Integrand size = 45, antiderivative size = 760 \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx=-\frac {(a-b) \sqrt {a+b} \left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{192 a b d \sqrt {\sec (c+d x)}}+\frac {\sqrt {a+b} \left (15 a^3 C+8 b^3 (12 A+16 B+9 C)+2 a^2 b (192 A+132 B+59 C)+4 a b^2 (108 A+52 B+71 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{192 b d \sqrt {\sec (c+d x)}}-\frac {\sqrt {a+b} \left (40 a^3 b B+160 a b^3 B-5 a^4 C+120 a^2 b^2 (2 A+C)+16 b^4 (4 A+3 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{64 b^2 d \sqrt {\sec (c+d x)}}+\frac {\left (16 A b^2+24 a b B+5 a^2 C+12 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{32 d \sqrt {\sec (c+d x)}}+\frac {(8 b B+5 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 d \sqrt {\sec (c+d x)}}+\frac {C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{4 d \sqrt {\sec (c+d x)}}+\frac {\left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) \sqrt {a+b \cos (c+d x)} \sqrt {\sec (c+d x)} \sin (c+d x)}{192 b d} \] Output:
-1/192*(a-b)*(a+b)^(1/2)*(264*B*a^2*b+128*B*b^3+15*a^3*C+4*a*b^2*(108*A+71 *C))*cos(d*x+c)^(1/2)*csc(d*x+c)*EllipticE((a+b*cos(d*x+c))^(1/2)/(a+b)^(1 /2)/cos(d*x+c)^(1/2),(-(a+b)/(a-b))^(1/2))*(a*(1-sec(d*x+c))/(a+b))^(1/2)* (a*(1+sec(d*x+c))/(a-b))^(1/2)/a/b/d/sec(d*x+c)^(1/2)+1/192*(a+b)^(1/2)*(1 5*a^3*C+8*b^3*(12*A+16*B+9*C)+2*a^2*b*(192*A+132*B+59*C)+4*a*b^2*(108*A+52 *B+71*C))*cos(d*x+c)^(1/2)*csc(d*x+c)*EllipticF((a+b*cos(d*x+c))^(1/2)/(a+ b)^(1/2)/cos(d*x+c)^(1/2),(-(a+b)/(a-b))^(1/2))*(a*(1-sec(d*x+c))/(a+b))^( 1/2)*(a*(1+sec(d*x+c))/(a-b))^(1/2)/b/d/sec(d*x+c)^(1/2)-1/64*(a+b)^(1/2)* (40*B*a^3*b+160*B*a*b^3-5*a^4*C+120*a^2*b^2*(2*A+C)+16*b^4*(4*A+3*C))*cos( d*x+c)^(1/2)*csc(d*x+c)*EllipticPi((a+b*cos(d*x+c))^(1/2)/(a+b)^(1/2)/cos( d*x+c)^(1/2),(a+b)/b,(-(a+b)/(a-b))^(1/2))*(a*(1-sec(d*x+c))/(a+b))^(1/2)* (a*(1+sec(d*x+c))/(a-b))^(1/2)/b^2/d/sec(d*x+c)^(1/2)+1/32*(16*A*b^2+24*B* a*b+5*C*a^2+12*C*b^2)*(a+b*cos(d*x+c))^(1/2)*sin(d*x+c)/d/sec(d*x+c)^(1/2) +1/24*(8*B*b+5*C*a)*(a+b*cos(d*x+c))^(3/2)*sin(d*x+c)/d/sec(d*x+c)^(1/2)+1 /4*C*(a+b*cos(d*x+c))^(5/2)*sin(d*x+c)/d/sec(d*x+c)^(1/2)+1/192*(264*B*a^2 *b+128*B*b^3+15*a^3*C+4*a*b^2*(108*A+71*C))*(a+b*cos(d*x+c))^(1/2)*sec(d*x +c)^(1/2)*sin(d*x+c)/b/d
Leaf count is larger than twice the leaf count of optimal. \(5541\) vs. \(2(760)=1520\).
