Integrand size = 23, antiderivative size = 177 \[ \int \frac {\tan ^2(e+f x)}{(a+a \sec (e+f x))^{9/2}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a^{9/2} f}+\frac {91 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)}}\right )}{32 \sqrt {2} a^{9/2} f}+\frac {\tan (e+f x)}{3 a f (a+a \sec (e+f x))^{7/2}}+\frac {11 \tan (e+f x)}{24 a^2 f (a+a \sec (e+f x))^{5/2}}+\frac {27 \tan (e+f x)}{32 a^3 f (a+a \sec (e+f x))^{3/2}} \]
[Out]
Time = 0.24 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.28, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3972, 482, 541, 536, 209} \[ \int \frac {\tan ^2(e+f x)}{(a+a \sec (e+f x))^{9/2}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a}}\right )}{a^{9/2} f}+\frac {91 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a \sec (e+f x)+a}}\right )}{32 \sqrt {2} a^{9/2} f}+\frac {27 \sin (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )}{64 a^4 f \sqrt {a \sec (e+f x)+a}}+\frac {\sin (e+f x) \cos ^2(e+f x) \sec ^6\left (\frac {1}{2} (e+f x)\right )}{24 a^4 f \sqrt {a \sec (e+f x)+a}}+\frac {11 \sin (e+f x) \cos (e+f x) \sec ^4\left (\frac {1}{2} (e+f x)\right )}{96 a^4 f \sqrt {a \sec (e+f x)+a}} \]
[In]
[Out]
Rule 209
Rule 482
Rule 536
Rule 541
Rule 3972
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \text {Subst}\left (\int \frac {x^2}{\left (1+a x^2\right ) \left (2+a x^2\right )^4} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a^3 f} \\ & = \frac {\cos ^2(e+f x) \sec ^6\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x)}{24 a^4 f \sqrt {a+a \sec (e+f x)}}+\frac {\text {Subst}\left (\int \frac {1-5 a x^2}{\left (1+a x^2\right ) \left (2+a x^2\right )^3} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{3 a^4 f} \\ & = \frac {11 \cos (e+f x) \sec ^4\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x)}{96 a^4 f \sqrt {a+a \sec (e+f x)}}+\frac {\cos ^2(e+f x) \sec ^6\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x)}{24 a^4 f \sqrt {a+a \sec (e+f x)}}+\frac {\text {Subst}\left (\int \frac {15 a-33 a^2 x^2}{\left (1+a x^2\right ) \left (2+a x^2\right )^2} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{24 a^5 f} \\ & = \frac {27 \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x)}{64 a^4 f \sqrt {a+a \sec (e+f x)}}+\frac {11 \cos (e+f x) \sec ^4\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x)}{96 a^4 f \sqrt {a+a \sec (e+f x)}}+\frac {\cos ^2(e+f x) \sec ^6\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x)}{24 a^4 f \sqrt {a+a \sec (e+f x)}}+\frac {\text {Subst}\left (\int \frac {111 a^2-81 a^3 x^2}{\left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{96 a^6 f} \\ & = \frac {27 \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x)}{64 a^4 f \sqrt {a+a \sec (e+f x)}}+\frac {11 \cos (e+f x) \sec ^4\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x)}{96 a^4 f \sqrt {a+a \sec (e+f x)}}+\frac {\cos ^2(e+f x) \sec ^6\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x)}{24 a^4 f \sqrt {a+a \sec (e+f x)}}+\frac {2 \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 )}{a^4 f}-\frac {91 \text {Subst}\left (\int \frac {1}{2+a x^2} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{32 a^4 f} \\ & = -\frac {2 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a^{9/2} f}+\frac {91 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)}}\right )}{32 \sqrt {2} a^{9/2} f}+\frac {27 \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x)}{64 a^4 f \sqrt {a+a \sec (e+f x)}}+\frac {11 \cos (e+f x) \sec ^4\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x)}{96 a^4 f \sqrt {a+a \sec (e+f x)}}+\frac {\cos ^2(e+f x) \sec ^6\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x)}{24 a^4 f \sqrt {a+a \sec (e+f x)}} \\ \end{align*}
Time = 6.36 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.36 \[ \int \frac {\tan ^2(e+f x)}{(a+a \sec (e+f x))^{9/2}} \, dx=\frac {\cos ^3\left (\frac {1}{2} (e+f x)\right ) \sec ^{\frac {9}{2}}(e+f x) \sqrt {a (1+\sec (e+f x))} \left (-384 \arctan \left (\frac {\tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {\frac {1}{1+\sec (e+f x)}}}\right ) \cos ^7\left (\frac {1}{2} (e+f x)\right ) \sqrt {\frac {\cos (e+f x)}{(1+\cos (e+f x))^2}} \sqrt {1+\sec (e+f x)}+273 \arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ) \cos ^7\left (\frac {1}{2} (e+f x)\right ) \sqrt {\sec ^2\left (\frac {1}{2} (e+f x)\right )} \sqrt {\frac {1}{1+\sec (e+f x)}} \sqrt {1+\sec (e+f x)}+\frac {(319+412 \cos (e+f x)+157 \cos (2 (e+f x))) \sin \left (\frac {1}{2} (e+f x)\right )}{8 \sqrt {\sec (e+f x)}}\right )}{6 a^5 f (1+\sec (e+f x))^5} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(330\) vs. \(2(148)=296\).
