Integrand size = 31, antiderivative size = 368 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c+d \sec (e+f x))^3} \, dx=-\frac {d^3 \left (20 c^2+30 c d+13 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{a^3 (c-d)^{11/2} (c+d)^{5/2} f}+\frac {d \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right ) \tan (e+f x)}{30 a^3 (c-d)^4 (c+d) f (c+d \sec (e+f x))^2}+\frac {\tan (e+f x)}{5 (c-d) f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2}+\frac {(2 c-11 d) \tan (e+f x)}{15 a (c-d)^2 f (a+a \sec (e+f x))^2 (c+d \sec (e+f x))^2}+\frac {\left (2 c^2-15 c d+76 d^2\right ) \tan (e+f x)}{15 (c-d)^3 f \left (a^3+a^3 \sec (e+f x)\right ) (c+d \sec (e+f x))^2}+\frac {d \left (4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right ) \tan (e+f x)}{30 a^3 (c-d)^5 (c+d)^2 f (c+d \sec (e+f x))} \]
-d^3*(20*c^2+30*c*d+13*d^2)*arctanh((c-d)^(1/2)*tan(1/2*f*x+1/2*e)/(c+d)^( 1/2))/a^3/(c-d)^(11/2)/(c+d)^(5/2)/f+1/30*d*(4*c^3-30*c^2*d+146*c*d^2+195* d^3)*tan(f*x+e)/a^3/(c-d)^4/(c+d)/f/(c+d*sec(f*x+e))^2+1/5*tan(f*x+e)/(c-d )/f/(a+a*sec(f*x+e))^3/(c+d*sec(f*x+e))^2+1/15*(2*c-11*d)*tan(f*x+e)/a/(c- d)^2/f/(a+a*sec(f*x+e))^2/(c+d*sec(f*x+e))^2+1/15*(2*c^2-15*c*d+76*d^2)*ta n(f*x+e)/(c-d)^3/f/(a^3+a^3*sec(f*x+e))/(c+d*sec(f*x+e))^2+1/30*d*(4*c^4-3 0*c^3*d+142*c^2*d^2+525*c*d^3+304*d^4)*tan(f*x+e)/a^3/(c-d)^5/(c+d)^2/f/(c +d*sec(f*x+e))
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} \]
(4*Cos[e/2 + (f*x)/2]^4*(d + c*Cos[e + f*x])^3*Sec[e/2]*Sec[e + f*x]^6*(-8 *c*Sin[e/2] + 23*d*Sin[e/2]))/(15*(-c + d)^4*f*(a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^3) + ((20*c^2 + 30*c*d + 13*d^2)*Cos[e/2 + (f*x)/2]^6*(d + c*Cos[e + f*x])^3*Sec[e + f*x]^6*(((-8*I)*d^3*ArcTan[Sec[(f*x)/2]*(Cos[e ]/(Sqrt[c^2 - d^2]*Sqrt[Cos[2*e] - I*Sin[2*e]]) - (I*Sin[e])/(Sqrt[c^2 - d ^2]*Sqrt[Cos[2*e] - I*Sin[2*e]]))*((-I)*d*Sin[(f*x)/2] + I*c*Sin[e + (f*x) /2])]*Cos[e])/(Sqrt[c^2 - d^2]*f*Sqrt[Cos[2*e] - I*Sin[2*e]]) - (8*d^3*Arc Tan[Sec[(f*x)/2]*(Cos[e]/(Sqrt[c^2 - d^2]*Sqrt[Cos[2*e] - I*Sin[2*e]]) - ( I*Sin[e])/(Sqrt[c^2 - d^2]*Sqrt[Cos[2*e] - I*Sin[2*e]]))*((-I)*d*Sin[(f*x) /2] + I*c*Sin[e + (f*x)/2])]*Sin[e])/(Sqrt[c^2 - d^2]*f*Sqrt[Cos[2*e] - I* Sin[2*e]])))/((-c + d)^5*(c + d)^2*(a + a*Sec[e + f*x])^3*(c + d*Sec[e + f *x])^3) - (2*Cos[e/2 + (f*x)/2]*(d + c*Cos[e + f*x])^3*Sec[e/2]*Sec[e + f* x]^6*Sin[(f*x)/2])/(5*(-c + d)^3*f*(a + a*Sec[e + f*x])^3*(c + d*Sec[e + f *x])^3) + (4*Cos[e/2 + (f*x)/2]^3*(d + c*Cos[e + f*x])^3*Sec[e/2]*Sec[e + f*x]^6*(-8*c*Sin[(f*x)/2] + 23*d*Sin[(f*x)/2]))/(15*(-c + d)^4*f*(a + a*Se c[e + f*x])^3*(c + d*Sec[e + f*x])^3) - (8*Cos[e/2 + (f*x)/2]^5*(d + c*Cos [e + f*x])^3*Sec[e/2]*Sec[e + f*x]^6*(7*c^2*Sin[(f*x)/2] - 44*c*d*Sin[(f*x )/2] + 127*d^2*Sin[(f*x)/2]))/(15*(-c + d)^5*f*(a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^3) + (4*Cos[e/2 + (f*x)/2]^6*(d + c*Cos[e + f*x])*Sec[e]* Sec[e + f*x]^6*(d^6*Sin[e] - c*d^5*Sin[f*x]))/(c^2*(-c + d)^4*(c + d)*f...
