Integrand size = 35, antiderivative size = 427 \[ \int \frac {A+C \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))^2} \, dx=\frac {\left (35 A b^4-3 a^2 b^2 (8 A-5 C)-2 a^4 (3 A+5 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^4 \left (a^2-b^2\right ) d}+\frac {b \left (7 A b^2-a^2 (4 A-3 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 a^3 \left (a^2-b^2\right ) d}+\frac {b \left (7 A b^4-3 a^2 b^2 (3 A-C)-5 a^4 C\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{a^4 (a-b) (a+b)^2 d}-\frac {\left (7 A b^2-a^2 (2 A-5 C)\right ) \sin (c+d x)}{5 a^2 \left (a^2-b^2\right ) d \cos ^{\frac {5}{2}}(c+d x)}+\frac {b \left (7 A b^2-a^2 (4 A-3 C)\right ) \sin (c+d x)}{3 a^3 \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\left (35 A b^4-3 a^2 b^2 (8 A-5 C)-2 a^4 (3 A+5 C)\right ) \sin (c+d x)}{5 a^4 \left (a^2-b^2\right ) d \sqrt {\cos (c+d x)}}+\frac {\left (A b^2+a^2 C\right ) \sin (c+d x)}{a \left (a^2-b^2\right ) d \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))} \]
1/5*(35*A*b^4-3*a^2*b^2*(8*A-5*C)-2*a^4*(3*A+5*C))*(cos(1/2*d*x+1/2*c)^2)^ (1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/a^4/(a^2-b^ 2)/d+1/3*b*(7*A*b^2-a^2*(4*A-3*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d* x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2))/a^3/(a^2-b^2)/d+b*(7*A*b^4- 3*a^2*b^2*(3*A-C)-5*a^4*C)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c) *EllipticPi(sin(1/2*d*x+1/2*c),2*b/(a+b),2^(1/2))/a^4/(a-b)/(a+b)^2/d-1/5* (7*A*b^2-a^2*(2*A-5*C))*sin(d*x+c)/a^2/(a^2-b^2)/d/cos(d*x+c)^(5/2)+1/3*b* (7*A*b^2-a^2*(4*A-3*C))*sin(d*x+c)/a^3/(a^2-b^2)/d/cos(d*x+c)^(3/2)+(A*b^2 +C*a^2)*sin(d*x+c)/a/(a^2-b^2)/d/cos(d*x+c)^(5/2)/(a+b*cos(d*x+c))-1/5*(35 *A*b^4-3*a^2*b^2*(8*A-5*C)-2*a^4*(3*A+5*C))*sin(d*x+c)/a^4/(a^2-b^2)/d/cos (d*x+c)^(1/2)
Time = 7.43 (sec) , antiderivative size = 492, normalized size of antiderivative = 1.15 \[ \int \frac {A+C \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))^2} \, dx=-\frac {\frac {2 \left (-58 a^4 A b-272 a^2 A b^3+315 A b^5-150 a^4 b C+135 a^2 b^3 C\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{a+b}+\frac {\left (-36 a^5 A-184 a^3 A b^2+280 a A b^4-60 a^5 C+120 a^3 b^2 C\right ) \left (2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-\frac {2 a \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{a+b}\right )}{b}+\frac {2 \left (-18 a^4 A b-72 a^2 A b^3+105 A b^5-30 a^4 b C+45 a^2 b^3 C\right ) \cos (2 (c+d x)) \left (-2 a b E\left (\left .\arcsin \left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 a (a+b) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )+\left (-2 a^2+b^2\right ) \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )\right ) \sin (c+d x)}{a b^2 \sqrt {1-\cos ^2(c+d x)} \left (-1+2 \cos ^2(c+d x)\right )}}{60 a^4 (-a+b) (a+b) d}+\frac {\sqrt {\cos (c+d x)} \left (\frac {2 \sec (c+d x) \left (3 a^2 A \sin (c+d x)+15 A b^2 \sin (c+d x)+5 a^2 C \sin (c+d x)\right )}{5 a^4}+\frac {-A b^5 \sin (c+d x)-a^2 b^3 C \sin (c+d x)}{a^4 \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {4 A b \sec (c+d x) \tan (c+d x)}{3 a^3}+\frac {2 A \sec ^2(c+d x) \tan (c+d x)}{5 a^2}\right )}{d} \]
-1/60*((2*(-58*a^4*A*b - 272*a^2*A*b^3 + 315*A*b^5 - 150*a^4*b*C + 135*a^2 *b^3*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + ((-36*a^5*A - 184*a^3*A*b^2 + 280*a*A*b^4 - 60*a^5*C + 120*a^3*b^2*C)*(2*EllipticF[(c + d*x)/2, 2] - (2*a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b)))/b + (2*(-18*a^4*A*b - 72*a^2*A*b^3 + 105*A*b^5 - 30*a^4*b*C + 45*a^2*b^3*C)* Cos[2*(c + d*x)]*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*( a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*Elliptic Pi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[1 - Cos[c + d*x]^2]*(-1 + 2*Cos[c + d*x]^2)))/(a^4*(-a + b)*(a + b)*d) + (Sqrt [Cos[c + d*x]]*((2*Sec[c + d*x]*(3*a^2*A*Sin[c + d*x] + 15*A*b^2*Sin[c + d *x] + 5*a^2*C*Sin[c + d*x]))/(5*a^4) + (-(A*b^5*Sin[c + d*x]) - a^2*b^3*C* Sin[c + d*x])/(a^4*(a^2 - b^2)*(a + b*Cos[c + d*x])) - (4*A*b*Sec[c + d*x] *Tan[c + d*x])/(3*a^3) + (2*A*Sec[c + d*x]^2*Tan[c + d*x])/(5*a^2)))/d
