Integrand size = 35, antiderivative size = 344 \[ \int \frac {A+C \cos ^2(c+d x)}{\cos ^{\frac {11}{2}}(c+d x) (a+b \cos (c+d x))} \, dx=-\frac {2 \left (15 A b^4+3 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 a^5 d}-\frac {2 b \left (7 A b^2+a^2 (5 A+7 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 a^4 d}-\frac {2 b^3 \left (A b^2+a^2 C\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{a^5 (a+b) d}+\frac {2 A \sin (c+d x)}{9 a d \cos ^{\frac {9}{2}}(c+d x)}-\frac {2 A b \sin (c+d x)}{7 a^2 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (9 A b^2+a^2 (7 A+9 C)\right ) \sin (c+d x)}{45 a^3 d \cos ^{\frac {5}{2}}(c+d x)}-\frac {2 b \left (7 A b^2+a^2 (5 A+7 C)\right ) \sin (c+d x)}{21 a^4 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (15 A b^4+3 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)\right ) \sin (c+d x)}{15 a^5 d \sqrt {\cos (c+d x)}} \] Output:
-2/15*(15*A*b^4+3*a^2*b^2*(3*A+5*C)+a^4*(7*A+9*C))*EllipticE(sin(1/2*d*x+1 /2*c),2^(1/2))/a^5/d-2/21*b*(7*A*b^2+a^2*(5*A+7*C))*InverseJacobiAM(1/2*d* x+1/2*c,2^(1/2))/a^4/d-2*b^3*(A*b^2+C*a^2)*EllipticPi(sin(1/2*d*x+1/2*c),2 *b/(a+b),2^(1/2))/a^5/(a+b)/d+2/9*A*sin(d*x+c)/a/d/cos(d*x+c)^(9/2)-2/7*A* b*sin(d*x+c)/a^2/d/cos(d*x+c)^(7/2)+2/45*(9*A*b^2+a^2*(7*A+9*C))*sin(d*x+c )/a^3/d/cos(d*x+c)^(5/2)-2/21*b*(7*A*b^2+a^2*(5*A+7*C))*sin(d*x+c)/a^4/d/c os(d*x+c)^(3/2)+2/15*(15*A*b^4+3*a^2*b^2*(3*A+5*C)+a^4*(7*A+9*C))*sin(d*x+ c)/a^5/d/cos(d*x+c)^(1/2)
Time = 5.43 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.24 \[ \int \frac {A+C \cos ^2(c+d x)}{\cos ^{\frac {11}{2}}(c+d x) (a+b \cos (c+d x))} \, dx=\frac {-3 \left (\frac {2 \left (315 A b^5+7 a^2 b^3 (19 A+45 C)+a^4 b (99 A+133 C)\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{a+b}+\frac {4 \left (140 a A b^4+7 a^5 (7 A+9 C)+4 a^3 b^2 (22 A+35 C)\right ) \left ((a+b) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-a \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )\right )}{b (a+b)}+\frac {14 \left (15 A b^4+3 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)\right ) \left (-2 a b E\left (\left .\arcsin \left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 a (a+b) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )+\left (-2 a^2+b^2\right ) \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )\right ) \sin (c+d x)}{a b \sqrt {\sin ^2(c+d x)}}\right )+\frac {2 \left (6 \left (-15 a^3 A b-5 a b \left (7 A b^2+a^2 (5 A+7 C)\right ) \cos ^2(c+d x)+7 \left (15 A b^4+3 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)\right ) \cos ^3(c+d x)\right ) \sin (c+d x)+7 \left (9 a^2 A b^2+a^4 (7 A+9 C)\right ) \sin (2 (c+d x))+70 a^4 A \tan (c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)}}{630 a^5 d} \] Input:
Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(11/2)*(a + b*Cos[c + d*x]) ),x]
Output:
(-3*((2*(315*A*b^5 + 7*a^2*b^3*(19*A + 45*C) + a^4*b*(99*A + 133*C))*Ellip ticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (4*(140*a*A*b^4 + 7*a^5*(7 *A + 9*C) + 4*a^3*b^2*(22*A + 35*C))*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(b*(a + b)) + (14*(15*A*b^4 + 3*a^2*b^2*(3*A + 5*C) + a^4*(7*A + 9*C))*(-2*a*b*EllipticE[ArcSin[Sqrt[C os[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*(6*(-15*a^3*A*b - 5*a*b*(7*A*b^ 2 + a^2*(5*A + 7*C))*Cos[c + d*x]^2 + 7*(15*A*b^4 + 3*a^2*b^2*(3*A + 5*C) + a^4*(7*A + 9*C))*Cos[c + d*x]^3)*Sin[c + d*x] + 7*(9*a^2*A*b^2 + a^4*(7* A + 9*C))*Sin[2*(c + d*x)] + 70*a^4*A*Tan[c + d*x]))/Cos[c + d*x]^(7/2))/( 630*a^5*d)
Time = 3.37 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.08, number of steps used = 24, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.686, Rules used = {3042, 3535, 27, 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.
