Integrand size = 26, antiderivative size = 78 \[ \int \frac {(a+a \sec (e+f x))^3}{c-c \sec (e+f x)} \, dx=\frac {a^3 x}{c}-\frac {4 a^3 \text {arctanh}(\sin (e+f x))}{c f}+\frac {8 a^3 \cot (e+f x)}{c f}+\frac {8 a^3 \csc (e+f x)}{c f}-\frac {a^3 \tan (e+f x)}{c f} \]
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Time = 0.25 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {3989, 3971, 3554, 8, 2686, 3852, 2701, 327, 213, 2700, 14} \[ \int \frac {(a+a \sec (e+f x))^3}{c-c \sec (e+f x)} \, dx=-\frac {4 a^3 \text {arctanh}(\sin (e+f x))}{c f}-\frac {a^3 \tan (e+f x)}{c f}+\frac {8 a^3 \cot (e+f x)}{c f}+\frac {8 a^3 \csc (e+f x)}{c f}+\frac {a^3 x}{c} \]
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Rule 8
Rule 14
Rule 213
Rule 327
Rule 2686
Rule 2700
Rule 2701
Rule 3554
Rule 3852
Rule 3971
Rule 3989
Rubi steps \begin{align*} \text {integral}& = -\frac {\int \cot ^2(e+f x) (a+a \sec (e+f x))^4 \, dx}{a c} \\ & = -\frac {\int \left (a^4 \cot ^2(e+f x)+4 a^4 \cot (e+f x) \csc (e+f x)+6 a^4 \csc ^2(e+f x)+4 a^4 \csc ^2(e+f x) \sec (e+f x)+a^4 \csc ^2(e+f x) \sec ^2(e+f x)\right ) \, dx}{a c} \\ & = -\frac {a^3 \int \cot ^2(e+f x) \, dx}{c}-\frac {a^3 \int \csc ^2(e+f x) \sec ^2(e+f x) \, dx}{c}-\frac {\left (4 a^3\right ) \int \cot (e+f x) \csc (e+f x) \, dx}{c}-\frac {\left (4 a^3\right ) \int \csc ^2(e+f x) \sec (e+f x) \, dx}{c}-\frac {\left (6 a^3\right ) \int \csc ^2(e+f x) \, dx}{c} \\ & = \frac {a^3 \cot (e+f x)}{c f}+\frac {a^3 \int 1 \, dx}{c}-\frac {a^3 \text {Subst}\left (\int \frac {1+x^2}{x^2} \, dx,x,\tan (e+f x)\right )}{c f}+\frac {\left (4 a^3\right ) \text {Subst}(\int 1 \, dx,x,\csc (e+f x))}{c f}+\frac {\left (4 a^3\right ) \text {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\csc (e+f x)\right )}{c f}+\frac {\left (6 a^3\right ) \text {Subst}(\int 1 \, dx,x,\cot (e+f x))}{c f} \\ & = \frac {a^3 x}{c}+\frac {7 a^3 \cot (e+f x)}{c f}+\frac {8 a^3 \csc (e+f x)}{c f}-\frac {a^3 \text {Subst}\left (\int \left (1+\frac {1}{x^2}\right ) \, dx,x,\tan (e+f x)\right )}{c f}+\frac {\left (4 a^3\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (e+f x)\right )}{c f} \\ & = \frac {a^3 x}{c}-\frac {4 a^3 \text {arctanh}(\sin (e+f x))}{c f}+\frac {8 a^3 \cot (e+f x)}{c f}+\frac {8 a^3 \csc (e+f x)}{c f}-\frac {a^3 \tan (e+f x)}{c f} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 4.39 (sec) , antiderivative size = 275, normalized size of antiderivative = 3.53 \[ \int \frac {(a+a \sec (e+f x))^3}{c-c \sec (e+f x)} \, dx=-\frac {\tan (e+f x) \left (-8 \sqrt {2} a^3 \sqrt {c} \cos ^6\left (\frac {1}{2} (e+f x)\right ) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {5}{2},\frac {7}{2},\frac {1}{2} (1+\sec (e+f x))\right ) \sec ^4(e+f x) \sin ^2\left (\frac {1}{2} (e+f x)\right )+5 a^{5/2} \left (4 \sqrt {c} \left (\sqrt {a} \sqrt {1-\sec (e+f x)} (1+\sec (e+f x))+\arcsin \left (\frac {\sqrt {a (1+\sec (e+f x))}}{\sqrt {2} \sqrt {a}}\right ) \sec (e+f x) \sqrt {a (1+\sec (e+f x))} \sin ^2\left (\frac {1}{2} (e+f x)\right )\right )-\text {arctanh}\left (\frac {\sqrt {-a c \tan ^2(e+f x)}}{\sqrt {a} \sqrt {c}}\right ) \sqrt {1-\sec (e+f x)} \sqrt {-a c \tan ^2(e+f x)}\right )\right )}{5 c^{3/2} f (1-\sec (e+f x))^{3/2} (1+\sec (e+f x))} \]
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Time = 0.65 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.21
method | result | size |
derivativedivides | \(\frac {8 a^{3} \left (\frac {1}{8 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+8}-\frac {\ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{2}+\frac {1}{8 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-8}+\frac {\ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{2}+\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}+\frac {\arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4}\right )}{f c}\) | \(94\) |
default | \(\frac {8 a^{3} \left (\frac {1}{8 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+8}-\frac {\ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{2}+\frac {1}{8 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-8}+\frac {\ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{2}+\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}+\frac {\arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4}\right )}{f c}\) | \(94\) |
parallelrisch | \(\frac {a^{3} \left (f x \cos \left (f x +e \right )+4 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right ) \cos \left (f x +e \right )-4 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right ) \cos \left (f x +e \right )+9 \cot \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (f x +e \right )-\cot \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f \cos \left (f x +e \right )}\) | \(97\) |
