Integrand size = 31, antiderivative size = 368 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c+d \sec (e+f x))^3} \, dx=-\frac {d^3 \left (20 c^2+30 c d+13 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{a^3 (c-d)^{11/2} (c+d)^{5/2} f}+\frac {d \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right ) \tan (e+f x)}{30 a^3 (c-d)^4 (c+d) f (c+d \sec (e+f x))^2}+\frac {\tan (e+f x)}{5 (c-d) f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2}+\frac {(2 c-11 d) \tan (e+f x)}{15 a (c-d)^2 f (a+a \sec (e+f x))^2 (c+d \sec (e+f x))^2}+\frac {\left (2 c^2-15 c d+76 d^2\right ) \tan (e+f x)}{15 (c-d)^3 f \left (a^3+a^3 \sec (e+f x)\right ) (c+d \sec (e+f x))^2}+\frac {d \left (4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right ) \tan (e+f x)}{30 a^3 (c-d)^5 (c+d)^2 f (c+d \sec (e+f x))} \]
[Out]
Time = 0.85 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.12, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {4072, 105, 156, 157, 12, 95, 211} \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c+d \sec (e+f x))^3} \, dx=\frac {\left (4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right ) \tan (e+f x)}{30 f (c-d)^5 (c+d)^2 \left (a^3 \sec (e+f x)+a^3\right )}+\frac {d^3 \left (20 c^2+30 c d+13 d^2\right ) \tan (e+f x) \arctan \left (\frac {\sqrt {c+d} \sqrt {a \sec (e+f x)+a}}{\sqrt {c-d} \sqrt {a-a \sec (e+f x)}}\right )}{a^2 f (c-d)^{11/2} (c+d)^{5/2} \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}-\frac {3 d (2 c+d) \tan (e+f x)}{2 f \left (c^2-d^2\right )^2 (a \sec (e+f x)+a)^3 (c+d \sec (e+f x))}-\frac {d \tan (e+f x)}{2 f \left (c^2-d^2\right ) (a \sec (e+f x)+a)^3 (c+d \sec (e+f x))^2}+\frac {\left (2 c^2+39 c d+22 d^2\right ) \tan (e+f x)}{10 f (c-d)^3 (c+d)^2 (a \sec (e+f x)+a)^3}+\frac {\left (4 c^3-26 c^2 d-184 c d^2-109 d^3\right ) \tan (e+f x)}{30 a f (c-d)^4 (c+d)^2 (a \sec (e+f x)+a)^2} \]
[In]
[Out]
Rule 12
Rule 95
Rule 105
Rule 156
Rule 157
Rule 211
Rule 4072
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a^2 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-a x} (a+a x)^{7/2} (c+d x)^3} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = -\frac {d \tan (e+f x)}{2 \left (c^2-d^2\right ) f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2}-\frac {\tan (e+f x) \text {Subst}\left (\int \frac {a^2 (2 c+3 d)-4 a^2 d x}{\sqrt {a-a x} (a+a x)^{7/2} (c+d x)^2} \, dx,x,\sec (e+f x)\right )}{2 \left (c^2-d^2\right ) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = -\frac {d \tan (e+f x)}{2 \left (c^2-d^2\right ) f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2}-\frac {3 d (2 c+d) \tan (e+f x)}{2 \left (c^2-d^2\right )^2 f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))}-\frac {\tan (e+f x) \text {Subst}\left (\int \frac {a^4 \left (2 c^2+21 c d+13 d^2\right )-9 a^4 d (2 c+d) x}{\sqrt {a-a x} (a+a x)^{7/2} (c+d x)} \, dx,x,\sec (e+f x)\right )}{2 a^2 \left (c^2-d^2\right )^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {\left (2 c^2+39 c d+22 d^2\right ) \tan (e+f x)}{10 (c-d)^3 (c+d)^2 f (a+a \sec (e+f x))^3}-\frac {d \tan (e+f x)}{2 \left (c^2-d^2\right ) f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2}-\frac {3 d (2 c+d) \tan (e+f x)}{2 \left (c^2-d^2\right )^2 f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))}+\frac {\tan (e+f x) \text {Subst}\left (\int \frac {-a^6 \left (4 c^3-22 c^2 d-106 c d^2-65 d^3\right )-2 a^6 d \left (2 c^2+39 c d+22 d^2\right ) x}{\sqrt {a-a