Integrand size = 31, antiderivative size = 288 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2} \, dx=-\frac {2 d^3 (4 c+3 d) \text {arctanh}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{a^3 (c-d)^{9/2} (c+d)^{3/2} f}+\frac {d \left (2 c^3-12 c^2 d+43 c d^2+72 d^3\right ) \tan (e+f x)}{15 a^3 (c-d)^4 (c+d) f (c+d \sec (e+f x))}+\frac {\tan (e+f x)}{5 (c-d) f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))}+\frac {(2 c-9 d) \tan (e+f x)}{15 a (c-d)^2 f (a+a \sec (e+f x))^2 (c+d \sec (e+f x))}+\frac {\left (2 c^2-12 c d+45 d^2\right ) \tan (e+f x)}{15 (c-d)^3 f \left (a^3+a^3 \sec (e+f x)\right ) (c+d \sec (e+f x))} \]
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Time = 0.65 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.13, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {4072, 105, 157, 12, 95, 211} \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2} \, dx=\frac {\left (2 c^3-12 c^2 d+43 c d^2+72 d^3\right ) \tan (e+f x)}{15 f (c-d)^4 (c+d) \left (a^3 \sec (e+f x)+a^3\right )}+\frac {2 d^3 (4 c+3 d) \tan (e+f x) \arctan \left (\frac {\sqrt {c+d} \sqrt {a \sec (e+f x)+a}}{\sqrt {c-d} \sqrt {a-a \sec (e+f x)}}\right )}{a^2 f (c-d)^{9/2} (c+d)^{3/2} \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}-\frac {d \tan (e+f x)}{f \left (c^2-d^2\right ) (a \sec (e+f x)+a)^3 (c+d \sec (e+f x))}+\frac {\left (2 c^2-10 c d-27 d^2\right ) \tan (e+f x)}{15 a f (c-d)^3 (c+d) (a \sec (e+f x)+a)^2}+\frac {(c+6 d) \tan (e+f x)}{5 f (c-d)^2 (c+d) (a \sec (e+f x)+a)^3} \]
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Rule 12
Rule 95
Rule 105
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)^2} \, 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)}{\left (c^2-d^2\right ) 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^2 (c+3 d)-3 a^2 d x}{\sqrt {a-a x} (a+a x)^{7/2} (c+d x)} \, dx,x,\sec (e+f x)\right )}{\left (c^2-d^2\right ) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {(c+6 d) \tan (e+f x)}{5 (c-d)^2 (c+d) f (a+a \sec (e+f x))^3}-\frac {d \tan (e+f x)}{\left (c^2-d^2\right ) 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^2-8 c d-15 d^2\right )-2 a^4 d (c+6 d) x}{\sqrt {a-a x} (a+a x)^{5/2} (c+d x)} \, dx,x,\sec (e+f x)\right )}{5 a^3 (c-d) \left (c^2-d^2\right ) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {(c+6 d) \tan (e+f x)}{5 (c-d)^2 (c+d) f (a+a \sec (e+f x))^3}+\frac {\left (2 c^2-10 c d-27 d^2\right ) \tan (e+f x)}{15 a (c-d)^3 (c+d) f (a+a \sec (e+f x))^2}-\frac {d \tan (e+f x)}{\left (c^2-d^2\right ) 