Integrand size = 43, antiderivative size = 311 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^{5/2} (1015 A+1132 B+1304 C) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{512 d}+\frac {a^3 (1015 A+1132 B+1304 C) \sin (c+d x)}{512 d \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (1015 A+1132 B+1304 C) \cos (c+d x) \sin (c+d x)}{768 d \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (545 A+628 B+680 C) \cos ^2(c+d x) \sin (c+d x)}{960 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (115 A+156 B+120 C) \cos ^3(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{480 d}+\frac {a (5 A+12 B) \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{60 d}+\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d} \]
1/512*a^(5/2)*(1015*A+1132*B+1304*C)*arctan(a^(1/2)*tan(d*x+c)/(a+a*sec(d* x+c))^(1/2))/d+1/60*a*(5*A+12*B)*cos(d*x+c)^4*(a+a*sec(d*x+c))^(3/2)*sin(d *x+c)/d+1/6*A*cos(d*x+c)^5*(a+a*sec(d*x+c))^(5/2)*sin(d*x+c)/d+1/512*a^3*( 1015*A+1132*B+1304*C)*sin(d*x+c)/d/(a+a*sec(d*x+c))^(1/2)+1/768*a^3*(1015* A+1132*B+1304*C)*cos(d*x+c)*sin(d*x+c)/d/(a+a*sec(d*x+c))^(1/2)+1/960*a^3* (545*A+628*B+680*C)*cos(d*x+c)^2*sin(d*x+c)/d/(a+a*sec(d*x+c))^(1/2)+1/480 *a^2*(115*A+156*B+120*C)*cos(d*x+c)^3*sin(d*x+c)*(a+a*sec(d*x+c))^(1/2)/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 2.57 (sec) , antiderivative size = 634, normalized size of antiderivative = 2.04 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^2 \left (862155 A \text {arctanh}\left (\sqrt {1-\sec (c+d x)}\right )+933660 B \text {arctanh}\left (\sqrt {1-\sec (c+d x)}\right )+1020600 C \text {arctanh}\left (\sqrt {1-\sec (c+d x)}\right )+449183 A \sqrt {1-\sec (c+d x)}+419436 B \sqrt {1-\sec (c+d x)}+349272 C \sqrt {1-\sec (c+d x)}+1358777 A \cos (c+d x) \sqrt {1-\sec (c+d x)}+1333044 B \cos (c+d x) \sqrt {1-\sec (c+d x)}+1283688 C \cos (c+d x) \sqrt {1-\sec (c+d x)}+505449 A \cos (2 (c+d x)) \sqrt {1-\sec (c+d x)}+455508 B \cos (2 (c+d x)) \sqrt {1-\sec (c+d x)}+352296 C \cos (2 (c+d x)) \sqrt {1-\sec (c+d x)}+190834 A \cos (3 (c+d x)) \sqrt {1-\sec (c+d x)}+137448 B \cos (3 (c+d x)) \sqrt {1-\sec (c+d x)}+87696 C \cos (3 (c+d x)) \sqrt {1-\sec (c+d x)}+57666 A \cos (4 (c+d x)) \sqrt {1-\sec (c+d x)}+36072 B \cos (4 (c+d x)) \sqrt {1-\sec (c+d x)}+3024 C \cos (4 (c+d x)) \sqrt {1-\sec (c+d x)}+15176 A \cos (5 (c+d x)) \sqrt {1-\sec (c+d x)}+2592 B \cos (5 (c+d x)) \sqrt {1-\sec (c+d x)}+1400 A \cos (6 (c+d x)) \sqrt {1-\sec (c+d x)}+774144 C \operatorname {Hypergeometric2F1}\left (\frac {1}{2},5,\frac {3}{2},1-\sec (c+d x)\right ) \sqrt {1-\sec (c+d x)}+552960 B \operatorname {Hypergeometric2F1}\left (\frac {1}{2},6,\frac {3}{2},1-\sec (c+d x)\right ) \sqrt {1-\sec (c+d x)}+430080 A \operatorname {Hypergeometric2F1}\left (\frac {1}{2},7,\frac {3}{2},1-\sec (c+d x)\right ) \sqrt {1-\sec (c+d x)}\right ) \sqrt {a (1+\sec (c+d x))} \sin (c+d x)}{483840 d (1+\cos (c+d x)) \sqrt {1-\sec (c+d x)}} \]
Integrate[Cos[c + d*x]^6*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
(a^2*(862155*A*ArcTanh[Sqrt[1 - Sec[c + d*x]]] + 933660*B*ArcTanh[Sqrt[1 - Sec[c + d*x]]] + 1020600*C*ArcTanh[Sqrt[1 - Sec[c + d*x]]] + 449183*A*Sqr t[1 - Sec[c + d*x]] + 419436*B*Sqrt[1 - Sec[c + d*x]] + 349272*C*Sqrt[1 - Sec[c + d*x]] + 1358777*A*Cos[c + d*x]*Sqrt[1 - Sec[c + d*x]] + 1333044*B* Cos[c + d*x]*Sqrt[1 - Sec[c + d*x]] + 1283688*C*Cos[c + d*x]*Sqrt[1 - Sec[ c + d*x]] + 505449*A*Cos[2*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 455508*B*Co s[2*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 352296*C*Cos[2*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 190834*A*Cos[3*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 137448 *B*Cos[3*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 87696*C*Cos[3*(c + d*x)]*Sqrt [1 - Sec[c + d*x]] + 57666*A*Cos[4*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 360 72*B*Cos[4*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 3024*C*Cos[4*(c + d*x)]*Sqr t[1 - Sec[c + d*x]] + 15176*A*Cos[5*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 25 92*B*Cos[5*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 1400*A*Cos[6*(c + d*x)]*Sqr t[1 - Sec[c + d*x]] + 774144*C*Hypergeometric2F1[1/2, 5, 3/2, 1 - Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]] + 552960*B*Hypergeometric2F1[1/2, 6, 3/2, 1 - Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]] + 430080*A*Hypergeometric2F1[1/2, 7, 3/2, 1 - Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]])*Sqrt[a*(1 + Sec[c + d*x])] *Sin[c + d*x])/(483840*d*(1 + Cos[c + d*x])*Sqrt[1 - Sec[c + d*x]])
