Optimal. Leaf size=32 \[ \frac{\tan (e+f x)}{f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}} \]
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Rubi [A] time = 0.0745116, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {4146, 191} \[ \frac{\tan (e+f x)}{f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}} \]
Antiderivative was successfully verified.
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Rule 4146
Rule 191
Rubi steps
\begin{align*} \int \frac{\sec ^2(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (a+b+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\tan (e+f x)}{(a+b) f \sqrt{a+b+b \tan ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.656438, size = 57, normalized size = 1.78 \[ \frac{\tan (e+f x) \sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b)}{2 f (a+b) \left (a+b \sec ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.23, size = 59, normalized size = 1.8 \begin{align*}{\frac{ \left ( b+a \left ( \cos \left ( fx+e \right ) \right ) ^{2} \right ) \sin \left ( fx+e \right ) }{f \left ( a+b \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{3}} \left ({\frac{b+a \left ( \cos \left ( fx+e \right ) \right ) ^{2}}{ \left ( \cos \left ( fx+e \right ) \right ) ^{2}}} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07825, size = 41, normalized size = 1.28 \begin{align*} \frac{\tan \left (f x + e\right )}{\sqrt{b \tan \left (f x + e\right )^{2} + a + b}{\left (a + b\right )} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.560278, size = 159, normalized size = 4.97 \begin{align*} \frac{\sqrt{\frac{a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right ) \sin \left (f x + e\right )}{{\left (a^{2} + a b\right )} f \cos \left (f x + e\right )^{2} +{\left (a b + b^{2}\right )} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec ^{2}{\left (e + f x \right )}}{\left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.94885, size = 157, normalized size = 4.91 \begin{align*} -\frac{2 \, a^{2} b^{2} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 1\right ) \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )}{{\left (a^{3} b^{2} + a^{2} b^{3}\right )} \sqrt{a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} + b \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 2 \, a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 2 \, b \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + a + b} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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