Integrand size = 43, antiderivative size = 318 \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))} \, dx=\frac {2 \left (5 A b^3-3 a^3 B-5 a b^2 B+a^2 b (3 A+5 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^4 d}+\frac {2 \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 a^3 d}+\frac {2 b^2 \left (A b^2-a (b B-a C)\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{a^4 (a+b) d}+\frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {2 (A b-a B) \sin (c+d x)}{5 a^2 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) \sin (c+d x)}{21 a^3 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (5 A b^3-3 a^3 B-5 a b^2 B+a^2 b (3 A+5 C)\right ) \sin (c+d x)}{5 a^4 d \sqrt {\cos (c+d x)}} \]
2/5*(5*A*b^3-3*B*a^3-5*B*a*b^2+a^2*b*(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/d+2/21*(7* A*b^2-7*B*a*b+a^2*(5*A+7*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/d+2*b^2*(A*b^2-a*(B*b-C*a))*( 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)/d+2/7*A*sin(d*x+c)/a/d/cos(d*x+c)^(7/2)-2/ 5*(A*b-B*a)*sin(d*x+c)/a^2/d/cos(d*x+c)^(5/2)+2/21*(7*A*b^2-7*B*a*b+a^2*(5 *A+7*C))*sin(d*x+c)/a^3/d/cos(d*x+c)^(3/2)-2/5*(5*A*b^3-3*B*a^3-5*B*a*b^2+ a^2*b*(3*A+5*C))*sin(d*x+c)/a^4/d/cos(d*x+c)^(1/2)
Time = 5.67 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.31 \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))} \, dx=\frac {\frac {2 \left (315 A b^4-133 a^3 b B-315 a b^3 B+10 a^4 (5 A+7 C)+7 a^2 b^2 (19 A+45 C)\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{a+b}+\frac {4 a \left (140 A b^3-63 a^3 B-140 a b^2 B+4 a^2 b (22 A+35 C)\right ) \left ((a+b) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-a \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )\right )}{b (a+b)}-\frac {42 \left (-5 A b^3+3 a^3 B+5 a b^2 B-a^2 b (3 A+5 C)\right ) \left (-2 a b E\left (\left .\arcsin \left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 a (a+b) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )+\left (-2 a^2+b^2\right ) \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )\right ) \sin (c+d x)}{a b \sqrt {\sin ^2(c+d x)}}+\frac {2 \left (42 \left (a^2 (-A b+a B)+\left (-5 A b^3+3 a^3 B+5 a b^2 B-a^2 b (3 A+5 C)\right ) \cos ^2(c+d x)\right ) \sin (c+d x)+5 \left (a \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) \sin (2 (c+d x))+6 a^3 A \tan (c+d x)\right )\right )}{\cos ^{\frac {5}{2}}(c+d x)}}{210 a^4 d} \]
Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(9/2)*(a + b*Cos[c + d*x])),x]
((2*(315*A*b^4 - 133*a^3*b*B - 315*a*b^3*B + 10*a^4*(5*A + 7*C) + 7*a^2*b^ 2*(19*A + 45*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (4*a *(140*A*b^3 - 63*a^3*B - 140*a*b^2*B + 4*a^2*b*(22*A + 35*C))*((a + b)*Ell ipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(b* (a + b)) - (42*(-5*A*b^3 + 3*a^3*B + 5*a*b^2*B - a^2*b*(3*A + 5*C))*(-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)*EllipticPi[-(b/a), ArcSin[Sqrt[ Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b*Sqrt[Sin[c + d*x]^2]) + (2*(42*(a^ 2*(-(A*b) + a*B) + (-5*A*b^3 + 3*a^3*B + 5*a*b^2*B - a^2*b*(3*A + 5*C))*Co s[c + d*x]^2)*Sin[c + d*x] + 5*(a*(7*A*b^2 - 7*a*b*B + a^2*(5*A + 7*C))*Si n[2*(c + d*x)] + 6*a^3*A*Tan[c + d*x])))/Cos[c + d*x]^(5/2))/(210*a^4*d)
Time = 2.99 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.07, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.488, Rules used = {3042, 3534, 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.
