Integrand size = 45, antiderivative size = 894 \[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=-\frac {(a-b) \sqrt {a+b} \left (150 a^3 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{1920 a b^2 d \sqrt {\sec (c+d x)}}-\frac {\sqrt {a+b} \left (45 a^4 C-30 a^3 b (5 B+C)-16 b^4 (80 A+45 B+64 C)-8 a b^3 (260 A+355 B+193 C)-4 a^2 b^2 (660 A+295 B+423 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{1920 b^2 d \sqrt {\sec (c+d x)}}+\frac {\sqrt {a+b} \left (10 a^4 b B-240 a^2 b^3 B-96 b^5 B-3 a^5 C-40 a^3 b^2 (2 A+C)-80 a b^4 (4 A+3 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{128 b^3 d \sqrt {\sec (c+d x)}}+\frac {\left (50 a^2 b B+120 b^3 B-15 a^3 C+4 a b^2 (60 A+43 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{320 b d \sqrt {\sec (c+d x)}}+\frac {\left (80 A b^2+50 a b B-15 a^2 C+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d \sqrt {\sec (c+d x)}}+\frac {(10 b B-3 a C) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d \sqrt {\sec (c+d x)}}+\frac {C (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d \sqrt {\sec (c+d x)}}+\frac {\left (150 a^3 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 C)\right ) \sqrt {a+b \cos (c+d x)} \sqrt {\sec (c+d x)} \sin (c+d x)}{1920 b^2 d} \] Output:
-1/1920*(a-b)*(a+b)^(1/2)*(150*B*a^3*b+2840*B*a*b^3-45*a^4*C+256*b^4*(5*A+ 4*C)+12*a^2*b^2*(220*A+141*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^2/d/sec(d*x+c )^(1/2)-1/1920*(a+b)^(1/2)*(45*a^4*C-30*a^3*b*(5*B+C)-16*b^4*(80*A+45*B+64 *C)-8*a*b^3*(260*A+355*B+193*C)-4*a^2*b^2*(660*A+295*B+423*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^2/d/sec(d*x+c)^(1/2)+1/128*(a+b)^(1/2)*(10*B*a^4*b-240*B* a^2*b^3-96*B*b^5-3*C*a^5-40*a^3*b^2*(2*A+C)-80*a*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^3/d/sec(d*x+c)^(1/2)+1/320*(50*B*a^2*b+120*B*b^ 3-15*a^3*C+4*a*b^2*(60*A+43*C))*(a+b*cos(d*x+c))^(1/2)*sin(d*x+c)/b/d/sec( d*x+c)^(1/2)+1/240*(80*A*b^2+50*B*a*b-15*C*a^2+64*C*b^2)*(a+b*cos(d*x+c))^ (3/2)*sin(d*x+c)/b/d/sec(d*x+c)^(1/2)+1/40*(10*B*b-3*C*a)*(a+b*cos(d*x+c)) ^(5/2)*sin(d*x+c)/b/d/sec(d*x+c)^(1/2)+1/5*C*(a+b*cos(d*x+c))^(7/2)*sin(d* x+c)/b/d/sec(d*x+c)^(1/2)+1/1920*(150*B*a^3*b+2840*B*a*b^3-45*a^4*C+256*b^ 4*(5*A+4*C)+12*a^2*b^2*(220*A+141*C))*(a+b*cos(d*x+c))^(1/2)*sec(d*x+c)^(1 /2)*sin(d*x+c)/b^2/d
Leaf count is larger than twice the leaf count of optimal. \(2908\) vs. \(2(894)=1788\).
