Integrand size = 25, antiderivative size = 164 \[ \int \frac {c+d \sec (e+f x)}{(a+a \sec (e+f x))^{5/2}} \, dx=\frac {2 c \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a^{5/2} f}-\frac {(43 c-3 d) \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)}}\right )}{16 \sqrt {2} a^{5/2} f}-\frac {(c-d) \tan (e+f x)}{4 f (a+a \sec (e+f x))^{5/2}}-\frac {(11 c-3 d) \tan (e+f x)}{16 a f (a+a \sec (e+f x))^{3/2}} \]
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Time = 0.32 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4007, 4005, 3859, 209, 3880} \[ \int \frac {c+d \sec (e+f x)}{(a+a \sec (e+f x))^{5/2}} \, dx=-\frac {(43 c-3 d) \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a \sec (e+f x)+a}}\right )}{16 \sqrt {2} a^{5/2} f}+\frac {2 c \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a}}\right )}{a^{5/2} f}-\frac {(11 c-3 d) \tan (e+f x)}{16 a f (a \sec (e+f x)+a)^{3/2}}-\frac {(c-d) \tan (e+f x)}{4 f (a \sec (e+f x)+a)^{5/2}} \]
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Rule 209
Rule 3859
Rule 3880
Rule 4005
Rule 4007
Rubi steps \begin{align*} \text {integral}& = -\frac {(c-d) \tan (e+f x)}{4 f (a+a \sec (e+f x))^{5/2}}-\frac {\int \frac {-4 a c+\frac {3}{2} a (c-d) \sec (e+f x)}{(a+a \sec (e+f x))^{3/2}} \, dx}{4 a^2} \\ & = -\frac {(c-d) \tan (e+f x)}{4 f (a+a \sec (e+f x))^{5/2}}-\frac {(11 c-3 d) \tan (e+f x)}{16 a f (a+a \sec (e+f x))^{3/2}}+\frac {\int \frac {8 a^2 c-\frac {1}{4} a^2 (11 c-3 d) \sec (e+f x)}{\sqrt {a+a \sec (e+f x)}} \, dx}{8 a^4} \\ & = -\frac {(c-d) \tan (e+f x)}{4 f (a+a \sec (e+f x))^{5/2}}-\frac {(11 c-3 d) \tan (e+f x)}{16 a f (a+a \sec (e+f x))^{3/2}}+\frac {c \int \sqrt {a+a \sec (e+f x)} \, dx}{a^3}-\frac {(43 c-3 d) \int \frac {\sec (e+f x)}{\sqrt {a+a \sec (e+f x)}} \, dx}{32 a^2} \\ & = -\frac {(c-d) \tan (e+f x)}{4 f (a+a \sec (e+f x))^{5/2}}-\frac {(11 c-3 d) \tan (e+f x)}{16 a f (a+a \sec (e+f x))^{3/2}}-\frac {(2 c) \text {Subst}\left (\int \frac {1}{a+x^2} \, dx,x,-\frac {a \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a^2 f}+\frac {(43 c-3 d) \text {Subst}\left (\int \frac {1}{2 a+x^2} \, dx,x,-\frac {a \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{16 a^2 f} \\ & = \frac {2 c \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a^{5/2} f}-\frac {(43 c-3 d) \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)}}\right )}{16 \sqrt {2} a^{5/2} f}-\frac {(c-d) \tan (e+f x)}{4 f (a+a \sec (e+f x))^{5/2}}-\frac {(11 c-3 d) \tan (e+f x)}{16 a f (a+a \sec (e+f x))^{3/2}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(343\) vs. \(2(164)=328\).
Time = 7.47 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.09 \[ \int \frac {c+d \sec (e+f x)}{(a+a \sec (e+f x))^{5/2}} \, dx=\frac {\left ((-43 c+3 d) \arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right )+32 \sqrt {2} c \arctan \left (\frac {\tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {\frac {\cos (e+f x)}{1+\cos (e+f x)}}}\right )\right ) \cos ^4\left (\frac {1}{2} (e+f x)\right ) \sqrt {\frac {\cos (e+f x)}{1+\cos (e+f x)}} \sec ^{\frac {3}{2}}(e+f x) \sqrt {1+\sec (e+f x)} (c+d \sec (e+f x))}{4 f (d+c \cos (e+f x)) \sqrt {\sec ^2\left (\frac {1}{2} (e+f x)\right )} (a (1+\sec (e+f x)))^{5/2}}+\frac {\cos ^5\left (\frac {1}{2} (e+f x)\right ) \sec ^2(e+f x) (c+d \sec (e+f x)) \left (\frac {1}{2} (-15 c+7 d) \sin \left (\frac {1}{2} (e+f x)\right )+\frac {1}{4} \sec ^2\left (\frac {1}{2} (e+f x)\right ) \left (19 c \sin \left (\frac {1}{2} (e+f x)\right )-11 d \sin \left (\frac {1}{2} (e+f x)\right )\right )+\frac {1}{2} \sec ^4\left (\frac {1}{2} (e+f x)\right ) \left (-c \sin \left (\frac {1}{2} (e+f x)\right )+d \sin \left (\frac {1}{2} (e+f x)\right )\right )\right )}{f (d+c \cos (e+f x)) (a (1+\sec (e+f x)))^{5/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(375\) vs. \(2(139)=278\).
