Integrand size = 42, antiderivative size = 617 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^{5/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {(a-b) \sqrt {a+b} \left (284 a^2 b B+15 b^3 B+128 a^3 C+264 a b^2 C\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{192 a b d}+\frac {\sqrt {a+b} \left (15 b^3 B+8 a^3 (9 B+16 C)+4 a^2 b (71 B+52 C)+2 a b^2 (59 B+132 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{192 a d}-\frac {\sqrt {a+b} \left (48 a^4 B+120 a^2 b^2 B-5 b^4 B+160 a^3 b C+40 a b^3 C\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{64 a^2 d}+\frac {\left (284 a^2 b B+15 b^3 B+128 a^3 C+264 a b^2 C\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{192 a d}+\frac {\left (36 a^2 B+59 b^2 B+104 a b C\right ) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{96 d}+\frac {a (11 b B+8 a C) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {a B \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d} \]
1/4*a*B*cos(d*x+c)^3*(a+b*sec(d*x+c))^(3/2)*sin(d*x+c)/d+1/192*(a-b)*(284* B*a^2*b+15*B*b^3+128*C*a^3+264*C*a*b^2)*cot(d*x+c)*EllipticE((a+b*sec(d*x+ c))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(a+b)^(1/2)*(b*(1-sec(d*x+c))/( a+b))^(1/2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/a/b/d+1/192*(15*B*b^3+8*a^3*(9 *B+16*C)+4*a^2*b*(71*B+52*C)+2*a*b^2*(59*B+132*C))*cot(d*x+c)*EllipticF((a +b*sec(d*x+c))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(a+b)^(1/2)*(b*(1-se c(d*x+c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/a/d-1/64*(48*B*a^4+ 120*B*a^2*b^2-5*B*b^4+160*C*a^3*b+40*C*a*b^3)*cot(d*x+c)*EllipticPi((a+b*s ec(d*x+c))^(1/2)/(a+b)^(1/2),(a+b)/a,((a+b)/(a-b))^(1/2))*(a+b)^(1/2)*(b*( 1-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/a^2/d+1/192*(28 4*B*a^2*b+15*B*b^3+128*C*a^3+264*C*a*b^2)*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2 )/a/d+1/96*(36*B*a^2+59*B*b^2+104*C*a*b)*cos(d*x+c)*sin(d*x+c)*(a+b*sec(d* x+c))^(1/2)/d+1/24*a*(11*B*b+8*C*a)*cos(d*x+c)^2*sin(d*x+c)*(a+b*sec(d*x+c ))^(1/2)/d
Leaf count is larger than twice the leaf count of optimal. \(5172\) vs. \(2(617)=1234\).
Time = 22.64 (sec) , antiderivative size = 5172, normalized size of antiderivative = 8.38 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^{5/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Result too large to show} \]
Time = 3.36 (sec) , antiderivative size = 624, normalized size of antiderivative = 1.01, number of steps used = 22, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {3042, 4560, 3042, 4513, 27, 3042, 4582, 27, 3042, 4592, 27, 3042, 4592, 27, 3042, 4546, 3042, 4409, 3042, 4271, 4319, 4492}
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.
