Integrand size = 31, antiderivative size = 68 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^2}{a+a \sec (e+f x)} \, dx=\frac {(2 c-d) d \text {arctanh}(\sin (e+f x))}{a f}+\frac {d^2 \tan (e+f x)}{a f}+\frac {(c-d)^2 \tan (e+f x)}{f (a+a \sec (e+f x))} \]
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Time = 0.17 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.84, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {4072, 91, 81, 65, 223, 209} \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^2}{a+a \sec (e+f x)} \, dx=\frac {2 d (2 c-d) \tan (e+f x) \arctan \left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a (\sec (e+f x)+1)}}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}+\frac {(c-d)^2 \tan (e+f x)}{f (a \sec (e+f x)+a)}+\frac {d^2 \tan (e+f x)}{a f} \]
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Rule 65
Rule 81
Rule 91
Rule 209
Rule 223
Rule 4072
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a^2 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {(c+d x)^2}{\sqrt {a-a x} (a+a x)^{3/2}} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {(c-d)^2 \tan (e+f x)}{f (a+a \sec (e+f x))}-\frac {\tan (e+f x) \text {Subst}\left (\int \frac {a^3 (2 c-d) d+a^3 d^2 x}{\sqrt {a-a x} \sqrt {a+a x}} \, dx,x,\sec (e+f x)\right )}{a^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {d^2 \tan (e+f x)}{a f}+\frac {(c-d)^2 \tan (e+f x)}{f (a+a \sec (e+f x))}-\frac {(a (2 c-d) d \tan (e+f x)) \text {Subst}\left (\int \frac {1}{\sqrt {a-a x} \sqrt {a+a x}} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {d^2 \tan (e+f x)}{a f}+\frac {(c-d)^2 \tan (e+f x)}{f (a+a \sec (e+f x))}+\frac {(2 (2 c-d) d \tan (e+f x)) \text {Subst}\left (\int \frac {1}{\sqrt {2 a-x^2}} \, dx,x,\sqrt {a-a \sec (e+f x)}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {d^2 \tan (e+f x)}{a f}+\frac {(c-d)^2 \tan (e+f x)}{f (a+a \sec (e+f x))}+\frac {(2 (2 c-d) d \tan (e+f x)) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a+a \sec (e+f x)}}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {d^2 \tan (e+f x)}{a f}+\frac {(c-d)^2 \tan (e+f x)}{f (a+a \sec (e+f x))}+\frac {2 (2 c-d) d \arctan \left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a+a \sec (e+f x)}}\right ) \tan (e+f x)}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(237\) vs. \(2(68)=136\).
Time = 2.66 (sec) , antiderivative size = 237, normalized size of antiderivative = 3.49 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^2}{a+a \sec (e+f x)} \, dx=\frac {2 \cos \left (\frac {1}{2} (e+f x)\right ) \cos (e+f x) (c+d \sec (e+f x))^2 \left ((c-d)^2 \sec \left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right )+d \cos \left (\frac {1}{2} (e+f x)\right ) \left (-\left ((2 c-d) \left (\log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )\right )\right )+\frac {d \sin (f x)}{\left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )}\right )\right )}{a f (d+c \cos (e+f x))^2 (1+\sec (e+f x))} \]
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Time = 0.71 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.51
method | result | size |
parallelrisch | \(\frac {-2 \left (c -\frac {d}{2}\right ) \cos \left (f x +e \right ) d \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )+2 \left (c -\frac {d}{2}\right ) \cos \left (f x +e \right ) d \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )+\tan \left (\frac {f x}{2}+\frac {e}{2}\right ) \left (\left (c^{2}-2 c d +2 d^{2}\right ) \cos \left (f x +e \right )+d^{2}\right )}{a f \cos \left (f x +e \right )}\) | \(103\) |
derivativedivides | \(\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c^{2}-2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c d +\tan \left (\frac {f x}{2}+\frac {e}{2}\right ) d^{2}-\frac {d^{2}}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-d \left (2 c -d \right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )-\frac {d^{2}}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}+d \left (2 c -d \right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{f a}\) | \(127\) |
default | \(\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c^{2}-2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c d +\tan \left (\frac {f x}{2}+\frac {e}{2}\right ) d^{2}-\frac {d^{2}}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-d \left (2 c -d \right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )-\frac {d^{2}}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}+d \left (2 c -d \right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{f a}\) | \(127\) |