| \(\displaystyle \int \frac {A+C \cos ^2(c+d x)}{\cos ^{\frac {11}{2}}(c+d x) (a+b \cos (c+d x))} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^{11/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx\) |
| \(\Big \downarrow \) 3535 |
| \(\displaystyle \frac {2 \int -\frac {-7 A b \cos ^2(c+d x)-a (7 A+9 C) \cos (c+d x)+9 A b}{2 \cos ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))}dx}{9 a}+\frac {2 A \sin (c+d x)}{9 a d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{9 a d \cos ^{\frac {9}{2}}(c+d x)}-\frac {\int \frac {-7 A b \cos ^2(c+d x)-a (7 A+9 C) \cos (c+d x)+9 A b}{\cos ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))}dx}{9 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{9 a d \cos ^{\frac {9}{2}}(c+d x)}-\frac {\int \frac {-7 A b \sin \left (c+d x+\frac {\pi }{2}\right )^2-a (7 A+9 C) \sin \left (c+d x+\frac {\pi }{2}\right )+9 A b}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{9 a}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{9 a d \cos ^{\frac {9}{2}}(c+d x)}-\frac {\frac {2 \int -\frac {-45 A b^2 \cos ^2(c+d x)+4 a A b \cos (c+d x)+7 \left ((7 A+9 C) a^2+9 A b^2\right )}{2 \cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))}dx}{7 a}+\frac {18 A b \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{9 a d \cos ^{\frac {9}{2}}(c+d x)}-\frac {\frac {18 A b \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\int \frac {-45 A b^2 \cos ^2(c+d x)+4 a A b \cos (c+d x)+7 \left ((7 A+9 C) a^2+9 A b^2\right )}{\cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))}dx}{7 a}}{9 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{9 a d \cos ^{\frac {9}{2}}(c+d x)}-\frac {\frac {18 A b \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\int \frac {-45 A b^2 \sin \left (c+d x+\frac {\pi }{2}\right )^2+4 a A b \sin \left (c+d x+\frac {\pi }{2}\right )+7 \left ((7 A+9 C) a^2+9 A b^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{7 a}}{9 a}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{9 a d \cos ^{\frac {9}{2}}(c+d x)}-\frac {\frac {18 A b \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 \int -\frac {3 \left (-7 b \left ((7 A+9 C) a^2+9 A b^2\right ) \cos ^2(c+d x)+a \left (12 A b^2-7 a^2 (7 A+9 C)\right ) \cos (c+d x)+15 b \left ((5 A+7 C) a^2+7 A b^2\right )\right )}{2 \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}dx}{5 a}+\frac {14 \left (a^2 (7 A+9 C)+9 A b^2\right ) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}}{9 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{9 a d \cos ^{\frac {9}{2}}(c+d x)}-\frac {\frac {18 A b \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 \left (a^2 (7 A+9 C)+9 A b^2\right ) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {3 \int \frac {-7 b \left ((7 A+9 C) a^2+9 A b^2\right ) \cos ^2(c+d x)+a \left (12 A b^2-7 a^2 (7 A+9 C)\right ) \cos (c+d x)+15 b \left ((5 A+7 C) a^2+7 A b^2\right )}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}dx}{5 a}}{7 a}}{9 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{9 a d \cos ^{\frac {9}{2}}(c+d x)}-\frac {\frac {18 A b \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 \left (a^2 (7 A+9 C)+9 A b^2\right ) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {3 \int \frac {-7 b \left ((7 A+9 C) a^2+9 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a \left (12 A b^2-7 a^2 (7 A+9 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+15 b \left ((5 A+7 C) a^2+7 A b^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{5 a}}{7 a}}{9 a}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{9 a d \cos ^{\frac {9}{2}}(c+d x)}-\frac {\frac {18 A b \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 \left (a^2 (7 A+9 C)+9 A b^2\right ) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {3 \left (\frac {2 \int -\frac {3 \left (-5 b^2 \left ((5 A+7 C) a^2+7 A b^2\right ) \cos ^2(c+d x)+4 a b \left ((6 A+7 