Integrand size = 35, antiderivative size = 297 \[ \int \frac {A+B \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^{7/2}} \, dx=\frac {(1015 A-363 B) \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right )}{64 \sqrt {2} a^{7/2} d}-\frac {(A-B) \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{7/2}}-\frac {(23 A-11 B) \sin (c+d x)}{48 a d \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2}}-\frac {(109 A-41 B) \sin (c+d x)}{64 a^2 d \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2}}+\frac {(579 A-199 B) \sin (c+d x)}{192 a^3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}-\frac {(1887 A-691 B) \sin (c+d x)}{192 a^3 d \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \]
-1/6*(A-B)*sin(d*x+c)/d/cos(d*x+c)^(3/2)/(a+a*cos(d*x+c))^(7/2)-1/48*(23*A -11*B)*sin(d*x+c)/a/d/cos(d*x+c)^(3/2)/(a+a*cos(d*x+c))^(5/2)-1/64*(109*A- 41*B)*sin(d*x+c)/a^2/d/cos(d*x+c)^(3/2)/(a+a*cos(d*x+c))^(3/2)+1/128*(1015 *A-363*B)*arctan(1/2*sin(d*x+c)*a^(1/2)*2^(1/2)/cos(d*x+c)^(1/2)/(a+a*cos( d*x+c))^(1/2))/a^(7/2)/d*2^(1/2)+1/192*(579*A-199*B)*sin(d*x+c)/a^3/d/cos( d*x+c)^(3/2)/(a+a*cos(d*x+c))^(1/2)-1/192*(1887*A-691*B)*sin(d*x+c)/a^3/d/ cos(d*x+c)^(1/2)/(a+a*cos(d*x+c))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 8.44 (sec) , antiderivative size = 1256, normalized size of antiderivative = 4.23 \[ \int \frac {A+B \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^{7/2}} \, dx =\text {Too large to display} \]
(2*B*Cos[c/2 + (d*x)/2]^7*Sec[(c + d*x)/2]^6*Sin[c/2 + (d*x)/2]*((16*Cos[( c + d*x)/2]^8*HypergeometricPFQ[{2, 2, 2, 2, 5/2}, {1, 1, 1, 13/2}, Sin[c/ 2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^2)/(3465* (-1 + 2*Sin[c/2 + (d*x)/2]^2)) - (Csc[c/2 + (d*x)/2]^10*(1 - 2*Sin[c/2 + ( d*x)/2]^2)^2*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*(105 *ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Cos[(c + d*x)/2]^6*(2187 - 12908*Sin[c/2 + (d*x)/2]^2 + 27986*Sin[c/2 + (d*x)/2]^ 4 - 26380*Sin[c/2 + (d*x)/2]^6 + 8752*Sin[c/2 + (d*x)/2]^8) + Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*(-229635 + 2120790*Sin[c/2 + (d*x)/2]^2 - 8267707*Sin[c/2 + (d*x)/2]^4 + 17646926*Sin[c/2 + (d*x)/2]^6 - 22251094*Sin[c/2 + (d*x)/2]^8 + 16548816*Sin[c/2 + (d*x)/2]^10 - 6712984 *Sin[c/2 + (d*x)/2]^12 + 1144608*Sin[c/2 + (d*x)/2]^14)))/1680))/(d*(a*(1 + Cos[c + d*x]))^(7/2)*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2)) + (A*Cot[c/2 + (d*x)/2]^7*Csc[c/2 + (d*x)/2]^4*Sec[(c + d*x)/2]^6*(-7680*Cos[(c + d*x)/2] ^10*HypergeometricPFQ[{2, 2, 2, 2, 2, 7/2}, {1, 1, 1, 1, 15/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^14 + 19200*Co s[(c + d*x)/2]^8*HypergeometricPFQ[{2, 2, 2, 2, 7/2}, {1, 1, 1, 15/2}, Sin [c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^14*(-7 + 6*Sin[c/2 + (d*x)/2]^2) + 143*(1 - 2*Sin[c/2 + (d*x)/2]^2)^3*Sqrt[Sin[c /2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*(315*ArcTanh[Sqrt[Sin[c/...
Time = 1.80 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.09, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.514, Rules used = {3042, 3457, 27, 3042, 3457, 27, 3042, 3457, 27, 3042, 3463, 27, 3042, 3463, 27, 3042, 3261, 218}
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.
| \(\displaystyle \int \frac {A+B \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^{7/2}}dx\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {\int \frac {3 a (5 A-B)-8 a (A-B) \cos (c+d x)}{2 \cos ^{\frac {5}{2}}(c+d x) (\cos (c+d x) a+a)^{5/2}}dx}{6 a^2}-\frac {(A-B) \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {3 a (5 A-B)-8 a (A-B) \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (\cos (c+d x) a+a)^{5/2}}dx}{12 a^2}-\frac {(A-B) \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {3 a (5 A-B)-8 a (A-B) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{5/2}}dx}{12 a^2}-\frac {(A-B) \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {\frac {\int \frac {3 \left (a^2 (63 A-19 B)-2 a^2 (23 A-11 B) \cos (c+d x)\right )}{2 \cos ^{\frac {5}{2}}(c+d x) (\cos (c+d x) a+a)^{3/2}}dx}{4 a^2}-\frac {a (23 A-11 B) \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {(A-B) \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 \int \frac {a^2 (63 A-19 B)-2 a^2 (23 A-11 B) \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (\cos (c+d x) a+a)^{3/2}}dx}{8 a^2}-\frac {a (23 A-11 B) \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {(A-B) \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 \int \frac {a^2 (63 A-19 B)-2 a^2 (23 A-11 B) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{3/2}}dx}{8 a^2}-\frac {a (23 A-11 B) \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {(A-B) \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {\frac {3 \left (\frac {\int \frac {a^3 (579 A-199 B)-4 a^3 (109 A-41 B) \cos (c+d x)}{2 \cos ^{\frac {5}{2}}(c+d x) \sqrt {\cos (c+d x) a+a}}dx}{2 a^2}-\frac {a^2 (109 A-41 B) \sin (c+d x)}{2 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}-\frac {a (23 A-11 B) \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {(A-B) \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 \left (\frac {\int \frac {a^3 (579 A-199 B)-4 a^3 (109 A-41 B) \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) \sqrt {\cos (c+d x) a+a}}dx}{4 a^2}-\frac {a^2 (109 A-41 B) \sin (c+d x)}{2 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}-\frac {a (23 A-11 B) \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {(A-B) \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 \left (\frac {\int \frac {a^3 (579 A-199 B)-4 a^3 (109 A-41 B) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{4 a^2}-\frac {a^2 (109 A-41 B) \sin (c+d x)}{2 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}-\frac {a (23 A-11 B) \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {(A-B) \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\) |
| \(\Big \downarrow \) 3463 |
| \(\displaystyle \frac {\frac {3 \left (\frac {\frac {2 \int -\frac {a^4 (1887 A-691 B)-2 a^4 (579 A-199 B) \cos (c+d x)}{2 \cos ^{\frac {3}{2}}(c+d x) \sqrt {\cos (c+d x) a+a}}dx}{3 a}+\frac {2 a^3 (579 A-199 B) \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}}{4 a^2}-\frac {a^2 (109 A-41 B) \sin (c+d x)}{2 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}-\frac {a (23 A-11 B) \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {(A-B) \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 \left (\frac {\frac {2 a^3 (579 A-199 B) \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}-\frac {\int \frac {a^4 (1887 A-691 B)-2 a^4 (579 A-199 B) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {\cos (c+d x) a+a}}dx}{3 a}}{4 a^2}-\frac {a^2 (109 A-41 B) \sin (c+d x)}{2 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}-\frac {a (23 A-11 B) \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {(A-B) \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 \left (\frac {\frac {2 a^3 (579 A-199 B) \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}-\frac {\int \frac {a^4 (1887 A-691 B)-2 a^4 (579 A-199 B) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{3 a}}{4 a^2}-\frac {a^2 (109 A-41 B) \sin (c+d x)}{2 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}-\frac {a (23 A-11 B) \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {(A-B) \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\) |
| \(\Big \downarrow \) 3463 |
