Optimal. Leaf size=43 \[ \frac {(c-d) \tan (e+f x)}{f (a \sec (e+f x)+a)}+\frac {d \tanh ^{-1}(\sin (e+f x))}{a f} \]
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Rubi [A] time = 0.08, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {3998, 3770, 3794} \[ \frac {(c-d) \tan (e+f x)}{f (a \sec (e+f x)+a)}+\frac {d \tanh ^{-1}(\sin (e+f x))}{a f} \]
Antiderivative was successfully verified.
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Rule 3770
Rule 3794
Rule 3998
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (c+d \sec (e+f x))}{a+a \sec (e+f x)} \, dx &=(c-d) \int \frac {\sec (e+f x)}{a+a \sec (e+f x)} \, dx+\frac {d \int \sec (e+f x) \, dx}{a}\\ &=\frac {d \tanh ^{-1}(\sin (e+f x))}{a f}+\frac {(c-d) \tan (e+f x)}{f (a+a \sec (e+f x))}\\ \end {align*}
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Mathematica [B] time = 0.27, size = 109, normalized size = 2.53 \[ \frac {2 \cos \left (\frac {1}{2} (e+f x)\right ) \left ((c-d) \sec \left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right )+d \cos \left (\frac {1}{2} (e+f x)\right ) \left (\log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )\right )\right )}{a f (\cos (e+f x)+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 74, normalized size = 1.72 \[ \frac {{\left (d \cos \left (f x + e\right ) + d\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) - {\left (d \cos \left (f x + e\right ) + d\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left (c - d\right )} \sin \left (f x + e\right )}{2 \, {\left (a f \cos \left (f x + e\right ) + a f\right )}} \]
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 = 0.74, size = 78, normalized size = 1.81 \[ -\frac {d \ln \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-1\right )}{a f}+\frac {\tan \left (\frac {e}{2}+\frac {f x}{2}\right ) c}{a f}-\frac {\tan \left (\frac {e}{2}+\frac {f x}{2}\right ) d}{a f}+\frac {d \ln \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )+1\right )}{a f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.38, size = 99, normalized size = 2.30 \[ \frac {d {\left (\frac {\log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a} - \frac {\log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a} - \frac {\sin \left (f x + e\right )}{a {\left (\cos \left (f x + e\right ) + 1\right )}}\right )} + \frac {c \sin \left (f x + e\right )}{a {\left (\cos \left (f x + e\right ) + 1\right )}}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.73, size = 41, normalized size = 0.95 \[ \frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (c-d\right )}{a\,f}+\frac {2\,d\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{a\,f} \]
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 \[ \frac {\int \frac {c \sec {\left (e + f x \right )}}{\sec {\left (e + f x \right )} + 1}\, dx + \int \frac {d \sec ^{2}{\left (e + f x \right )}}{\sec {\left (e + f x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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