C) a^2+7 A b^2\right ) \cos (c+d x)+7 \left ((7 A+9 C) a^4+3 b^2 (3 A+5 C) a^2+15 A b^4\right )\right )}{2 \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}dx}{3 a}+\frac {10 b \left (a^2 (5 A+7 C)+7 A b^2\right ) \sin (c+d x)}{a d \cos ^{\frac {3}{2}}(c+d x)}\right )}{5 a}}{7 a}}{9 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{9 a d \cos ^{\frac {9}{2}}(c+d x)}-\frac {\frac {18 A b \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 \left (a^2 (7 A+9 C)+9 A b^2\right ) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {3 \left (\frac {10 b \left (a^2 (5 A+7 C)+7 A b^2\right ) \sin (c+d x)}{a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-5 b^2 \left ((5 A+7 C) a^2+7 A b^2\right ) \cos ^2(c+d x)+4 a b \left ((6 A+7 C) a^2+7 A b^2\right ) \cos (c+d x)+7 \left ((7 A+9 C) a^4+3 b^2 (3 A+5 C) a^2+15 A b^4\right )}{\cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}dx}{a}\right )}{5 a}}{7 a}}{9 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{9 a d \cos ^{\frac {9}{2}}(c+d x)}-\frac {\frac {18 A b \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 \left (a^2 (7 A+9 C)+9 A b^2\right ) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {3 \left (\frac {10 b \left (a^2 (5 A+7 C)+7 A b^2\right ) \sin (c+d x)}{a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-5 b^2 \left ((5 A+7 C) a^2+7 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+4 a b \left ((6 A+7 C) a^2+7 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+7 \left ((7 A+9 C) a^4+3 b^2 (3 A+5 C) a^2+15 A b^4\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}\right )}{5 a}}{7 a}}{9 a}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{9 a d \cos ^{\frac {9}{2}}(c+d x)}-\frac {\frac {18 A b \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 \left (a^2 (7 A+9 C)+9 A b^2\right ) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {3 \left (\frac {10 b \left (a^2 (5 A+7 C)+7 A b^2\right ) \sin (c+d x)}{a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \int -\frac {7 b \left ((7 A+9 C) a^4+3 b^2 (3 A+5 C) a^2+15 A b^4\right ) \cos ^2(c+d x)+a \left (7 (7 A+9 C) a^4+4 b^2 (22 A+35 C) a^2+140 A b^4\right ) \cos (c+d x)+5 b \left ((5 A+7 C) a^4+7 b^2 (A+3 C) a^2+21 A b^4\right )}{2 \sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{a}+\frac {14 \left (a^4 (7 A+9 C)+3 a^2 b^2 (3 A+5 C)+15 A b^4\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}}{a}\right )}{5 a}}{7 a}}{9 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{9 a d \cos ^{\frac {9}{2}}(c+d x)}-\frac {\frac {18 A b \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 \left (a^2 (7 A+9 C)+9 A b^2\right ) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {3 \left (\frac {10 b \left (a^2 (5 A+7 C)+7 A b^2\right ) \sin (c+d x)}{a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {14 \left (a^4 (7 A+9 C)+3 a^2 b^2 (3 A+5 C)+15 A b^4\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {\int \frac {7 b \left ((7 A+9 C) a^4+3 b^2 (3 A+5 C) a^2+15 A b^4\right ) \cos ^2(c+d x)+a \left (7 (7 A+9 C) a^4+4 b^2 (22 A+35 C) a^2+140 A b^4\right ) \cos (c+d x)+5 b \left ((5 A+7 C) a^4+7 b^2 (A+3 C) a^2+21 A b^4\right )}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{a}}{a}\right )}{5 a}}{7 a}}{9 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{9 a d \cos ^{\frac {9}{2}}(c+d x)}-\frac {\frac {18 A b \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 \left (a^2 (7 A+9 C)+9 A b^2\right ) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {3 \left (\frac {10 b \left (a^2 (5 A+7 C)+7 A b^2\right ) \sin (c+d x)}{a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {14 \left (a^4 (7 A+9 C)+3 a^2 b^2 (3 A+5 C)+15 A b^4\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {\int \frac {7 b \left ((7 A+9 C) a^4+3 b^2 (3 A+5 C) a^2+15 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a \left (7 (7 A+9 C) a^4+4 b^2 (22 