| \(\displaystyle \frac {\frac {3 \left (\frac {\frac {2 a^3 (579 A-199 B) \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}-\frac {\frac {2 \int -\frac {3 a^5 (1015 A-363 B)}{2 \sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}}dx}{a}+\frac {2 a^4 (1887 A-691 B) \sin (c+d x)}{d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}}{3 a}}{4 a^2}-\frac {a^2 (109 A-41 B) \sin (c+d x)}{2 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}-\frac {a (23 A-11 B) \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {(A-B) \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 \left (\frac {\frac {2 a^3 (579 A-199 B) \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}-\frac {\frac {2 a^4 (1887 A-691 B) \sin (c+d x)}{d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}-3 a^4 (1015 A-363 B) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}}dx}{3 a}}{4 a^2}-\frac {a^2 (109 A-41 B) \sin (c+d x)}{2 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}-\frac {a (23 A-11 B) \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {(A-B) \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 \left (\frac {\frac {2 a^3 (579 A-199 B) \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}-\frac {\frac {2 a^4 (1887 A-691 B) \sin (c+d x)}{d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}-3 a^4 (1015 A-363 B) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{3 a}}{4 a^2}-\frac {a^2 (109 A-41 B) \sin (c+d x)}{2 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}-\frac {a (23 A-11 B) \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {(A-B) \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\) |
| \(\Big \downarrow \) 3261 |
| \(\displaystyle \frac {\frac {3 \left (\frac {\frac {2 a^3 (579 A-199 B) \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}-\frac {\frac {6 a^5 (1015 A-363 B) \int \frac {1}{\frac {\sin (c+d x) \tan (c+d x) a^3}{\cos (c+d x) a+a}+2 a^2}d\left (-\frac {a \sin (c+d x)}{\sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}}\right )}{d}+\frac {2 a^4 (1887 A-691 B) \sin (c+d x)}{d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}}{3 a}}{4 a^2}-\frac {a^2 (109 A-41 B) \sin (c+d x)}{2 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}-\frac {a (23 A-11 B) \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {(A-B) \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {3 \left (\frac {\frac {2 a^3 (579 A-199 B) \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}-\frac {\frac {2 a^4 (1887 A-691 B) \sin (c+d x)}{d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}-\frac {3 \sqrt {2} a^{7/2} (1015 A-363 B) \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{d}}{3 a}}{4 a^2}-\frac {a^2 (109 A-41 B) \sin (c+d x)}{2 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}-\frac {a (23 A-11 B) \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {(A-B) \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\) |
-1/6*((A - B)*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(7/ 2)) + (-1/4*(a*(23*A - 11*B)*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)*(a + a*Co s[c + d*x])^(5/2)) + (3*(-1/2*(a^2*(109*A - 41*B)*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)) + ((2*a^3*(579*A - 199*B)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - ((-3*Sqrt[2]*a^( 7/2)*(1015*A - 363*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/d + (2*a^4*(1887*A - 691*B)*Sin[c + d*x] )/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]))/(3*a))/(4*a^2)))/(8*a^2 ))/(12*a^2)
3.3.14.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_.) + (d_.)*sin[(e _.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*(a/f) Subst[Int[1/(2*b^2 - (a*c - b*d)*x^2), x], x, b*(Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*S in[e + f*x]]))], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[b*(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^( n + 1)/(a*f*(2*m + 1)*(b*c - a*d))), x] + Simp[1/(a*(2*m + 1)*(b*c - a*d)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[B*(a*c*m + b *d*(n + 1)) + A*(b*c*(m + 1) - a*d*(2*m + n + 2)) + d*(A*b - a*B)*(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ [b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] && !GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(b*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*(a*d*m + b*c*(n + 1)) - B*(a*c*m + b*d*(n + 1)) + b*(B*c - A*d)*(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && Eq Q[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1] && (IntegerQ[n] || EqQ[m + 1/2, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(666\) vs. \(2(254)=508\).