Time = 1.98 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {3042, 4574, 27, 3042, 4505, 27, 3042, 4505, 27, 3042, 4503, 3042, 4292, 3042, 4292, 3042, 4261, 216}
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.
| \(\displaystyle \int \cos ^6(c+d x) (a \sec (c+d x)+a)^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^{5/2} \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^6}dx\) |
| \(\Big \downarrow \) 4574 |
| \(\displaystyle \frac {\int \frac {1}{2} \cos ^5(c+d x) (\sec (c+d x) a+a)^{5/2} (a (5 A+12 B)+a (5 A+12 C) \sec (c+d x))dx}{6 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \cos ^5(c+d x) (\sec (c+d x) a+a)^{5/2} (a (5 A+12 B)+a (5 A+12 C) \sec (c+d x))dx}{12 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{5/2} \left (a (5 A+12 B)+a (5 A+12 C) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^5}dx}{12 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\) |
| \(\Big \downarrow \) 4505 |
| \(\displaystyle \frac {\frac {1}{5} \int \frac {1}{2} \cos ^4(c+d x) (\sec (c+d x) a+a)^{3/2} \left ((115 A+156 B+120 C) a^2+15 (5 A+4 B+8 C) \sec (c+d x) a^2\right )dx+\frac {a^2 (5 A+12 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}}{12 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{10} \int \cos ^4(c+d x) (\sec (c+d x) a+a)^{3/2} \left ((115 A+156 B+120 C) a^2+15 (5 A+4 B+8 C) \sec (c+d x) a^2\right )dx+\frac {a^2 (5 A+12 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}}{12 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{10} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{3/2} \left ((115 A+156 B+120 C) a^2+15 (5 A+4 B+8 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {a^2 (5 A+12 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}}{12 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\) |
| \(\Big \downarrow \) 4505 |
| \(\displaystyle \frac {\frac {1}{10} \left (\frac {1}{4} \int \frac {1}{2} \cos ^3(c+d x) \sqrt {\sec (c+d x) a+a} \left (3 (545 A+628 B+680 C) a^3+5 (235 A+252 B+312 C) \sec (c+d x) a^3\right )dx+\frac {a^3 (115 A+156 B+120 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \sec (c+d x)+a}}{4 d}\right )+\frac {a^2 (5 A+12 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}}{12 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{10} \left (\frac {1}{8} \int \cos ^3(c+d x) \sqrt {\sec (c+d x) a+a} \left (3 (545 A+628 B+680 C) a^3+5 (235 A+252 B+312 C) \sec (c+d x) a^3\right )dx+\frac {a^3 (115 A+156 B+120 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \sec (c+d x)+a}}{4 d}\right )+\frac {a^2 (5 A+12 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}}{12 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{10} \left (\frac {1}{8} \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a} \left (3 (545 A+628 B+680 C) a^3+5 (235 A+252 B+312 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {a^3 (115 A+156 B+120 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \sec (c+d x)+a}}{4 d}\right )+\frac {a^2 (5 A+12 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}}{12 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\) |
| \(\Big \downarrow \) 4503 |
| \(\displaystyle \frac {\frac {1}{10} \left (\frac {1}{8} \left (\frac {5}{2} a^3 (1015 A+1132 B+1304 C) \int \cos ^2(c+d x) \sqrt {\sec (c+d x) a+a}dx+\frac {a^4 (545 A+628 B+680 C) \sin (c+d x) \cos ^2(c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^3 (115 A+156 B+120 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \sec (c+d x)+a}}{4 d}\right )+\frac {a^2 (5 A+12 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}}{12 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{10} \left (\frac {1}{8} \left (\frac {5}{2} a^3 (1015 A+1132 B+1304 C) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {a^4 (545 A+628 B+680 C) \sin (c+d x) \cos ^2(c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^3 (115 A+156 B+120 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \sec (c+d x)+a}}{4 d}\right )+\frac {a^2 (5 A+12 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}}{12 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\) |