Time = 3.31 (sec) , antiderivative size = 403, normalized size of antiderivative = 0.94, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 3535, 27, 3042, 3534, 27, 3042, 3534, 27, 3042, 3534, 27, 3042, 3538, 27, 3042, 3119, 3481, 3042, 3120, 3284}
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.
| \(\displaystyle \int \frac {A+C \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 3535 |
| \(\displaystyle \frac {\int -\frac {-2 \left (A-\frac {5 C}{2}\right ) a^2+2 b (A+C) \cos (c+d x) a+7 A b^2-5 \left (C a^2+A b^2\right ) \cos ^2(c+d x)}{2 \cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))}dx}{a \left (a^2-b^2\right )}+\frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\int \frac {-\left ((2 A-5 C) a^2\right )+2 b (A+C) \cos (c+d x) a+7 A b^2-5 \left (C a^2+A b^2\right ) \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))}dx}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\int \frac {-\left ((2 A-5 C) a^2\right )+2 b (A+C) \sin \left (c+d x+\frac {\pi }{2}\right ) a+7 A b^2-5 \left (C a^2+A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \int -\frac {-3 b \left (7 A b^2-a^2 (2 A-5 C)\right ) \cos ^2(c+d x)+2 a \left ((3 A+5 C) a^2+2 A b^2\right ) \cos (c+d x)+5 b \left (7 A b^2-a^2 (4 A-3 C)\right )}{2 \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}dx}{5 a}+\frac {2 \left (7 A b^2-a^2 (2 A-5 C)\right ) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \left (7 A b^2-a^2 (2 A-5 C)\right ) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\int \frac {-3 b \left (7 A b^2-a^2 (2 A-5 C)\right ) \cos ^2(c+d x)+2 a \left ((3 A+5 C) a^2+2 A b^2\right ) \cos (c+d x)+5 b \left (7 A b^2-a^2 (4 A-3 C)\right )}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}dx}{5 a}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \left (7 A b^2-a^2 (2 A-5 C)\right ) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\int \frac {-3 b \left (7 A b^2-a^2 (2 A-5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 a \left ((3 A+5 C) a^2+2 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+5 b \left (7 A b^2-a^2 (4 A-3 C)\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{5 a}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \left (7 A b^2-a^2 (2 A-5 C)\right ) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \int -\frac {-5 b^2 \left (7 A b^2-a^2 (4 A-3 C)\right ) \cos ^2(c+d x)+2 a b \left ((A+15 C) a^2+14 A b^2\right ) \cos (c+d x)+3 \left (-2 (3 A+5 C) a^4-3 b^2 (8 A-5 C) a^2+35 A b^4\right )}{2 \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}dx}{3 a}+\frac {10 b \left (7 A b^2-a^2 (4 A-3 C)\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \left (7 A b^2-a^2 (2 A-5 C)\right ) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {10 b \left (7 A b^2-a^2 (4 A-3 C)\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-5 b^2 \left (7 A b^2-a^2 (4 A-3 C)\right ) \cos ^2(c+d x)+2 a b \left ((A+15 C) a^2+14 A b^2\right ) \cos (c+d x)+3 \left (-2 (3 A+5 C) a^4-3 b^2 (8 A-5 C) a^2+35 A b^4\right )}{\cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}dx}{3 a}}{5 a}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \left (7 A b^2-a^2 (2 A-5 C)\right ) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {10 b \left (7 A b^2-a^2 (4 A-3 C)\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-5 b^2 \left (7 A b^2-a^2 (4 A-3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 a b \left ((A+15 C) a^2+14 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 \left (-2 (3 A+5 C) a^4-3 b^2 (8 A-5 C) a^2+35 A b^4\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{3 a}}{5 a}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \left (7 A b^2-a^2 (2 A-5 C)\right ) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {10 b \left (7 A b^2-a^2 (4 A-3 C)\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \int -\frac {3 b \left (-2 (3 A+5 C) a^4-3 b^2 (8 A-5 C) a^2+35 A b^4\right ) \cos ^2(c+d x)+2 a \left (-3 (3 A+5 C) a^4-2 b^2 (23 A-15 C) a^2+70 A b^4\right ) \cos (c+d x)+5 b \left (-4 (A+3 C) a^4-b^2 (20 A-9 C) a^2+21 A b^4\right )}{2 \sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{a}+\frac {6 \left (-2 a^4 (3 A+5 C)-3 a^2 b^2 (8 A-5 C)+35 A b^4\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}}{3 a}}{5 a}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \left (7 A b^2-a^2 (2 A-5 C)\right ) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {10 b \left (7 A b^2-a^2 (4 A-3 C)\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \left (-2 a^4 (3 A+5 C)-3 a^2 b^2 (8 A-5 C)+35 A b^4\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {\int \frac {3 b \left (-2 (3 A+5 C) a^4-3 b^2 (8 A-5 C) a^2+35 A b^4\right ) \cos ^2(c+d x)+2 a \left (-3 (3 A+5 C) a^4-2 b^2 (23 A-15 C) a^2+70 A b^4\right ) \cos (c+d x)+5 b \left (-4 (A+3 C) a^4-b^2 (20 A-9 C) a^2+21 A b^4\right )}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{a}}{3 a}}{5 a}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \left (7 A b^2-a^2 (2 A-5 C)\right ) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {10 b \left (7 A b^2-a^2 (4 A-3 C)\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \left (-2 a^4 (3 A+5 C)-3 a^2 b^2 (8 A-5 C)+35 A b^4\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {\int \frac {3 b \left (-2 (3 A+5 C) a^4-3 b^2 (8 A-5 C) a^2+35 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 a \left (-3 (3 A+5 C) a^4-2 b^2 (23 A-15 C) a^2+70 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+5 b \left (-4 (A+3 C) a^4-b^2 (20 A-9 C) a^2+21 A b^4\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}}{3 a}}{5 a}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \left (7 A b^2-a^2 (2 A-5 C)\right ) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {10 b \left (7 A b^2-a^2 (4 A-3 C)\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \left (-2 a^4 (3 A+5 C)-3 a^2 b^2 (8 A-5 C)+35 A b^4\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {3 \left (-2 a^4 (3 A+5 C)-3 a^2 b^2 (8 A-5 C)+35 A b^4\right ) \int \sqrt {\cos (c+d x)}dx-\frac {\int -\frac {5 \left (a \left (7 A b^2-a^2 (4 A-3 C)\right ) \cos (c+d x) b^3+\left (-4 (A+3 C) a^4-b^2 (20 A-9 C) a^2+21 A b^4\right ) b^2\right )}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}}{a}}{3 a}}{5 a}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \left (7 A b^2-a^2 (2 A-5 C)\right ) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {10 b \left (7 A b^2-a^2 (4 A-3 C)\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \left (-2 a^4 (3 A+5 C)-3 a^2 b^2 (8 A-5 C)+35 A b^4\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {3 \left (-2 a^4 (3 A+5 C)-3 a^2 b^2 (8 A-5 C)+35 A b^4\right ) \int \sqrt {\cos (c+d x)}dx+\frac {5 \int \frac {a \left (7 A b^2-a^2 (4 A-3 C)\right ) \cos (c+d x) b^3+\left (-4 (A+3 C) a^4-b^2 (20 A-9 C) a^2+21 A b^4\right ) b^2}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}}{a}}{3 a}}{5 a}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \left (7 A b^2-a^2 (2 A-5 C)\right ) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {10 b \left (7 A b^2-a^2 (4 A-3 C)\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \left (-2 a^4 (3 A+5 C)-3 a^2 b^2 (8 A-5 C)+35 A b^4\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {3 \left (-2 a^4 (3 A+5 C)-3 a^2 b^2 (8 A-5 C)+35 A b^4\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {5 \int \frac {a \left (7 A b^2-a^2 (4 A-3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) b^3+\left (-4 (A+3 C) a^4-b^2 (20 A-9 C) a^2+21 A b^4\right ) b^2}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}}{a}}{3 a}}{5 a}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \left (7 A b^2-a^2 (2 A-5 C)\right ) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {10 b \left (7 A b^2-a^2 (4 A-3 C)\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \left (-2 