Optimal. Leaf size=47 \[ \frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {b}}\right )}{a^{3/2} f}-\frac {\cos (e+f x)}{a f} \]
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Rubi [A] time = 0.04, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4133, 321, 205} \[ \frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {b}}\right )}{a^{3/2} f}-\frac {\cos (e+f x)}{a f} \]
Antiderivative was successfully verified.
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Rule 205
Rule 321
Rule 4133
Rubi steps
\begin {align*} \int \frac {\sin (e+f x)}{a+b \sec ^2(e+f x)} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {x^2}{b+a x^2} \, dx,x,\cos (e+f x)\right )}{f}\\ &=-\frac {\cos (e+f x)}{a f}+\frac {b \operatorname {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\cos (e+f x)\right )}{a f}\\ &=\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {b}}\right )}{a^{3/2} f}-\frac {\cos (e+f x)}{a f}\\ \end {align*}
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Mathematica [C] time = 0.55, size = 329, normalized size = 7.00 \[ \frac {(a \cos (2 (e+f x))+a+2 b) \left (-4 \sqrt {a} \sqrt {b} \cos (e+f x)-a \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {a+b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {b}}\right )-a \tan ^{-1}\left (\frac {\sqrt {a+b} \tan \left (\frac {1}{2} (e+f x)\right )+\sqrt {a}}{\sqrt {b}}\right )+(a+4 b) \tan ^{-1}\left (\frac {\sin (e) \tan \left (\frac {f x}{2}\right ) \left (-\sqrt {a}-i \sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2}\right )+\cos (e) \left (\sqrt {a}-\sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2} \tan \left (\frac {f x}{2}\right )\right )}{\sqrt {b}}\right )+(a+4 b) \tan ^{-1}\left (\frac {\sin (e) \tan \left (\frac {f x}{2}\right ) \left (-\sqrt {a}+i \sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2}\right )+\cos (e) \left (\sqrt {a}+\sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2} \tan \left (\frac {f x}{2}\right )\right )}{\sqrt {b}}\right )\right )}{8 a^{3/2} \sqrt {b} f \left (a \cos ^2(e+f x)+b\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 118, normalized size = 2.51 \[ \left [\frac {\sqrt {-\frac {b}{a}} \log \left (-\frac {a \cos \left (f x + e\right )^{2} + 2 \, a \sqrt {-\frac {b}{a}} \cos \left (f x + e\right ) - b}{a \cos \left (f x + e\right )^{2} + b}\right ) - 2 \, \cos \left (f x + e\right )}{2 \, a f}, \frac {\sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}} \cos \left (f x + e\right )}{b}\right ) - \cos \left (f x + e\right )}{a f}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.54, size = 44, normalized size = 0.94 \[ \frac {b \arctan \left (\frac {a \cos \left (f x + e\right )}{\sqrt {a b}}\right )}{\sqrt {a b} a f} - \frac {\cos \left (f x + e\right )}{a f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.40, size = 46, normalized size = 0.98 \[ -\frac {b \arctan \left (\frac {\sec \left (f x +e \right ) b}{\sqrt {a b}}\right )}{f a \sqrt {a b}}-\frac {1}{f a \sec \left (f x +e \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 40, normalized size = 0.85 \[ \frac {\frac {b \arctan \left (\frac {a \cos \left (f x + e\right )}{\sqrt {a b}}\right )}{\sqrt {a b} a} - \frac {\cos \left (f x + e\right )}{a}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 39, normalized size = 0.83 \[ \frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\cos \left (e+f\,x\right )}{\sqrt {b}}\right )}{a^{3/2}\,f}-\frac {\cos \left (e+f\,x\right )}{a\,f} \]
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 \[ \int \frac {\sin {\left (e + f x \right )}}{a + b \sec ^{2}{\left (e + f x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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