norman | \(\frac {\frac {\left (c^{2}-2 c d +d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{a f}+\frac {\left (c^{2}-2 c d +3 d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a f}-\frac {2 \left (c^{2}-2 c d +2 d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{a f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{2}}+\frac {d \left (2 c -d \right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{a f}-\frac {d \left (2 c -d \right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{a f}\) | \(164\) |
risch | \(\frac {2 i \left (c^{2} {\mathrm e}^{2 i \left (f x +e \right )}-2 c d \,{\mathrm e}^{2 i \left (f x +e \right )}+d^{2} {\mathrm e}^{2 i \left (f x +e \right )}+d^{2} {\mathrm e}^{i \left (f x +e \right )}+c^{2}-2 c d +2 d^{2}\right )}{f a \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right ) \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}+\frac {2 d \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) c}{a f}-\frac {d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{a f}-\frac {2 d \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) c}{a f}+\frac {d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{a f}\) | \(195\) |
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Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (68) = 136\).
Time = 0.27 (sec) , antiderivative size = 155, normalized size of antiderivative = 2.28 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^2}{a+a \sec (e+f x)} \, dx=\frac {{\left ({\left (2 \, c d - d^{2}\right )} \cos \left (f x + e\right )^{2} + {\left (2 \, c d - d^{2}\right )} \cos \left (f x + e\right )\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) - {\left ({\left (2 \, c d - d^{2}\right )} \cos \left (f x + e\right )^{2} + {\left (2 \, c d - d^{2}\right )} \cos \left (f x + e\right )\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left (d^{2} + {\left (c^{2} - 2 \, c d + 2 \, d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{2 \, {\left (a f \cos \left (f x + e\right )^{2} + a f \cos \left (f x + e\right )\right )}} \]
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\[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^2}{a+a \sec (e+f x)} \, dx=\frac {\int \frac {c^{2} \sec {\left (e + f x \right )}}{\sec {\left (e + f x \right )} + 1}\, dx + \int \frac {d^{2} \sec ^{3}{\left (e + f x \right )}}{\sec {\left (e + f x \right )} + 1}\, dx + \int \frac {2 c d \sec ^{2}{\left (e + f x \right )}}{\sec {\left (e + f x \right )} + 1}\, dx}{a} \]
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Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (68) = 136\).
Time = 0.22 (sec) , antiderivative size = 223, normalized size of antiderivative = 3.28 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^2}{a+a \sec (e+f x)} \, dx=-\frac {d^{2} {\left (\frac {\log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a} - \frac {\log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a} - \frac {2 \, \sin \left (f x + e\right )}{{\left (a - \frac {a \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (f x + e\right ) + 1\right )}} - \frac {\sin \left (f x + e\right )}{a {\left (\cos \left (f x + e\right ) + 1\right )}}\right )} - 2 \, c d {\left (\frac {\log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a} - \frac {\log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a} - \frac {\sin \left (f x + e\right )}{a {\left (\cos \left (f x + e\right ) + 1\right )}}\right )} - \frac {c^{2} \sin \left (f x + e\right )}{a {\left (\cos \left (f x + e\right ) + 1\right )}}}{f} \]
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Time = 0.31 (sec) , antiderivative size = 136, normalized size of antiderivative = 2.00 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^2}{a+a \sec (e+f x)} \, dx=\frac {\frac {{\left (2 \, c d - d^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{a} - \frac {{\left (2 \, c d - d^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{a} - \frac {2 \, d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )} a} + \frac {c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2 \, c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{a}}{f} \]
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Time = 13.53 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.25 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^2}{a+a \sec (e+f x)} \, dx=\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,{\left (c-d\right )}^2}{a\,f}+\frac {2\,d^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,\left (a-a\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\right )}+\frac {2\,d\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )\,\left (2\,c-d\right )}{a\,f} \]
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