a^4 (3 A+5 C)-3 a^2 b^2 (8 A-5 C)+35 A b^4\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {5 \int \frac {a \left (7 A b^2-a^2 (4 A-3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) b^3+\left (-4 (A+3 C) a^4-b^2 (20 A-9 C) a^2+21 A b^4\right ) b^2}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}+\frac {6 \left (-2 a^4 (3 A+5 C)-3 a^2 b^2 (8 A-5 C)+35 A b^4\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{a}}{3 a}}{5 a}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \left (7 A b^2-a^2 (2 A-5 C)\right ) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {10 b \left (7 A b^2-a^2 (4 A-3 C)\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \left (-2 a^4 (3 A+5 C)-3 a^2 b^2 (8 A-5 C)+35 A b^4\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {5 \left (a b^2 \left (7 A b^2-a^2 (4 A-3 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx+3 b^2 \left (-5 a^4 C-3 a^2 b^2 (3 A-C)+7 A b^4\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx\right )}{b}+\frac {6 \left (-2 a^4 (3 A+5 C)-3 a^2 b^2 (8 A-5 C)+35 A b^4\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{a}}{3 a}}{5 a}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \left (7 A b^2-a^2 (2 A-5 C)\right ) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {10 b \left (7 A b^2-a^2 (4 A-3 C)\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \left (-2 a^4 (3 A+5 C)-3 a^2 b^2 (8 A-5 C)+35 A b^4\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {5 \left (a b^2 \left (7 A b^2-a^2 (4 A-3 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+3 b^2 \left (-5 a^4 C-3 a^2 b^2 (3 A-C)+7 A b^4\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx\right )}{b}+\frac {6 \left (-2 a^4 (3 A+5 C)-3 a^2 b^2 (8 A-5 C)+35 A b^4\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{a}}{3 a}}{5 a}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \left (7 A b^2-a^2 (2 A-5 C)\right ) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {10 b \left (7 A b^2-a^2 (4 A-3 C)\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \left (-2 a^4 (3 A+5 C)-3 a^2 b^2 (8 A-5 C)+35 A b^4\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {5 \left (3 b^2 \left (-5 a^4 C-3 a^2 b^2 (3 A-C)+7 A b^4\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx+\frac {2 a b^2 \left (7 A b^2-a^2 (4 A-3 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}\right )}{b}+\frac {6 \left (-2 a^4 (3 A+5 C)-3 a^2 b^2 (8 A-5 C)+35 A b^4\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{a}}{3 a}}{5 a}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \left (7 A b^2-a^2 (2 A-5 C)\right ) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {10 b \left (7 A b^2-a^2 (4 A-3 C)\right ) \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \left (-2 a^4 (3 A+5 C)-3 a^2 b^2 (8 A-5 C)+35 A b^4\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {6 \left (-2 a^4 (3 A+5 C)-3 a^2 b^2 (8 A-5 C)+35 A b^4\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {5 \left (\frac {2 a b^2 \left (7 A b^2-a^2 (4 A-3 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+\frac {6 b^2 \left (-5 a^4 C-3 a^2 b^2 (3 A-C)+7 A b^4\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{d (a+b)}\right )}{b}}{a}}{3 a}}{5 a}}{2 a \left (a^2-b^2\right )}\) |
((A*b^2 + a^2*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Cos[c + d*x]^(5/2)*(a + b* Cos[c + d*x])) - ((2*(7*A*b^2 - a^2*(2*A - 5*C))*Sin[c + d*x])/(5*a*d*Cos[ c + d*x]^(5/2)) - ((10*b*(7*A*b^2 - a^2*(4*A - 3*C))*Sin[c + d*x])/(3*a*d* Cos[c + d*x]^(3/2)) - (-(((6*(35*A*b^4 - 3*a^2*b^2*(8*A - 5*C) - 2*a^4*(3* A + 5*C))*EllipticE[(c + d*x)/2, 2])/d + (5*((2*a*b^2*(7*A*b^2 - a^2*(4*A - 3*C))*EllipticF[(c + d*x)/2, 2])/d + (6*b^2*(7*A*b^4 - 3*a^2*b^2*(3*A - C) - 5*a^4*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/((a + b)*d)))/b)/ a) + (6*(35*A*b^4 - 3*a^2*b^2*(8*A - 5*C) - 2*a^4*(3*A + 5*C))*Sin[c + d*x ])/(a*d*Sqrt[Cos[c + d*x]]))/(3*a))/(5*a))/(2*a*(a^2 - b^2))