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 (c+d) \left (2 c^2-12 c d+45 d^2\right )+a^6 d \left (2 c^2-10 c d-27 d^2\right ) x}{\sqrt {a-a x} (a+a x)^{3/2} (c+d x)} \, dx,x,\sec (e+f x)\right )}{15 a^6 (c-d)^2 \left (c^2-d^2\right ) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {(c+6 d) \tan (e+f x)}{5 (c-d)^2 (c+d) f (a+a \sec (e+f x))^3}+\frac {\left (2 c^2-10 c d-27 d^2\right ) \tan (e+f x)}{15 a (c-d)^3 (c+d) f (a+a \sec (e+f x))^2}+\frac {\left (2 c^3-12 c^2 d+43 c d^2+72 d^3\right ) \tan (e+f x)}{15 (c-d)^4 (c+d) f \left (a^3+a^3 \sec (e+f x)\right )}-\frac {d \tan (e+f x)}{\left (c^2-d^2\right ) 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^8 d^3 (4 c+3 d)}{\sqrt {a-a x} \sqrt {a+a x} (c+d x)} \, dx,x,\sec (e+f x)\right )}{15 a^9 (c-d)^3 \left (c^2-d^2\right ) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {(c+6 d) \tan (e+f x)}{5 (c-d)^2 (c+d) f (a+a \sec (e+f x))^3}+\frac {\left (2 c^2-10 c d-27 d^2\right ) \tan (e+f x)}{15 a (c-d)^3 (c+d) f (a+a \sec (e+f x))^2}+\frac {\left (2 c^3-12 c^2 d+43 c d^2+72 d^3\right ) \tan (e+f x)}{15 (c-d)^4 (c+d) f \left (a^3+a^3 \sec (e+f x)\right )}-\frac {d \tan (e+f x)}{\left (c^2-d^2\right ) f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))}+\frac {\left (d^3 (4 c+3 d) \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 )}{a (c-d)^3 \left (c^2-d^2\right ) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {(c+6 d) \tan (e+f x)}{5 (c-d)^2 (c+d) f (a+a \sec (e+f x))^3}+\frac {\left (2 c^2-10 c d-27 d^2\right ) \tan (e+f x)}{15 a (c-d)^3 (c+d) f (a+a \sec (e+f x))^2}+\frac {\left (2 c^3-12 c^2 d+43 c d^2+72 d^3\right ) \tan (e+f x)}{15 (c-d)^4 (c+d) f \left (a^3+a^3 \sec (e+f x)\right )}-\frac {d \tan (e+f x)}{\left (c^2-d^2\right ) f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))}+\frac {\left (2 d^3 (4 c+3 d) \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 ) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {(c+6 d) \tan (e+f x)}{5 (c-d)^2 (c+d) f (a+a \sec (e+f x))^3}+\frac {\left (2 c^2-10 c d-27 d^2\right ) \tan (e+f x)}{15 a (c-d)^3 (c+d) f (a+a \sec (e+f x))^2}+\frac {2 d^3 (4 c+3 d) \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)^{9/2} (c+d)^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {\left (2 c^3-12 c^2 d+43 c d^2+72 d^3\right ) \tan (e+f x)}{15 (c-d)^4 (c+d) f \left (a^3+a^3 \sec (e+f x)\right )}-\frac {d \tan (e+f x)}{\left (c^2-d^2\right ) f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))} \\ \end{align*}
Result contains complex when optimal does not.