Time = 31.51 (sec) , antiderivative size = 5541, normalized size of antiderivative = 7.29 \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx=\text {Result too large to show} \] Input:
Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^ 2)*Sqrt[Sec[c + d*x]],x]
Output:
Result too large to show
Time = 4.18 (sec) , antiderivative size = 729, normalized size of antiderivative = 0.96, number of steps used = 22, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.489, Rules used = {3042, 4709, 3042, 3528, 27, 3042, 3528, 27, 3042, 3528, 27, 3042, 3540, 25, 3042, 3532, 3042, 3288, 3477, 3042, 3295, 3473}
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.
| \(\displaystyle \int \sqrt {\sec (c+d x)} (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {\sec (c+d x)} (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos (c+d x)^2\right )dx\) |
| \(\Big \downarrow \) 4709 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {(a+b \cos (c+d x))^{5/2} \left (C \cos ^2(c+d x)+B \cos (c+d x)+A\right )}{\sqrt {\cos (c+d x)}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (C \sin \left (c+d x+\frac {\pi }{2}\right )^2+B \sin \left (c+d x+\frac {\pi }{2}\right )+A\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{4} \int \frac {(a+b \cos (c+d x))^{3/2} \left ((8 b B+5 a C) \cos ^2(c+d x)+2 (4 A b+3 C b+4 a B) \cos (c+d x)+a (8 A+C)\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{8} \int \frac {(a+b \cos (c+d x))^{3/2} \left ((8 b B+5 a C) \cos ^2(c+d x)+2 (4 A b+3 C b+4 a B) \cos (c+d x)+a (8 A+C)\right )}{\sqrt {\cos (c+d x)}}dx+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{8} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left ((8 b B+5 a C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 (4 A b+3 C b+4 a B) \sin \left (c+d x+\frac {\pi }{2}\right )+a (8 A+C)\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}\right )\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{8} \left (\frac {1}{3} \int \frac {\sqrt {a+b \cos (c+d x)} \left (3 \left (5 C a^2+24 b B a+16 A b^2+12 b^2 C\right ) \cos ^2(c+d x)+2 \left (24 B a^2+b (48 A+31 C) a+16 b^2 B\right ) \cos (c+d x)+a (48 a A+8 b B+11 a C)\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {(5 a C+8 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{8} \left (\frac {1}{6} \int \frac {\sqrt {a+b \cos (c+d x)} \left (3 \left (5 C a^2+24 b B a+16 A b^2+12 b^2 C\right ) \cos ^2(c+d x)+2 \left (24 B a^2+b (48 A+31 C) a+16 b^2 B\right ) \cos (c+d x)+a (48 a A+8 b B+11 a C)\right )}{\sqrt {\cos (c+d x)}}dx+\frac {(5 a C+8 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{8} \left (\frac {1}{6} \int \frac {\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (3 \left (5 C a^2+24 b B a+16 A b^2+12 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (24 B a^2+b (48 A+31 C) a+16 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a (48 a A+8 b B+11 a C)\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {(5 a C+8 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}\right )\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {\left (15 C a^3+264 b B a^2+4 b^2 (108 A+71 C) a+128 b^3 B\right ) \cos ^2(c+d x)+2 \left (96 B a^3+b (288 A+161 C) a^2+152 b^2 B a+12 b^3 (4 A+3 C)\right ) \cos (c+d x)+a \left (2 \left (96 A+\frac {59 C}{2}\right ) a^2+104 b B a+12 b^2 (4 A+3 C)\right )}{2 \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {(5 a C+8 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \int \frac {\left (15 C a^3+264 b B a^2+4 b^2 (108 A+71 C) a+128 b^3 B\right ) \cos ^2(c+d x)+2 \left (96 B a^3+b (288 A+161 C) a^2+152 b^2 B a+12 b^3 (4 A+3 C)\right ) \cos (c+d x)+a \left ((192 A+59 C) a^2+104 b B a+12 b^2 (4 A+3 C)\right )}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {(5 a C+8 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \int \frac {\left (15 C a^3+264 b B a^2+4 b^2 (108 A+71 C) a+128 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (96 B a^3+b (288 A+161 C) a^2+152 b^2 B a+12 b^3 (4 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left ((192 A+59 C) a^2+104 b B a+12 b^2 (4 A+3 C)\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {(5 a C+8 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}\right )\) |