Time = 2.89 (sec) , antiderivative size = 376, normalized size of antiderivative = 2.29
method | result | size |
default | \(-\frac {\sqrt {-\frac {2 a}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\, \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \left (2 c \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right )^{\frac {3}{2}} \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )-2 d \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right )^{\frac {3}{2}} \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )-32 c \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\right )-11 c \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+3 d \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+43 c \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\right )-3 d \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\right )\right )}{32 a^{3} f}\) | \(376\) |
parts | \(\frac {c \sqrt {-\frac {2 a}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\, \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \left (-2 \left (1-\cos \left (f x +e \right )\right )^{3} \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \csc \left (f x +e \right )^{3}+32 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\right )+13 \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )-43 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\right )\right )}{32 f \,a^{3}}+\frac {d \sqrt {-\frac {2 a}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\, \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \left (2 \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right )^{\frac {3}{2}} \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )-3 \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+3 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\right )\right )}{32 f \,a^{3}}\) | \(441\) |
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Leaf count of result is larger than twice the leaf count of optimal. 292 vs. \(2 (139) = 278\).
Time = 3.61 (sec) , antiderivative size = 670, normalized size of antiderivative = 4.09 \[ \int \frac {c+d \sec (e+f x)}{(a+a \sec (e+f x))^{5/2}} \, dx=\left [\frac {\sqrt {2} {\left ({\left (43 \, c - 3 \, d\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (43 \, c - 3 \, d\right )} \cos \left (f x + e\right )^{2} + 3 \, {\left (43 \, c - 3 \, d\right )} \cos \left (f x + e\right ) + 43 \, c - 3 \, d\right )} \sqrt {-a} \log \left (\frac {2 \, \sqrt {2} \sqrt {-a} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 3 \, a \cos \left (f x + e\right )^{2} + 2 \, a \cos \left (f x + e\right ) - a}{\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1}\right ) - 64 \, {\left (c \cos \left (f x + e\right )^{3} + 3 \, c \cos \left (f x + e\right )^{2} + 3 \, c \cos \left (f x + e\right ) + c\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (f x + e\right )^{2} + 2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + a \cos \left (f x + e\right ) - a}{\cos \left (f x + e\right ) + 1}\right ) - 4 \, {\left ({\left (15 \, c - 7 \, d\right )} \cos \left (f x + e\right )^{2} + {\left (11 \, c - 3 \, d\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{64 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} + 3 \, a^{3} f \cos \left (f x + e\right ) + a^{3} f\right )}}, \frac {\sqrt {2} {\left ({\left (43 \, c - 3 \, d\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (43 \, c - 3 \, d\right )} \cos \left (f x + e\right )^{2} + 3 \, {\left (43 \, c - 3 \, d\right )} \cos \left (f x + e\right ) + 43 \, c - 3 \, d\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt {a} \sin \left (f x + e\right )}\right ) - 64 \, {\left (c \cos \left (f x + e\right )^{3} + 3 \, c \cos \left (f x + e\right )^{2} + 3 \, c \cos \left (f x + e\right ) + c\right )} \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt {a} \sin \left (f x + e\right )}\right ) - 2 \, {\left ({\left (15 \, c - 7 \, d\right )} \cos \left (f x + e\right )^{2} + {\left (11 \, c - 3 \, d\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{32 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} + 3 \, a^{3} f \cos \left (f x + e\right ) + a^{3} f\right )}}\right ] \]
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\[ \int \frac {c+d \sec (e+f x)}{(a+a \sec (e+f x))^{5/2}} \, dx=\int \frac {c + d \sec {\left (e + f x \right )}}{\left (a \left (\sec {\left (e + f x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {c+d \sec (e+f x)}{(a+a \sec (e+f x))^{5/2}} \, dx=\int { \frac {d \sec \left (f x + e\right ) + c}{{\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {c+d \sec (e+f x)}{(a+a \sec (e+f x))^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {c+d \sec (e+f x)}{(a+a \sec (e+f x))^{5/2}} \, dx=\int \frac {c+\frac {d}{\cos \left (e+f\,x\right )}}{{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{5/2}} \,d x \]
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