Time = 7.96 (sec) , antiderivative size = 1772, normalized size of antiderivative = 6.15 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2} \, dx=\frac {(4 c+3 d) \cos ^6\left (\frac {e}{2}+\frac {f x}{2}\right ) (d+c \cos (e+f x))^2 \sec ^5(e+f x) \left (\frac {16 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 {16 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)^4 (c+d) (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2}+\frac {\cos \left (\frac {e}{2}+\frac {f x}{2}\right ) (d+c \cos (e+f x)) \sec \left (\frac {e}{2}\right ) \sec (e) \sec ^5(e+f x) \left (-55 c^5 \sin \left (\frac {f x}{2}\right )+135 c^4 d \sin \left (\frac {f x}{2}\right )-20 c^3 d^2 \sin \left (\frac {f x}{2}\right )-810 c^2 d^3 \sin \left (\frac {f x}{2}\right )-450 c d^4 \sin \left (\frac {f x}{2}\right )+150 d^5 \sin \left (\frac {f x}{2}\right )+47 c^5 \sin \left (\frac {3 f x}{2}\right )-137 c^4 d \sin \left (\frac {3 f x}{2}\right )+88 c^3 d^2 \sin \left (\frac {3 f x}{2}\right )+812 c^2 d^3 \sin \left (\frac {3 f x}{2}\right )+690 c d^4 \sin \left (\frac {3 f x}{2}\right )+75 d^5 \sin \left (\frac {3 f x}{2}\right )-50 c^5 \sin \left (e-\frac {f x}{2}\right )+130 c^4 d \sin \left (e-\frac {f x}{2}\right )-10 c^3 d^2 \sin \left (e-\frac {f x}{2}\right )-1030 c^2 d^3 \sin \left (e-\frac {f x}{2}\right )-990 c d^4 \sin \left (e-\frac {f x}{2}\right )-150 d^5 \sin \left (e-\frac {f x}{2}\right )+50 c^5 \sin \left (e+\frac {f x}{2}\right )-130 c^4 d \sin \left (e+\frac {f x}{2}\right )+10 c^3 d^2 \sin \left (e+\frac {f x}{2}\right )+1030 c^2 d^3 \sin \left (e+\frac {f x}{2}\right )+765 c d^4 \sin \left (e+\frac {f x}{2}\right )-150 d^5 \sin \left (e+\frac {f x}{2}\right )-55 c^5 \sin \left (2 e+\frac {f x}{2}\right )+135 c^4 d \sin \left (2 e+\frac {f x}{2}\right )-20 c^3 d^2 \sin \left (2 e+\frac {f x}{2}\right )-810 c^2 d^3 \sin \left (2 e+\frac {f x}{2}\right )-675 c d^4 \sin \left (2 e+\frac {f x}{2}\right )-150 d^5 \sin \left (2 e+\frac {f x}{2}\right )-30 c^5 \sin \left (e+\frac {3 f x}{2}\right )+90 c^4 d \sin \left (e+\frac {3 f x}{2}\right )-60 c^3 d^2 \sin \left (e+\frac {3 f x}{2}\right )-360 c^2 d^3 \sin \left (e+\frac {3 f x}{2}\right )-30 c d^4 \sin \left (e+\frac {3 f x}{2}\right )+75 d^5 \sin \left (e+\frac {3 f x}{2}\right )+47 c^5 \sin \left (2 e+\frac {3 f x}{2}\right )-137 c^4 d \sin \left (2 e+\frac {3 f x}{2}\right )+88 c^3 d^2 \sin \left (2 e+\frac {3 f x}{2}\right )+812 c^2 d^3 \sin \left (2 e+\frac {3 f x}{2}\right )+525 c d^4 \sin \left (2 e+\frac {3 f x}{2}\right )-75 d^5 \sin \left (2 e+\frac {3 f x}{2}\right )-30 c^5 \sin \left (3 e+\frac {3 f x}{2}\right )+90 c^4 d \sin \left (3 e+\frac {3 f x}{2}\right )-60 c^3 d^2 \sin \left (3 e+\frac {3 f x}{2}\right )-360 c^2 d^3 \sin \left (3 e+\frac {3 f x}{2}\right )-195 c d^4 \sin \left (3 e+\frac {3 f x}{2}\right )-75 d^5 \sin \left (3 e+\frac {3 f x}{2}\right )+20 c^5 \sin \left (e+\frac {5 f x}{2}\right )-76 c^4 d \sin \left (e+\frac {5 f x}{2}\right )+106 c^3 d^2 \sin \left (e+\frac {5 f x}{2}\right )+346 c^2 d^3 \sin \left (e+\frac {5 f x}{2}\right )+219 c d^4 \sin \left (e+\frac {5 f x}{2}\right )+15 d^5 \sin \left (e+\frac {5 f x}{2}\right )-15 c^5 \sin \left (2 e+\frac {5 f x}{2}\right )+45 c^4 d \sin \left (2 e+\frac {5 f x}{2}\right )-30 c^3 d^2 \sin \left (2 e+\frac {5 f x}{2}\right )-90 c^2 d^3 \sin \left (2 e+\frac {5 f x}{2}\right )+75 c d^4 \sin \left (2 e+\frac {5 f