| \(\displaystyle \int \cos ^5(c+d x) (a+b \sec (c+d x))^{5/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^5}dx\) |
| \(\Big \downarrow \) 4560 |
| \(\displaystyle \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} (B+C \sec (c+d x))dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (B+C \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^4}dx\) |
| \(\Big \downarrow \) 4513 |
| \(\displaystyle \frac {a B \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}-\frac {1}{4} \int -\frac {1}{2} \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left (b (3 a B+8 b C) \sec ^2(c+d x)+2 \left (3 B a^2+8 b C a+4 b^2 B\right ) \sec (c+d x)+a (11 b B+8 a C)\right )dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \int \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left (b (3 a B+8 b C) \sec ^2(c+d x)+2 \left (3 B a^2+8 b C a+4 b^2 B\right ) \sec (c+d x)+a (11 b B+8 a C)\right )dx+\frac {a B \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )} \left (b (3 a B+8 b C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (3 B a^2+8 b C a+4 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a (11 b B+8 a C)\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {a B \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{3} \int \frac {\cos ^2(c+d x) \left (3 b \left (8 C a^2+17 b B a+16 b^2 C\right ) \sec ^2(c+d x)+2 \left (16 C a^3+49 b B a^2+72 b^2 C a+24 b^3 B\right ) \sec (c+d x)+a \left (36 B a^2+104 b C a+59 b^2 B\right )\right )}{2 \sqrt {a+b \sec (c+d x)}}dx+\frac {a (8 a C+11 b B) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {a B \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \int \frac {\cos ^2(c+d x) \left (3 b \left (8 C a^2+17 b B a+16 b^2 C\right ) \sec ^2(c+d x)+2 \left (16 C a^3+49 b B a^2+72 b^2 C a+24 b^3 B\right ) \sec (c+d x)+a \left (36 B a^2+104 b C a+59 b^2 B\right )\right )}{\sqrt {a+b \sec (c+d x)}}dx+\frac {a (8 a C+11 b B) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {a B \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \int \frac {3 b \left (8 C a^2+17 b B a+16 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (16 C a^3+49 b B a^2+72 b^2 C a+24 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a \left (36 B a^2+104 b C a+59 b^2 B\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {a (8 a C+11 b B) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {a B \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\left (36 a^2 B+104 a b C+59 b^2 B\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}-\frac {\int -\frac {\cos (c+d x) \left (a b \left (36 B a^2+104 b C a+59 b^2 B\right ) \sec ^2(c+d x)+2 a \left (36 B a^3+152 b C a^2+161 b^2 B a+96 b^3 C\right ) \sec (c+d x)+a \left (128 C a^3+284 b B a^2+264 b^2 C a+15 b^3 B\right )\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{2 a}\right )+\frac {a (8 a C+11 b B) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {a B \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\int \frac {\cos (c+d x) \left (a b \left (36 B a^2+104 b C a+59 b^2 B\right ) \sec ^2(c+d x)+2 a \left (36 B a^3+152 b C a^2+161 b^2 B a+96 b^3 C\right ) \sec (c+d x)+a \left (128 C a^3+284 b B a^2+264 b^2 C a+15 b^3 B\right )\right )}{\sqrt {a+b \sec (c+d x)}}dx}{4 a}+\frac {\left (36 a^2 B+104 a b C+59 b^2 B\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {a (8 a C+11 b B) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {a B \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\int \frac {a b \left (36 B a^2+104 b C a+59 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 a \left (36 B a^3+152 b C a^2+161 b^2 B a+96 b^3 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a \left (128 C a^3+284 b B a^2+264 b^2 C a+15 b^3 B\right )}{\csc \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{4 a}+\frac {\left (36 a^2 B+104 a b C+59 b^2 B\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {a (8 a C+11 b B) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {a B \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\left (128 a^3 C+284 a^2 b B+264 a b^2 C+15 b^3 B\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}-\frac {\int -\frac {2 b \left (36 B a^2+104 