| \(\Big \downarrow \) 3540 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\int -\frac {-3 \left (-5 C a^4+40 b B a^3+120 b^2 (2 A+C) a^2+160 b^3 B a+16 b^4 (4 A+3 C)\right ) \cos ^2(c+d x)-2 a b \left ((192 A+59 C) a^2+104 b B a+12 b^2 (4 A+3 C)\right ) \cos (c+d x)+a \left (15 C a^3+264 b B a^2+4 b^2 (108 A+71 C) a+128 b^3 B\right )}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{2 b}+\frac {\sin (c+d x) \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {(5 a C+8 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {\int \frac {-3 \left (-5 C a^4+40 b B a^3+120 b^2 (2 A+C) a^2+160 b^3 B a+16 b^4 (4 A+3 C)\right ) \cos ^2(c+d x)-2 a b \left ((192 A+59 C) a^2+104 b B a+12 b^2 (4 A+3 C)\right ) \cos (c+d x)+a \left (15 C a^3+264 b B a^2+4 b^2 (108 A+71 C) a+128 b^3 B\right )}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {(5 a C+8 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {\int \frac {-3 \left (-5 C a^4+40 b B a^3+120 b^2 (2 A+C) a^2+160 b^3 B a+16 b^4 (4 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-2 a b \left ((192 A+59 C) a^2+104 b B a+12 b^2 (4 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left (15 C a^3+264 b B a^2+4 b^2 (108 A+71 C) a+128 b^3 B\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {(5 a C+8 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}\right )\) |
| \(\Big \downarrow \) 3532 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {\int \frac {a \left (15 C a^3+264 b B a^2+4 b^2 (108 A+71 C) a+128 b^3 B\right )-2 a b \left ((192 A+59 C) a^2+104 b B a+12 b^2 (4 A+3 C)\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx-3 \left (-5 a^4 C+40 a^3 b B+120 a^2 b^2 (2 A+C)+160 a b^3 B+16 b^4 (4 A+3 C)\right ) \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {a+b \cos (c+d x)}}dx}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {(5 a C+8 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {\int \frac {a \left (15 C a^3+264 b B a^2+4 b^2 (108 A+71 C) a+128 b^3 B\right )-2 a b \left ((192 A+59 C) a^2+104 b B a+12 b^2 (4 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-3 \left (-5 a^4 C+40 a^3 b B+120 a^2 b^2 (2 A+C)+160 a b^3 B+16 b^4 (4 A+3 C)\right ) \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {(5 a C+8 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}\right )\) |
| \(\Big \downarrow \) 3288 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {\int \frac {a \left (15 C a^3+264 b B a^2+4 b^2 (108 A+71 C) a+128 b^3 B\right )-2 a b \left ((192 A+59 C) a^2+104 b B a+12 b^2 (4 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 \sqrt {a+b} \cot (c+d x) \left (-5 a^4 C+40 a^3 b B+120 a^2 b^2 (2 A+C)+160 a b^3 B+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b d}}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {(5 a C+8 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}\right )\) |
| \(\Big \downarrow \) 3477 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {a \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \int \frac {\cos (c+d x)+1}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx-a \left (15 a^3 C+2 a^2 b (192 A+132 B+59 C)+4 a b^2 (108 A+52 B+71 C)+8 b^3 (12 A+16 B+9 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx+\frac {6 \sqrt {a+b} \cot (c+d x) \left (-5 a^4 C+40 a^3 b B+120 a^2 b^2 (2 A+C)+160 a b^3 B+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b d}}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {(5 a C+8 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {-a \left (15 a^3 C+2 a^2 b (192 A+132 B+59 C)+4 a b^2 (108 A+52 B+71 C)+8 b^3 (12 A+16 B+9 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+a \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 \sqrt {a+b} \cot (c+d x) \left (-5 a^4 C+40 a^3 b B+120 a^2 b^2 (2 A+C)+160 a b^3 B+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b d}}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {(5 a C+8 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}\right )\) |
| \(\Big \downarrow \) 3295 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {a \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 \sqrt {a+b} \cot (c+d x) \left (15 a^3 C+2 a^2 b (192 A+132 B+59 C)+4 a b^2 (108 A+52 B+71 C)+8 b^3 (12 A+16 B+9 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}+\frac {6 \sqrt {a+b} \cot (c+d x) \left (-5 a^4 C+40 a^3 b B+120 a^2 b^2 (2 A+C)+160 a b^3 B+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b d}}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {(5 a C+8 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}\right )\) |
| \(\Big \downarrow \) 3473 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{2 d}+\frac {1}{4} \left (\frac {\sin (c+d x) \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {-\frac {2 \sqrt {a+b} \cot (c+d x) \left (15 a^3 C+2 a^2 b (192 A+132 B+59 C)+4 a b^2 (108 A+52 B+71 C)+8 b^3 (12 A+16 B+9 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}+\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{a d}+\frac {6 \sqrt {a+b} \cot (c+d x) \left (-5 a^4 C+40 a^3 b B+120 a^2 b^2 (2 A+C)+160 a b^3 B+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b d}}{2 b}\right )\right )+\frac {(5 a C+8 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}\right )\) |