Time = 0.70 (sec) , antiderivative size = 508, normalized size of antiderivative = 1.38, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.516, Rules used = {3042, 4475, 114, 27, 168, 27, 169, 25, 27, 169, 25, 27, 169, 27, 104, 218}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sec (e+f x)}{(a \sec (e+f x)+a)^3 (c+d \sec (e+f x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right )}{\left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^3 \left (c+d \csc \left (e+f x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 4475 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \int \frac {1}{\sqrt {a-a \sec (e+f x)} (\sec (e+f x) a+a)^{7/2} (c+d \sec (e+f x))^3}d\sec (e+f x)}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\int \frac {a^2 (2 c+3 d-4 d \sec (e+f x))}{\sqrt {a-a \sec (e+f x)} (\sec (e+f x) a+a)^{7/2} (c+d \sec (e+f x))^2}d\sec (e+f x)}{2 a^2 \left (c^2-d^2\right )}+\frac {d \sqrt {a-a \sec (e+f x)}}{2 a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))^2}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\int \frac {2 c+3 d-4 d \sec (e+f x)}{\sqrt {a-a \sec (e+f x)} (\sec (e+f x) a+a)^{7/2} (c+d \sec (e+f x))^2}d\sec (e+f x)}{2 \left (c^2-d^2\right )}+\frac {d \sqrt {a-a \sec (e+f x)}}{2 a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))^2}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\frac {\int \frac {a^2 \left (2 c^2+21 d c+13 d^2-9 d (2 c+d) \sec (e+f x)\right )}{\sqrt {a-a \sec (e+f x)} (\sec (e+f x) a+a)^{7/2} (c+d \sec (e+f x))}d\sec (e+f x)}{a^2 \left (c^2-d^2\right )}+\frac {3 d (2 c+d) \sqrt {a-a \sec (e+f x)}}{a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))}}{2 \left (c^2-d^2\right )}+\frac {d \sqrt {a-a \sec (e+f x)}}{2 a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))^2}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\frac {\int \frac {2 c^2+21 d c+13 d^2-9 d (2 c+d) \sec (e+f x)}{\sqrt {a-a \sec (e+f x)} (\sec (e+f x) a+a)^{7/2} (c+d \sec (e+f x))}d\sec (e+f x)}{c^2-d^2}+\frac {3 d (2 c+d) \sqrt {a-a \sec (e+f x)}}{a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))}}{2 \left (c^2-d^2\right )}+\frac {d \sqrt {a-a \sec (e+f x)}}{2 a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))^2}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\frac {-\frac {\int -\frac {a^2 \left ((2 c+5 d) \left (2 c^2-16 d c-13 d^2\right )+2 d \left (2 c^2+39 d c+22 d^2\right ) \sec (e+f x)\right )}{\sqrt {a-a \sec (e+f x)} (\sec (e+f x) a+a)^{5/2} (c+d \sec (e+f x))}d\sec (e+f x)}{5 a^3 (c-d)}-\frac {\left (2 c^2+39 c d+22 d^2\right ) \sqrt {a-a \sec (e+f x)}}{5 a^2 (c-d) (a \sec (e+f x)+a)^{5/2}}}{c^2-d^2}+\frac {3 d (2 c+d) \sqrt {a-a \sec (e+f x)}}{a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))}}{2 \left (c^2-d^2\right )}+\frac {d \sqrt {a-a \sec (e+f x)}}{2 a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))^2}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\frac {\frac {\int \frac {a^2 \left ((2 c+5 d) \left (2 c^2-16 d c-13 d^2\right )+2 d \left (2 c^2+39 d c+22 d^2\right ) \sec (e+f x)\right )}{\sqrt {a-a \sec (e+f x)} (\sec (e+f x) a+a)^{5/2} (c+d \sec (e+f x))}d\sec (e+f x)}{5 a^3 (c-d)}-\frac {\left (2 c^2+39 c d+22 d^2\right ) \sqrt {a-a \sec (e+f x)}}{5 a^2 (c-d) (a \sec (e+f x)+a)^{5/2}}}{c^2-d^2}+\frac {3 d (2 c+d) \sqrt {a-a \sec (e+f x)}}{a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))}}{2 \left (c^2-d^2\right )}+\frac {d \sqrt {a-a \sec (e+f x)}}{2 a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))^2}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\frac {\frac {\int \frac {(2 c+5 d) \left (2 c^2-16 d c-13 d^2\right )+2 d \left (2 c^2+39 d c+22 d^2\right ) \sec (e+f x)}{\sqrt {a-a \sec (e+f x)} (\sec (e+f x) a+a)^{5/2} (c+d \sec (e+f x))}d\sec (e+f x)}{5 a (c-d)}-\frac {\left (2 c^2+39 c d+22 d^2\right ) \sqrt {a-a \sec (e+f x)}}{5 a^2 (c-d) (a \sec (e+f x)+a)^{5/2}}}{c^2-d^2}+\frac {3 d (2 c+d) \sqrt {a-a \sec (e+f x)}}{a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))}}{2 \left (c^2-d^2\right )}+\frac {d \sqrt {a-a \sec (e+f x)}}{2 a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))^2}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\frac {\frac {-\frac {\int -\frac {a^2 \left ((c+d) \left (4 c^3-30 d c^2+146 d^2 c+195 d^3\right )+d \left (4 c^3-26 d c^2-184 d^2 c-109 d^3\right ) \sec (e+f x)\right )}{\sqrt {a-a \sec (e+f x)} (\sec (e+f x) a+a)^{3/2} (c+d \sec (e+f x))}d\sec (e+f x)}{3 a^3 (c-d)}-\frac {\left (4 c^3-26 c^2 d-184 c d^2-109 d^3\right ) \sqrt {a-a \sec (e+f x)}}{3 a^2 (c-d) (a \sec (e+f x)+a)^{3/2}}}{5 a (c-d)}-\frac {\left (2 c^2+39 c d+22 d^2\right ) \sqrt {a-a \sec (e+f x)}}{5 a^2 (c-d) (a \sec (e+f x)+a)^{5/2}}}{c^2-d^2}+\frac {3 d (2 c+d) \sqrt {a-a \sec (e+f x)}}{a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))}}{2 \left (c^2-d^2\right )}+\frac {d \sqrt {a-a \sec (e+f x)}}{2 a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))^2}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\frac {\frac {\frac {\int \frac {a^2 \left ((c+d) \left (4 c^3-30 d c^2+146 d^2 c+195 d^3\right )+d \left (4 c^3-26 d c^2-184 d^2 c-109 d^3\right ) \sec (e+f x)\right )}{\sqrt {a-a \sec (e+f x)} (\sec (e+f x) a+a)^{3/2} (c+d \sec (e+f x))}d\sec (e+f x)}{3 a^3 (c-d)}-\frac {\left (4 c^3-26 c^2 d-184 c d^2-109 d^3\right ) \sqrt {a-a \sec (e+f x)}}{3 a^2 (c-d) (a \sec (e+f x)+a)^{3/2}}}{5 a (c-d)}-\frac {\left (2 c^2+39 c d+22 d^2\right ) \sqrt {a-a \sec (e+f x)}}{5 a^2 (c-d) (a \sec (e+f x)+a)^{5/2}}}{c^2-d^2}+\frac {3 d (2 c+d) \sqrt {a-a \sec (e+f x)}}{a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))}}{2 \left (c^2-d^2\right )}+\frac {d \sqrt {a-a \sec (e+f x)}}{2 a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))^2}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\frac {\frac {\frac {\int \frac {(c+d) \left (4 c^3-30 d c^2+146 d^2 c+195 d^3\right )+d \left (4 c^3-26 d c^2-184 d^2 c-109 d^3\right ) \sec (e+f x)}{\sqrt {a-a \sec (e+f x)} (\sec (e+f x) a+a)^{3/2} (c+d \sec (e+f x))}d\sec (e+f x)}{3 a (c-d)}-\frac {\left (4 c^3-26 c^2 d-184 c d^2-109 d^3\right ) \sqrt {a-a \sec (e+f x)}}{3 a^2 (c-d) (a \sec (e+f x)+a)^{3/2}}}{5 a (c-d)}-\frac {\left (2 c^2+39 c d+22 d^2\right ) \sqrt {a-a \sec (e+f x)}}{5 a^2 (c-d) (a \sec (e+f x)+a)^{5/2}}}{c^2-d^2}+\frac {3 d (2 c+d) \sqrt {a-a \sec (e+f x)}}{a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))}}{2 \left (c^2-d^2\right )}+\frac {d \sqrt {a-a \sec (e+f x)}}{2 a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))^2}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\frac {\frac {\frac {-\frac {\int \frac {15 a^2 d^3 \left (20 c^2+30 d c+13 d^2\right )}{\sqrt {a-a \sec (e+f x)} \sqrt {\sec (e+f x) a+a} (c+d \sec (e+f x))}d\sec (e+f x)}{a^3 (c-d)}-\frac {\left (4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right ) \sqrt {a-a \sec (e+f x)}}{a^2 (c-d) \sqrt {a \sec (e+f x)+a}}}{3 a (c-d)}-\frac {\left (4 c^3-26 c^2 d-184 c d^2-109 d^3\right ) \sqrt {a-a \sec (e+f x)}}{3 a^2 (c-d) (a \sec (e+f x)+a)^{3/2}}}{5 a (c-d)}-\frac {\left (2 c^2+39 c d+22 d^2\right ) \sqrt {a-a \sec (e+f x)}}{5 a^2 (c-d) (a \sec (e+f x)+a)^{5/2}}}{c^2-d^2}+\frac {3 d (2 c+d) \sqrt {a-a \sec (e+f x)}}{a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))}}{2 \left (c^2-d^2\right )}+\frac {d \sqrt {a-a \sec (e+f x)}}{2 a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))^2}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\frac {\frac {\frac {-\frac {15 d^3 \left (20 c^2+30 c d+13 d^2\right ) \int \frac {1}{\sqrt {a-a \sec (e+f x)} \sqrt {\sec (e+f x) a+a} (c+d \sec (e+f x))}d\sec (e+f x)}{a (c-d)}-\frac {\left (4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right ) \sqrt {a-a \sec (e+f x)}}{a^2 (c-d) \sqrt {a \sec (e+f x)+a}}}{3 a (c-d)}-\frac {\left (4 c^3-26 c^2 d-184 c d^2-109 d^3\right ) \sqrt {a-a \sec (e+f x)}}{3 a^2 (c-d) (a \sec (e+f x)+a)^{3/2}}}{5 a (c-d)}-\frac {\left (2 c^2+39 c d+22 d^2\right ) \sqrt {a-a \sec (e+f x)}}{5 a^2 (c-d) (a \sec (e+f x)+a)^{5/2}}}{c^2-d^2}+\frac {3 d (2 c+d) \sqrt {a-a \sec (e+f x)}}{a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))}}{2 \left (c^2-d^2\right )}+\frac {d \sqrt {a-a \sec (e+f x)}}{2 a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))^2}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\frac {\frac {\frac {-\frac {30 d^3 \left (20 c^2+30 c d+13 d^2\right ) \int \frac {1}{a (c-d)+\frac {a (c+d) (\sec (e+f x) a+a)}{a-a \sec (e+f x)}}d\frac {\sqrt {\sec (e+f x) a+a}}{\sqrt {a-a \sec (e+f x)}}}{a (c-d)}-\frac {\left (4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right ) \sqrt {a-a \sec (e+f x)}}{a^2 (c-d) \sqrt {a \sec (e+f x)+a}}}{3 a (c-d)}-\frac {\left (4 c^3-26 c^2 d-184 c d^2-109 d^3\right ) \sqrt {a-a \sec (e+f x)}}{3 a^2 (c-d) (a \sec (e+f x)+a)^{3/2}}}{5 a (c-d)}-\frac {\left (2 c^2+39 c d+22 d^2\right ) \sqrt {a-a \sec (e+f x)}}{5 a^2 (c-d) (a \sec (e+f x)+a)^{5/2}}}{c^2-d^2}+\frac {3 d (2 c+d) \sqrt {a-a \sec (e+f x)}}{a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))}}{2 \left (c^2-d^2\right )}+\frac {d \sqrt {a-a \sec (e+f x)}}{2 a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))^2}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\frac {\frac {\frac {-\frac {30 d^3 \left (20 c^2+30 c d+13 d^2\right ) \arctan \left (\frac {\sqrt {c+d} \sqrt {a \sec (e+f x)+a}}{\sqrt {c-d} \sqrt {a-a \sec (e+f x)}}\right )}{a^2 (c-d)^{3/2} \sqrt {c+d}}-\frac {\left (4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right ) \sqrt {a-a \sec (e+f x)}}{a^2 (c-d) \sqrt {a \sec (e+f x)+a}}}{3 a (c-d)}-\frac {\left (4 c^3-26 c^2 d-184 c d^2-109 d^3\right ) \sqrt {a-a \sec (e+f x)}}{3 a^2 (c-d) (a \sec (e+f x)+a)^{3/2}}}{5 a (c-d)}-\frac {\left (2 c^2+39 c d+22 d^2\right ) \sqrt {a-a \sec (e+f x)}}{5 a^2 (c-d) (a \sec (e+f x)+a)^{5/2}}}{c^2-d^2}+\frac {3 d (2 c+d) \sqrt {a-a \sec (e+f x)}}{a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))}}{2 \left (c^2-d^2\right )}+\frac {d \sqrt {a-a \sec (e+f x)}}{2 a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))^2}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
-((a^2*((d*Sqrt[a - a*Sec[e + f*x]])/(2*a^2*(c^2 - d^2)*(a + a*Sec[e + f*x ])^(5/2)*(c + d*Sec[e + f*x])^2) + ((3*d*(2*c + d)*Sqrt[a - a*Sec[e + f*x] ])/(a^2*(c^2 - d^2)*(a + a*Sec[e + f*x])^(5/2)*(c + d*Sec[e + f*x])) + (-1 /5*((2*c^2 + 39*c*d + 22*d^2)*Sqrt[a - a*Sec[e + f*x]])/(a^2*(c - d)*(a + a*Sec[e + f*x])^(5/2)) + (-1/3*((4*c^3 - 26*c^2*d - 184*c*d^2 - 109*d^3)*S qrt[a - a*Sec[e + f*x]])/(a^2*(c - d)*(a + a*Sec[e + f*x])^(3/2)) + ((-30* d^3*(20*c^2 + 30*c*d + 13*d^2)*ArcTan[(Sqrt[c + d]*Sqrt[a + a*Sec[e + f*x] ])/(Sqrt[c - d]*Sqrt[a - a*Sec[e + f*x]])])/(a^2*(c - d)^(3/2)*Sqrt[c + d] ) - ((4*c^4 - 30*c^3*d + 142*c^2*d^2 + 525*c*d^3 + 304*d^4)*Sqrt[a - a*Sec [e + f*x]])/(a^2*(c - d)*Sqrt[a + a*Sec[e + f*x]]))/(3*a*(c - d)))/(5*a*(c - d)))/(c^2 - d^2))/(2*(c^2 - d^2)))*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]))