Time = 3.07 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.87
method | result | size |
default | \(\frac {\sqrt {-\frac {2 a}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\, \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \left (-8 \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right )^{\frac {5}{2}} \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+10 \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right )^{\frac {3}{2}} \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+12 \left (1-\cos \left (f x +e \right )\right )^{3} \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \csc \left (f x +e \right )^{3}-192 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\right )-93 \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+273 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\right )\right )}{192 f \,a^{5}}\) | \(331\) |
[In]
[Out]
none
Time = 0.42 (sec) , antiderivative size = 674, normalized size of antiderivative = 3.81 \[ \int \frac {\tan ^2(e+f x)}{(a+a \sec (e+f x))^{9/2}} \, dx=\left [-\frac {273 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{4} + 4 \, \cos \left (f x + e\right )^{3} + 6 \, \cos \left (f x + e\right )^{2} + 4 \, \cos \left (f x + e\right ) + 1\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 ) + 384 \, {\left (\cos \left (f x + e\right )^{4} + 4 \, \cos \left (f x + e\right )^{3} + 6 \, \cos \left (f x + e\right )^{2} + 4 \, \cos \left (f x + e\right ) + 1\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 (157 \, \cos \left (f x + e\right )^{3} + 206 \, \cos \left (f x + e\right )^{2} + 81 \, \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 )}{384 \, {\left (a^{5} f \cos \left (f x + e\right )^{4} + 4 \, a^{5} f \cos \left (f x + e\right )^{3} + 6 \, a^{5} f \cos \left (f x + e\right )^{2} + 4 \, a^{5} f \cos \left (f x + e\right ) + a^{5} f\right )}}, -\frac {273 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{4} + 4 \, \cos \left (f x + e\right )^{3} + 6 \, \cos \left (f x + e\right )^{2} + 4 \, \cos \left (f x + e\right ) + 1\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 ) - 384 \, {\left (\cos \left (f x + e\right )^{4} + 4 \, \cos \left (f x + e\right )^{3} + 6 \, \cos \left (f x + e\right )^{2} + 4 \, \cos \left (f x + e\right ) + 1\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 (157 \, \cos \left (f x + e\right )^{3} + 206 \, \cos \left (f x + e\right )^{2} + 81 \, \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 )}{192 \, {\left (a^{5} f \cos \left (f x + e\right )^{4} + 4 \, a^{5} f \cos \left (f x + e\right )^{3} + 6 \, a^{5} f \cos \left (f x + e\right )^{2} + 4 \, a^{5} f \cos \left (f x + e\right ) + a^{5} f\right )}}\right ] \]
[In]
[Out]
Timed out. \[ \int \frac {\tan ^2(e+f x)}{(a+a \sec (e+f x))^{9/2}} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {\tan ^2(e+f x)}{(a+a \sec (e+f x))^{9/2}} \, dx=\int { \frac {\tan \left (f x + e\right )^{2}}{{\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {9}{2}}} \,d x } \]
[In]
[Out]
none
Time = 1.34 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.62 \[ \int \frac {\tan ^2(e+f x)}{(a+a \sec (e+f x))^{9/2}} \, dx=\frac {\sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a} {\left (2 \, {\left (\frac {4 \, \sqrt {2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}}{a^{5} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )} - \frac {19 \, \sqrt {2}}{a^{5} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}\right )} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + \frac {111 \, \sqrt {2}}{a^{5} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}\right )} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{192 \, f} \]
[In]
[Out]
Timed out. \[ \int \frac {\tan ^2(e+f x)}{(a+a \sec (e+f x))^{9/2}} \, dx=\int \frac {{\mathrm {tan}\left (e+f\,x\right )}^2}{{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{9/2}} \,d x \]
[In]
[Out]