Optimal. Leaf size=52 \[ \frac {\sin (e+f x)}{a f}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )}{a^{3/2} f \sqrt {a+b}} \]
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Rubi [A] time = 0.06, antiderivative size = 52, 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 = {4147, 388, 208} \[ \frac {\sin (e+f x)}{a f}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )}{a^{3/2} f \sqrt {a+b}} \]
Antiderivative was successfully verified.
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Rule 208
Rule 388
Rule 4147
Rubi steps
\begin {align*} \int \frac {\cos (e+f x)}{a+b \sec ^2(e+f x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1-x^2}{a+b-a x^2} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac {\sin (e+f x)}{a f}-\frac {b \operatorname {Subst}\left (\int \frac {1}{a+b-a x^2} \, dx,x,\sin (e+f x)\right )}{a f}\\ &=-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )}{a^{3/2} \sqrt {a+b} f}+\frac {\sin (e+f x)}{a f}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 52, normalized size = 1.00 \[ \frac {\sqrt {a} \sin (e+f x)-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )}{\sqrt {a+b}}}{a^{3/2} f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 164, normalized size = 3.15 \[ \left [\frac {\sqrt {a^{2} + a b} b \log \left (-\frac {a \cos \left (f x + e\right )^{2} + 2 \, \sqrt {a^{2} + a b} \sin \left (f x + e\right ) - 2 \, a - b}{a \cos \left (f x + e\right )^{2} + b}\right ) + 2 \, {\left (a^{2} + a b\right )} \sin \left (f x + e\right )}{2 \, {\left (a^{3} + a^{2} b\right )} f}, \frac {\sqrt {-a^{2} - a b} b \arctan \left (\frac {\sqrt {-a^{2} - a b} \sin \left (f x + e\right )}{a + b}\right ) + {\left (a^{2} + a b\right )} \sin \left (f x + e\right )}{{\left (a^{3} + a^{2} b\right )} f}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 55, normalized size = 1.06 \[ \frac {\frac {b \arctan \left (\frac {a \sin \left (f x + e\right )}{\sqrt {-a^{2} - a b}}\right )}{\sqrt {-a^{2} - a b} a} + \frac {\sin \left (f x + e\right )}{a}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.40, size = 45, normalized size = 0.87 \[ \frac {\frac {\sin \left (f x +e \right )}{a}-\frac {b \arctanh \left (\frac {a \sin \left (f x +e \right )}{\sqrt {\left (a +b \right ) a}}\right )}{a \sqrt {\left (a +b \right ) a}}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 67, normalized size = 1.29 \[ \frac {\frac {b \log \left (\frac {a \sin \left (f x + e\right ) - \sqrt {{\left (a + b\right )} a}}{a \sin \left (f x + e\right ) + \sqrt {{\left (a + b\right )} a}}\right )}{\sqrt {{\left (a + b\right )} a} a} + \frac {2 \, \sin \left (f x + e\right )}{a}}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.40, size = 44, normalized size = 0.85 \[ \frac {\sin \left (e+f\,x\right )}{a\,f}-\frac {b\,\mathrm {atanh}\left (\frac {\sqrt {a}\,\sin \left (e+f\,x\right )}{\sqrt {a+b}}\right )}{a^{3/2}\,f\,\sqrt {a+b}} \]
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 \[ \int \frac {\cos {\left (e + f x \right )}}{a + b \sec ^{2}{\left (e + f x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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