Time = 19.06 (sec) , antiderivative size = 2908, normalized size of antiderivative = 3.25 \[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\text {Result too large to show} \] 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:
(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(((80*A*b^2 + 170*a*b*B + 93* a^2*C + 88*b^2*C)*Sin[c + d*x])/960 + ((1040*a*A*b^2 + 590*a^2*b*B + 480*b ^3*B + 15*a^3*C + 1024*a*b^2*C)*Sin[2*(c + d*x)])/(1920*b) + ((80*A*b^2 + 170*a*b*B + 93*a^2*C + 100*b^2*C)*Sin[3*(c + d*x)])/960 + (b*(10*b*B + 21* a*C)*Sin[4*(c + d*x)])/320 + (b^2*C*Sin[5*(c + d*x)])/80))/d + (Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)] *(2640*a^3*A*b^2*Tan[(c + d*x)/2] + 2640*a^2*A*b^3*Tan[(c + d*x)/2] + 1280 *a*A*b^4*Tan[(c + d*x)/2] + 1280*A*b^5*Tan[(c + d*x)/2] + 150*a^4*b*B*Tan[ (c + d*x)/2] + 150*a^3*b^2*B*Tan[(c + d*x)/2] + 2840*a^2*b^3*B*Tan[(c + d* x)/2] + 2840*a*b^4*B*Tan[(c + d*x)/2] - 45*a^5*C*Tan[(c + d*x)/2] - 45*a^4 *b*C*Tan[(c + d*x)/2] + 1692*a^3*b^2*C*Tan[(c + d*x)/2] + 1692*a^2*b^3*C*T an[(c + d*x)/2] + 1024*a*b^4*C*Tan[(c + d*x)/2] + 1024*b^5*C*Tan[(c + d*x) /2] - 5280*a^2*A*b^3*Tan[(c + d*x)/2]^3 - 2560*A*b^5*Tan[(c + d*x)/2]^3 - 300*a^3*b^2*B*Tan[(c + d*x)/2]^3 - 5680*a*b^4*B*Tan[(c + d*x)/2]^3 + 90*a^ 4*b*C*Tan[(c + d*x)/2]^3 - 3384*a^2*b^3*C*Tan[(c + d*x)/2]^3 - 2048*b^5*C* Tan[(c + d*x)/2]^3 - 2640*a^3*A*b^2*Tan[(c + d*x)/2]^5 + 2640*a^2*A*b^3*Ta n[(c + d*x)/2]^5 - 1280*a*A*b^4*Tan[(c + d*x)/2]^5 + 1280*A*b^5*Tan[(c + d *x)/2]^5 - 150*a^4*b*B*Tan[(c + d*x)/2]^5 + 150*a^3*b^2*B*Tan[(c + d*x)/2] ^5 - 2840*a^2*b^3*B*Tan[(c + d*x)/2]^5 + 2840*a*b^4*B*Tan[(c + d*x)/2]^5 + 45*a^5*C*Tan[(c + d*x)/2]^5 - 45*a^4*b*C*Tan[(c + d*x)/2]^5 - 1692*a^3...
Time = 5.07 (sec) , antiderivative size = 862, normalized size of antiderivative = 0.96, number of steps used = 25, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {3042, 4709, 3042, 3528, 27, 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.
| \(\displaystyle \int \frac {(a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos (c+d x)^2\right )}{\sqrt {\sec (c+d x)}}dx\) |
| \(\Big \downarrow \) 4709 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \left (C \cos ^2(c+d x)+B \cos (c+d x)+A\right )dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (C \sin \left (c+d x+\frac {\pi }{2}\right )^2+B \sin \left (c+d x+\frac {\pi }{2}\right )+A\right )dx\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \frac {(a+b \cos (c+d x))^{5/2} \left ((10 b B-3 a C) \cos ^2(c+d x)+2 b (5 A+4 C) \cos (c+d x)+a C\right )}{2 \sqrt {\cos (c+d x)}}dx}{5 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \frac {(a+b \cos (c+d x))^{5/2} \left ((10 b B-3 a C) \cos ^2(c+d x)+2 b (5 A+4 C) \cos (c+d x)+a C\right )}{\sqrt {\cos (c+d x)}}dx}{10 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left ((10 b B-3 a C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 b (5 A+4 C) \sin \left (c+d x+\frac {\pi }{2}\right )+a C\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{10 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\right )\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{4} \int \frac {(a+b \cos (c+d x))^{3/2} \left (\left (-15 C a^2+50 b B a+80 A b^2+64 b^2 C\right ) \cos ^2(c+d x)+2 b (40 a A+30 b B+27 a C) \cos (c+d x)+5 a (2 b B+a C)\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {(10 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}}{10 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{8} \int \frac {(a+b \cos (c+d x))^{3/2} \left (\left (-15 C a^2+50 b B a+80 A b^2+64 b^2 C\right ) \cos ^2(c+d x)+2 b (40 a A+30 b B+27 a C) \cos (c+d x)+5 a (2 b B+a C)\right )}{\sqrt {\cos (c+d x)}}dx+\frac {(10 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}}{10 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{8} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (\left (-15 C a^2+50 b B a+80 A b^2+64 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 b (40 a A+30 b B+27 a C) \sin \left (c+d x+\frac {\pi }{2}\right )+5 a (2 b B+a C)\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {(10 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}}{10 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\right )\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{8} \left (\frac {1}{3} \int \frac {\sqrt {a+b \cos (c+d x)} \left (3 \left (-15 C a^3+50 b B a^2+4 b^2 (60 A+43 C) a+120 b^3 B\right ) \cos ^2(c+d x)+2 b \left (3 (80 A+49 C) a^2+310 b B a+32 b^2 (5 A+4 C)\right ) \cos (c+d x)+a \left (15 C a^2+110 b B a+80 A b^2+64 b^2 C\right )\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^2 C+50 a b B+80 A b^2+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {(10 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}}{10 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{8} \left (\frac {1}{6} \int \frac {\sqrt {a+b \cos (c+d x)} \left (3 \left (-15 C a^3+50 b B a^2+4 b^2 (60 A+43 C) a+120 b^3 B\right ) \cos ^2(c+d x)+2 b \left (3 (80 A+49 C) a^2+310 b B a+32 b^2 (5 A+4 C)\right ) \cos (c+d x)+a \left (15 C a^2+110 b B a+80 A b^2+64 b^2 C\right )\right )}{\sqrt {\cos (c+d x)}}dx+\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^2 C+50 a b B+80 A b^2+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {(10 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}}{10 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{8} \left (\frac {1}{6} \int \frac {\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (3 \left (-15 C a^3+50 b B a^2+4 b^2 (60 A+43 C) a+120 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 b \left (3 (80 A+49 C) a^2+310 b B a+32 b^2 (5 A+4 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left (15 C a^2+110 b B a+80 A b^2+64 b^2 C\right )\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^2 C+50 a b B+80 A b^2+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {(10 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}}{10 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\right )\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {\left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right ) \cos ^2(c+d x)+2 b \left ((960 A+573 C) a^3+1610 b B a^2+4 b^2 (380 A+289 C) a+360 b^3 B\right ) \cos (c+d x)+a \left (15 C a^3+590 b B a^2+4 b^2 (260 A+193 C) a+360 b^3 B\right )}{2 \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^3 C+50 a^2 b B+4 a b^2 (60 A+43 C)+120 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^2 C+50 a b B+80 A b^2+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {(10 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}}{10 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \int \frac {\left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right ) \cos ^2(c+d x)+2 b \left ((960 A+573 C) a^3+1610 b B a^2+4 b^2 (380 A+289 C) a+360 b^3 B\right ) \cos (c+d x)+a \left (15 C a^3+590 b B a^2+4 b^2 (260 A+193 C) a+360 b^3 B\right )}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^3 C+50 a^2 b B+4 a b^2 (60 A+43 C)+120 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^2 C+50 a b B+80 A b^2+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {(10 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}}{10 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \int \frac {\left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 b \left ((960 A+573 C) a^3+1610 b B a^2+4 b^2 (380 A+289 C) a+360 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left (15 C a^3+590 b B a^2+4 b^2 (260 A+193 C) a+360 b^3 B\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^3 C+50 a^2 b B+4 a b^2 (60 A+43 C)+120 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^2 C+50 a b B+80 A b^2+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {(10 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}}{10 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\right )\) |
| \(\Big \downarrow \) 3540 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\int -\frac {15 \left (-3 C a^5+10 b B a^4-40 b^2 (2 A+C) a^3-240 b^3 B a^2-80 b^4 (4 A+3 C) a-96 b^5 B\right ) \cos ^2(c+d x)-2 a b \left (15 C a^3+590 b B a^2+4 b^2 (260 A+193 C) a+360 b^3 B\right ) \cos (c+d x)+a \left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right )}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{2 b}+\frac {\sin (c+d x) \left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^3 C+50 a^2 b B+4 a b^2 (60 A+43 C)+120 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^2 C+50 a b B+80 A b^2+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {(10 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}}{10 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {\int \frac {15 \left (-3 C a^5+10 b B a^4-40 b^2 (2 A+C) a^3-240 b^3 B a^2-80 b^4 (4 A+3 C) a-96 b^5 B\right ) \cos ^2(c+d x)-2 a b \left (15 C a^3+590 b B a^2+4 b^2 (260 A+193 C) a+360 b^3 B\right ) \cos (c+d x)+a \left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right )}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^3 C+50 a^2 b B+4 a b^2 (60 A+43 C)+120 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^2 C+50 a b B+80 A b^2+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {(10 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}}{10 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {\int \frac {15 \left (-3 C a^5+10 b B a^4-40 b^2 (2 A+C) a^3-240 b^3 B a^2-80 b^4 (4 A+3 C) a-96 b^5 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-2 a b \left (15 C a^3+590 b B a^2+4 b^2 (260 A+193 C) a+360 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^3 C+50 a^2 b B+4 a b^2 (60 A+43 C)+120 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^2 C+50 a b B+80 A b^2+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {(10 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}}{10 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\right )\) |
| \(\Big \downarrow \) 3532 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {\int \frac {a \left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right )-2 a b \left (15 C a^3+590 b B a^2+4 b^2 (260 A+193 C) a+360 b^3 B\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx+15 \left (-3 a^5 C+10 a^4 b B-40 a^3 b^2 (2 A+C)-240 a^2 b^3 B-80 a b^4 (4 A+3 C)-96 b^5 B\right ) \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {a+b \cos (c+d x)}}dx}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^3 C+50 a^2 b B+4 a b^2 (60 A+43 C)+120 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^2 C+50 a b B+80 A b^2+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {(10 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}}{10 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {\int \frac {a \left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right )-2 a b \left (15 C a^3+590 b B a^2+4 b^2 (260 A+193 C) a+360 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+15 \left (-3 a^5 C+10 a^4 b B-40 a^3 b^2 (2 A+C)-240 a^2 b^3 B-80 a b^4 (4 A+3 C)-96 b^5 B\right ) \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^3 C+50 a^2 b B+4 a b^2 (60 A+43 C)+120 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^2 C+50 a b B+80 A b^2+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {(10 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}}{10 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\right )\) |
| \(\Big \downarrow \) 3288 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {\int \frac {a \left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right )-2 a b \left (15 C a^3+590 b B a^2+4 b^2 (260 A+193 C) a+360 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {30 \sqrt {a+b} \cot (c+d x) \left (-3 a^5 C+10 a^4 b B-40 a^3 b^2 (2 A+C)-240 a^2 b^3 B-80 a b^4 (4 A+3 C)-96 b^5 B\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b d}}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^3 C+50 a^2 b B+4 a b^2 (60 A+43 C)+120 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^2 C+50 a b B+80 A b^2+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {(10 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}}{10 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\right )\) |
| \(\Big \downarrow \) 3477 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {a \left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \int \frac {\cos (c+d x)+1}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx+a \left (45 a^4 C-30 a^3 b (5 B+C)-4 a^2 b^2 (660 A+295 B+423 C)-8 a b^3 (260 A+355 B+193 C)-16 b^4 (80 A+45 B+64 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx-\frac {30 \sqrt {a+b} \cot (c+d x) \left (-3 a^5 C+10 a^4 b B-40 a^3 b^2 (2 A+C)-240 a^2 b^3 B-80 a b^4 (4 A+3 C)-96 b^5 B\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b d}}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^3 C+50 a^2 b B+4 a b^2 (60 A+43 C)+120 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^2 C+50 a b B+80 A b^2+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {(10 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}}{10 