Integrand size = 33, antiderivative size = 232 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {7 a^4 (7 A+10 C) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {4 a^4 (18 A+25 C) \tan (c+d x)}{15 d}+\frac {a^4 (417 A+550 C) \sec (c+d x) \tan (c+d x)}{240 d}+\frac {(43 A+50 C) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^2(c+d x) \tan (c+d x)}{60 d}+\frac {(37 A+30 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {2 a A (a+a \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac {A (a+a \cos (c+d x))^4 \sec ^5(c+d x) \tan (c+d x)}{6 d} \] Output:
7/16*a^4*(7*A+10*C)*arctanh(sin(d*x+c))/d+4/15*a^4*(18*A+25*C)*tan(d*x+c)/ d+1/240*a^4*(417*A+550*C)*sec(d*x+c)*tan(d*x+c)/d+1/60*(43*A+50*C)*(a^4+a^ 4*cos(d*x+c))*sec(d*x+c)^2*tan(d*x+c)/d+1/120*(37*A+30*C)*(a^2+a^2*cos(d*x +c))^2*sec(d*x+c)^3*tan(d*x+c)/d+2/15*a*A*(a+a*cos(d*x+c))^3*sec(d*x+c)^4* tan(d*x+c)/d+1/6*A*(a+a*cos(d*x+c))^4*sec(d*x+c)^5*tan(d*x+c)/d
Time = 5.18 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.11 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {a^4 C \coth ^{-1}(\sin (c+d x))}{d}+\frac {49 a^4 A \text {arctanh}(\sin (c+d x))}{16 d}+\frac {27 a^4 C \text {arctanh}(\sin (c+d x))}{8 d}+\frac {8 a^4 A \tan (c+d x)}{d}+\frac {8 a^4 C \tan (c+d x)}{d}+\frac {49 a^4 A \sec (c+d x) \tan (c+d x)}{16 d}+\frac {27 a^4 C \sec (c+d x) \tan (c+d x)}{8 d}+\frac {41 a^4 A \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac {a^4 C \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {a^4 A \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {4 a^4 A \tan ^3(c+d x)}{d}+\frac {4 a^4 C \tan ^3(c+d x)}{3 d}+\frac {4 a^4 A \tan ^5(c+d x)}{5 d} \] Input:
Integrate[(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]
Output:
(a^4*C*ArcCoth[Sin[c + d*x]])/d + (49*a^4*A*ArcTanh[Sin[c + d*x]])/(16*d) + (27*a^4*C*ArcTanh[Sin[c + d*x]])/(8*d) + (8*a^4*A*Tan[c + d*x])/d + (8*a ^4*C*Tan[c + d*x])/d + (49*a^4*A*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (27*a ^4*C*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (41*a^4*A*Sec[c + d*x]^3*Tan[c + d *x])/(24*d) + (a^4*C*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (a^4*A*Sec[c + d *x]^5*Tan[c + d*x])/(6*d) + (4*a^4*A*Tan[c + d*x]^3)/d + (4*a^4*C*Tan[c + d*x]^3)/(3*d) + (4*a^4*A*Tan[c + d*x]^5)/(5*d)
Time = 1.88 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.08, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.576, Rules used = {3042, 3523, 3042, 3454, 3042, 3454, 3042, 3454, 27, 3042, 3447, 3042, 3500, 3042, 3227, 3042, 4254, 24, 4257}
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.