A+35 C) a^2+140 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+5 b \left ((5 A+7 C) a^4+7 b^2 (A+3 C) a^2+21 A b^4\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}}{a}\right )}{5 a}}{7 a}}{9 a}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{9 a d \cos ^{\frac {9}{2}}(c+d x)}-\frac {\frac {18 A b \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 \left (a^2 (7 A+9 C)+9 A b^2\right ) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {3 \left (\frac {10 b \left (a^2 (5 A+7 C)+7 A b^2\right ) \sin (c+d x)}{a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {14 \left (a^4 (7 A+9 C)+3 a^2 b^2 (3 A+5 C)+15 A b^4\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {7 \left (a^4 (7 A+9 C)+3 a^2 b^2 (3 A+5 C)+15 A b^4\right ) \int \sqrt {\cos (c+d x)}dx-\frac {\int -\frac {5 \left (a \left ((5 A+7 C) a^2+7 A b^2\right ) \cos (c+d x) b^3+\left ((5 A+7 C) a^4+7 b^2 (A+3 C) a^2+21 A b^4\right ) b^2\right )}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}}{a}}{a}\right )}{5 a}}{7 a}}{9 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{9 a d \cos ^{\frac {9}{2}}(c+d x)}-\frac {\frac {18 A b \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 \left (a^2 (7 A+9 C)+9 A b^2\right ) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {3 \left (\frac {10 b \left (a^2 (5 A+7 C)+7 A b^2\right ) \sin (c+d x)}{a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {14 \left (a^4 (7 A+9 C)+3 a^2 b^2 (3 A+5 C)+15 A b^4\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {7 \left (a^4 (7 A+9 C)+3 a^2 b^2 (3 A+5 C)+15 A b^4\right ) \int \sqrt {\cos (c+d x)}dx+\frac {5 \int \frac {a \left ((5 A+7 C) a^2+7 A b^2\right ) \cos (c+d x) b^3+\left ((5 A+7 C) a^4+7 b^2 (A+3 C) a^2+21 A b^4\right ) b^2}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}}{a}}{a}\right )}{5 a}}{7 a}}{9 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{9 a d \cos ^{\frac {9}{2}}(c+d x)}-\frac {\frac {18 A b \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 \left (a^2 (7 A+9 C)+9 A b^2\right ) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {3 \left (\frac {10 b \left (a^2 (5 A+7 C)+7 A b^2\right ) \sin (c+d x)}{a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {14 \left (a^4 (7 A+9 C)+3 a^2 b^2 (3 A+5 C)+15 A b^4\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {7 \left (a^4 (7 A+9 C)+3 a^2 b^2 (3 A+5 C)+15 A b^4\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {5 \int \frac {a \left ((5 A+7 C) a^2+7 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) b^3+\left ((5 A+7 C) a^4+7 b^2 (A+3 C) a^2+21 A b^4\right ) b^2}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}}{a}}{a}\right )}{5 a}}{7 a}}{9 a}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{9 a d \cos ^{\frac {9}{2}}(c+d x)}-\frac {\frac {18 A b \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 \left (a^2 (7 A+9 C)+9 A b^2\right ) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {3 \left (\frac {10 b \left (a^2 (5 A+7 C)+7 A b^2\right ) \sin (c+d x)}{a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {14 \left (a^4 (7 A+9 C)+3 a^2 b^2 (3 A+5 C)+15 A b^4\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {5 \int \frac {a \left ((5 A+7 C) a^2+7 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) b^3+\left ((5 A+7 C) a^4+7 b^2 (A+3 C) a^2+21 A b^4\right ) b^2}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}+\frac {14 \left (a^4 (7 A+9 C)+3 a^2 b^2 (3 A+5 C)+15 A b^4\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{a}}{a}\right )}{5 a}}{7 a}}{9 a}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{9 a d \cos ^{\frac {9}{2}}(c+d x)}-\frac {\frac {18 A b \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 \left (a^2 (7 A+9 C)+9 A b^2\right ) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {3 \left (\frac {10 b \left (a^2 (5 A+7 C)+7 A b^2\right ) \sin (c+d x)}{a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {14 \left (a^4 (7 A+9 C)+3 a^2 b^2 (3 A+5 C)+15 A b^4\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {5 \left (a b^2 \left (a^2 (5 A+7 C)+7 A b^2\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx+21 b^4 \left (a^2 C+A b^2\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx\right )}{b}+\frac {14 \left (a^4 (7 A+9 C)+3 a^2 b^2 (3 A+5 C)+15 A b^4\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{a}}{a}\right )}{5 a}}{7 a}}{9 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{9 a d \cos ^{\frac {9}{2}}(c+d x)}-\frac {\frac {18 A b \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 \left (a^2 (7 A+9 C)+9 A b^2\right ) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {3 \left (\frac {10 b \left (a^2 (5 A+7 C)+7 A b^2\right ) \sin (c+d x)}{a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {14 \left (a^4 (7 A+9 C)+3 a^2 b^2 (3 A+5 C)+15 A b^4\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {5 \left (a b^2 \left (a^2 (5 A+7 C)+7 A b^2\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+21 b^4 \left (a^2 C+A b^2\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx\right )}{b}+\frac {14 \left (a^4 (7 A+9 C)+3 a^2 b^2 (3 A+5 C)+15 A b^4\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{a}}{a}\right )}{5 a}}{7 a}}{9 a}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{9 a d \cos ^{\frac {9}{2}}(c+d x)}-\frac {\frac {18 A b \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 \left (a^2 (7 A+9 C)+9 A b^2\right ) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {3 \left (\frac {10 b \left (a^2 (5 A+7 C)+7 A b^2\right ) \sin (c+d x)}{a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {14 \left (a^4 (7 A+9 C)+3 a^2 b^2 (3 A+5 C)+15 A b^4\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {5 \left (21 b^4 \left (a^2 C+A b^2\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx+\frac {2 a b^2 \left (a^2 (5 A+7 C)+7 A b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}\right )}{b}+\frac {14 \left (a^4 (7 A+9 C)+3 a^2 b^2 (3 A+5 C)+15 A b^4\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{a}}{a}\right )}{5 a}}{7 a}}{9 a}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{9 a d \cos ^{\frac {9}{2}}(c+d x)}-\frac {\frac {18 A b \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {14 \left (a^2 (7 A+9 C)+9 A b^2\right ) \sin (c+d x)}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {3 \left (\frac {10 b \left (a^2 (5 A+7 C)+7 A b^2\right ) \sin (c+d x)}{a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {14 \left (a^4 (7 A+9 C)+3 a^2 b^2 (3 A+5 C)+15 A b^4\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {5 \left (\frac {2 a b^2 \left (a^2 (5 A+7 C)+7 A b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+\frac {42 b^4 \left (a^2 C+A b^2\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{d (a+b)}\right )}{b}+\frac {14 \left (a^4 (7 A+9 C)+3 a^2 b^2 (3 A+5 C)+15 A b^4\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{a}}{a}\right )}{5 a}}{7 a}}{9 a}\) |
Input:
Int[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(11/2)*(a + b*Cos[c + d*x])),x]
Output:
(2*A*Sin[c + d*x])/(9*a*d*Cos[c + d*x]^(9/2)) - ((18*A*b*Sin[c + d*x])/(7* a*d*Cos[c + d*x]^(7/2)) - ((14*(9*A*b^2 + a^2*(7*A + 9*C))*Sin[c + d*x])/( 5*a*d*Cos[c + d*x]^(5/2)) - (3*((10*b*(7*A*b^2 + a^2*(5*A + 7*C))*Sin[c + d*x])/(a*d*Cos[c + d*x]^(3/2)) - (-(((14*(15*A*b^4 + 3*a^2*b^2*(3*A + 5*C) + a^4*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2])/d + (5*((2*a*b^2*(7*A*b^2 + a^2*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/d + (42*b^4*(A*b^2 + a^2*C)*E llipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/((a + b)*d)))/b)/a) + (14*(15*A* b^4 + 3*a^2*b^2*(3*A + 5*C) + a^4*(7*A + 9*C))*Sin[c + d*x])/(a*d*Sqrt[Cos [c + d*x]]))/a))/(5*a))/(7*a))/(9*a)
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. \(1292\) vs. \(2(325)=650\).