b C a+59 b^2 B\right ) \sec (c+d x) a^2-b \left (128 C a^3+284 b B a^2+264 b^2 C a+15 b^3 B\right ) \sec ^2(c+d x) a+3 \left (48 B a^4+160 b C a^3+120 b^2 B a^2+40 b^3 C a-5 b^4 B\right ) a}{2 \sqrt {a+b \sec (c+d x)}}dx}{a}}{4 a}+\frac {\left (36 a^2 B+104 a b C+59 b^2 B\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {a (8 a C+11 b B) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {a B \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\int \frac {2 b \left (36 B a^2+104 b C a+59 b^2 B\right ) \sec (c+d x) a^2-b \left (128 C a^3+284 b B a^2+264 b^2 C a+15 b^3 B\right ) \sec ^2(c+d x) a+3 \left (48 B a^4+160 b C a^3+120 b^2 B a^2+40 b^3 C a-5 b^4 B\right ) a}{\sqrt {a+b \sec (c+d x)}}dx}{2 a}+\frac {\left (128 a^3 C+284 a^2 b B+264 a b^2 C+15 b^3 B\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}}{4 a}+\frac {\left (36 a^2 B+104 a b C+59 b^2 B\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {a (8 a C+11 b B) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {a B \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\int \frac {2 b \left (36 B a^2+104 b C a+59 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2-b \left (128 C a^3+284 b B a^2+264 b^2 C a+15 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2 a+3 \left (48 B a^4+160 b C a^3+120 b^2 B a^2+40 b^3 C a-5 b^4 B\right ) a}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}+\frac {\left (128 a^3 C+284 a^2 b B+264 a b^2 C+15 b^3 B\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}}{4 a}+\frac {\left (36 a^2 B+104 a b C+59 b^2 B\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {a (8 a C+11 b B) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {a B \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 4546 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\int \frac {3 a \left (48 B a^4+160 b C a^3+120 b^2 B a^2+40 b^3 C a-5 b^4 B\right )+\left (2 b \left (36 B a^2+104 b C a+59 b^2 B\right ) a^2+b \left (128 C a^3+284 b B a^2+264 b^2 C a+15 b^3 B\right ) a\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx-a b \left (128 a^3 C+284 a^2 b B+264 a b^2 C+15 b^3 B\right ) \int \frac {\sec (c+d x) (\sec (c+d x)+1)}{\sqrt {a+b \sec (c+d x)}}dx}{2 a}+\frac {\left (128 a^3 C+284 a^2 b B+264 a b^2 C+15 b^3 B\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}}{4 a}+\frac {\left (36 a^2 B+104 a b C+59 b^2 B\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {a (8 a C+11 b B) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {a B \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\int \frac {3 a \left (48 B a^4+160 b C a^3+120 b^2 B a^2+40 b^3 C a-5 b^4 B\right )+\left (2 b \left (36 B a^2+104 b C a+59 b^2 B\right ) a^2+b \left (128 C a^3+284 b B a^2+264 b^2 C a+15 b^3 B\right ) a\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-a b \left (128 a^3 C+284 a^2 b B+264 a b^2 C+15 b^3 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}+\frac {\left (128 a^3 C+284 a^2 b B+264 a b^2 C+15 b^3 B\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}}{4 a}+\frac {\left (36 a^2 B+104 a b C+59 b^2 B\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {a (8 a C+11 b B) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {a B \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 4409 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {-a b \left (128 a^3 C+284 a^2 b B+264 a b^2 C+15 b^3 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+a b \left (8 a^3 (9 B+16 C)+4 a^2 b (71 B+52 C)+2 a b^2 (59 B+132 C)+15 b^3 B\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx+3 a \left (48 a^4 B+160 a^3 b C+120 a^2 b^2 B+40 a b^3 C-5 b^4 B\right ) \int \frac {1}{\sqrt {a+b \sec (c+d x)}}dx}{2 a}+\frac {\left (128 a^3 C+284 a^2 b B+264 a b^2 C+15 b^3 B\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}}{4 a}+\frac {\left (36 a^2 B+104 a b C+59 b^2 B\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {a (8 a C+11 b B) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {a B \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {a b \left (8 a^3 (9 B+16 C)+4 a^2 b (71 B+52 C)+2 a b^2 (59 B+132 C)+15 b^3 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-a b \left (128 a^3 C+284 a^2 b B+264 a b^2 C+15 b^3 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+3 a \left (48 a^4 B+160 a^3 b C+120 a^2 b^2 B+40 a b^3 C-5 b^4 B\right ) \int \frac {1}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}+\frac {\left (128 a^3 C+284 a^2 b B+264 a b^2 C+15 b^3 B\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}}{4 a}+\frac {\left (36 a^2 B+104 a b C+59 b^2 B\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {a (8 a C+11 b B) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {a B \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 4271 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {a b \left (8 a^3 (9 B+16 C)+4 a^2 b (71 B+52 C)+2 a b^2 (59 B+132 C)+15 b^3 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-a b \left (128 a^3 C+284 a^2 b B+264 a b^2 C+15 b^3 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 \sqrt {a+b} \left (48 a^4 B+160 a^3 b C+120 a^2 b^2 B+40 a b^3 C-5 b^4 B\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}}{2 a}+\frac {\left (128 a^3 C+284 a^2 b B+264 a b^2 C+15 b^3 B\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}}{4 a}+\frac {\left (36 a^2 B+104 a b C+59 b^2 B\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {a (8 a C+11 b B) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {a B \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 4319 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {-a b \left (128 a^3 C+284 a^2 b B+264 a b^2 C+15 b^3 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a \sqrt {a+b} \left (8 a^3 (9 B+16 C)+4 a^2 b (71 B+52 C)+2 a b^2 (59 B+132 C)+15 b^3 B\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}-\frac {6 \sqrt {a+b} \left (48 a^4 B+160 a^3 b C+120 a^2 b^2 B+40 a b^3 C-5 b^4 B\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}}{2 a}+\frac {\left (128 a^3 C+284 a^2 b B+264 a b^2 C+15 b^3 B\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}}{4 a}+\frac {\left (36 a^2 B+104 a b C+59 b^2 B\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {a (8 a C+11 b B) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {a B \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 4492 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\left (36 a^2 B+104 a b C+59 b^2 B\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}+\frac {\frac {\left (128 a^3 C+284 a^2 b B+264 a b^2 C+15 b^3 B\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}+\frac {\frac {2 a \sqrt {a+b} \left (8 a^3 (9 B+16 C)+4 a^2 b (71 B+52 C)+2 a b^2 (59 B+132 C)+15 b^3 B\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}+\frac {2 a (a-b) \sqrt {a+b} \left (128 a^3 C+284 a^2 b B+264 a b^2 C+15 b^3 B\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{b d}-\frac {6 \sqrt {a+b} \left (48 a^4 B+160 a^3 b C+120 a^2 b^2 B+40 a b^3 C-5 b^4 B\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}}{2 a}}{4 a}\right )+\frac {a (8 a C+11 b B) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {a B \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
(a*B*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d) + ((a*( 11*b*B + 8*a*C)*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d ) + (((36*a^2*B + 59*b^2*B + 104*a*b*C)*Cos[c + d*x]*Sqrt[a + b*Sec[c + d* x]]*Sin[c + d*x])/(2*d) + (((2*a*(a - b)*Sqrt[a + b]*(284*a^2*b*B + 15*b^3 *B + 128*a^3*C + 264*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec [c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) + (2*a*Sqrt[a + b]*(15 *b^3*B + 8*a^3*(9*B + 16*C) + 4*a^2*b*(71*B + 52*C) + 2*a*b^2*(59*B + 132* C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], ( a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - (6*Sqrt[a + b]*(48*a^4*B + 120*a^2*b^2*B - 5*b^4*B + 160*a^3*b*C + 40*a*b^3*C)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqr t[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d)/(2*a) + ((284* a^2*b*B + 15*b^3*B + 128*a^3*C + 264*a*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Sin [c + d*x])/d)/(4*a))/6)/8