x} (a+a x)^{5/2} (c+d x)} \, dx,x,\sec (e+f x)\right )}{10 a^5 (c-d) \left (c^2-d^2\right )^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {\left (2 c^2+39 c d+22 d^2\right ) \tan (e+f x)}{10 (c-d)^3 (c+d)^2 f (a+a \sec (e+f x))^3}+\frac {\left (4 c^3-26 c^2 d-184 c d^2-109 d^3\right ) \tan (e+f x)}{30 a (c-d)^4 (c+d)^2 f (a+a \sec (e+f x))^2}-\frac {d \tan (e+f x)}{2 \left (c^2-d^2\right ) f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2}-\frac {3 d (2 c+d) \tan (e+f x)}{2 \left (c^2-d^2\right )^2 f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))}-\frac {\tan (e+f x) \text {Subst}\left (\int \frac {a^8 (c+d) \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right )+a^8 d \left (4 c^3-26 c^2 d-184 c d^2-109 d^3\right ) x}{\sqrt {a-a x} (a+a x)^{3/2} (c+d x)} \, dx,x,\sec (e+f x)\right )}{30 a^8 (c-d)^2 \left (c^2-d^2\right )^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {\left (2 c^2+39 c d+22 d^2\right ) \tan (e+f x)}{10 (c-d)^3 (c+d)^2 f (a+a \sec (e+f x))^3}+\frac {\left (4 c^3-26 c^2 d-184 c d^2-109 d^3\right ) \tan (e+f x)}{30 a (c-d)^4 (c+d)^2 f (a+a \sec (e+f x))^2}+\frac {\left (4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right ) \tan (e+f x)}{30 (c-d)^5 (c+d)^2 f \left (a^3+a^3 \sec (e+f x)\right )}-\frac {d \tan (e+f x)}{2 \left (c^2-d^2\right ) f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2}-\frac {3 d (2 c+d) \tan (e+f x)}{2 \left (c^2-d^2\right )^2 f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))}+\frac {\tan (e+f x) \text {Subst}\left (\int \frac {15 a^{10} d^3 \left (20 c^2+30 c d+13 d^2\right )}{\sqrt {a-a x} \sqrt {a+a x} (c+d x)} \, dx,x,\sec (e+f x)\right )}{30 a^{11} (c-d)^3 \left (c^2-d^2\right )^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {\left (2 c^2+39 c d+22 d^2\right ) \tan (e+f x)}{10 (c-d)^3 (c+d)^2 f (a+a \sec (e+f x))^3}+\frac {\left (4 c^3-26 c^2 d-184 c d^2-109 d^3\right ) \tan (e+f x)}{30 a (c-d)^4 (c+d)^2 f (a+a \sec (e+f x))^2}+\frac {\left (4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right ) \tan (e+f x)}{30 (c-d)^5 (c+d)^2 f \left (a^3+a^3 \sec (e+f x)\right )}-\frac {d \tan (e+f x)}{2 \left (c^2-d^2\right ) f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2}-\frac {3 d (2 c+d) \tan (e+f x)}{2 \left (c^2-d^2\right )^2 f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))}+\frac {\left (d^3 \left (20 c^2+30 c d+13 d^2\right ) \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-a x} \sqrt {a+a x} (c+d x)} \, dx,x,\sec (e+f x)\right )}{2 a (c-d)^3 \left (c^2-d^2\right )^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {\left (2 c^2+39 c d+22 d^2\right ) \tan (e+f x)}{10 (c-d)^3 (c+d)^2 f (a+a \sec (e+f x))^3}+\frac {\left (4 c^3-26 c^2 d-184 c d^2-109 d^3\right ) \tan (e+f x)}{30 a (c-d)^4 (c+d)^2 f (a+a \sec (e+f x))^2}+\frac {\left (4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right ) \tan (e+f x)}{30 (c-d)^5 (c+d)^2 f \left (a^3+a^3 \sec (e+f x)\right )}-\frac {d \tan (e+f x)}{2 \left (c^2-d^2\right ) f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2}-\frac {3 d (2 c+d) \tan (e+f x)}{2 \left (c^2-d^2\right )^2 f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))}+\frac {\left (d^3 \left (20 c^2+30 c d+13 d^2\right ) \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{a c-a d-(-a c-a d) x^2} \, dx,x,\frac {\sqrt {a+a \sec (e+f x)}}{\sqrt {a-a \sec (e+f x)}}\right )}{a (c-d)^3 \left (c^2-d^2\right )^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {\left (2 c^2+39 c d+22 d^2\right ) \tan (e+f x)}{10 (c-d)^3 (c+d)^2 f (a+a \sec (e+f x))^3}+\frac {\left (4 c^3-26 c^2 d-184 c d^2-109 d^3\right ) \tan (e+f x)}{30 a (c-d)^4 (c+d)^2 f (a+a \sec (e+f x))^2}+\frac {d^3 \left (20 c^2+30 c d+13 d^2\right ) \arctan \left (\frac {\sqrt {c+d} \sqrt {a+a \sec (e+f x)}}{\sqrt {c-d} \sqrt {a-a \sec (e+f x)}}\right ) \tan (e+f x)}{a^2 (c-d)^{11/2} (c+d)^{5/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {\left (4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right ) \tan (e+f x)}{30 (c-d)^5 (c+d)^2 f \left (a^3+a^3 \sec (e+f x)\right )}-\frac {d \tan (e+f x)}{2 \left (c^2-d^2\right ) f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2}-\frac {3 d (2 c+d) \tan (e+f x)}{2 \left (c^2-d^2\right )^2 f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))} \\ \end{align*}
Result contains complex when optimal does not.
Time = 9.05 (sec) , antiderivative size = 1096, normalized size of antiderivative = 2.98 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c+d \sec (e+f x))^3} \, dx=\frac {4 \cos ^4\left (\frac {e}{2}+\frac {f x}{2}\right ) (d+c \cos (e+f x))^3 \sec \left (\frac {e}{2}\right ) \sec ^6(e+f x) \left (-8 c \sin \left (\frac {e}{2}\right )+23 d \sin \left (\frac {e}{2}\right )\right )}{15 (-c+d)^4 f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^3}+\frac {\left (20 c^2+30 c d+13 d^2\right ) \cos ^6\left (\frac {e}{2}+\frac {f x}{2}\right ) (d+c \cos (e+f x))^3 \sec ^6(e+f x) \left (-\frac {8 i d^3 \arctan \left (\sec \left (\frac {f x}{2}\right ) \left (\frac {\cos (e)}{\sqrt {c^2-d^2} \sqrt {\cos (2 e)-i \sin (2 e)}}-\frac {i \sin (e)}{\sqrt {c^2-d^2} \sqrt {\cos (2 e)-i \sin (2 e)}}\right ) \left (-i d \sin \left (\frac {f x}{2}\right )+i c \sin \left (e+\frac {f x}{2}\right )\right )\right ) \cos (e)}{\sqrt {c^2-d^2} f \sqrt {\cos (2 e)-i \sin (2 e)}}-\frac {8 d^3 \arctan \left (\sec \left (\frac {f x}{2}\right ) \left (\frac {\cos (e)}{\sqrt {c^2-d^2} \sqrt {\cos (2 e)-i \sin (2 e)}}-\frac {i \sin (e)}{\sqrt {c^2-d^2} \sqrt {\cos (2 e)-i \sin (2 e)}}\right ) \left (-i d \sin \left (\frac {f x}{2}\right )+i c \sin \left (e+\frac {f x}{2}\right )\right )\right ) \sin (e)}{\sqrt {c^2-d^2} f \sqrt {\cos (2 e)-i \sin (2 e)}}\right )}{(-c+d)^5 (c+d)^2 (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^3}-\frac {2 \cos \left (\frac {e}{2}+\frac {f x}{2}\right ) (d+c \cos (e+f x))^3 \sec \left (\frac {e}{2}\right ) \sec ^6(e+f x) \sin \left (\frac {f x}{2}\right )}{5 (-c+d)^3 f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^3}+\frac {4 \cos ^3\left (\frac {e}{2}+\frac {f x}{2}\right ) (d+c \cos (e+f x))^3 \sec \left (\frac {e}{2}\right ) \sec ^6(e+f x) \left (-8 c \sin \left (\frac {f x}{2}\right )+23 d \sin \left (\frac {f x}{2}\right )\right )}{15 (-c+d)^4 f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^3}-\frac {8 \cos ^5\left (\frac {e}{2}+\frac {f x}{2}\right ) (d+c \cos (e+f x))^3 \sec \left (\frac {e}{2}\right ) \sec ^6(e+f x) \left (7 c^2 \sin \left (\frac {f x}{2}\right )-44 c d \sin \left (\frac {f x}{2}\right )+127 d^2 \sin \left (\frac {f x}{2}\right )\right )}{15 (-c+d)^5 f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^3}+\frac {4 \cos ^6\left (\frac {e}{2}+\frac {f x}{2}\right ) (d+c \cos (e+f x)) \sec (e) \sec ^6(e+f x) \left (d^6 \sin (e)-c d^5 \sin (f x)\right )}{c^2 (-c+d)^4 (c+d) f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^3}-\frac {4 \cos ^6\left (\frac {e}{2}+\frac {f x}{2}\right ) (d+c \cos (e+f x))^2 \sec (e) \sec ^6(e+f x) \left (-11 c^2 d^5 \sin (e)-6 c d^6 \sin (e)+2 d^7 \sin (e)+10 c^3 d^4 \sin (f x)+6 c^2 d^5 \sin (f x)-c d^6 \sin (f x)\right )}{c^2 (-c+d)^5 (c+d)^2 f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^3}-\frac {2 \cos ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) (d+c \cos (e+f x))^3 \sec ^6(e+f x) \tan \left (\frac {e}{2}\right )}{5 (-c+d)^3 f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^3} \]
