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 = 0.07, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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 191
Rule 4146
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*}
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Mathematica [A] time = 0.63, 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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fricas [B] time = 0.55, size = 65, normalized size = 2.03 \[ \frac {\sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right ) \sin \left (f x + e\right )}{{\left (a^{2} + a b\right )} f \cos \left (f x + e\right )^{2} + {\left (a b + b^{2}\right )} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.45, size = 59, normalized size = 1.84 \[ \frac {\left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right ) \sin \left (f x +e \right )}{f \cos \left (f x +e \right )^{3} \left (\frac {b +a \left (\cos ^{2}\left (f x +e \right )\right )}{\cos \left (f x +e \right )^{2}}\right )^{\frac {3}{2}} \left (a +b \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 30, normalized size = 0.94 \[ \frac {\tan \left (f x + e\right )}{\sqrt {b \tan \left (f x + e\right )^{2} + a + b} {\left (a + b\right )} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.29, size = 199, normalized size = 6.22 \[ \frac {\sqrt {\frac {a+2\,b+a\,\cos \left (2\,e+2\,f\,x\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (5\,a\,\sin \left (2\,e+2\,f\,x\right )+4\,a\,\sin \left (4\,e+4\,f\,x\right )+a\,\sin \left (6\,e+6\,f\,x\right )+8\,b\,\sin \left (2\,e+2\,f\,x\right )+4\,b\,\sin \left (4\,e+4\,f\,x\right )\right )}{f\,\left (a+b\right )\,\left (24\,a\,b+10\,a^2+16\,b^2+15\,a^2\,\cos \left (2\,e+2\,f\,x\right )+6\,a^2\,\cos \left (4\,e+4\,f\,x\right )+a^2\,\cos \left (6\,e+6\,f\,x\right )+16\,b^2\,\cos \left (2\,e+2\,f\,x\right )+32\,a\,b\,\cos \left (2\,e+2\,f\,x\right )+8\,a\,b\,\cos \left (4\,e+4\,f\,x\right )\right )} \]
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 \[ \int \frac {\sec ^{2}{\left (e + f x \right )}}{\left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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