x}{2}\right )+15 d^5 \sin \left (2 e+\frac {5 f x}{2}\right )+20 c^5 \sin \left (3 e+\frac {5 f x}{2}\right )-76 c^4 d \sin \left (3 e+\frac {5 f x}{2}\right )+106 c^3 d^2 \sin \left (3 e+\frac {5 f x}{2}\right )+346 c^2 d^3 \sin \left (3 e+\frac {5 f x}{2}\right )+144 c d^4 \sin \left (3 e+\frac {5 f x}{2}\right )-15 d^5 \sin \left (3 e+\frac {5 f x}{2}\right )-15 c^5 \sin \left (4 e+\frac {5 f x}{2}\right )+45 c^4 d \sin \left (4 e+\frac {5 f x}{2}\right )-30 c^3 d^2 \sin \left (4 e+\frac {5 f x}{2}\right )-90 c^2 d^3 \sin \left (4 e+\frac {5 f x}{2}\right )-15 d^5 \sin \left (4 e+\frac {5 f x}{2}\right )+7 c^5 \sin \left (2 e+\frac {7 f x}{2}\right )-27 c^4 d \sin \left (2 e+\frac {7 f x}{2}\right )+38 c^3 d^2 \sin \left (2 e+\frac {7 f x}{2}\right )+72 c^2 d^3 \sin \left (2 e+\frac {7 f x}{2}\right )+15 c d^4 \sin \left (2 e+\frac {7 f x}{2}\right )+15 c d^4 \sin \left (3 e+\frac {7 f x}{2}\right )+7 c^5 \sin \left (4 e+\frac {7 f x}{2}\right )-27 c^4 d \sin \left (4 e+\frac {7 f x}{2}\right )+38 c^3 d^2 \sin \left (4 e+\frac {7 f x}{2}\right )+72 c^2 d^3 \sin \left (4 e+\frac {7 f x}{2}\right )\right )}{120 c (-c+d)^4 (c+d) f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2} \]
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Time = 0.89 (sec) , antiderivative size = 284, normalized size of antiderivative = 0.99
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} c^{2}}{5}-\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} c d}{5}+\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} d^{2}}{5}-\frac {2 c^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}+\frac {8 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} c d}{3}-2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} d^{2}+\tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c^{2}-6 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c d +17 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) d^{2}}{\left (c^{2}-2 c d +d^{2}\right ) \left (c -d \right )^{2}}+\frac {16 d^{3} \left (-\frac {d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 \left (c +d \right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} c -\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} d -c -d \right )}-\frac {\left (4 c +3 d \right ) \operatorname {arctanh}\left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{2 \left (c +d \right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c -d \right )^{4}}}{4 f \,a^{3}}\) | \(284\) |
default | \(\frac {\frac {\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} c^{2}}{5}-\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} c d}{5}+\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} d^{2}}{5}-\frac {2 c^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}+\frac {8 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} c d}{3}-2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} d^{2}+\tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c^{2}-6 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c d +17 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) d^{2}}{\left (c^{2}-2 c d +d^{2}\right ) \left (c -d \right )^{2}}+\frac {16 d^{3} \left (-\frac {d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 \left (c +d \right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} c -\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} d -c -d \right )}-\frac {\left (4 c +3 d \right ) \operatorname {arctanh}\left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{2 \left (c +d \right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c -d \right )^{4}}}{4 f \,a^{3}}\) | \(284\) |