[In]
[Out]
Time = 1.21 (sec) , antiderivative size = 365, normalized size of antiderivative = 0.99
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} c^{2}}{5}-\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} c d}{5}+\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} d^{2}}{5}-\frac {2 c^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}+\frac {10 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} c d}{3}-\frac {8 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} d^{2}}{3}+\tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c^{2}-8 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c d +31 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) d^{2}}{\left (c^{3}-3 c^{2} d +3 c \,d^{2}-d^{3}\right ) \left (c^{2}-2 c d +d^{2}\right )}+\frac {16 d^{3} \left (\frac {-\frac {d \left (10 c^{2}-3 c d -7 d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{4 \left (c^{2}+2 c d +d^{2}\right )}+\frac {5 d \left (2 c +d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 \left (c +d \right )}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} c -\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} d -c -d \right )^{2}}-\frac {\left (20 c^{2}+30 c d +13 d^{2}\right ) \operatorname {arctanh}\left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{4 \left (c^{2}+2 c d +d^{2}\right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c -d \right )^{5}}}{4 f \,a^{3}}\) | \(365\) |
default | \(\frac {\frac {\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} c^{2}}{5}-\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} c d}{5}+\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} d^{2}}{5}-\frac {2 c^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}+\frac {10 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} c d}{3}-\frac {8 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} d^{2}}{3}+\tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c^{2}-8 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c d +31 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) d^{2}}{\left (c^{3}-3 c^{2} d +3 c \,d^{2}-d^{3}\right ) \left (c^{2}-2 c d +d^{2}\right )}+\frac {16 d^{3} \left (\frac {-\frac {d \left (10 c^{2}-3 c d -7 d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{4 \left (c^{2}+2 c d +d^{2}\right )}+\frac {5 d \left (2 c +d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 \left (c +d \right )}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} c -\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} d -c -d \right )^{2}}-\frac {\left (20 c^{2}+30 c d +13 d^{2}\right ) \operatorname {arctanh}\left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{4 \left (c^{2}+2 c d +d^{2}\right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c -d \right )^{5}}}{4 f \,a^{3}}\) | \(365\) |
risch | \(\text {Expression too large to display}\) | \(1764\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 1310 vs. \(2 (349) = 698\).