3.9.36.3.1 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[1/Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[2*(Rt[a + b, 2]/(a*d*Cot[c + d*x]))*Sqrt[b*((1 - Csc[c + d*x])/(a + b))]*Sqrt[(-b) *((1 + Csc[c + d*x])/(a - b))]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Csc[ c + d*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S ymbol] :> Simp[-2*(Rt[a + b, 2]/(b*f*Cot[e + f*x]))*Sqrt[(b*(1 - Csc[e + f* x]))/(a + b)]*Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]*EllipticF[ArcSin[Sqrt [a + b*Csc[e + f*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_ .) + (a_)], x_Symbol] :> Simp[c Int[1/Sqrt[a + b*Csc[e + f*x]], x], x] + Simp[d Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[c sc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(A*b - a*B)*Rt[a + b*(B/A), 2]*Sqrt[b*((1 - Csc[e + f*x])/(a + b))]*(Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]/(b^2*f*Cot[e + f*x]))*EllipticE[ArcSin[Sqrt[a + b*Csc[e + f*x]]/Rt[a + b*(B/A), 2]], (a*A + b*B)/(a*A - b*B)], x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 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[1/(d*n) Int[(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^(n + 1)*Simp[ a*(a*B*n - A*b*(m - n - 1)) + (2*a*b*B*n + A*(b^2*n + a^2*(1 + n)))*Csc[e + f*x] + b*(b*B*n + a*A*(m + n))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] & & LeQ[n, -1]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Int[(A + (B - C )*Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x] + Simp[C Int[Csc[e + f*x]*(( 1 + Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]]), x], x] /; FreeQ[{a, b, e, f, A , B, C}, x] && NeQ[a^2 - b^2, 0]
Int[((a_.) + csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_. )*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*((c_.) + csc[(e_.) + (f_.) *(x_)]*(d_.))^(n_.), x_Symbol] :> Simp[1/b^2 Int[(a + b*Csc[e + f*x])^(m + 1)*(c + d*Csc[e + f*x])^n*(b*B - a*C + b*C*Csc[e + f*x]), x], x] /; FreeQ [{a, b, c, d, e, f, A, B, C, m, n}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0]
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/(d*n) Int[(a + b*Csc[e + f*x])^(m - 1)*(d* Csc[e + f*x])^(n + 1)*Simp[A*b*m - a*B*n - (b*B*n + a*(C*n + A*(n + 1)))*Cs c[e + f*x] - b*(C*n + A*(m + n + 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a , b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && LeQ[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 + 1)*((d *Csc[e + f*x])^n/(a*f*n)), x] + Simp[1/(a*d*n) Int[(a + b*Csc[e + f*x])^m *(d*Csc[e + f*x])^(n + 1)*Simp[a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)* Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d , e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(5804\) vs. \(2(568)=1136\).
Time = 506.52 (sec) , antiderivative size = 5805, normalized size of antiderivative = 9.41
int(cos(d*x+c)^5*(a+b*sec(d*x+c))^(5/2)*(B*sec(d*x+c)+C*sec(d*x+c)^2),x,me thod=_RETURNVERBOSE)
\[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^{5/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right )\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{5} \,d x } \]
integrate(cos(d*x+c)^5*(a+b*sec(d*x+c))^(5/2)*(B*sec(d*x+c)+C*sec(d*x+c)^2 ),x, algorithm="fricas")
integral((C*b^2*cos(d*x + c)^5*sec(d*x + c)^4 + B*a^2*cos(d*x + c)^5*sec(d *x + c) + (2*C*a*b + B*b^2)*cos(d*x + c)^5*sec(d*x + c)^3 + (C*a^2 + 2*B*a *b)*cos(d*x + c)^5*sec(d*x + c)^2)*sqrt(b*sec(d*x + c) + a), x)
Timed out. \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^{5/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
\[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^{5/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right )\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{5} \,d x } \]
integrate(cos(d*x+c)^5*(a+b*sec(d*x+c))^(5/2)*(B*sec(d*x+c)+C*sec(d*x+c)^2 ),x, algorithm="maxima")
\[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^{5/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right )\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{5} \,d x } \]
integrate(cos(d*x+c)^5*(a+b*sec(d*x+c))^(5/2)*(B*sec(d*x+c)+C*sec(d*x+c)^2 ),x, algorithm="giac")
Timed out. \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^{5/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^5\,\left (\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2} \,d x \]