| \(\displaystyle \int \sec ^7(c+d x) (a \cos (c+d x)+a)^4 \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4 \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^7}dx\) |
| \(\Big \downarrow \) 3523 |
| \(\displaystyle \frac {\int (\cos (c+d x) a+a)^4 (4 a A+a (A+6 C) \cos (c+d x)) \sec ^6(c+d x)dx}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4 \left (4 a A+a (A+6 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^6}dx}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\frac {1}{5} \int (\cos (c+d x) a+a)^3 \left ((37 A+30 C) a^2+3 (3 A+10 C) \cos (c+d x) a^2\right ) \sec ^5(c+d x)dx+\frac {4 a^2 A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left ((37 A+30 C) a^2+3 (3 A+10 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^5}dx+\frac {4 a^2 A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \int (\cos (c+d x) a+a)^2 \left (6 (43 A+50 C) a^3+(73 A+150 C) \cos (c+d x) a^3\right ) \sec ^4(c+d x)dx+\frac {a^3 (37 A+30 C) \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^2}{4 d}\right )+\frac {4 a^2 A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (6 (43 A+50 C) a^3+(73 A+150 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {a^3 (37 A+30 C) \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^2}{4 d}\right )+\frac {4 a^2 A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \int 3 (\cos (c+d x) a+a) \left ((417 A+550 C) a^4+(159 A+250 C) \cos (c+d x) a^4\right ) \sec ^3(c+d x)dx+\frac {2 (43 A+50 C) \tan (c+d x) \sec ^2(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {a^3 (37 A+30 C) \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^2}{4 d}\right )+\frac {4 a^2 A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \left (\int (\cos (c+d x) a+a) \left ((417 A+550 C) a^4+(159 A+250 C) \cos (c+d x) a^4\right ) \sec ^3(c+d x)dx+\frac {2 (43 A+50 C) \tan (c+d x) \sec ^2(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {a^3 (37 A+30 C) \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^2}{4 d}\right )+\frac {4 a^2 A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \left (\int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left ((417 A+550 C) a^4+(159 A+250 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^4\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {2 (43 A+50 C) \tan (c+d x) \sec ^2(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {a^3 (37 A+30 C) \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^2}{4 d}\right )+\frac {4 a^2 A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \left (\int \left ((159 A+250 C) \cos ^2(c+d x) a^5+(417 A+550 C) a^5+\left ((159 A+250 C) a^5+(417 A+550 C) a^5\right ) \cos (c+d x)\right ) \sec ^3(c+d x)dx+\frac {2 (43 A+50 C) \tan (c+d x) \sec ^2(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {a^3 (37 A+30 C) \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^2}{4 d}\right )+\frac {4 a^2 A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \left (\int \frac {(159 A+250 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^5+(417 A+550 C) a^5+\left ((159 A+250 C) a^5+(417 A+550 C) a^5\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {2 (43 A+50 C) \tan (c+d x) \sec ^2(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {a^3 (37 A+30 C) \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^2}{4 d}\right )+\frac {4 a^2 A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{2} \int \left (64 (18 A+25 C) a^5+105 (7 A+10 C) \cos (c+d x) a^5\right ) \sec ^2(c+d x)dx+\frac {a^5 (417 A+550 C) \tan (c+d x) \sec (c+d x)}{2 d}+\frac {2 (43 A+50 C) \tan (c+d x) \sec ^2(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {a^3 (37 A+30 C) \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^2}{4 d}\right )+\frac {4 a^2 A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{2} \int \frac {64 (18 A+25 C) a^5+105 (7 A+10 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^5}{\sin \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {a^5 (417 A+550 C) \tan (c+d x) \sec (c+d x)}{2 d}+\frac {2 (43 A+50 C) \tan (c+d x) \sec ^2(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {a^3 (37 A+30 C) \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^2}{4 d}\right )+\frac {4 a^2 A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{2} \left (64 a^5 (18 A+25 C) \int \sec ^2(c+d x)dx+105 a^5 (7 A+10 C) \int \sec (c+d x)dx\right )+\frac {a^5 (417 A+550 C) \tan (c+d x) \sec (c+d x)}{2 d}+\frac {2 (43 A+50 C) \tan (c+d x) \sec ^2(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {a^3 (37 A+30 C) \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^2}{4 d}\right )+\frac {4 a^2 A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{2} \left (105 a^5 (7 A+10 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+64 a^5 (18 A+25 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^2dx\right )+\frac {a^5 (417 A+550 C) \tan (c+d x) \sec (c+d x)}{2 d}+\frac {2 (43 A+50 C) \tan (c+d x) \sec ^2(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {a^3 (37 A+30 C) \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^2}{4 d}\right )+\frac {4 a^2 A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{2} \left (105 a^5 (7 A+10 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx-\frac {64 a^5 (18 A+25 C) \int 1d(-\tan (c+d x))}{d}\right )+\frac {a^5 (417 A+550 C) \tan (c+d x) \sec (c+d x)}{2 d}+\frac {2 (43 A+50 C) \tan (c+d x) \sec ^2(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {a^3 (37 A+30 C) \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^2}{4 d}\right )+\frac {4 a^2 A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{2} \left (105 a^5 (7 A+10 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {64 a^5 (18 A+25 C) \tan (c+d x)}{d}\right )+\frac {a^5 (417 A+550 C) \tan (c+d x) \sec (c+d x)}{2 d}+\frac {2 (43 A+50 C) \tan (c+d x) \sec ^2(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {a^3 (37 A+30 C) \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^2}{4 d}\right )+\frac {4 a^2 A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\frac {4 a^2 A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}+\frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {105 a^5 (7 A+10 C) \text {arctanh}(\sin (c+d x))}{d}+\frac {64 a^5 (18 A+25 C) \tan (c+d x)}{d}\right )+\frac {a^5 (417 A+550 C) \tan (c+d x) \sec (c+d x)}{2 d}+\frac {2 (43 A+50 C) \tan (c+d x) \sec ^2(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {a^3 (37 A+30 C) \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^2}{4 d}\right )}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}\) |
Input:
Int[(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]
Output:
(A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]^5*Tan[c + d*x])/(6*d) + ((4*a^2*A*( a + a*Cos[c + d*x])^3*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + ((a^3*(37*A + 3 0*C)*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + ((a^5*(41 7*A + 550*C)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (2*(43*A + 50*C)*(a^5 + a^ 5*Cos[c + d*x])*Sec[c + d*x]^2*Tan[c + d*x])/d + ((105*a^5*(7*A + 10*C)*Ar cTanh[Sin[c + d*x]])/d + (64*a^5*(18*A + 25*C)*Tan[c + d*x])/d)/2)/4)/5)/( 6*a)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(-b^2)*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[ e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c + a*d))), x] - Simp[b/(d*(n + 1)*(b*c + a*d)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp [a*A*d*(m - n - 2) - B*(a*c*(m - 1) + b*d*(n + 1)) - (A*b*d*(m + n + 1) - B *(b*c*m - a*d*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f , A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0 ])
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 1)* (a^2 - b^2))), x] + Simp[1/(b*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x ])^(m + 1)*Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A *b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C + A*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(b*d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(a *d*m + b*c*(n + 1)) + c*C*(a*c*m + b*d*(n + 1)) - b*(A*d^2*(m + n + 2) + C* (c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ[m, -2^(-1)] && (LtQ[n, -1] || EqQ[m + n + 2, 0])
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 12.78 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.06
| method | result | size |
| parallelrisch | \(\frac {125 \left (-\frac {147 \left (\frac {\cos \left (6 d x +6 c \right )}{15}+\frac {2 \cos \left (4 d x +4 c \right )}{5}+\cos \left (2 d x +2 c \right )+\frac {2}{3}\right ) \left (A +\frac {10 C}{7}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{100}+\frac {147 \left (\frac {\cos \left (6 d x +6 c \right )}{15}+\frac {2 \cos \left (4 d x +4 c \right )}{5}+\cos \left (2 d x +2 c \right )+\frac {2}{3}\right ) \left (A +\frac {10 C}{7}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{100}+\left (\frac {224 A}{125}+\frac {176 C}{125}\right ) \sin \left (2 d x +2 c \right )+\left (\frac {89 C}{125}+\frac {769 A}{750}\right ) \sin \left (3 d x +3 c \right )+\left (\frac {576 A}{625}+\frac {128 C}{125}\right ) \sin \left (4 d x +4 c \right )+\left (\frac {27 C}{125}+\frac {49 A}{250}\right ) \sin \left (5 d x +5 c \right )+\left (\frac {96 A}{625}+\frac {16 C}{75}\right ) \sin \left (6 d x +6 c \right )+\sin \left (d x +c \right ) \left (A +\frac {62 C}{125}\right )\right ) a^{4}}{4 d \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right )}\) | \(246\) |
| parts | \(\frac {a^{4} A \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}+\frac {\left (a^{4} A +6 a^{4} C \right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}-\frac {\left (4 a^{4} A +4 a^{4} C \right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (6 a^{4} A +a^{4} C \right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}+\frac {C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right ) a^{4}}{d}-\frac {4 a^{4} A \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {4 a^{4} C \tan \left (d x +c \right )}{d}\) | \(281\) |
| derivativedivides | \(\frac {a^{4} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+a^{4} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )-4 a^{4} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 a^{4} C \tan \left (d x +c \right )+6 a^{4} A \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+6 a^{4} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-4 a^{4} A \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )-4 a^{4} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+a^{4} A \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(350\) |
| default | \(\frac {a^{4} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+a^{4} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )-4 a^{4} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 a^{4} C \tan \left (d x +c \right )+6 a^{4} A \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+6 a^{4} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-4 a^{4} A \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )-4 a^{4} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+a^{4} A \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(350\) |
| risch | \(-\frac {i a^{4} \left (735 A \,{\mathrm e}^{11 i \left (d x +c \right )}+810 C \,{\mathrm e}^{11 i \left (d x +c \right )}-960 C \,{\mathrm e}^{10 i \left (d x +c \right )}+3845 A \,{\mathrm e}^{9 i \left (d x +c \right )}+2670 C \,{\mathrm e}^{9 i \left (d x +c \right )}-1920 A \,{\mathrm e}^{8 i \left (d x +c \right )}-6720 C \,{\mathrm e}^{8 i \left (d x +c \right )}+3750 A \,{\mathrm e}^{7 i \left (d x +c \right )}+1860 C \,{\mathrm e}^{7 i \left (d x +c \right )}-11520 A \,{\mathrm e}^{6 i \left (d x +c \right )}-16000 C \,{\mathrm e}^{6 i \left (d x +c \right )}-3750 A \,{\mathrm e}^{5 i \left (d x +c \right )}-1860 C \,{\mathrm e}^{5 i \left (d x +c \right )}-15360 A \,{\mathrm e}^{4 i \left (d x +c \right )}-17280 C \,{\mathrm e}^{4 i \left (d x +c \right )}-3845 A \,{\mathrm e}^{3 i \left (d x +c \right )}-2670 C \,{\mathrm e}^{3 i \left (d x +c \right )}-6912 A \,{\mathrm e}^{2 i \left (d x +c \right )}-8640 C \,{\mathrm e}^{2 i \left (d x +c \right )}-735 A \,{\mathrm e}^{i \left (d x +c \right )}-810 C \,{\mathrm e}^{i \left (d x +c \right )}-1152 A -1600 C \right )}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{6}}+\frac {49 a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{16 d}+\frac {35 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{8 d}-\frac {49 a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{16 d}-\frac {35 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{8 d}\) | \(371\) |
Input:
int((a+a*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c)^7,x,method=_RETURNVER BOSE)
Output:
125/4*(-147/100*(1/15*cos(6*d*x+6*c)+2/5*cos(4*d*x+4*c)+cos(2*d*x+2*c)+2/3 )*(A+10/7*C)*ln(tan(1/2*d*x+1/2*c)-1)+147/100*(1/15*cos(6*d*x+6*c)+2/5*cos (4*d*x+4*c)+cos(2*d*x+2*c)+2/3)*(A+10/7*C)*ln(tan(1/2*d*x+1/2*c)+1)+(224/1 25*A+176/125*C)*sin(2*d*x+2*c)+(89/125*C+769/750*A)*sin(3*d*x+3*c)+(576/62 5*A+128/125*C)*sin(4*d*x+4*c)+(27/125*C+49/250*A)*sin(5*d*x+5*c)+(96/625*A +16/75*C)*sin(6*d*x+6*c)+sin(d*x+c)*(A+62/125*C))*a^4/d/(cos(6*d*x+6*c)+6* cos(4*d*x+4*c)+15*cos(2*d*x+2*c)+10)
Time = 0.11 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.78 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {105 \, {\left (7 \, A + 10 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (7 \, A + 10 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (64 \, {\left (18 \, A + 25 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} + 15 \, {\left (49 \, A + 54 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 64 \, {\left (9 \, A + 5 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 10 \, {\left (41 \, A + 6 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 192 \, A a^{4} \cos \left (d x + c\right ) + 40 \, A a^{4}\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \] Input:
integrate((a+a*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c)^7,x, algorithm= "fricas")
Output:
1/480*(105*(7*A + 10*C)*a^4*cos(d*x + c)^6*log(sin(d*x + c) + 1) - 105*(7* A + 10*C)*a^4*cos(d*x + c)^6*log(-sin(d*x + c) + 1) + 2*(64*(18*A + 25*C)* a^4*cos(d*x + c)^5 + 15*(49*A + 54*C)*a^4*cos(d*x + c)^4 + 64*(9*A + 5*C)* a^4*cos(d*x + c)^3 + 10*(41*A + 6*C)*a^4*cos(d*x + c)^2 + 192*A*a^4*cos(d* x + c) + 40*A*a^4)*sin(d*x + c))/(d*cos(d*x + c)^6)
Timed out. \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\text {Timed out} \] Input:
integrate((a+a*cos(d*x+c))**4*(A+C*cos(d*x+c)**2)*sec(d*x+c)**7,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 456 vs. \(2 (218) = 436\).
Time = 0.05 (sec) , antiderivative size = 456, normalized size of antiderivative = 1.97 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {128 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} A a^{4} + 640 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{4} + 640 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{4} - 5 \, A a^{4} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 180 \, A a^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 30 \, C a^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 120 \, A a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 720 \, C a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 240 \, C a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 1920 \, C a^{4} \tan \left (d x + c\right )}{480 \, d} \] Input:
integrate((a+a*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c)^7,x, algorithm= "maxima")
Output:
1/480*(128*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*A*a^4 + 640*(tan(d*x + c)^3 + 3*tan(d*x + c))*A*a^4 + 640*(tan(d*x + c)^3 + 3*ta n(d*x + c))*C*a^4 - 5*A*a^4*(2*(15*sin(d*x + c)^5 - 40*sin(d*x + c)^3 + 33 *sin(d*x + c))/(sin(d*x + c)^6 - 3*sin(d*x + c)^4 + 3*sin(d*x + c)^2 - 1) - 15*log(sin(d*x + c) + 1) + 15*log(sin(d*x + c) - 1)) - 180*A*a^4*(2*(3*s in(d*x + c)^3 - 5*sin(d*x + c))/(sin(d*x + c)^4 - 2*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1)) - 30*C*a^4*(2*(3*sin(d* x + c)^3 - 5*sin(d*x + c))/(sin(d*x + c)^4 - 2*sin(d*x + c)^2 + 1) - 3*log (sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1)) - 120*A*a^4*(2*sin(d*x + c)/ (sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) - 72 0*C*a^4*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log (sin(d*x + c) - 1)) + 240*C*a^4*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 1920*C*a^4*tan(d*x + c))/d
Time = 0.41 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.21 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {105 \, {\left (7 \, A a^{4} + 10 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 105 \, {\left (7 \, A a^{4} + 10 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (735 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1050 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 4165 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 5950 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 9702 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 13860 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 11802 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 16860 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 7355 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 10690 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3105 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2790 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{6}}}{240 \, d} \] Input:
integrate((a+a*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c)^7,x, algorithm= "giac")
Output:
1/240*(105*(7*A*a^4 + 10*C*a^4)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 105*( 7*A*a^4 + 10*C*a^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(735*A*a^4*tan( 1/2*d*x + 1/2*c)^11 + 1050*C*a^4*tan(1/2*d*x + 1/2*c)^11 - 4165*A*a^4*tan( 1/2*d*x + 1/2*c)^9 - 5950*C*a^4*tan(1/2*d*x + 1/2*c)^9 + 9702*A*a^4*tan(1/ 2*d*x + 1/2*c)^7 + 13860*C*a^4*tan(1/2*d*x + 1/2*c)^7 - 11802*A*a^4*tan(1/ 2*d*x + 1/2*c)^5 - 16860*C*a^4*tan(1/2*d*x + 1/2*c)^5 + 7355*A*a^4*tan(1/2 *d*x + 1/2*c)^3 + 10690*C*a^4*tan(1/2*d*x + 1/2*c)^3 - 3105*A*a^4*tan(1/2* d*x + 1/2*c) - 2790*C*a^4*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^6)/d
Time = 1.73 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.13 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {\left (-\frac {49\,A\,a^4}{8}-\frac {35\,C\,a^4}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {833\,A\,a^4}{24}+\frac {595\,C\,a^4}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (-\frac {1617\,A\,a^4}{20}-\frac {231\,C\,a^4}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {1967\,A\,a^4}{20}+\frac {281\,C\,a^4}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {1471\,A\,a^4}{24}-\frac {1069\,C\,a^4}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {207\,A\,a^4}{8}+\frac {93\,C\,a^4}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {7\,a^4\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (7\,A+10\,C\right )}{8\,d} \] Input:
int(((A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^4)/cos(c + d*x)^7,x)
Output:
(tan(c/2 + (d*x)/2)*((207*A*a^4)/8 + (93*C*a^4)/4) - tan(c/2 + (d*x)/2)^11 *((49*A*a^4)/8 + (35*C*a^4)/4) + tan(c/2 + (d*x)/2)^9*((833*A*a^4)/24 + (5 95*C*a^4)/12) - tan(c/2 + (d*x)/2)^7*((1617*A*a^4)/20 + (231*C*a^4)/2) + t an(c/2 + (d*x)/2)^5*((1967*A*a^4)/20 + (281*C*a^4)/2) - tan(c/2 + (d*x)/2) ^3*((1471*A*a^4)/24 + (1069*C*a^4)/12))/(d*(15*tan(c/2 + (d*x)/2)^4 - 6*ta n(c/2 + (d*x)/2)^2 - 20*tan(c/2 + (d*x)/2)^6 + 15*tan(c/2 + (d*x)/2)^8 - 6 *tan(c/2 + (d*x)/2)^10 + tan(c/2 + (d*x)/2)^12 + 1)) + (7*a^4*atanh(tan(c/ 2 + (d*x)/2))*(7*A + 10*C))/(8*d)
Time = 0.21 (sec) , antiderivative size = 663, normalized size of antiderivative = 2.86 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx =\text {Too large to display} \] Input:
int((a+a*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c)^7,x)
Output:
(a**4*( - 735*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*a - 1 050*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*c + 2205*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*a + 3150*cos(c + d*x)*log (tan((c + d*x)/2) - 1)*sin(c + d*x)**4*c - 2205*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*a - 3150*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*c + 735*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*a + 1050 *cos(c + d*x)*log(tan((c + d*x)/2) - 1)*c + 735*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**6*a + 1050*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**6*c - 2205*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**4*a - 3150*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**4*c + 2205*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2*a + 3150*co s(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2*c - 735*cos(c + d*x)* log(tan((c + d*x)/2) + 1)*a - 1050*cos(c + d*x)*log(tan((c + d*x)/2) + 1)* c - 735*cos(c + d*x)*sin(c + d*x)**5*a - 810*cos(c + d*x)*sin(c + d*x)**5* c + 1880*cos(c + d*x)*sin(c + d*x)**3*a + 1680*cos(c + d*x)*sin(c + d*x)** 3*c - 1185*cos(c + d*x)*sin(c + d*x)*a - 870*cos(c + d*x)*sin(c + d*x)*c + 1152*sin(c + d*x)**7*a + 1600*sin(c + d*x)**7*c - 4032*sin(c + d*x)**5*a - 5120*sin(c + d*x)**5*c + 4800*sin(c + d*x)**3*a + 5440*sin(c + d*x)**3*c - 1920*sin(c + d*x)*a - 1920*sin(c + d*x)*c))/(240*cos(c + d*x)*d*(sin(c + d*x)**6 - 3*sin(c + d*x)**4 + 3*sin(c + d*x)**2 - 1))