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {a \left (45 a^4 C-30 a^3 b (5 B+C)-4 a^2 b^2 (660 A+295 B+423 C)-8 a b^3 (260 A+355 B+193 C)-16 b^4 (80 A+45 B+64 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+a \left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {30 \sqrt {a+b} \cot (c+d x) \left (-3 a^5 C+10 a^4 b B-40 a^3 b^2 (2 A+C)-240 a^2 b^3 B-80 a b^4 (4 A+3 C)-96 b^5 B\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b d}}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^3 C+50 a^2 b B+4 a b^2 (60 A+43 C)+120 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^2 C+50 a b B+80 A b^2+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {(10 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}}{10 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\right )\) |
| \(\Big \downarrow \) 3295 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {C \sqrt {\cos (c+d x)} \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{5 b d}+\frac {\frac {(10 b B-3 a C) \sqrt {\cos (c+d x)} \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{4 d}+\frac {1}{8} \left (\frac {\left (-15 C a^2+50 b B a+80 A b^2+64 b^2 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d}+\frac {1}{6} \left (\frac {3 \left (-15 C a^3+50 b B a^2+4 b^2 (60 A+43 C) a+120 b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{2 d}+\frac {1}{4} \left (\frac {\left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{b d \sqrt {\cos (c+d x)}}-\frac {\frac {2 \sqrt {a+b} \left (45 C a^4-30 b (5 B+C) a^3-4 b^2 (660 A+295 B+423 C) a^2-8 b^3 (260 A+355 B+193 C) a-16 b^4 (80 A+45 B+64 C)\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}+a \left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {30 \sqrt {a+b} \left (-3 C a^5+10 b B a^4-40 b^2 (2 A+C) a^3-240 b^3 B a^2-80 b^4 (4 A+3 C) a-96 b^5 B\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}}}{b d}}{2 b}\right )\right )\right )}{10 b}\right )\) |
| \(\Big \downarrow \) 3473 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {C \sqrt {\cos (c+d x)} \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{5 b d}+\frac {\frac {(10 b B-3 a C) \sqrt {\cos (c+d x)} \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{4 d}+\frac {1}{8} \left (\frac {\left (-15 C a^2+50 b B a+80 A b^2+64 b^2 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d}+\frac {1}{6} \left (\frac {3 \left (-15 C a^3+50 b B a^2+4 b^2 (60 A+43 C) a+120 b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{2 d}+\frac {1}{4} \left (\frac {\left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{b d \sqrt {\cos (c+d x)}}-\frac {\frac {2 (a-b) \sqrt {a+b} \left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{a d}+\frac {2 \sqrt {a+b} \left (45 C a^4-30 b (5 B+C) a^3-4 b^2 (660 A+295 B+423 C) a^2-8 b^3 (260 A+355 B+193 C) a-16 b^4 (80 A+45 B+64 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}}}{d}-\frac {30 \sqrt {a+b} \left (-3 C a^5+10 b B a^4-40 b^2 (2 A+C) a^3-240 b^3 B a^2-80 b^4 (4 A+3 C) a-96 b^5 B\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}}}{b d}}{2 b}\right )\right )\right )}{10 b}\right )\) |
Input:
Int[((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/S qrt[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])^(7/2)*Sin[c + d*x])/(5*b*d) + (((10*b*B - 3*a*C)*Sqrt[Cos[c + d*x] ]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*d) + (((80*A*b^2 + 50*a*b*B - 15*a^2*C + 64*b^2*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d) + ((3*(50*a^2*b*B + 120*b^3*B - 15*a^3*C + 4*a*b^2*(60*A + 43*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]*(150*a^3*b*B + 2840*a*b^3*B - 45*a^4*C + 256* b^4*(5*A + 4*C) + 12*a^2*b^2*(220*A + 141*C))*Cot[c + d*x]*EllipticE[ArcSi n[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)])/(a*d) + (2*Sqrt[a + b]*(45*a^4*C - 30*a^3*b*(5*B + C) - 16*b^4*(8 0*A + 45*B + 64*C) - 8*a*b^3*(260*A + 355*B + 193*C) - 4*a^2*b^2*(660*A + 295*B + 423*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sq rt[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 - (30*Sqrt[a + b]*( 10*a^4*b*B - 240*a^2*b^3*B - 96*b^5*B - 3*a^5*C - 40*a^3*b^2*(2*A + C) - 8 0*a*b^4*(4*A + 3*C))*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[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 + ((150*a^3*b*B + 2840*a*b^3*B - 45*a^4*C + 256*b^4*(5*A + 4*C) + ...
