Integrand size = 18, antiderivative size = 140 \[ \int \frac {c+d x}{(a+a \sec (e+f x))^2} \, dx=\frac {(c+d x)^2}{2 a^2 d}-\frac {10 d \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{3 a^2 f^2}-\frac {d \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f^2}-\frac {5 (c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f} \]
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Time = 0.22 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {4276, 3399, 4270, 4269, 3556} \[ \int \frac {c+d x}{(a+a \sec (e+f x))^2} \, dx=-\frac {5 (c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {(c+d x)^2}{2 a^2 d}-\frac {d \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f^2}-\frac {10 d \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{3 a^2 f^2} \]
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Rule 3399
Rule 3556
Rule 4269
Rule 4270
Rule 4276
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 1.98 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.23 \[ \int \frac {c+d x}{(a+a \sec (e+f x))^2} \, dx=\frac {2 \cos \left (\frac {1}{2} (e+f x)\right ) \sec ^2(e+f x) \left (f (c+d x) \sec \left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right )-10 f (c+d x) \cos ^2\left (\frac {1}{2} (e+f x)\right ) \sec \left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right )+\cos ^3\left (\frac {1}{2} (e+f x)\right ) \left (3 f^2 x (2 c+d x)-20 d \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )-10 d f x \tan \left (\frac {e}{2}\right )\right )+\cos \left (\frac {1}{2} (e+f x)\right ) \left (-d+f (c+d x) \tan \left (\frac {e}{2}\right )\right )\right )}{3 a^2 f^2 (1+\sec (e+f x))^2} \]
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Time = 0.62 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.61
method | result | size |
parallelrisch | \(\frac {10 d \ln \left (\sec \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )+\left (d x +c \right ) f \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}-d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-9 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) f \left (d x +c \right )+6 f^{2} \left (\frac {d x}{2}+c \right ) x}{6 a^{2} f^{2}}\) | \(86\) |
norman | \(\frac {\frac {c x}{a}+\frac {d \,x^{2}}{2 a}-\frac {3 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 a f}+\frac {c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{6 a f}-\frac {d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{6 a \,f^{2}}-\frac {3 d x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 a f}+\frac {d x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{6 a f}}{a}+\frac {5 d \ln \left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )}{3 a^{2} f^{2}}\) | \(143\) |
risch | \(\frac {d \,x^{2}}{2 a^{2}}+\frac {x c}{a^{2}}+\frac {10 i d x}{3 a^{2} f}+\frac {10 i d e}{3 a^{2} f^{2}}-\frac {2 i \left (6 d f x \,{\mathrm e}^{2 i \left (f x +e \right )}-i d \,{\mathrm e}^{2 i \left (f x +e \right )}+6 c f \,{\mathrm e}^{2 i \left (f x +e \right )}+9 d f x \,{\mathrm e}^{i \left (f x +e \right )}-i d \,{\mathrm e}^{i \left (f x +e \right )}+9 c f \,{\mathrm e}^{i \left (f x +e \right )}+5 d f x +5 c f \right )}{3 f^{2} a^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{3}}-\frac {10 d \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{3 a^{2} f^{2}}\) | \(172\) |
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Time = 0.29 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.31 \[ \int \frac {c+d x}{(a+a \sec (e+f x))^2} \, dx=\frac {3 \, d f^{2} x^{2} + 6 \, c f^{2} x + 3 \, {\left (d f^{2} x^{2} + 2 \, c f^{2} x\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (3 \, d f^{2} x^{2} + 6 \, c f^{2} x - d\right )} \cos \left (f x + e\right ) - 10 \, {\left (d \cos \left (f x + e\right )^{2} + 2 \, d \cos \left (f x + e\right ) + d\right )} \log \left (\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) - 2 \, {\left (4 \, d f x + 4 \, c f + 5 \, {\left (d f x + c f\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right ) - 2 \, d}{6 \, {\left (a^{2} f^{2} \cos \left (f x + e\right )^{2} + 2 \, a^{2} f^{2} \cos \left (f x + e\right ) + a^{2} f^{2}\right )}} \]
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\[ \int \frac {c+d x}{(a+a \sec (e+f x))^2} \, dx=\frac {\int \frac {c}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {d x}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx}{a^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1058 vs. \(2 (110) = 220\).
Time = 0.36 (sec) , antiderivative size = 1058, normalized size of antiderivative = 7.56 \[ \int \frac {c+d x}{(a+a \sec (e+f x))^2} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 798 vs. \(2 (110) = 220\).
Time = 0.56 (sec) , antiderivative size = 798, normalized size of antiderivative = 5.70 \[ \int \frac {c+d x}{(a+a \sec (e+f x))^2} \, dx=\text {Too large to display} \]
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Time = 19.32 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.76 \[ \int \frac {c+d x}{(a+a \sec (e+f x))^2} \, dx=\frac {d\,x^2}{2\,a^2}-\frac {\frac {\left (c+d\,x\right )\,4{}\mathrm {i}}{3\,a^2\,f}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,\left (c+d\,x\right )\,4{}\mathrm {i}}{3\,a^2\,f}+\frac {{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\left (c+d\,x\right )\,4{}\mathrm {i}}{3\,a^2\,f}}{3\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+3\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}+1}-\frac {10\,d\,\ln \left ({\mathrm {e}}^{e\,1{}\mathrm {i}}\,{\mathrm {e}}^{f\,x\,1{}\mathrm {i}}+1\right )}{3\,a^2\,f^2}-\frac {\left (4\,c\,f+4\,d\,f\,x-d\,1{}\mathrm {i}\right )\,2{}\mathrm {i}}{3\,a^2\,f^2\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+1\right )}+\frac {\left (c\,f+d\,f\,x-d\,1{}\mathrm {i}\right )\,2{}\mathrm {i}}{3\,a^2\,f^2\,\left (2\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}+\frac {x\,\left (3\,c\,f+d\,10{}\mathrm {i}\right )}{3\,a^2\,f} \]
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