Optimal. Leaf size=73 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{2 \sqrt {b} f (a+b)^{3/2}}+\frac {\tan (e+f x)}{2 f (a+b) \left (a+b \tan ^2(e+f x)+b\right )} \]
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Rubi [A] time = 0.07, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4146, 199, 205} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{2 \sqrt {b} f (a+b)^{3/2}}+\frac {\tan (e+f x)}{2 f (a+b) \left (a+b \tan ^2(e+f x)+b\right )} \]
Antiderivative was successfully verified.
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Rule 199
Rule 205
Rule 4146
Rubi steps
\begin {align*} \int \frac {\sec ^2(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\left (a+b+b x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\tan (e+f x)}{2 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {1}{a+b+b x^2} \, dx,x,\tan (e+f x)\right )}{2 (a+b) f}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{2 \sqrt {b} (a+b)^{3/2} f}+\frac {\tan (e+f x)}{2 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )}\\ \end {align*}
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Mathematica [C] time = 0.89, size = 211, normalized size = 2.89 \[ \frac {\sec ^4(e+f x) (a \cos (2 (e+f x))+a+2 b) \left (\frac {a \sin (2 f x)-(a+2 b) \sin (2 e)}{a (\cos (e)-\sin (e)) (\sin (e)+\cos (e))}-\frac {(\cos (2 e)-i \sin (2 e)) (a \cos (2 (e+f x))+a+2 b) \tan ^{-1}\left (\frac {(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt {a+b} \sqrt {b (\cos (e)-i \sin (e))^4}}\right )}{\sqrt {a+b} \sqrt {b (\cos (e)-i \sin (e))^4}}\right )}{8 f (a+b) \left (a+b \sec ^2(e+f x)\right )^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.57, size = 368, normalized size = 5.04 \[ \left [\frac {4 \, {\left (a b + b^{2}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - {\left (a \cos \left (f x + e\right )^{2} + b\right )} \sqrt {-a b - b^{2}} \log \left (\frac {{\left (a^{2} + 8 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (3 \, a b + 4 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 4 \, {\left ({\left (a + 2 \, b\right )} \cos \left (f x + e\right )^{3} - b \cos \left (f x + e\right )\right )} \sqrt {-a b - b^{2}} \sin \left (f x + e\right ) + b^{2}}{a^{2} \cos \left (f x + e\right )^{4} + 2 \, a b \cos \left (f x + e\right )^{2} + b^{2}}\right )}{8 \, {\left ({\left (a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right )} f\right )}}, \frac {2 \, {\left (a b + b^{2}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - {\left (a \cos \left (f x + e\right )^{2} + b\right )} \sqrt {a b + b^{2}} \arctan \left (\frac {{\left (a + 2 \, b\right )} \cos \left (f x + e\right )^{2} - b}{2 \, \sqrt {a b + b^{2}} \cos \left (f x + e\right ) \sin \left (f x + e\right )}\right )}{4 \, {\left ({\left (a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right )} f\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.67, size = 87, normalized size = 1.19 \[ \frac {\frac {\pi \left \lfloor \frac {f x + e}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (b) + \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b + b^{2}}}\right )}{\sqrt {a b + b^{2}} {\left (a + b\right )}} + \frac {\tan \left (f x + e\right )}{{\left (b \tan \left (f x + e\right )^{2} + a + b\right )} {\left (a + b\right )}}}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.57, size = 66, normalized size = 0.90 \[ \frac {\tan \left (f x +e \right )}{2 \left (a +b \right ) f \left (a +b +b \left (\tan ^{2}\left (f x +e \right )\right )\right )}+\frac {\arctan \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {\left (a +b \right ) b}}\right )}{2 f \left (a +b \right ) \sqrt {\left (a +b \right ) b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 71, normalized size = 0.97 \[ \frac {\frac {\tan \left (f x + e\right )}{{\left (a b + b^{2}\right )} \tan \left (f x + e\right )^{2} + a^{2} + 2 \, a b + b^{2}} + \frac {\arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {{\left (a + b\right )} b}}\right )}{\sqrt {{\left (a + b\right )} b} {\left (a + b\right )}}}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.47, size = 69, normalized size = 0.95 \[ \frac {\mathrm {tan}\left (e+f\,x\right )}{2\,f\,\left (a+b\right )\,\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a+b\right )}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\mathrm {tan}\left (e+f\,x\right )\,\left (2\,a+2\,b\right )}{2\,{\left (a+b\right )}^{3/2}}\right )}{2\,\sqrt {b}\,f\,{\left (a+b\right )}^{3/2}} \]
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 \[ \int \frac {\sec ^{2}{\left (e + f x \right )}}{\left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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