| \(\Big \downarrow \) 4292 |
| \(\displaystyle \frac {\frac {1}{10} \left (\frac {1}{8} \left (\frac {5}{2} a^3 (1015 A+1132 B+1304 C) \left (\frac {3}{4} \int \cos (c+d x) \sqrt {\sec (c+d x) a+a}dx+\frac {a \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^4 (545 A+628 B+680 C) \sin (c+d x) \cos ^2(c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^3 (115 A+156 B+120 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \sec (c+d x)+a}}{4 d}\right )+\frac {a^2 (5 A+12 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}}{12 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{10} \left (\frac {1}{8} \left (\frac {5}{2} a^3 (1015 A+1132 B+1304 C) \left (\frac {3}{4} \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {a \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^4 (545 A+628 B+680 C) \sin (c+d x) \cos ^2(c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^3 (115 A+156 B+120 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \sec (c+d x)+a}}{4 d}\right )+\frac {a^2 (5 A+12 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}}{12 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\) |
| \(\Big \downarrow \) 4292 |
| \(\displaystyle \frac {\frac {1}{10} \left (\frac {1}{8} \left (\frac {5}{2} a^3 (1015 A+1132 B+1304 C) \left (\frac {3}{4} \left (\frac {1}{2} \int \sqrt {\sec (c+d x) a+a}dx+\frac {a \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^4 (545 A+628 B+680 C) \sin (c+d x) \cos ^2(c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^3 (115 A+156 B+120 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \sec (c+d x)+a}}{4 d}\right )+\frac {a^2 (5 A+12 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}}{12 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{10} \left (\frac {1}{8} \left (\frac {5}{2} a^3 (1015 A+1132 B+1304 C) \left (\frac {3}{4} \left (\frac {1}{2} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}dx+\frac {a \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^4 (545 A+628 B+680 C) \sin (c+d x) \cos ^2(c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^3 (115 A+156 B+120 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \sec (c+d x)+a}}{4 d}\right )+\frac {a^2 (5 A+12 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}}{12 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\) |
| \(\Big \downarrow \) 4261 |
| \(\displaystyle \frac {\frac {1}{10} \left (\frac {1}{8} \left (\frac {5}{2} a^3 (1015 A+1132 B+1304 C) \left (\frac {3}{4} \left (\frac {a \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {a \int \frac {1}{\frac {a^2 \tan ^2(c+d x)}{\sec (c+d x) a+a}+a}d\left (-\frac {a \tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{d}\right )+\frac {a \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^4 (545 A+628 B+680 C) \sin (c+d x) \cos ^2(c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^3 (115 A+156 B+120 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \sec (c+d x)+a}}{4 d}\right )+\frac {a^2 (5 A+12 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}}{12 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {a^2 (5 A+12 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}+\frac {1}{10} \left (\frac {a^3 (115 A+156 B+120 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \sec (c+d x)+a}}{4 d}+\frac {1}{8} \left (\frac {a^4 (545 A+628 B+680 C) \sin (c+d x) \cos ^2(c+d x)}{d \sqrt {a \sec (c+d x)+a}}+\frac {5}{2} a^3 (1015 A+1132 B+1304 C) \left (\frac {3}{4} \left (\frac {\sqrt {a} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{d}+\frac {a \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a \sec (c+d x)+a}}\right )\right )\right )}{12 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\) |