risch | \(\frac {a^{3} x}{c}+\frac {2 i a^{3} \left (8 \,{\mathrm e}^{2 i \left (f x +e \right )}-{\mathrm e}^{i \left (f x +e \right )}+9\right )}{f c \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right ) \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}-\frac {4 a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{c f}+\frac {4 a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{c f}\) | \(120\) |
norman | \(\frac {\frac {a^{3} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{c}+\frac {a^{3} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{c}+\frac {8 a^{3}}{c f}-\frac {18 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{c f}+\frac {10 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{c f}-\frac {2 a^{3} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{c}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}+\frac {4 a^{3} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{c f}-\frac {4 a^{3} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{c f}\) | \(187\) |
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Time = 0.25 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.60 \[ \int \frac {(a+a \sec (e+f x))^3}{c-c \sec (e+f x)} \, dx=\frac {a^{3} f x \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, a^{3} \cos \left (f x + e\right ) \log \left (\sin \left (f x + e\right ) + 1\right ) \sin \left (f x + e\right ) + 2 \, a^{3} \cos \left (f x + e\right ) \log \left (-\sin \left (f x + e\right ) + 1\right ) \sin \left (f x + e\right ) + 9 \, a^{3} \cos \left (f x + e\right )^{2} + 8 \, a^{3} \cos \left (f x + e\right ) - a^{3}}{c f \cos \left (f x + e\right ) \sin \left (f x + e\right )} \]
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\[ \int \frac {(a+a \sec (e+f x))^3}{c-c \sec (e+f x)} \, dx=- \frac {a^{3} \left (\int \frac {3 \sec {\left (e + f x \right )}}{\sec {\left (e + f x \right )} - 1}\, dx + \int \frac {3 \sec ^{2}{\left (e + f x \right )}}{\sec {\left (e + f x \right )} - 1}\, dx + \int \frac {\sec ^{3}{\left (e + f x \right )}}{\sec {\left (e + f x \right )} - 1}\, dx + \int \frac {1}{\sec {\left (e + f x \right )} - 1}\, dx\right )}{c} \]
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Leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (78) = 156\).
Time = 0.29 (sec) , antiderivative size = 274, normalized size of antiderivative = 3.51 \[ \int \frac {(a+a \sec (e+f x))^3}{c-c \sec (e+f x)} \, dx=-\frac {a^{3} {\left (\frac {\frac {3 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - 1}{\frac {c \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {c \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}} + \frac {\log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{c} - \frac {\log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{c}\right )} - a^{3} {\left (\frac {2 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{c} + \frac {\cos \left (f x + e\right ) + 1}{c \sin \left (f x + e\right )}\right )} + 3 \, a^{3} {\left (\frac {\log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{c} - \frac {\log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{c} - \frac {\cos \left (f x + e\right ) + 1}{c \sin \left (f x + e\right )}\right )} - \frac {3 \, a^{3} {\left (\cos \left (f x + e\right ) + 1\right )}}{c \sin \left (f x + e\right )}}{f} \]
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Time = 0.34 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.42 \[ \int \frac {(a+a \sec (e+f x))^3}{c-c \sec (e+f x)} \, dx=\frac {\frac {{\left (f x + e\right )} a^{3}}{c} - \frac {4 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{c} + \frac {4 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{c} + \frac {2 \, {\left (5 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 4 \, a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} c}}{f} \]
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Time = 14.38 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.09 \[ \int \frac {(a+a \sec (e+f x))^3}{c-c \sec (e+f x)} \, dx=\frac {a^3\,x}{c}-\frac {10\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-8\,a^3}{f\,\left (c\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )-c\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\right )}-\frac {8\,a^3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{c\,f} \]
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