Optimal. Leaf size=144 \[ -\frac {3 b (2 a+b) \sin (e+f x)}{8 a^2 f (a+b)^2 \left (-a \sin ^2(e+f x)+a+b\right )}+\frac {\left (8 a^2+8 a b+3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )}{8 a^{5/2} f (a+b)^{5/2}}-\frac {b \sin (e+f x) \cos ^2(e+f x)}{4 a f (a+b) \left (-a \sin ^2(e+f x)+a+b\right )^2} \]
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Rubi [A] time = 0.13, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {4147, 413, 385, 208} \[ \frac {\left (8 a^2+8 a b+3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )}{8 a^{5/2} f (a+b)^{5/2}}-\frac {3 b (2 a+b) \sin (e+f x)}{8 a^2 f (a+b)^2 \left (-a \sin ^2(e+f x)+a+b\right )}-\frac {b \sin (e+f x) \cos ^2(e+f x)}{4 a f (a+b) \left (-a \sin ^2(e+f x)+a+b\right )^2} \]
Antiderivative was successfully verified.
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Rule 208
Rule 385
Rule 413
Rule 4147
Rubi steps
\begin {align*} \int \frac {\sec (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx &=\frac {\operatorname {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 {\operatorname {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 ) \operatorname {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 ) \tanh ^{-1}\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*}
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Mathematica [C] time = 6.82, size = 927, normalized size = 6.44 \[ \frac {(\cos (2 (e+f x)) a+a+2 b) \sec ^5(e+f x) \left (32 \sqrt {a} (a+b)^{3/2} \sqrt {(\cos (e)-i \sin (e))^2} \tan (e+f x) b^2-8 \sqrt {a} \sqrt {a+b} (8 a+5 b) (\cos (2 (e+f x)) a+a+2 b) \sqrt {(\cos (e)-i \sin (e))^2} \tan (e+f x) b-2 i \left (8 a^2+8 b a+3 b^2\right ) \tan ^{-1}\left (\frac {(a+b) \sin (e)}{(a+b) \cos (e)-\sqrt {a} \sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2} (\cos (2 e)+i \sin (2 e)) \sin (e+f x)}\right ) (\cos (2 (e+f x)) a+a+2 b)^2 \sec (e+f x) (\cos (e)-i \sin (e))+\left (8 a^2+8 b a+3 b^2\right ) (\cos (2 (e+f x)) a+a+2 b)^2 \log \left (-\cos (2 (e+f x)) a-2 i \sin (2 e) a+a+2 \sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2} \sin (f x) \sqrt {a}+2 \sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2} \sin (2 e+f x) \sqrt {a}+2 (a+b) \cos (2 e)-2 i b \sin (2 e)\right ) \sec (e+f x) (\cos (e)-i \sin (e))-\left (8 a^2+8 b a+3 b^2\right ) (\cos (2 (e+f x)) a+a+2 b)^2 \log \left (\cos (2 (e+f x)) a+2 i \sin (2 e) a-a+2 \sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2} \sin (f x) \sqrt {a}+2 \sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2} \sin (2 e+f x) \sqrt {a}-2 (a+b) \cos (2 e)+2 i b \sin (2 e)\right ) \sec (e+f x) (\cos (e)-i \sin (e))+2 \left (8 a^2+8 b a+3 b^2\right ) \tan ^{-1}\left (\frac {2 \sin (e) \left (\sin (2 e) a+i a-i \sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2} \sin (f x) \sqrt {a}-i \sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2} \sin (2 e+f x) \sqrt {a}+\sqrt {a+b} \cos (f x) \sqrt {(\cos (e)-i \sin (e))^2} \sqrt {a}-\sqrt {a+b} \cos (2 e+f x) \sqrt {(\cos (e)-i \sin (e))^2} \sqrt {a}+i b+i (a+b) \cos (2 e)+b \sin (2 e)\right )}{i (a+3 b) \cos (e)+i (a+b) \cos (3 e)+i a \cos (e+2 f x)+i a \cos (3 e+2 f x)+3 a \sin (e)+b \sin (e)+a \sin (3 e)+b \sin (3 e)+a \sin (e+2 f x)-a \sin (3 e+2 f x)}\right ) (\cos (2 (e+f x)) a+a+2 b)^2 \sec (e+f x) (i \cos (e)+\sin (e))\right )}{256 a^{5/2} (a+b)^{5/2} f \left (b \sec ^2(e+f x)+a\right )^3 \sqrt {(\cos (e)-i \sin (e))^2}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 1.20, size = 613, normalized size = 4.26 \[ \left [\frac {{\left ({\left (8 \, a^{4} + 8 \, a^{3} b + 3 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{4} + 8 \, a^{2} b^{2} + 8 \, a b^{3} + 3 \, b^{4} + 2 \, {\left (8 \, a^{3} b + 8 \, a^{2} b^{2} + 3 \, a b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {a^{2} + a b} \log \left (-\frac {a \cos \left (f x + e\right )^{2} - 2 \, \sqrt {a^{2} + a b} \sin \left (f x + e\right ) - 2 \, a - b}{a \cos \left (f x + e\right )^{2} + b}\right ) - 2 \, {\left (6 \, a^{3} b^{2} + 9 \, a^{2} b^{3} + 3 \, a b^{4} + {\left (8 \, a^{4} b + 13 \, a^{3} b^{2} + 5 \, a^{2} b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}{16 \, {\left ({\left (a^{8} + 3 \, a^{7} b + 3 \, a^{6} b^{2} + a^{5} b^{3}\right )} f \cos \left (f x + e\right )^{4} + 2 \, {\left (a^{7} b + 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} + a^{4} b^{4}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{6} b^{2} + 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} + a^{3} b^{5}\right )} f\right )}}, -\frac {{\left ({\left (8 \, a^{4} + 8 \, a^{3} b + 3 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{4} + 8 \, a^{2} b^{2} + 8 \, a b^{3} + 3 \, b^{4} + 2 \, {\left (8 \, a^{3} b + 8 \, a^{2} b^{2} + 3 \, a b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-a^{2} - a b} \arctan \left (\frac {\sqrt {-a^{2} - a b} \sin \left (f x + e\right )}{a + b}\right ) + {\left (6 \, a^{3} b^{2} + 9 \, a^{2} b^{3} + 3 \, a b^{4} + {\left (8 \, a^{4} b + 13 \, a^{3} b^{2} + 5 \, a^{2} b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}{8 \, {\left ({\left (a^{8} + 3 \, a^{7} b + 3 \, a^{6} b^{2} + a^{5} b^{3}\right )} f \cos \left (f x + e\right )^{4} + 2 \, {\left (a^{7} b + 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} + a^{4} b^{4}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{6} b^{2} + 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} + a^{3} b^{5}\right )} f\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.10, size = 185, normalized size = 1.28 \[ -\frac {\frac {{\left (8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right )} \arctan \left (\frac {a \sin \left (f x + e\right )}{\sqrt {-a^{2} - a b}}\right )}{{\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} \sqrt {-a^{2} - a b}} - \frac {8 \, a^{2} b \sin \left (f x + e\right )^{3} + 5 \, a b^{2} \sin \left (f x + e\right )^{3} - 8 \, a^{2} b \sin \left (f x + e\right ) - 11 \, a b^{2} \sin \left (f x + e\right ) - 3 \, b^{3} \sin \left (f x + e\right )}{{\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} {\left (a \sin \left (f x + e\right )^{2} - a - b\right )}^{2}}}{8 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.96, size = 142, normalized size = 0.99 \[ \frac {-\frac {-\frac {b \left (8 a +5 b \right ) \left (\sin ^{3}\left (f x +e \right )\right )}{8 a \left (a^{2}+2 a b +b^{2}\right )}+\frac {\left (8 a +3 b \right ) b \sin \left (f x +e \right )}{8 a^{2} \left (a +b \right )}}{\left (-a -b +a \left (\sin ^{2}\left (f x +e \right )\right )\right )^{2}}+\frac {\left (8 a^{2}+8 a b +3 b^{2}\right ) \arctanh \left (\frac {a \sin \left (f x +e \right )}{\sqrt {\left (a +b \right ) a}}\right )}{8 \left (a^{2}+2 a b +b^{2}\right ) a^{2} \sqrt {\left (a +b \right ) a}}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 233, normalized size = 1.62 \[ -\frac {\frac {{\left (8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right )} \log \left (\frac {a \sin \left (f x + e\right ) - \sqrt {{\left (a + b\right )} a}}{a \sin \left (f x + e\right ) + \sqrt {{\left (a + b\right )} a}}\right )}{{\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} \sqrt {{\left (a + b\right )} a}} - \frac {2 \, {\left ({\left (8 \, a^{2} b + 5 \, a b^{2}\right )} \sin \left (f x + e\right )^{3} - {\left (8 \, a^{2} b + 11 \, a b^{2} + 3 \, b^{3}\right )} \sin \left (f x + e\right )\right )}}{a^{6} + 4 \, a^{5} b + 6 \, a^{4} b^{2} + 4 \, a^{3} b^{3} + a^{2} b^{4} + {\left (a^{6} + 2 \, a^{5} b + a^{4} b^{2}\right )} \sin \left (f x + e\right )^{4} - 2 \, {\left (a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right )} \sin \left (f x + e\right )^{2}}}{16 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.25, size = 149, normalized size = 1.03 \[ \frac {\frac {{\sin \left (e+f\,x\right )}^3\,\left (5\,b^2+8\,a\,b\right )}{8\,a\,{\left (a+b\right )}^2}-\frac {\sin \left (e+f\,x\right )\,\left (3\,b^2+8\,a\,b\right )}{8\,a^2\,\left (a+b\right )}}{f\,\left (2\,a\,b+a^2+b^2-{\sin \left (e+f\,x\right )}^2\,\left (2\,a^2+2\,b\,a\right )+a^2\,{\sin \left (e+f\,x\right )}^4\right )}+\frac {\mathrm {atanh}\left (\frac {\sqrt {a}\,\sin \left (e+f\,x\right )}{\sqrt {a+b}}\right )\,\left (8\,a^2+8\,a\,b+3\,b^2\right )}{8\,a^{5/2}\,f\,{\left (a+b\right )}^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec {\left (e + f x \right )}}{\left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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