(A*Cos[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(6*d) + ((a^2*( 5*A + 12*B)*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d) + ((a^3*(115*A + 156*B + 120*C)*Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Si n[c + d*x])/(4*d) + ((a^4*(545*A + 628*B + 680*C)*Cos[c + d*x]^2*Sin[c + d *x])/(d*Sqrt[a + a*Sec[c + d*x]]) + (5*a^3*(1015*A + 1132*B + 1304*C)*((a* Cos[c + d*x]*Sin[c + d*x])/(2*d*Sqrt[a + a*Sec[c + d*x]]) + (3*((Sqrt[a]*A rcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a*Sin[c + d*x ])/(d*Sqrt[a + a*Sec[c + d*x]])))/4))/2)/8)/10)/(12*a)
3.6.9.3.1 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_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(b/d) Subst[Int[1/(a + x^2), x], x, b*(Cot[c + d*x]/Sqrt[a + b*Csc[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[a*Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*n*Sqrt[a + b*Csc[e + f*x]])), x] + Simp[a*((2*n + 1)/(2*b*d*n)) Int[Sqrt[a + b*Csc [e + f*x]]*(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, -2^(-1)] && IntegerQ[2*n]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[A*b^2*Co t[e + f*x]*((d*Csc[e + f*x])^n/(a*f*n*Sqrt[a + b*Csc[e + f*x]])), x] + Simp [(A*b*(2*n + 1) + 2*a*B*n)/(2*a*d*n) Int[Sqrt[a + b*Csc[e + f*x]]*(d*Csc[ e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a *B, 0] && EqQ[a^2 - b^2, 0] && NeQ[A*b*(2*n + 1) + 2*a*B*n, 0] && LtQ[n, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[a*A*Cot [e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*((d*Csc[e + f*x])^n/(f*n)), x] - Sim p[b/(a*d*n) Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)*Sim p[a*A*(m - n - 1) - b*B*n - (a*B*n + A*b*(m + n))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0 ] && GtQ[m, 1/2] && LtQ[n, -1]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(f*n)), x] - Simp[1/(b*d*n) Int[(a + b*Csc[e + f*x])^m*(d*Csc[ e + f*x])^(n + 1)*Simp[a*A*m - b*B*n - b*(A*(m + n + 1) + C*n)*Csc[e + f*x] , x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, m}, x] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)] && (LtQ[n, -2^(-1)] || EqQ[m + n + 1, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(664\) vs. \(2(279)=558\).
Time = 0.57 (sec) , antiderivative size = 665, normalized size of antiderivative = 2.14
\[\frac {a^{2} \left (1280 A \cos \left (d x +c \right )^{6} \sin \left (d x +c \right )+4480 A \cos \left (d x +c \right )^{5} \sin \left (d x +c \right )+1536 B \cos \left (d x +c \right )^{5} \sin \left (d x +c \right )+6960 A \cos \left (d x +c \right )^{4} \sin \left (d x +c \right )+5568 B \cos \left (d x +c \right )^{4} \sin \left (d x +c \right )+1920 C \cos \left (d x +c \right )^{4} \sin \left (d x +c \right )+8120 A \cos \left (d x +c \right )^{3} \sin \left (d x +c \right )+9056 B \sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}+7360 C \cos \left (d x +c \right )^{3} \sin \left (d x +c \right )+15225 A \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )+10150 A \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )+16980 B \,\operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right )+11320 B \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}+19560 C \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )+13040 C \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )+15225 A \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+15225 A \cos \left (d x +c \right ) \sin \left (d x +c \right )+16980 B \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+16980 B \cos \left (d x +c \right ) \sin \left (d x +c \right )+19560 C \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+19560 C \cos \left (d x +c \right ) \sin \left (d x +c \right )\right ) \sqrt {a \left (1+\sec \left (d x +c \right )\right )}}{7680 d \left (\cos \left (d x +c \right )+1\right )}\]
1/7680*a^2/d*(1280*A*cos(d*x+c)^6*sin(d*x+c)+4480*A*cos(d*x+c)^5*sin(d*x+c )+1536*B*cos(d*x+c)^5*sin(d*x+c)+6960*A*cos(d*x+c)^4*sin(d*x+c)+5568*B*cos (d*x+c)^4*sin(d*x+c)+1920*C*cos(d*x+c)^4*sin(d*x+c)+8120*A*cos(d*x+c)^3*si n(d*x+c)+9056*B*sin(d*x+c)*cos(d*x+c)^3+7360*C*cos(d*x+c)^3*sin(d*x+c)+152 25*A*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(sin(d*x+c)/(cos(d*x+c)+1)/ (-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)+10150*A*cos(d*x+c)^2*sin(d* x+c)+16980*B*arctanh(sin(d*x+c)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1) )^(1/2))*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)+11320*B*sin(d*x+c)* cos(d*x+c)^2+19560*C*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(sin(d*x+c) /(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)+13040*C*cos (d*x+c)^2*sin(d*x+c)+15225*A*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(si n(d*x+c)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+15225*A*cos(d* x+c)*sin(d*x+c)+16980*B*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(sin(d*x +c)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+16980*B*cos(d*x+c)* sin(d*x+c)+19560*C*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(sin(d*x+c)/( cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+19560*C*cos(d*x+c)*sin(d *x+c))*(a*(1+sec(d*x+c)))^(1/2)/(cos(d*x+c)+1)
Time = 0.45 (sec) , antiderivative size = 536, normalized size of antiderivative = 1.72 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\left [\frac {15 \, {\left ({\left (1015 \, A + 1132 \, B + 1304 \, C\right )} a^{2} \cos \left (d x + c\right ) + {\left (1015 \, A + 1132 \, B + 1304 \, C\right )} a^{2}\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) + 1}\right ) + 2 \, {\left (1280 \, A a^{2} \cos \left (d x + c\right )^{6} + 128 \, {\left (35 \, A + 12 \, B\right )} a^{2} \cos \left (d x + c\right )^{5} + 48 \, {\left (145 \, A + 116 \, B + 40 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 8 \, {\left (1015 \, A + 1132 \, B + 920 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 10 \, {\left (1015 \, A + 1132 \, B + 1304 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 15 \, {\left (1015 \, A + 1132 \, B + 1304 \, C\right )} a^{2} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{15360 \, {\left (d \cos \left (d x + c\right ) + d\right )}}, -\frac {15 \, {\left ({\left (1015 \, A + 1132 \, B + 1304 \, C\right )} a^{2} \cos \left (d x + c\right ) + {\left (1015 \, A + 1132 \, B + 1304 \, C\right )} a^{2}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) - {\left (1280 \, A a^{2} \cos \left (d x + c\right )^{6} + 128 \, {\left (35 \, A + 12 \, B\right )} a^{2} \cos \left (d x + c\right )^{5} + 48 \, {\left (145 \, A + 116 \, B + 40 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 8 \, {\left (1015 \, A + 1132 \, B + 920 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 10 \, {\left (1015 \, A + 1132 \, B + 1304 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 15 \, {\left (1015 \, A + 1132 \, B + 1304 \, C\right )} a^{2} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{7680 \, {\left (d \cos \left (d x + c\right ) + d\right )}}\right ] \]
integrate(cos(d*x+c)^6*(a+a*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="fricas")
[1/15360*(15*((1015*A + 1132*B + 1304*C)*a^2*cos(d*x + c) + (1015*A + 1132 *B + 1304*C)*a^2)*sqrt(-a)*log((2*a*cos(d*x + c)^2 - 2*sqrt(-a)*sqrt((a*co s(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + a*cos(d*x + c) - a)/(cos(d*x + c) + 1)) + 2*(1280*A*a^2*cos(d*x + c)^6 + 128*(35*A + 12*B) *a^2*cos(d*x + c)^5 + 48*(145*A + 116*B + 40*C)*a^2*cos(d*x + c)^4 + 8*(10 15*A + 1132*B + 920*C)*a^2*cos(d*x + c)^3 + 10*(1015*A + 1132*B + 1304*C)* a^2*cos(d*x + c)^2 + 15*(1015*A + 1132*B + 1304*C)*a^2*cos(d*x + c))*sqrt( (a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c) + d), -1/ 7680*(15*((1015*A + 1132*B + 1304*C)*a^2*cos(d*x + c) + (1015*A + 1132*B + 1304*C)*a^2)*sqrt(a)*arctan(sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d *x + c)/(sqrt(a)*sin(d*x + c))) - (1280*A*a^2*cos(d*x + c)^6 + 128*(35*A + 12*B)*a^2*cos(d*x + c)^5 + 48*(145*A + 116*B + 40*C)*a^2*cos(d*x + c)^4 + 8*(1015*A + 1132*B + 920*C)*a^2*cos(d*x + c)^3 + 10*(1015*A + 1132*B + 13 04*C)*a^2*cos(d*x + c)^2 + 15*(1015*A + 1132*B + 1304*C)*a^2*cos(d*x + c)) *sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c) + d )]
Timed out. \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
Timed out. \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
integrate(cos(d*x+c)^6*(a+a*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="maxima")
\[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{6} \,d x } \]
integrate(cos(d*x+c)^6*(a+a*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="giac")
Timed out. \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^6\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \]