}+\frac {\mathrm {atanh}\left (\frac {\sqrt {a}\,\sin \left (e+f\,x\right )}{\sqrt {a+b}}\right )\,\left (8\,a^2+8\,a\,b+3\,b^2\right )}{8\,a^{5/2}\,f\,{\left (a+b\right )}^{5/2}} \]
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