Optimal. Leaf size=67 \[ -\frac {(c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}+\frac {(c+d x)^2}{2 a d}-\frac {2 d \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{a f^2} \]
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Rubi [A] time = 0.10, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4191, 3318, 4184, 3475} \[ -\frac {(c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}+\frac {(c+d x)^2}{2 a d}-\frac {2 d \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{a f^2} \]
Antiderivative was successfully verified.
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Rule 3318
Rule 3475
Rule 4184
Rule 4191
Rubi steps
\begin {align*} \int \frac {c+d x}{a+a \sec (e+f x)} \, dx &=\int \left (\frac {c+d x}{a}-\frac {c+d x}{a+a \cos (e+f x)}\right ) \, dx\\ &=\frac {(c+d x)^2}{2 a d}-\int \frac {c+d x}{a+a \cos (e+f x)} \, dx\\ &=\frac {(c+d x)^2}{2 a d}-\frac {\int (c+d x) \csc ^2\left (\frac {e+\pi }{2}+\frac {f x}{2}\right ) \, dx}{2 a}\\ &=\frac {(c+d x)^2}{2 a d}-\frac {(c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}+\frac {d \int \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{a f}\\ &=\frac {(c+d x)^2}{2 a d}-\frac {2 d \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{a f^2}-\frac {(c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}\\ \end {align*}
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Mathematica [A] time = 0.75, size = 104, normalized size = 1.55 \[ \frac {\cos \left (\frac {1}{2} (e+f x)\right ) \sec (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right ) \left (f^2 x (2 c+d x)-2 d f x \tan \left (\frac {e}{2}\right )-4 d \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )\right )-2 f \sec \left (\frac {e}{2}\right ) (c+d x) \sin \left (\frac {f x}{2}\right )\right )}{a f^2 (\sec (e+f x)+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 99, normalized size = 1.48 \[ \frac {d f^{2} x^{2} + 2 \, c f^{2} x + {\left (d f^{2} x^{2} + 2 \, c f^{2} x\right )} \cos \left (f x + e\right ) - 2 \, {\left (d \cos \left (f x + e\right ) + d\right )} \log \left (\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) - 2 \, {\left (d f x + c f\right )} \sin \left (f x + e\right )}{2 \, {\left (a f^{2} \cos \left (f x + e\right ) + a f^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.03, size = 290, normalized size = 4.33 \[ \frac {d f^{2} x^{2} \tan \left (\frac {1}{2} \, f x\right ) \tan \left (\frac {1}{2} \, e\right ) + 2 \, c f^{2} x \tan \left (\frac {1}{2} \, f x\right ) \tan \left (\frac {1}{2} \, e\right ) - d f^{2} x^{2} - 2 \, c f^{2} x + 2 \, d f x \tan \left (\frac {1}{2} \, f x\right ) + 2 \, d f x \tan \left (\frac {1}{2} \, e\right ) - 2 \, d \log \left (\frac {4 \, {\left (\tan \left (\frac {1}{2} \, f x\right )^{4} \tan \left (\frac {1}{2} \, e\right )^{2} - 2 \, \tan \left (\frac {1}{2} \, f x\right )^{3} \tan \left (\frac {1}{2} \, e\right ) + \tan \left (\frac {1}{2} \, f x\right )^{2} \tan \left (\frac {1}{2} \, e\right )^{2} + \tan \left (\frac {1}{2} \, f x\right )^{2} - 2 \, \tan \left (\frac {1}{2} \, f x\right ) \tan \left (\frac {1}{2} \, e\right ) + 1\right )}}{\tan \left (\frac {1}{2} \, e\right )^{2} + 1}\right ) \tan \left (\frac {1}{2} \, f x\right ) \tan \left (\frac {1}{2} \, e\right ) + 2 \, c f \tan \left (\frac {1}{2} \, f x\right ) + 2 \, c f \tan \left (\frac {1}{2} \, e\right ) + 2 \, d \log \left (\frac {4 \, {\left (\tan \left (\frac {1}{2} \, f x\right )^{4} \tan \left (\frac {1}{2} \, e\right )^{2} - 2 \, \tan \left (\frac {1}{2} \, f x\right )^{3} \tan \left (\frac {1}{2} \, e\right ) + \tan \left (\frac {1}{2} \, f x\right )^{2} \tan \left (\frac {1}{2} \, e\right )^{2} + \tan \left (\frac {1}{2} \, f x\right )^{2} - 2 \, \tan \left (\frac {1}{2} \, f x\right ) \tan \left (\frac {1}{2} \, e\right ) + 1\right )}}{\tan \left (\frac {1}{2} \, e\right )^{2} + 1}\right )}{2 \, {\left (a f^{2} \tan \left (\frac {1}{2} \, f x\right ) \tan \left (\frac {1}{2} \, e\right ) - a f^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.53, size = 76, normalized size = 1.13 \[ \frac {c x}{a}+\frac {d \,x^{2}}{2 a}-\frac {c \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}-\frac {x d \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{f a}+\frac {d \ln \left (1+\tan ^{2}\left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{a \,f^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.04, size = 273, normalized size = 4.07 \[ -\frac {2 \, d e {\left (\frac {2 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a f} - \frac {\sin \left (f x + e\right )}{a f {\left (\cos \left (f x + e\right ) + 1\right )}}\right )} - 2 \, c {\left (\frac {2 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a} - \frac {\sin \left (f x + e\right )}{a {\left (\cos \left (f x + e\right ) + 1\right )}}\right )} - \frac {{\left ({\left (f x + e\right )}^{2} \cos \left (f x + e\right )^{2} + {\left (f x + e\right )}^{2} \sin \left (f x + e\right )^{2} + 2 \, {\left (f x + e\right )}^{2} \cos \left (f x + e\right ) + {\left (f x + e\right )}^{2} - 2 \, {\left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1\right )} \log \left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1\right ) - 4 \, {\left (f x + e\right )} \sin \left (f x + e\right )\right )} d}{a f \cos \left (f x + e\right )^{2} + a f \sin \left (f x + e\right )^{2} + 2 \, a f \cos \left (f x + e\right ) + a f}}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.76, size = 79, normalized size = 1.18 \[ \frac {d\,x^2}{2\,a}-\frac {2\,d\,\ln \left ({\mathrm {e}}^{e\,1{}\mathrm {i}}\,{\mathrm {e}}^{f\,x\,1{}\mathrm {i}}+1\right )}{a\,f^2}-\frac {\left (c+d\,x\right )\,2{}\mathrm {i}}{a\,f\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+1\right )}+\frac {x\,\left (c\,f+d\,2{}\mathrm {i}\right )}{a\,f} \]
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 \[ \frac {\int \frac {c}{\sec {\left (e + f x \right )} + 1}\, dx + \int \frac {d x}{\sec {\left (e + f x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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