risch | \(\frac {2 i \left (7 c^{5}+10 c^{3} d^{2} {\mathrm e}^{3 i \left (f x +e \right )}-137 c^{4} d \,{\mathrm e}^{2 i \left (f x +e \right )}+106 c^{3} d^{2} {\mathrm e}^{i \left (f x +e \right )}-76 c^{4} d \,{\mathrm e}^{i \left (f x +e \right )}+195 c \,d^{4} {\mathrm e}^{5 i \left (f x +e \right )}+990 c \,d^{4} {\mathrm e}^{3 i \left (f x +e \right )}+60 c^{3} d^{2} {\mathrm e}^{5 i \left (f x +e \right )}+360 c^{2} d^{3} {\mathrm e}^{5 i \left (f x +e \right )}+20 c^{3} d^{2} {\mathrm e}^{4 i \left (f x +e \right )}+810 c^{2} d^{3} {\mathrm e}^{4 i \left (f x +e \right )}+88 c^{3} d^{2} {\mathrm e}^{2 i \left (f x +e \right )}+812 c^{2} d^{3} {\mathrm e}^{2 i \left (f x +e \right )}+1030 c^{2} d^{3} {\mathrm e}^{3 i \left (f x +e \right )}-130 c^{4} d \,{\mathrm e}^{3 i \left (f x +e \right )}-45 c^{4} d \,{\mathrm e}^{6 i \left (f x +e \right )}-135 c^{4} d \,{\mathrm e}^{4 i \left (f x +e \right )}+72 c^{2} d^{3}+15 c \,d^{4}+38 c^{3} d^{2}+47 c^{5} {\mathrm e}^{2 i \left (f x +e \right )}+75 d^{5} {\mathrm e}^{2 i \left (f x +e \right )}+75 d^{5} {\mathrm e}^{5 i \left (f x +e \right )}+15 d^{5} {\mathrm e}^{6 i \left (f x +e \right )}-27 c^{4} d +219 c \,d^{4} {\mathrm e}^{i \left (f x +e \right )}-90 c^{4} d \,{\mathrm e}^{5 i \left (f x +e \right )}+346 c^{2} d^{3} {\mathrm e}^{i \left (f x +e \right )}+675 c \,d^{4} {\mathrm e}^{4 i \left (f x +e \right )}+690 c \,d^{4} {\mathrm e}^{2 i \left (f x +e \right )}+90 c^{2} d^{3} {\mathrm e}^{6 i \left (f x +e \right )}+30 c^{3} d^{2} {\mathrm e}^{6 i \left (f x +e \right )}+50 c^{5} {\mathrm e}^{3 i \left (f x +e \right )}+150 d^{5} {\mathrm e}^{4 i \left (f x +e \right )}+15 c^{5} {\mathrm e}^{6 i \left (f x +e \right )}+30 c^{5} {\mathrm e}^{5 i \left (f x +e \right )}+55 c^{5} {\mathrm e}^{4 i \left (f x +e \right )}+15 d^{5} {\mathrm e}^{i \left (f x +e \right )}+150 d^{5} {\mathrm e}^{3 i \left (f x +e \right )}+20 c^{5} {\mathrm e}^{i \left (f x +e \right )}\right )}{15 \left ({\mathrm e}^{2 i \left (f x +e \right )} c +2 d \,{\mathrm e}^{i \left (f x +e \right )}+c \right ) \left (-c^{2}+d^{2}\right ) c \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{5} \left (-c +d \right )^{3} a^{3} f}+\frac {4 d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {i c^{2}-i d^{2}-\sqrt {c^{2}-d^{2}}\, d}{\sqrt {c^{2}-d^{2}}\, c}\right ) c}{\sqrt {c^{2}-d^{2}}\, \left (c +d \right ) \left (c -d \right )^{4} f \,a^{3}}+\frac {3 d^{4} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {i c^{2}-i d^{2}-\sqrt {c^{2}-d^{2}}\, d}{\sqrt {c^{2}-d^{2}}\, c}\right )}{\sqrt {c^{2}-d^{2}}\, \left (c +d \right ) \left (c -d \right )^{4} f \,a^{3}}-\frac {4 d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c^{2}-i d^{2}+\sqrt {c^{2}-d^{2}}\, d}{\sqrt {c^{2}-d^{2}}\, c}\right ) c}{\sqrt {c^{2}-d^{2}}\, \left (c +d \right ) \left (c -d \right )^{4} f \,a^{3}}-\frac {3 d^{4} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c^{2}-i d^{2}+\sqrt {c^{2}-d^{2}}\, d}{\sqrt {c^{2}-d^{2}}\, c}\right )}{\sqrt {c^{2}-d^{2}}\, \left (c +d \right ) \left (c -d \right )^{4} f \,a^{3}}\) | \(993\) |
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Leaf count of result is larger than twice the leaf count of optimal. 818 vs. \(2 (271) = 542\).
Time = 0.34 (sec) , antiderivative size = 1693, normalized size of antiderivative = 5.88 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2} \, dx=\text {Too large to display} \]
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\[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2} \, dx=\frac {\int \frac {\sec {\left (e + f x \right )}}{c^{2} \sec ^{3}{\left (e + f x \right )} + 3 c^{2} \sec ^{2}{\left (e + f x \right )} + 3 c^{2} \sec {\left (e + f x \right )} + c^{2} + 2 c d \sec ^{4}{\left (e + f x \right )} + 6 c d \sec ^{3}{\left (e + f x \right )} + 6 c d \sec ^{2}{\left (e + f x \right )} + 2 c d \sec {\left (e + f x \right )} + d^{2} \sec ^{5}{\left (e + f x \right )} + 3 d^{2} \sec ^{4}{\left (e + f x \right )} + 3 d^{2} \sec ^{3}{\left (e + f x \right )} + d^{2} \sec ^{2}{\left (e + f x \right )}}\, dx}{a^{3}} \]
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Exception generated. \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 918 vs. \(2 (271) = 542\).
Time = 0.39 (sec) , antiderivative size = 918, normalized size of antiderivative = 3.19 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2} \, dx=\text {Too large to display} \]
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Time = 13.88 (sec) , antiderivative size = 464, normalized size of antiderivative = 1.61 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2} \, dx=\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{20\,a^3\,f\,{\left (c-d\right )}^2}-\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {2\,\left (c^2-d^2\right )\,\left (\frac {1}{a^3\,{\left (c-d\right )}^2}-\frac {c^2-d^2}{2\,a^3\,{\left (c-d\right )}^4}\right )}{{\left (c-d\right )}^2}-\frac {3}{2\,a^3\,{\left (c-d\right )}^2}+\frac {{\left (c+d\right )}^2}{4\,a^3\,{\left (c-d\right )}^4}\right )}{f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (\frac {1}{3\,a^3\,{\left (c-d\right )}^2}-\frac {c^2-d^2}{6\,a^3\,{\left (c-d\right )}^4}\right )}{f}+\frac {2\,d^4\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,\left (c+d\right )\,\left (a^3\,c^5-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (a^3\,c^5-5\,a^3\,c^4\,d+10\,a^3\,c^3\,d^2-10\,a^3\,c^2\,d^3+5\,a^3\,c\,d^4-a^3\,d^5\right )+a^3\,d^5-3\,a^3\,c\,d^4-3\,a^3\,c^4\,d+2\,a^3\,c^2\,d^3+2\,a^3\,c^3\,d^2\right )}+\frac {d^3\,\mathrm {atan}\left (\frac {1{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c^5-5{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c^4\,d+10{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c^3\,d^2-10{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c^2\,d^3+5{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c\,d^4-1{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,d^5}{\sqrt {c+d}\,{\left (c-d\right )}^{9/2}}\right )\,\left (4\,c+3\,d\right )\,2{}\mathrm {i}}{a^3\,f\,{\left (c+d\right )}^{3/2}\,{\left (c-d\right )}^{9/2}} \]
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