| \(\displaystyle \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {2 \int -\frac {-5 A b \cos ^2(c+d x)-a (5 A+7 C) \cos (c+d x)+7 (A b-a B)}{2 \cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))}dx}{7 a}+\frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\int \frac {-5 A b \cos ^2(c+d x)-a (5 A+7 C) \cos (c+d x)+7 (A b-a B)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))}dx}{7 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\int \frac {-5 A b \sin \left (c+d x+\frac {\pi }{2}\right )^2-a (5 A+7 C) \sin \left (c+d x+\frac {\pi }{2}\right )+7 (A b-a B)}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{7 a}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 \int -\frac {-21 b (A b-a B) \cos ^2(c+d x)+a (4 A b+21 a B) \cos (c+d x)+5 \left ((5 A+7 C) a^2-7 b B a+7 A b^2\right )}{2 \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}dx}{5 a}+\frac {14 (A b-a B) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 (A b-a B) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\int \frac {-21 b (A b-a B) \cos ^2(c+d x)+a (4 A b+21 a B) \cos (c+d x)+5 \left ((5 A+7 C) a^2-7 b B a+7 A b^2\right )}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}dx}{5 a}}{7 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 (A b-a B) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\int \frac {-21 b (A b-a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a (4 A b+21 a B) \sin \left (c+d x+\frac {\pi }{2}\right )+5 \left ((5 A+7 C) a^2-7 b B a+7 A b^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{5 a}}{7 a}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 (A b-a B) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \int -\frac {-5 b \left ((5 A+7 C) a^2-7 b B a+7 A b^2\right ) \cos ^2(c+d x)+a \left (-5 (5 A+7 C) a^2-28 b B a+28 A b^2\right ) \cos (c+d x)+21 \left (-3 B a^3+b (3 A+5 C) a^2-5 b^2 B a+5 A b^3\right )}{2 \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}dx}{3 a}+\frac {10 \sin (c+d x) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}}{7 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 (A b-a B) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {10 \sin (c+d x) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-5 b \left ((5 A+7 C) a^2-7 b B a+7 A b^2\right ) \cos ^2(c+d x)+a \left (-5 (5 A+7 C) a^2-28 b B a+28 A b^2\right ) \cos (c+d x)+21 \left (-3 B a^3+b (3 A+5 C) a^2-5 b^2 B a+5 A b^3\right )}{\cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}dx}{3 a}}{5 a}}{7 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 (A b-a B) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {10 \sin (c+d x) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-5 b \left ((5 A+7 C) a^2-7 b B a+7 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a \left (-5 (5 A+7 C) a^2-28 b B a+28 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+21 \left (-3 B a^3+b (3 A+5 C) a^2-5 b^2 B a+5 A b^3\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{3 a}}{5 a}}{7 a}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 (A b-a B) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {10 \sin (c+d x) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \int -\frac {21 b \left (-3 B a^3+b (3 A+5 C) a^2-5 b^2 B a+5 A b^3\right ) \cos ^2(c+d x)+a \left (-63 B a^3+4 b (22 A+35 C) a^2-140 b^2 B a+140 A b^3\right ) \cos (c+d x)+5 \left ((5 A+7 C) a^4-7 b B a^3+7 b^2 (A+3 C) a^2-21 b^3 B a+21 A b^4\right )}{2 \sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{a}+\frac {42 \sin (c+d x) \left (-3 a^3 B+a^2 b (3 A+5 C)-5 a b^2 B+5 A b^3\right )}{a d \sqrt {\cos (c+d x)}}}{3 a}}{5 a}}{7 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 (A b-a B) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {10 \sin (c+d x) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {42 \sin (c+d x) \left (-3 a^3 B+a^2 b (3 A+5 C)-5 a b^2 B+5 A b^3\right )}{a d \sqrt {\cos (c+d x)}}-\frac {\int \frac {21 b \left (-3 B a^3+b (3 A+5 C) a^2-5 b^2 B a+5 A b^3\right ) \cos ^2(c+d x)+a \left (-63 B a^3+4 b (22 A+35 C) a^2-140 b^2 B a+140 A b^3\right ) \cos (c+d x)+5 \left ((5 A+7 C) a^4-7 b B a^3+7 b^2 (A+3 C) a^2-21 b^3 B a+21 A b^4\right )}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{a}}{3 a}}{5 a}}{7 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 (A b-a B) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {10 \sin (c+d x) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {42 \sin (c+d x) \left (-3 a^3 B+a^2 b (3 A+5 C)-5 a b^2 B+5 A b^3\right )}{a d \sqrt {\cos (c+d x)}}-\frac {\int \frac {21 b \left (-3 B a^3+b (3 A+5 C) a^2-5 b^2 B a+5 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a \left (-63 B a^3+4 b (22 A+35 C) a^2-140 b^2 B a+140 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+5 \left ((5 A+7 C) a^4-7 b B a^3+7 b^2 (A+3 C) a^2-21 b^3 B a+21 A b^4\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}}{3 a}}{5 a}}{7 a}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 (A b-a B) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {10 \sin (c+d x) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {42 \sin (c+d x) \left (-3 a^3 B+a^2 b (3 A+5 C)-5 a b^2 B+5 A b^3\right )}{a d \sqrt {\cos (c+d x)}}-\frac {21 \left (-3 a^3 B+a^2 b (3 A+5 C)-5 a b^2 B+5 A b^3\right ) \int \sqrt {\cos (c+d x)}dx-\frac {\int -\frac {5 \left (a \left ((5 A+7 C) a^2-7 b B a+7 A b^2\right ) \cos (c+d x) b^2+\left ((5 A+7 C) a^4-7 b B a^3+7 b^2 (A+3 C) a^2-21 b^3 B a+21 A b^4\right ) b\right )}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}}{a}}{3 a}}{5 a}}{7 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 (A b-a B) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {10 \sin (c+d x) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {42 \sin (c+d x) \left (-3 a^3 B+a^2 b (3 A+5 C)-5 a b^2 B+5 A b^3\right )}{a d \sqrt {\cos (c+d x)}}-\frac {21 \left (-3 a^3 B+a^2 b (3 A+5 C)-5 a b^2 B+5 A b^3\right ) \int \sqrt {\cos (c+d x)}dx+\frac {5 \int \frac {a \left ((5 A+7 C) a^2-7 b B a+7 A b^2\right ) \cos (c+d x) b^2+\left ((5 A+7 C) a^4-7 b B a^3+7 b^2 (A+3 C) a^2-21 b^3 B a+21 A b^4\right ) b}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}}{a}}{3 a}}{5 a}}{7 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 (A b-a B) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {10 \sin (c+d x) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {42 \sin (c+d x) \left (-3 a^3 B+a^2 b (3 A+5 C)-5 a b^2 B+5 A b^3\right )}{a d \sqrt {\cos (c+d x)}}-\frac {21 \left (-3 a^3 B+a^2 b (3 A+5 C)-5 a b^2 B+5 A b^3\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {5 \int \frac {a \left ((5 A+7 C) a^2-7 b B a+7 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) b^2+\left ((5 A+7 C) a^4-7 b B a^3+7 b^2 (A+3 C) a^2-21 b^3 B a+21 A b^4\right ) b}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}}{a}}{3 a}}{5 a}}{7 a}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 (A b-a B) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {10 \sin (c+d x) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {42 \sin (c+d x) \left (-3 a^3 B+a^2 b (3 A+5 C)-5 a b^2 B+5 A b^3\right )}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {5 \int \frac {a \left ((5 A+7 C) a^2-7 b B a+7 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) b^2+\left ((5 A+7 C) a^4-7 b B a^3+7 b^2 (A+3 C) a^2-21 b^3 B a+21 A b^4\right ) b}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}+\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-3 a^3 B+a^2 b (3 A+5 C)-5 a b^2 B+5 A b^3\right )}{d}}{a}}{3 a}}{5 a}}{7 a}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 (A b-a B) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {10 \sin (c+d x) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {42 \sin (c+d x) \left (-3 a^3 B+a^2 b (3 A+5 C)-5 a b^2 B+5 A b^3\right )}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {5 \left (a b \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx+21 b^3 \left (A b^2-a (b B-a C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx\right )}{b}+\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-3 a^3 B+a^2 b (3 A+5 C)-5 a b^2 B+5 A b^3\right )}{d}}{a}}{3 a}}{5 a}}{7 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 (A b-a B) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {10 \sin (c+d x) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {42 \sin (c+d x) \left (-3 a^3 B+a^2 b (3 A+5 C)-5 a b^2 B+5 A b^3\right )}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {5 \left (a b \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+21 b^3 \left (A b^2-a (b B-a C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx\right )}{b}+\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-3 a^3 B+a^2 b (3 A+5 C)-5 a b^2 B+5 A b^3\right )}{d}}{a}}{3 a}}{5 a}}{7 a}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 (A b-a B) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {10 \sin (c+d x) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {42 \sin (c+d x) \left (-3 a^3 B+a^2 b (3 A+5 C)-5 a b^2 B+5 A b^3\right )}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {5 \left (21 b^3 \left (A b^2-a (b B-a C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx+\frac {2 a b \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{d}\right )}{b}+\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-3 a^3 B+a^2 b (3 A+5 C)-5 a b^2 B+5 A b^3\right )}{d}}{a}}{3 a}}{5 a}}{7 a}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 (A b-a B) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {10 \sin (c+d x) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {42 \sin (c+d x) \left (-3 a^3 B+a^2 b (3 A+5 C)-5 a b^2 B+5 A b^3\right )}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {5 \left (\frac {2 a b \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{d}+\frac {42 b^3 \left (A b^2-a (b B-a C)\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{d (a+b)}\right )}{b}+\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-3 a^3 B+a^2 b (3 A+5 C)-5 a b^2 B+5 A b^3\right )}{d}}{a}}{3 a}}{5 a}}{7 a}\) |
(2*A*Sin[c + d*x])/(7*a*d*Cos[c + d*x]^(7/2)) - ((14*(A*b - a*B)*Sin[c + d *x])/(5*a*d*Cos[c + d*x]^(5/2)) - ((10*(7*A*b^2 - 7*a*b*B + a^2*(5*A + 7*C ))*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) - (-(((42*(5*A*b^3 - 3*a^3*B - 5*a*b^2*B + a^2*b*(3*A + 5*C))*EllipticE[(c + d*x)/2, 2])/d + (5*((2*a*b* (7*A*b^2 - 7*a*b*B + a^2*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/d + (42*b ^3*(A*b^2 - a*(b*B - a*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/((a + b)*d)))/b)/a) + (42*(5*A*b^3 - 3*a^3*B - 5*a*b^2*B + a^2*b*(3*A + 5*C))* Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]))/(3*a))/(5*a))/(7*a)
3.12.2.3.1 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_)] + (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. \(975\) vs. \(2(376)=752\).
Time = 5.49 (sec) , antiderivative size = 976, normalized size of antiderivative = 3.07
int((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(9/2)/(a+b*cos(d*x+c)),x,me thod=_RETURNVERBOSE)
-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*A/a*(-1/56*c os(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)/(co s(1/2*d*x+1/2*c)^2-1/2)^4-5/42*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+5/21*(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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)))+2/5* (-A*b+B*a)/a^2/(8*sin(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)*Ell ipticE(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)*EllipticE(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)+2*(A*b^2-B*a*b+C*a^2)/a ^3*(-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*c os(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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)))-2*(A*b^2-B*a*b+C*a^2)/a^4*b /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)...
Timed out. \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))} \, dx=\text {Timed out} \]
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(9/2)/(a+b*cos(d*x+c) ),x, algorithm="fricas")
Timed out. \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))} \, dx=\text {Timed out} \]
\[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))} \, dx=\int { \frac {C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A}{{\left (b \cos \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac {9}{2}}} \,d x } \]
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(9/2)/(a+b*cos(d*x+c) ),x, algorithm="maxima")
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)/((b*cos(d*x + c) + a)*co s(d*x + c)^(9/2)), x)
\[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))} \, dx=\int { \frac {C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A}{{\left (b \cos \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac {9}{2}}} \,d x } \]
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(9/2)/(a+b*cos(d*x+c) ),x, algorithm="giac")
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)/((b*cos(d*x + c) + a)*co s(d*x + c)^(9/2)), x)
Timed out. \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))} \, dx=\int \frac {C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A}{{\cos \left (c+d\,x\right )}^{9/2}\,\left (a+b\,\cos \left (c+d\,x\right )\right )} \,d x \]