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. \(3913\) vs. \(2(813)=1626\).
Time = 43.30 (sec) , antiderivative size = 3914, normalized size of antiderivative = 4.38
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(3914\) |
| parts | \(\text {Expression too large to display}\) | \(3958\) |
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:
-1/1920/d/b^2*(a+b*cos(d*x+c))^(1/2)/(b*cos(d*x+c)^2+a*cos(d*x+c)+b*cos(d* x+c)+a)/sec(d*x+c)^(1/2)*(-2640*A*a^3*b^2*sin(d*x+c)-150*B*a^4*b*sin(d*x+c )+B*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x +c)))^(1/2)*b^5*EllipticF(-csc(d*x+c)+cot(d*x+c),(-(a-b)/(a+b))^(1/2))*(-1 440*cos(d*x+c)-2880-1440*sec(d*x+c))+A*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*( 1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*b^5*EllipticE(-csc(d*x+c)+c ot(d*x+c),(-(a-b)/(a+b))^(1/2))*(1280*cos(d*x+c)+2560+1280*sec(d*x+c))+C*( cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c))) ^(1/2)*a^5*EllipticE(-csc(d*x+c)+cot(d*x+c),(-(a-b)/(a+b))^(1/2))*(-45*cos (d*x+c)-90-45*sec(d*x+c))+C*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+ b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*b^5*EllipticE(-csc(d*x+c)+cot(d*x+c),( -(a-b)/(a+b))^(1/2))*(1024*cos(d*x+c)+2048+1024*sec(d*x+c))+B*(cos(d*x+c)/ (1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*b^5* EllipticPi(-csc(d*x+c)+cot(d*x+c),-1,(-(a-b)/(a+b))^(1/2))*(2880*cos(d*x+c )+5760+2880*sec(d*x+c))+C*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b* cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a^5*EllipticPi(-csc(d*x+c)+cot(d*x+c),-1 ,(-(a-b)/(a+b))^(1/2))*(90*cos(d*x+c)+180+90*sec(d*x+c))+C*(cos(d*x+c)/(1+ cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a*b^4*E llipticPi(-csc(d*x+c)+cot(d*x+c),-1,(-(a-b)/(a+b))^(1/2))*(7200*cos(d*x+c) +14400+7200*sec(d*x+c))+A*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a...
\[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\sqrt {\sec \left (d x + c\right )}} \,d x } \] 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. \[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\text {Timed out} \] 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
\[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\sqrt {\sec \left (d x + c\right )}} \,d x } \] 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)
\[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\sqrt {\sec \left (d x + c\right )}} \,d x } \] 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. \[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\int \frac {{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \,d x \] Input:
int(((a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/( 1/cos(c + d*x))^(1/2),x)
Output:
int(((a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/( 1/cos(c + d*x))^(1/2), x)
\[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=3 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )}{\sec \left (d x +c \right )}d x \right ) a^{2} b +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{4}}{\sec \left (d x +c \right )}d x \right ) b^{2} c +2 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3}}{\sec \left (d x +c \right )}d x \right ) a b c +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3}}{\sec \left (d x +c \right )}d x \right ) b^{3}+\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2}}{\sec \left (d x +c \right )}d x \right ) a^{2} c +3 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2}}{\sec \left (d x +c \right )}d x \right ) a \,b^{2}+\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right ) b +a}}{\sec \left (d x +c \right )}d x \right ) a^{3} \] 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))/sec(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)/sec(c + d*x),x)*b**2*c + 2*int((sqrt(sec(c + d*x))*sqrt(cos(c + d* x)*b + a)*cos(c + d*x)**3)/sec(c + d*x),x)*a*b*c + int((sqrt(sec(c + d*x)) *sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**3)/sec(c + d*x),x)*b**3 + int((sqr t(sec(c + d*x))*sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**2)/sec(c + d*x),x)* a**2*c + 3*int((sqrt(sec(c + d*x))*sqrt(cos(c + d*x)*b + a)*cos(c + d*x)** 2)/sec(c + d*x),x)*a*b**2 + int((sqrt(sec(c + d*x))*sqrt(cos(c + d*x)*b + a))/sec(c + d*x),x)*a**3