Time = 0.39 (sec) , antiderivative size = 2677, normalized size of antiderivative = 7.27 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c+d \sec (e+f x))^3} \, dx=\text {Too large to display} \]
[In]
[Out]
\[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c+d \sec (e+f x))^3} \, dx=\frac {\int \frac {\sec {\left (e + f x \right )}}{c^{3} \sec ^{3}{\left (e + f x \right )} + 3 c^{3} \sec ^{2}{\left (e + f x \right )} + 3 c^{3} \sec {\left (e + f x \right )} + c^{3} + 3 c^{2} d \sec ^{4}{\left (e + f x \right )} + 9 c^{2} d \sec ^{3}{\left (e + f x \right )} + 9 c^{2} d \sec ^{2}{\left (e + f x \right )} + 3 c^{2} d \sec {\left (e + f x \right )} + 3 c d^{2} \sec ^{5}{\left (e + f x \right )} + 9 c d^{2} \sec ^{4}{\left (e + f x \right )} + 9 c d^{2} \sec ^{3}{\left (e + f x \right )} + 3 c d^{2} \sec ^{2}{\left (e + f x \right )} + d^{3} \sec ^{6}{\left (e + f x \right )} + 3 d^{3} \sec ^{5}{\left (e + f x \right )} + 3 d^{3} \sec ^{4}{\left (e + f x \right )} + d^{3} \sec ^{3}{\left (e + f x \right )}}\, dx}{a^{3}} \]
[In]
[Out]
Exception generated. \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c+d \sec (e+f x))^3} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 1369 vs. \(2 (349) = 698\).
Time = 0.47 (sec) , antiderivative size = 1369, normalized size of antiderivative = 3.72 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c+d \sec (e+f x))^3} \, dx=\text {Too large to display} \]
[In]
[Out]
Time = 14.09 (sec) , antiderivative size = 655, normalized size of antiderivative = 1.78 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c+d \sec (e+f x))^3} \, dx=\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{20\,a^3\,f\,{\left (c-d\right )}^3}-\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {3\,{\left (c+d\right )}^2}{4\,a^3\,{\left (c-d\right )}^5}-\frac {5}{2\,a^3\,{\left (c-d\right )}^3}+\frac {3\,\left (c+d\right )\,\left (\frac {5}{4\,a^3\,{\left (c-d\right )}^3}-\frac {3\,\left (c+d\right )}{4\,a^3\,{\left (c-d\right )}^4}\right )}{c-d}\right )}{f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (\frac {5}{12\,a^3\,{\left (c-d\right )}^3}-\frac {c+d}{4\,a^3\,{\left (c-d\right )}^4}\right )}{f}-\frac {\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (-10\,c^2\,d^4+3\,c\,d^5+7\,d^6\right )}{{\left (c+d\right )}^2}+\frac {5\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (d^5+2\,c\,d^4\right )}{c+d}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (2\,a^3\,c^7-10\,a^3\,c^6\,d+18\,a^3\,c^5\,d^2-10\,a^3\,c^4\,d^3-10\,a^3\,c^3\,d^4+18\,a^3\,c^2\,d^5-10\,a^3\,c\,d^6+2\,a^3\,d^7\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (a^3\,c^7-7\,a^3\,c^6\,d+21\,a^3\,c^5\,d^2-35\,a^3\,c^4\,d^3+35\,a^3\,c^3\,d^4-21\,a^3\,c^2\,d^5+7\,a^3\,c\,d^6-a^3\,d^7\right )-a^3\,c^7+a^3\,d^7-3\,a^3\,c\,d^6+3\,a^3\,c^6\,d+a^3\,c^2\,d^5+5\,a^3\,c^3\,d^4-5\,a^3\,c^4\,d^3-a^3\,c^5\,d^2\right )}+\frac {d^3\,\mathrm {atan}\left (\frac {1{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c^6-6{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c^5\,d+15{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c^4\,d^2-20{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c^3\,d^3+15{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c^2\,d^4-6{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c\,d^5+1{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,d^6}{\sqrt {c+d}\,{\left (c-d\right )}^{11/2}}\right )\,\left (20\,c^2+30\,c\,d+13\,d^2\right )\,1{}\mathrm {i}}{a^3\,f\,{\left (c+d\right )}^{5/2}\,{\left (c-d\right )}^{11/2}} \]
[In]
[Out]