Optimal. Leaf size=31 \[ \frac {1}{2} x (a+2 b)+\frac {a \sin (e+f x) \cos (e+f x)}{2 f} \]
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Rubi [A] time = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {4045, 8} \[ \frac {1}{2} x (a+2 b)+\frac {a \sin (e+f x) \cos (e+f x)}{2 f} \]
Antiderivative was successfully verified.
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Rule 8
Rule 4045
Rubi steps
\begin {align*} \int \cos ^2(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx &=\frac {a \cos (e+f x) \sin (e+f x)}{2 f}+\frac {1}{2} (a+2 b) \int 1 \, dx\\ &=\frac {1}{2} (a+2 b) x+\frac {a \cos (e+f x) \sin (e+f x)}{2 f}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 33, normalized size = 1.06 \[ \frac {a (e+f x)}{2 f}+\frac {a \sin (2 (e+f x))}{4 f}+b x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 28, normalized size = 0.90 \[ \frac {{\left (a + 2 \, b\right )} f x + a \cos \left (f x + e\right ) \sin \left (f x + e\right )}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 40, normalized size = 1.29 \[ \frac {{\left (f x + e\right )} {\left (a + 2 \, b\right )} + \frac {a \tan \left (f x + e\right )}{\tan \left (f x + e\right )^{2} + 1}}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.69, size = 37, normalized size = 1.19 \[ \frac {a \left (\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+\left (f x +e \right ) b}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 37, normalized size = 1.19 \[ \frac {{\left (f x + e\right )} {\left (a + 2 \, b\right )} + \frac {a \tan \left (f x + e\right )}{\tan \left (f x + e\right )^{2} + 1}}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.31, size = 25, normalized size = 0.81 \[ \frac {\frac {a\,\sin \left (2\,e+2\,f\,x\right )}{4}+f\,x\,\left (\frac {a}{2}+b\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.79, size = 51, normalized size = 1.65 \[ a \left (\begin {cases} \frac {x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {x \cos ^{2}{\left (e + f x \right )}}{2} + \frac {\sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} & \text {for}\: f \neq 0 \\x \cos ^{2}{\relax (e )} & \text {otherwise} \end {cases}\right ) + b x \]
Verification of antiderivative is not currently implemented for this CAS.
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