3.8.20.3.1 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[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[ 2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a, b, c , d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]
Int[(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]))/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[ B/d Int[(a + b*Sin[e + f*x])^m, x], x] - Simp[(B*c - A*d)/d Int[(a + b* Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, 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_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int [(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A *b^2 - a*b*B + a^2*C)*(m + n + 3)*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] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ [n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) | | EqQ[a, 0])))
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 + a^2*C))*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*S in[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin [e + f*x])^n*Simp[a*(m + 1)*(b*c - a*d)*(A + C) + d*(A*b^2 + a^2*C)*(m + n + 2) - (c*(A*b^2 + a^2*C) + b*(m + 1)*(b*c - a*d)*(A + C))*Sin[e + f*x] - d *(A*b^2 + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ[n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) || EqQ[a, 0])))
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[C/(b*d) Int[Sqrt[a + b*Sin[e + f*x]], x] , x] - Simp[1/(b*d) Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[ e + f*x], x]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; Fre eQ[{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]
Leaf count of result is larger than twice the leaf count of optimal. \(1325\) vs. \(2(487)=974\).
Time = 25.26 (sec) , antiderivative size = 1326, normalized size of antiderivative = 3.11
-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2/5/a^2*A/(8*s in(1/2*d*x+1/2*c)^6-12*sin(1/2*d*x+1/2*c)^4+6*sin(1/2*d*x+1/2*c)^2-1)/sin( 1/2*d*x+1/2*c)^2*(24*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6-12*(2*sin(1/2 *d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+ 1/2*c),2^(1/2))*sin(1/2*d*x+1/2*c)^4-24*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1 /2*c)+12*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*Ell ipticE(cos(1/2*d*x+1/2*c),2^(1/2))*sin(1/2*d*x+1/2*c)^2+8*sin(1/2*d*x+1/2* c)^2*cos(1/2*d*x+1/2*c)-3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2* c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))*(-2*sin(1/2*d*x+1/2*c )^4+sin(1/2*d*x+1/2*c)^2)^(1/2)-4/a^3*b*A*(-1/6*cos(1/2*d*x+1/2*c)*(-2*sin (1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^2 +1/3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*si n(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c ),2^(1/2)))+2*(3*A*b^2+C*a^2)/a^4/sin(1/2*d*x+1/2*c)^2/(2*sin(1/2*d*x+1/2* c)^2-1)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d* x+1/2*c)^2*cos(1/2*d*x+1/2*c)-(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+ 1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))+4*b^2*(3*A*b^2+C* a^2)/a^4/(-2*a*b+2*b^2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c )^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*Elliptic Pi(cos(1/2*d*x+1/2*c),-2*b/(a-b),2^(1/2))-2*(A*b^2+C*a^2)*b/a^3*(-1/a*b...
Timed out. \[ \int \frac {A+C \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {A+C \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {A+C \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))^2} \, dx=\text {Timed out} \]
\[ \int \frac {A+C \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))^2} \, dx=\int { \frac {C \cos \left (d x + c\right )^{2} + A}{{\left (b \cos \left (d x + c\right ) + a\right )}^{2} \cos \left (d x + c\right )^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {A+C \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))^2} \, dx=\int \frac {C\,{\cos \left (c+d\,x\right )}^2+A}{{\cos \left (c+d\,x\right )}^{7/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^2} \,d x \]