Optimal. Leaf size=140 \[ -\frac {5 (c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {(c+d x)^2}{2 a^2 d}-\frac {d \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f^2}-\frac {10 d \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{3 a^2 f^2} \]
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Rubi [A] time = 0.20, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {4191, 3318, 4185, 4184, 3475} \[ -\frac {5 (c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {(c+d x)^2}{2 a^2 d}-\frac {d \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f^2}-\frac {10 d \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{3 a^2 f^2} \]
Antiderivative was successfully verified.
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Rule 3318
Rule 3475
Rule 4184
Rule 4185
Rule 4191
Rubi steps
\begin {align*} \int \frac {c+d x}{(a+a \sec (e+f x))^2} \, dx &=\int \left (\frac {c+d x}{a^2}+\frac {c+d x}{a^2 (1+\cos (e+f x))^2}-\frac {2 (c+d x)}{a^2 (1+\cos (e+f x))}\right ) \, dx\\ &=\frac {(c+d x)^2}{2 a^2 d}+\frac {\int \frac {c+d x}{(1+\cos (e+f x))^2} \, dx}{a^2}-\frac {2 \int \frac {c+d x}{1+\cos (e+f x)} \, dx}{a^2}\\ &=\frac {(c+d x)^2}{2 a^2 d}+\frac {\int (c+d x) \csc ^4\left (\frac {e+\pi }{2}+\frac {f x}{2}\right ) \, dx}{4 a^2}-\frac {\int (c+d x) \csc ^2\left (\frac {e+\pi }{2}+\frac {f x}{2}\right ) \, dx}{a^2}\\ &=\frac {(c+d x)^2}{2 a^2 d}-\frac {d \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f^2}-\frac {2 (c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a^2 f}+\frac {(c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {\int (c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{6 a^2}+\frac {(2 d) \int \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{a^2 f}\\ &=\frac {(c+d x)^2}{2 a^2 d}-\frac {4 d \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{a^2 f^2}-\frac {d \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f^2}-\frac {5 (c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}-\frac {d \int \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{3 a^2 f}\\ &=\frac {(c+d x)^2}{2 a^2 d}-\frac {10 d \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{3 a^2 f^2}-\frac {d \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f^2}-\frac {5 (c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}\\ \end {align*}
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Mathematica [A] time = 1.65, size = 172, normalized size = 1.23 \[ \frac {2 \cos \left (\frac {1}{2} (e+f x)\right ) \sec ^2(e+f x) \left (\cos ^3\left (\frac {1}{2} (e+f x)\right ) \left (3 f^2 x (2 c+d x)-10 d f x \tan \left (\frac {e}{2}\right )-20 d \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )\right )+\cos \left (\frac {1}{2} (e+f x)\right ) \left (f \tan \left (\frac {e}{2}\right ) (c+d x)-d\right )+f \sec \left (\frac {e}{2}\right ) (c+d x) \sin \left (\frac {f x}{2}\right )-10 f \sec \left (\frac {e}{2}\right ) (c+d x) \sin \left (\frac {f x}{2}\right ) \cos ^2\left (\frac {1}{2} (e+f x)\right )\right )}{3 a^2 f^2 (\sec (e+f x)+1)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.35, size = 183, normalized size = 1.31 \[ \frac {3 \, d f^{2} x^{2} + 6 \, c f^{2} x + 3 \, {\left (d f^{2} x^{2} + 2 \, c f^{2} x\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (3 \, d f^{2} x^{2} + 6 \, c f^{2} x - d\right )} \cos \left (f x + e\right ) - 10 \, {\left (d \cos \left (f x + e\right )^{2} + 2 \, d \cos \left (f x + e\right ) + d\right )} \log \left (\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) - 2 \, {\left (4 \, d f x + 4 \, c f + 5 \, {\left (d f x + c f\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right ) - 2 \, d}{6 \, {\left (a^{2} f^{2} \cos \left (f x + e\right )^{2} + 2 \, a^{2} f^{2} \cos \left (f x + e\right ) + a^{2} f^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.25, size = 900, normalized size = 6.43 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.64, size = 138, normalized size = 0.99 \[ \frac {c x}{a^{2}}+\frac {d \,x^{2}}{2 a^{2}}-\frac {3 c \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{2 a^{2} f}+\frac {c \left (\tan ^{3}\left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{6 a^{2} f}-\frac {d \left (\tan ^{2}\left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{6 a^{2} f^{2}}+\frac {d x \left (\tan ^{3}\left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{6 a^{2} f}-\frac {3 x d \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{2 a^{2} f}+\frac {5 d \ln \left (1+\tan ^{2}\left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{3 a^{2} f^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.62, size = 1058, normalized size = 7.56 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.98, size = 247, normalized size = 1.76 \[ \frac {d\,x^2}{2\,a^2}-\frac {\frac {\left (c+d\,x\right )\,4{}\mathrm {i}}{3\,a^2\,f}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,\left (c+d\,x\right )\,4{}\mathrm {i}}{3\,a^2\,f}+\frac {{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\left (c+d\,x\right )\,4{}\mathrm {i}}{3\,a^2\,f}}{3\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+3\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}+1}-\frac {10\,d\,\ln \left ({\mathrm {e}}^{e\,1{}\mathrm {i}}\,{\mathrm {e}}^{f\,x\,1{}\mathrm {i}}+1\right )}{3\,a^2\,f^2}-\frac {\left (4\,c\,f+4\,d\,f\,x-d\,1{}\mathrm {i}\right )\,2{}\mathrm {i}}{3\,a^2\,f^2\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+1\right )}+\frac {\left (c\,f+d\,f\,x-d\,1{}\mathrm {i}\right )\,2{}\mathrm {i}}{3\,a^2\,f^2\,\left (2\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}+\frac {x\,\left (3\,c\,f+d\,10{}\mathrm {i}\right )}{3\,a^2\,f} \]
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 \[ \frac {\int \frac {c}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {d x}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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