Integrand size = 28, antiderivative size = 140 \[ \int \frac {(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^4} \, dx=\frac {2 a^{5/2} \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{c^4 f}+\frac {2 a^2 \cot (e+f x) \sqrt {a+a \sec (e+f x)}}{c^4 f}-\frac {2 a \cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{3 c^4 f}-\frac {8 \cot ^7(e+f x) (a+a \sec (e+f x))^{7/2}}{7 a c^4 f} \]
[Out]
Time = 0.23 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3989, 3972, 472, 209} \[ \int \frac {(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^4} \, dx=\frac {2 a^{5/2} \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a}}\right )}{c^4 f}+\frac {2 a^2 \cot (e+f x) \sqrt {a \sec (e+f x)+a}}{c^4 f}-\frac {8 \cot ^7(e+f x) (a \sec (e+f x)+a)^{7/2}}{7 a c^4 f}-\frac {2 a \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 c^4 f} \]
[In]
[Out]
Rule 209
Rule 472
Rule 3972
Rule 3989
Rubi steps \begin{align*} \text {integral}& = \frac {\int \cot ^8(e+f x) (a+a \sec (e+f x))^{13/2} \, dx}{a^4 c^4} \\ & = -\frac {2 \text {Subst}\left (\int \frac {\left (2+a x^2\right )^2}{x^8 \left (1+a x^2\right )} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a c^4 f} \\ & = -\frac {2 \text {Subst}\left (\int \left (\frac {4}{x^8}+\frac {a^2}{x^4}-\frac {a^3}{x^2}+\frac {a^4}{1+a x^2}\right ) \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a c^4 f} \\ & = \frac {2 a^2 \cot (e+f x) \sqrt {a+a \sec (e+f x)}}{c^4 f}-\frac {2 a \cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{3 c^4 f}-\frac {8 \cot ^7(e+f x) (a+a \sec (e+f x))^{7/2}}{7 a c^4 f}-\frac {\left (2 a^3\right ) \text {Subst}\left (\int \frac {1}{1+a x^2} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{c^4 f} \\ & = \frac {2 a^{5/2} \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{c^4 f}+\frac {2 a^2 \cot (e+f x) \sqrt {a+a \sec (e+f x)}}{c^4 f}-\frac {2 a \cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{3 c^4 f}-\frac {8 \cot ^7(e+f x) (a+a \sec (e+f x))^{7/2}}{7 a c^4 f} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 4.70 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.52 \[ \int \frac {(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^4} \, dx=-\frac {2 a^3 \left (8+5 \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},1,-\frac {5}{2},1-\sec (e+f x)\right )+7 \sec (e+f x)\right ) \tan (e+f x)}{35 c^4 f (-1+\sec (e+f x))^4 \sqrt {a (1+\sec (e+f x))}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(350\) vs. \(2(124)=248\).
Time = 1.64 (sec) , antiderivative size = 351, normalized size of antiderivative = 2.51
\[\frac {2 a^{2} \sqrt {a \left (\sec \left (f x +e \right )+1\right )}\, \left (21 \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}}\right ) \cos \left (f x +e \right )^{3}-63 \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}}\right ) \cos \left (f x +e \right )^{2}+40 \cos \left (f x +e \right )^{3} \cot \left (f x +e \right )+63 \,\operatorname {arctanh}\left (\frac {\sin \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}}\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \cos \left (f x +e \right )-77 \cos \left (f x +e \right )^{2} \cot \left (f x +e \right )-21 \,\operatorname {arctanh}\left (\frac {\sin \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}}\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}+70 \cos \left (f x +e \right ) \cot \left (f x +e \right )-21 \cot \left (f x +e \right )\right )}{21 c^{4} f \left (\cos \left (f x +e \right )-1\right )^{3}}\]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 250 vs. \(2 (124) = 248\).
Time = 0.34 (sec) , antiderivative size = 527, normalized size of antiderivative = 3.76 \[ \int \frac {(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^4} \, dx=\left [\frac {21 \, {\left (a^{2} \cos \left (f x + e\right )^{3} - 3 \, a^{2} \cos \left (f x + e\right )^{2} + 3 \, a^{2} \cos \left (f x + e\right ) - a^{2}\right )} \sqrt {-a} \log \left (-\frac {8 \, a \cos \left (f x + e\right )^{3} - 4 \, {\left (2 \, \cos \left (f x + e\right )^{2} - \cos \left (f x + e\right )\right )} \sqrt {-a} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) - 7 \, a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right ) + 1}\right ) \sin \left (f x + e\right ) + 4 \, {\left (40 \, a^{2} \cos \left (f x + e\right )^{4} - 77 \, a^{2} \cos \left (f x + e\right )^{3} + 70 \, a^{2} \cos \left (f x + e\right )^{2} - 21 \, a^{2} \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}}}{42 \, {\left (c^{4} f \cos \left (f x + e\right )^{3} - 3 \, c^{4} f \cos \left (f x + e\right )^{2} + 3 \, c^{4} f \cos \left (f x + e\right ) - c^{4} f\right )} \sin \left (f x + e\right )}, \frac {21 \, {\left (a^{2} \cos \left (f x + e\right )^{3} - 3 \, a^{2} \cos \left (f x + e\right )^{2} + 3 \, a^{2} \cos \left (f x + e\right ) - a^{2}\right )} \sqrt {a} \arctan \left (\frac {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 )}{2 \, a \cos \left (f x + e\right )^{2} + a \cos \left (f x + e\right ) - a}\right ) \sin \left (f x + e\right ) + 2 \, {\left (40 \, a^{2} \cos \left (f x + e\right )^{4} - 77 \, a^{2} \cos \left (f x + e\right )^{3} + 70 \, a^{2} \cos \left (f x + e\right )^{2} - 21 \, a^{2} \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}}}{21 \, {\left (c^{4} f \cos \left (f x + e\right )^{3} - 3 \, c^{4} f \cos \left (f x + e\right )^{2} + 3 \, c^{4} f \cos \left (f x + e\right ) - c^{4} f\right )} \sin \left (f x + e\right )}\right ] \]
[In]
[Out]
Timed out. \[ \int \frac {(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^4} \, dx=\text {Timed out} \]
[In]
[Out]
Timed out. \[ \int \frac {(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^4} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^4} \, dx=\int { \frac {{\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{{\left (c \sec \left (f x + e\right ) - c\right )}^{4}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^4} \, dx=\int \frac {{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{5/2}}{{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^4} \,d x \]
[In]
[Out]