Time = 7.90 (sec) , antiderivative size = 667, normalized size of antiderivative = 2.25
| method | result | size |
| default | \(-\frac {\left (3045 A \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )-1089 B \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )+12180 A \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )-4356 B \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )+18270 A \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}\, \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )-6534 B \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}\, \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )+12180 A \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}\, \left (\cos ^{2}\left (d x +c \right )\right ) \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )+3774 A \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right )-4356 B \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}\, \left (\cos ^{2}\left (d x +c \right )\right ) \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )-1382 B \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right )+3045 A \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right ) \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \sqrt {2}+10164 A \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )-1089 B \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right ) \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \sqrt {2}-3748 B \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )+8502 A \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )-3198 B \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )+1792 A \sin \left (d x +c \right ) \cos \left (d x +c \right )-768 B \sin \left (d x +c \right ) \cos \left (d x +c \right )-256 A \sin \left (d x +c \right )\right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}}{384 a^{4} d \left (1+\cos \left (d x +c \right )\right )^{4} \cos \left (d x +c \right )^{\frac {3}{2}}}\) | \(667\) |
| parts | \(-\frac {A \left (3045 \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )+1887 \sqrt {2}\, \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right )+12180 \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )+5082 \sqrt {2}\, \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )+18270 \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )+4251 \sqrt {2}\, \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+12180 \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \left (\cos ^{2}\left (d x +c \right )\right ) \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )+896 \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {2}+3045 \cos \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )-128 \sqrt {2}\, \sin \left (d x +c \right )\right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, \sqrt {2}}{384 d \left (1+\cos \left (d x +c \right )\right )^{4} \cos \left (d x +c \right )^{\frac {3}{2}} a^{4}}+\frac {B \left (1089 \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )+691 \sqrt {2}\, \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )+4356 \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )+1874 \sqrt {2}\, \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+6534 \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \left (\cos ^{2}\left (d x +c \right )\right ) \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )+1599 \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {2}+4356 \cos \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )+384 \sqrt {2}\, \sin \left (d x +c \right )+1089 \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )\right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, \sqrt {2}}{384 d \left (1+\cos \left (d x +c \right )\right )^{4} \sqrt {\cos \left (d x +c \right )}\, a^{4}}\) | \(677\) |
-1/384/a^4/d*(3045*A*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arcsin(cot( d*x+c)-csc(d*x+c))*cos(d*x+c)^5-1089*B*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c))) ^(1/2)*arcsin(cot(d*x+c)-csc(d*x+c))*cos(d*x+c)^5+12180*A*2^(1/2)*(cos(d*x +c)/(1+cos(d*x+c)))^(1/2)*arcsin(cot(d*x+c)-csc(d*x+c))*cos(d*x+c)^4-4356* B*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arcsin(cot(d*x+c)-csc(d*x+c))* cos(d*x+c)^4+18270*A*cos(d*x+c)^3*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2 )*arcsin(cot(d*x+c)-csc(d*x+c))-6534*B*cos(d*x+c)^3*(cos(d*x+c)/(1+cos(d*x +c)))^(1/2)*2^(1/2)*arcsin(cot(d*x+c)-csc(d*x+c))+12180*A*(cos(d*x+c)/(1+c os(d*x+c)))^(1/2)*2^(1/2)*cos(d*x+c)^2*arcsin(cot(d*x+c)-csc(d*x+c))+3774* A*cos(d*x+c)^4*sin(d*x+c)-4356*B*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2) *cos(d*x+c)^2*arcsin(cot(d*x+c)-csc(d*x+c))-1382*B*cos(d*x+c)^4*sin(d*x+c) +3045*A*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*cos(d*x+c)*arcsin(cot(d*x+c)-csc (d*x+c))*2^(1/2)+10164*A*cos(d*x+c)^3*sin(d*x+c)-1089*B*(cos(d*x+c)/(1+cos (d*x+c)))^(1/2)*cos(d*x+c)*arcsin(cot(d*x+c)-csc(d*x+c))*2^(1/2)-3748*B*si n(d*x+c)*cos(d*x+c)^3+8502*A*sin(d*x+c)*cos(d*x+c)^2-3198*B*sin(d*x+c)*cos (d*x+c)^2+1792*A*sin(d*x+c)*cos(d*x+c)-768*B*sin(d*x+c)*cos(d*x+c)-256*A*s in(d*x+c))*(a*(1+cos(d*x+c)))^(1/2)/(1+cos(d*x+c))^4/cos(d*x+c)^(3/2)
Time = 0.36 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.07 \[ \int \frac {A+B \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^{7/2}} \, dx=\frac {3 \, \sqrt {2} {\left ({\left (1015 \, A - 363 \, B\right )} \cos \left (d x + c\right )^{6} + 4 \, {\left (1015 \, A - 363 \, B\right )} \cos \left (d x + c\right )^{5} + 6 \, {\left (1015 \, A - 363 \, B\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (1015 \, A - 363 \, B\right )} \cos \left (d x + c\right )^{3} + {\left (1015 \, A - 363 \, B\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{2 \, {\left (a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right )\right )}}\right ) - 2 \, {\left ({\left (1887 \, A - 691 \, B\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (2541 \, A - 937 \, B\right )} \cos \left (d x + c\right )^{3} + 39 \, {\left (109 \, A - 41 \, B\right )} \cos \left (d x + c\right )^{2} + 128 \, {\left (7 \, A - 3 \, B\right )} \cos \left (d x + c\right ) - 128 \, A\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{384 \, {\left (a^{4} d \cos \left (d x + c\right )^{6} + 4 \, a^{4} d \cos \left (d x + c\right )^{5} + 6 \, a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + a^{4} d \cos \left (d x + c\right )^{2}\right )}} \]
1/384*(3*sqrt(2)*((1015*A - 363*B)*cos(d*x + c)^6 + 4*(1015*A - 363*B)*cos (d*x + c)^5 + 6*(1015*A - 363*B)*cos(d*x + c)^4 + 4*(1015*A - 363*B)*cos(d *x + c)^3 + (1015*A - 363*B)*cos(d*x + c)^2)*sqrt(a)*arctan(1/2*sqrt(2)*sq rt(a*cos(d*x + c) + a)*sqrt(a)*sqrt(cos(d*x + c))*sin(d*x + c)/(a*cos(d*x + c)^2 + a*cos(d*x + c))) - 2*((1887*A - 691*B)*cos(d*x + c)^4 + 2*(2541*A - 937*B)*cos(d*x + c)^3 + 39*(109*A - 41*B)*cos(d*x + c)^2 + 128*(7*A - 3 *B)*cos(d*x + c) - 128*A)*sqrt(a*cos(d*x + c) + a)*sqrt(cos(d*x + c))*sin( d*x + c))/(a^4*d*cos(d*x + c)^6 + 4*a^4*d*cos(d*x + c)^5 + 6*a^4*d*cos(d*x + c)^4 + 4*a^4*d*cos(d*x + c)^3 + a^4*d*cos(d*x + c)^2)
Timed out. \[ \int \frac {A+B \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^{7/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {A+B \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^{7/2}} \, dx=\text {Timed out} \]
\[ \int \frac {A+B \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^{7/2}} \, dx=\int { \frac {B \cos \left (d x + c\right ) + A}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {7}{2}} \cos \left (d x + c\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {A+B \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^{7/2}} \, dx=\int \frac {A+B\,\cos \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{5/2}\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{7/2}} \,d x \]