Input:
Int[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqr t[Sec[c + d*x]],x]
Output:
Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*d) + (((8*b*B + 5*a*C)*Sqrt[Cos[c + d*x]]*( a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d) + ((3*(16*A*b^2 + 24*a*b*B + 5*a^2*C + 12*b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d *x])/(2*d) + (-1/2*((2*(a - b)*Sqrt[a + b]*(264*a^2*b*B + 128*b^3*B + 15*a ^3*C + 4*a*b^2*(108*A + 71*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Co s[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a *(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d) - (2*Sqrt[a + b]*(15*a^3*C + 8*b^3*(12*A + 16*B + 9*C) + 2*a^2*b*(192*A + 132*B + 59*C) + 4*a*b^2*(108*A + 52*B + 71*C))*Cot[c + d*x]*EllipticF[ArcS in[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/( a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/ (a - b)])/d + (6*Sqrt[a + b]*(40*a^3*b*B + 160*a*b^3*B - 5*a^4*C + 120*a^2 *b^2*(2*A + C) + 16*b^4*(4*A + 3*C))*Cot[c + d*x]*EllipticPi[(a + b)/b, Ar cSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b) /(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]) )/(a - b)])/(b*d))/b + ((264*a^2*b*B + 128*b^3*B + 15*a^3*C + 4*a*b^2*(108 *A + 71*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]] ))/4)/6)/8)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.) *(x_)]], x_Symbol] :> Simp[2*b*(Tan[e + f*x]/(d*f))*Rt[(c + d)/b, 2]*Sqrt[c *((1 + Csc[e + f*x])/(c - d))]*Sqrt[c*((1 - Csc[e + f*x])/(c + d))]*Ellipti cPi[(c + d)/d, ArcSin[Sqrt[c + d*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]]/Rt[(c + d)/b, 2]], -(c + d)/(c - d)], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] && PosQ[(c + d)/b]
Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f _.)*(x_)]]), x_Symbol] :> Simp[-2*(Tan[e + f*x]/(a*f))*Rt[(a + b)/d, 2]*Sqr t[a*((1 - Csc[e + f*x])/(a + b))]*Sqrt[a*((1 + Csc[e + f*x])/(a - b))]*Elli pticF[ArcSin[Sqrt[a + b*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]/Rt[(a + b)/d, 2] ], -(a + b)/(a - b)], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && PosQ[(a + b)/d]
Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((b_.)*sin[(e_.) + (f_.)*(x_)]) ^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*A* (c - d)*(Tan[e + f*x]/(f*b*c^2))*Rt[(c + d)/b, 2]*Sqrt[c*((1 + Csc[e + f*x] )/(c - d))]*Sqrt[c*((1 - Csc[e + f*x])/(c + d))]*EllipticE[ArcSin[Sqrt[c + d*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]]/Rt[(c + d)/b, 2]], -(c + d)/(c - d)], x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && PosQ[(c + d)/b]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ .)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> S imp[(A - B)/(a - b) Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f* x]]), x], x] - Simp[(A*b - a*B)/(a - b) Int[(1 + Sin[e + f*x])/((a + b*Si n[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e , f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && NeQ[A, B]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_ .) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a *c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n} , x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[ m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_.) + (d_.)*sin[(e _.) + (f_.)*(x_)]]), x_Symbol] :> Simp[C/b^2 Int[Sqrt[a + b*Sin[e + f*x]] /Sqrt[c + d*Sin[e + f*x]], x], x] + Simp[1/b^2 Int[(A*b^2 - a^2*C + b*(b* B - 2*a*C)*Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x ]]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] & & NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(Sqrt[c + d*Sin[e + f *x]]/(d*f*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[1/(2*d) Int[(1/((a + b*Si n[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]))*Simp[2*a*A*d - C*(b*c - a*d) - 2*(a*c*C - d*(A*b + a*B))*Sin[e + f*x] + (2*b*B*d - C*(b*c + a*d))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a *d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[(u_)*((c_.)*sec[(a_.) + (b_.)*(x_)])^(m_.), x_Symbol] :> Simp[(c*Sec[a + b*x])^m*(c*Cos[a + b*x])^m Int[ActivateTrig[u]/(c*Cos[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownSineIntegrandQ[u, x]
Leaf count of result is larger than twice the leaf count of optimal. \(3346\) vs. \(2(685)=1370\).
Time = 37.44 (sec) , antiderivative size = 3347, normalized size of antiderivative = 4.40
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(3347\) |
| parts | \(\text {Expression too large to display}\) | \(3404\) |
Input:
int((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(1/2 ),x,method=_RETURNVERBOSE)
Output:
1/192/d/b*(sin(d*x+c)*cos(d*x+c)*(272*cos(d*x+c)^2+272*cos(d*x+c)+128)*B*a *b^3+sin(d*x+c)*cos(d*x+c)*(133*cos(d*x+c)+118)*a^3*b*C+15*C*a^4*cos(d*x+c )*sin(d*x+c)+264*B*a^3*b*cos(d*x+c)*sin(d*x+c)+sin(d*x+c)*cos(d*x+c)^2*(96 *cos(d*x+c)+96)*A*b^4+sin(d*x+c)*cos(d*x+c)^2*(64*cos(d*x+c)^2+64*cos(d*x+ c)+128)*B*b^4+sin(d*x+c)*cos(d*x+c)^2*(48*cos(d*x+c)^3+48*cos(d*x+c)^2+72* cos(d*x+c)+72)*C*b^4+432*A*a^2*b^2*cos(d*x+c)*sin(d*x+c)+sin(d*x+c)*cos(d* x+c)*(254*cos(d*x+c)^2+254*cos(d*x+c)+284)*a^2*b^2*C+sin(d*x+c)*cos(d*x+c) *(184*cos(d*x+c)^3+184*cos(d*x+c)^2+356*cos(d*x+c)+72)*C*a*b^3+(192*cos(d* x+c)^2+384*cos(d*x+c)+192)*A*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/ 2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*b^4*EllipticF(-csc(d*x+c)+cot(d*x+c), (-(a-b)/(a+b))^(1/2))+(144*cos(d*x+c)^2+288*cos(d*x+c)+144)*C*(1/(a+b)*(a+ b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*b^4* EllipticF(-csc(d*x+c)+cot(d*x+c),(-(a-b)/(a+b))^(1/2))+(-128*cos(d*x+c)^2- 256*cos(d*x+c)-128)*B*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(cos (d*x+c)/(1+cos(d*x+c)))^(1/2)*b^4*EllipticE(-csc(d*x+c)+cot(d*x+c),(-(a-b) /(a+b))^(1/2))+(-15*cos(d*x+c)^2-30*cos(d*x+c)-15)*C*(1/(a+b)*(a+b*cos(d*x +c))/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*a^4*EllipticE (-csc(d*x+c)+cot(d*x+c),(-(a-b)/(a+b))^(1/2))+(-384*cos(d*x+c)^2-768*cos(d *x+c)-384)*A*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/( 1+cos(d*x+c)))^(1/2)*b^4*EllipticPi(-csc(d*x+c)+cot(d*x+c),-1,(-(a-b)/(...
\[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sqrt {\sec \left (d x + c\right )} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c )^(1/2),x, algorithm="fricas")
Output:
integral((C*b^2*cos(d*x + c)^4 + (2*C*a*b + B*b^2)*cos(d*x + c)^3 + A*a^2 + (C*a^2 + 2*B*a*b + A*b^2)*cos(d*x + c)^2 + (B*a^2 + 2*A*a*b)*cos(d*x + c ))*sqrt(b*cos(d*x + c) + a)*sqrt(sec(d*x + c)), x)
Timed out. \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))**(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x +c)**(1/2),x)
Output:
Timed out
\[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sqrt {\sec \left (d x + c\right )} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c )^(1/2),x, algorithm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(5/ 2)*sqrt(sec(d*x + c)), x)
\[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sqrt {\sec \left (d x + c\right )} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c )^(1/2),x, algorithm="giac")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(5/ 2)*sqrt(sec(d*x + c)), x)
Timed out. \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx=\int \sqrt {\frac {1}{\cos \left (c+d\,x\right )}}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \] Input:
int((1/cos(c + d*x))^(1/2)*(a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2),x)
Output:
int((1/cos(c + d*x))^(1/2)*(a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2), x)
\[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx=3 \left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )d x \right ) a^{2} b +\left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{4}d x \right ) b^{2} c +2 \left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3}d x \right ) a b c +\left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3}d x \right ) b^{3}+\left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2}d x \right ) a^{2} c +3 \left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2}d x \right ) a \,b^{2}+\left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right ) b +a}d x \right ) a^{3} \] Input:
int((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(1/2 ),x)
Output:
3*int(sqrt(sec(c + d*x))*sqrt(cos(c + d*x)*b + a)*cos(c + d*x),x)*a**2*b + int(sqrt(sec(c + d*x))*sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**4,x)*b**2*c + 2*int(sqrt(sec(c + d*x))*sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**3,x)*a* b*c + int(sqrt(sec(c + d*x))*sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**3,x)*b **3 + int(sqrt(sec(c + d*x))*sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**2,x)*a **2*c + 3*int(sqrt(sec(c + d*x))*sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**2, x)*a*b**2 + int(sqrt(sec(c + d*x))*sqrt(cos(c + d*x)*b + a),x)*a**3