3.3.33.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n, 2*p]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[(csc[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_), x_Symbol] :> Simp[a ^2*g*(Cot[e + f*x]/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[a - b*Csc[e + f*x]])) Subst[Int[(g*x)^(p - 1)*(a + b*x)^(m - 1/2)*((c + d*x)^n/Sqrt[a - b*x]), x ], x, Csc[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[ b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && (EqQ[p, 1] || In tegerQ[m - 1/2])
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 | \(\text {Expression too large to display}\) | \(1764\) |
1/4/f/a^3*(1/(c^3-3*c^2*d+3*c*d^2-d^3)/(c^2-2*c*d+d^2)*(1/5*tan(1/2*f*x+1/ 2*e)^5*c^2-2/5*tan(1/2*f*x+1/2*e)^5*c*d+1/5*tan(1/2*f*x+1/2*e)^5*d^2-2/3*c ^2*tan(1/2*f*x+1/2*e)^3+10/3*tan(1/2*f*x+1/2*e)^3*c*d-8/3*tan(1/2*f*x+1/2* e)^3*d^2+tan(1/2*f*x+1/2*e)*c^2-8*tan(1/2*f*x+1/2*e)*c*d+31*tan(1/2*f*x+1/ 2*e)*d^2)+16*d^3/(c-d)^5*((-1/4*d*(10*c^2-3*c*d-7*d^2)/(c^2+2*c*d+d^2)*tan (1/2*f*x+1/2*e)^3+5/4*d*(2*c+d)/(c+d)*tan(1/2*f*x+1/2*e))/(tan(1/2*f*x+1/2 *e)^2*c-tan(1/2*f*x+1/2*e)^2*d-c-d)^2-1/4*(20*c^2+30*c*d+13*d^2)/(c^2+2*c* d+d^2)/((c+d)*(c-d))^(1/2)*arctanh((c-d)*tan(1/2*f*x+1/2*e)/((c+d)*(c-d))^ (1/2))))
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 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c+d \sec (e+f x))^3} \, dx=\text {Too large to display} \]
[-1/60*(15*(20*c^2*d^5 + 30*c*d^6 + 13*d^7 + (20*c^4*d^3 + 30*c^3*d^4 + 13 *c^2*d^5)*cos(f*x + e)^5 + (60*c^4*d^3 + 130*c^3*d^4 + 99*c^2*d^5 + 26*c*d ^6)*cos(f*x + e)^4 + (60*c^4*d^3 + 210*c^3*d^4 + 239*c^2*d^5 + 108*c*d^6 + 13*d^7)*cos(f*x + e)^3 + (20*c^4*d^3 + 150*c^3*d^4 + 253*c^2*d^5 + 168*c* d^6 + 39*d^7)*cos(f*x + e)^2 + (40*c^3*d^4 + 120*c^2*d^5 + 116*c*d^6 + 39* d^7)*cos(f*x + e))*sqrt(c^2 - d^2)*log((2*c*d*cos(f*x + e) - (c^2 - 2*d^2) *cos(f*x + e)^2 + 2*sqrt(c^2 - d^2)*(d*cos(f*x + e) + c)*sin(f*x + e) + 2* c^2 - d^2)/(c^2*cos(f*x + e)^2 + 2*c*d*cos(f*x + e) + d^2)) - 2*(4*c^6*d^2 - 30*c^5*d^3 + 138*c^4*d^4 + 555*c^3*d^5 + 162*c^2*d^6 - 525*c*d^7 - 304* d^8 + (14*c^8 - 60*c^7*d + 78*c^6*d^2 + 480*c^5*d^3 + 312*c^4*d^4 - 330*c^ 3*d^5 - 419*c^2*d^6 - 90*c*d^7 + 15*d^8)*cos(f*x + e)^4 + (12*c^8 - 62*c^7 *d + 114*c^6*d^2 + 1056*c^5*d^3 + 1626*c^4*d^4 - 81*c^3*d^5 - 1707*c^2*d^6 - 913*c*d^7 - 45*d^8)*cos(f*x + e)^3 + (4*c^8 - 6*c^7*d - 28*c^6*d^2 + 82 8*c^5*d^3 + 2400*c^4*d^4 + 1197*c^3*d^5 - 1897*c^2*d^6 - 2019*c*d^7 - 479* d^8)*cos(f*x + e)^2 + (8*c^7*d - 48*c^6*d^2 + 186*c^5*d^3 + 1224*c^4*d^4 + 1539*c^3*d^5 - 459*c^2*d^6 - 1733*c*d^7 - 717*d^8)*cos(f*x + e))*sin(f*x + e))/((a^3*c^11 - 3*a^3*c^10*d + 8*a^3*c^8*d^3 - 6*a^3*c^7*d^4 - 6*a^3*c^ 6*d^5 + 8*a^3*c^5*d^6 - 3*a^3*c^3*d^8 + a^3*c^2*d^9)*f*cos(f*x + e)^5 + (3 *a^3*c^11 - 7*a^3*c^10*d - 6*a^3*c^9*d^2 + 24*a^3*c^8*d^3 - 2*a^3*c^7*d^4 - 30*a^3*c^6*d^5 + 12*a^3*c^5*d^6 + 16*a^3*c^4*d^7 - 9*a^3*c^3*d^8 - 3*...
\[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c+d \sec (e+f x))^3} \, dx=\frac {\int \frac {\sec {\left (e + f x \right )}}{c^{3} \sec ^{3}{\left (e + f x \right )} + 3 c^{3} \sec ^{2}{\left (e + f x \right )} + 3 c^{3} \sec {\left (e + f x \right )} + c^{3} + 3 c^{2} d \sec ^{4}{\left (e + f x \right )} + 9 c^{2} d \sec ^{3}{\left (e + f x \right )} + 9 c^{2} d \sec ^{2}{\left (e + f x \right )} + 3 c^{2} d \sec {\left (e + f x \right )} + 3 c d^{2} \sec ^{5}{\left (e + f x \right )} + 9 c d^{2} \sec ^{4}{\left (e + f x \right )} + 9 c d^{2} \sec ^{3}{\left (e + f x \right )} + 3 c d^{2} \sec ^{2}{\left (e + f x \right )} + d^{3} \sec ^{6}{\left (e + f x \right )} + 3 d^{3} \sec ^{5}{\left (e + f x \right )} + 3 d^{3} \sec ^{4}{\left (e + f x \right )} + d^{3} \sec ^{3}{\left (e + f x \right )}}\, dx}{a^{3}} \]
Integral(sec(e + f*x)/(c**3*sec(e + f*x)**3 + 3*c**3*sec(e + f*x)**2 + 3*c **3*sec(e + f*x) + c**3 + 3*c**2*d*sec(e + f*x)**4 + 9*c**2*d*sec(e + f*x) **3 + 9*c**2*d*sec(e + f*x)**2 + 3*c**2*d*sec(e + f*x) + 3*c*d**2*sec(e + f*x)**5 + 9*c*d**2*sec(e + f*x)**4 + 9*c*d**2*sec(e + f*x)**3 + 3*c*d**2*s ec(e + f*x)**2 + d**3*sec(e + f*x)**6 + 3*d**3*sec(e + f*x)**5 + 3*d**3*se c(e + f*x)**4 + d**3*sec(e + f*x)**3), x)/a**3
Exception generated. \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c+d \sec (e+f x))^3} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*c^2-4*d^2>0)', see `assume?` f or more de
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 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c+d \sec (e+f x))^3} \, dx=\text {Too large to display} \]
-1/60*(60*(20*c^2*d^3 + 30*c*d^4 + 13*d^5)*(pi*floor(1/2*(f*x + e)/pi + 1/ 2)*sgn(-2*c + 2*d) + arctan(-(c*tan(1/2*f*x + 1/2*e) - d*tan(1/2*f*x + 1/2 *e))/sqrt(-c^2 + d^2)))/((a^3*c^7 - 3*a^3*c^6*d + a^3*c^5*d^2 + 5*a^3*c^4* d^3 - 5*a^3*c^3*d^4 - a^3*c^2*d^5 + 3*a^3*c*d^6 - a^3*d^7)*sqrt(-c^2 + d^2 )) - (3*a^12*c^12*tan(1/2*f*x + 1/2*e)^5 - 36*a^12*c^11*d*tan(1/2*f*x + 1/ 2*e)^5 + 198*a^12*c^10*d^2*tan(1/2*f*x + 1/2*e)^5 - 660*a^12*c^9*d^3*tan(1 /2*f*x + 1/2*e)^5 + 1485*a^12*c^8*d^4*tan(1/2*f*x + 1/2*e)^5 - 2376*a^12*c ^7*d^5*tan(1/2*f*x + 1/2*e)^5 + 2772*a^12*c^6*d^6*tan(1/2*f*x + 1/2*e)^5 - 2376*a^12*c^5*d^7*tan(1/2*f*x + 1/2*e)^5 + 1485*a^12*c^4*d^8*tan(1/2*f*x + 1/2*e)^5 - 660*a^12*c^3*d^9*tan(1/2*f*x + 1/2*e)^5 + 198*a^12*c^2*d^10*t an(1/2*f*x + 1/2*e)^5 - 36*a^12*c*d^11*tan(1/2*f*x + 1/2*e)^5 + 3*a^12*d^1 2*tan(1/2*f*x + 1/2*e)^5 - 10*a^12*c^12*tan(1/2*f*x + 1/2*e)^3 + 150*a^12* c^11*d*tan(1/2*f*x + 1/2*e)^3 - 990*a^12*c^10*d^2*tan(1/2*f*x + 1/2*e)^3 + 3850*a^12*c^9*d^3*tan(1/2*f*x + 1/2*e)^3 - 9900*a^12*c^8*d^4*tan(1/2*f*x + 1/2*e)^3 + 17820*a^12*c^7*d^5*tan(1/2*f*x + 1/2*e)^3 - 23100*a^12*c^6*d^ 6*tan(1/2*f*x + 1/2*e)^3 + 21780*a^12*c^5*d^7*tan(1/2*f*x + 1/2*e)^3 - 148 50*a^12*c^4*d^8*tan(1/2*f*x + 1/2*e)^3 + 7150*a^12*c^3*d^9*tan(1/2*f*x + 1 /2*e)^3 - 2310*a^12*c^2*d^10*tan(1/2*f*x + 1/2*e)^3 + 450*a^12*c*d^11*tan( 1/2*f*x + 1/2*e)^3 - 40*a^12*d^12*tan(1/2*f*x + 1/2*e)^3 + 15*a^12*c^12*ta n(1/2*f*x + 1/2*e) - 270*a^12*c^11*d*tan(1/2*f*x + 1/2*e) + 2340*a^12*c...
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}} \]
tan(e/2 + (f*x)/2)^5/(20*a^3*f*(c - d)^3) - (tan(e/2 + (f*x)/2)*((3*(c + d )^2)/(4*a^3*(c - d)^5) - 5/(2*a^3*(c - d)^3) + (3*(c + d)*(5/(4*a^3*(c - d )^3) - (3*(c + d))/(4*a^3*(c - d)^4)))/(c - d)))/f - (tan(e/2 + (f*x)/2)^3 *(5/(12*a^3*(c - d)^3) - (c + d)/(4*a^3*(c - d)^4)))/f - ((tan(e/2 + (f*x) /2)^3*(3*c*d^5 + 7*d^6 - 10*c^2*d^4))/(c + d)^2 + (5*tan(e/2 + (f*x)/2)*(2 *c*d^4 + d^5))/(c + d))/(f*(tan(e/2 + (f*x)/2)^2*(2*a^3*c^7 + 2*a^3*d^7 - 10*a^3*c*d^6 - 10*a^3*c^6*d + 18*a^3*c^2*d^5 - 10*a^3*c^3*d^4 - 10*a^3*c^4 *d^3 + 18*a^3*c^5*d^2) - tan(e/2 + (f*x)/2)^4*(a^3*c^7 - a^3*d^7 + 7*a^3*c *d^6 - 7*a^3*c^6*d - 21*a^3*c^2*d^5 + 35*a^3*c^3*d^4 - 35*a^3*c^4*d^3 + 21 *a^3*c^5*d^2) - 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)) + (d^3*atan((c^6*tan(e/2 + (f*x)/2)*1i + d^6*tan(e/2 + (f*x)/2)*1i - c*d^5*tan(e/2 + (f*x)/2)*6i - c^5*d*tan(e/2 + (f*x)/2)*6i + c^2*d^4*tan(e/2 + (f*x)/2)*15i - c^3*d^3*ta n(e/2 + (f*x)/2)*20i + c^4*d^2*tan(e/2 + (f*x)/2)*15i)/((c + d)^(1/2)*(c - d)^(11/2)))*(30*c*d + 20*c^2 + 13*d^2)*1i)/(a^3*f*(c + d)^(5/2)*(c - d)^( 11/2))