Time = 14.43 (sec) , antiderivative size = 1293, normalized size of antiderivative = 3.76
Input:
int((A+C*cos(d*x+c)^2)/cos(d*x+c)^(11/2)/(a+b*cos(d*x+c)),x,method=_RETURN VERBOSE)
Output:
-(-(1-2*cos(1/2*d*x+1/2*c)^2)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*A/a*(-1/144*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)^5-7/180*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)^3-14/15*sin(1/2*d *x+1/2*c)^2*cos(1/2*d*x+1/2*c)/(-(1-2*cos(1/2*d*x+1/2*c)^2)*sin(1/2*d*x+1/ 2*c)^2)^(1/2)+7/15*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(1-2*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)*EllipticF(cos( 1/2*d*x+1/2*c),2^(1/2))-7/15*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(1-2*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)*(Ell ipticF(cos(1/2*d*x+1/2*c),2^(1/2))-EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))) +2/5*(A*b^2+C*a^2)/a^3/(8*sin(1/2*d*x+1/2*c)^6-12*sin(1/2*d*x+1/2*c)^4+6*s in(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)*EllipticE(cos(1/2*d*x+1/ 2*c),2^(1/2))*(sin(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)*Ellip ticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(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+C*a^2) /a^5*b^2/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*...
Timed out. \[ \int \frac {A+C \cos ^2(c+d x)}{\cos ^{\frac {11}{2}}(c+d x) (a+b \cos (c+d x))} \, dx=\text {Timed out} \] Input:
integrate((A+C*cos(d*x+c)^2)/cos(d*x+c)^(11/2)/(a+b*cos(d*x+c)),x, algorit hm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {A+C \cos ^2(c+d x)}{\cos ^{\frac {11}{2}}(c+d x) (a+b \cos (c+d x))} \, dx=\text {Timed out} \] Input:
integrate((A+C*cos(d*x+c)**2)/cos(d*x+c)**(11/2)/(a+b*cos(d*x+c)),x)
Output:
Timed out
\[ \int \frac {A+C \cos ^2(c+d x)}{\cos ^{\frac {11}{2}}(c+d x) (a+b \cos (c+d x))} \, dx=\int { \frac {C \cos \left (d x + c\right )^{2} + A}{{\left (b \cos \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac {11}{2}}} \,d x } \] Input:
integrate((A+C*cos(d*x+c)^2)/cos(d*x+c)^(11/2)/(a+b*cos(d*x+c)),x, algorit hm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + A)/((b*cos(d*x + c) + a)*cos(d*x + c)^(11/2) ), x)
\[ \int \frac {A+C \cos ^2(c+d x)}{\cos ^{\frac {11}{2}}(c+d x) (a+b \cos (c+d x))} \, dx=\int { \frac {C \cos \left (d x + c\right )^{2} + A}{{\left (b \cos \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac {11}{2}}} \,d x } \] Input:
integrate((A+C*cos(d*x+c)^2)/cos(d*x+c)^(11/2)/(a+b*cos(d*x+c)),x, algorit hm="giac")
Output:
integrate((C*cos(d*x + c)^2 + A)/((b*cos(d*x + c) + a)*cos(d*x + c)^(11/2) ), x)
Timed out. \[ \int \frac {A+C \cos ^2(c+d x)}{\cos ^{\frac {11}{2}}(c+d x) (a+b \cos (c+d x))} \, dx=\int \frac {C\,{\cos \left (c+d\,x\right )}^2+A}{{\cos \left (c+d\,x\right )}^{11/2}\,\left (a+b\,\cos \left (c+d\,x\right )\right )} \,d x \] Input:
int((A + C*cos(c + d*x)^2)/(cos(c + d*x)^(11/2)*(a + b*cos(c + d*x))),x)
Output:
int((A + C*cos(c + d*x)^2)/(cos(c + d*x)^(11/2)*(a + b*cos(c + d*x))), x)
\[ \int \frac {A+C \cos ^2(c+d x)}{\cos ^{\frac {11}{2}}(c+d x) (a+b \cos (c+d x))} \, dx=\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{7} b +\cos \left (d x +c \right )^{6} a}d x \right ) a +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{5} b +\cos \left (d x +c \right )^{4} a}d x \right ) c \] Input:
int((A+C*cos(d*x+c)^2)/cos(d*x+c)^(11/2)/(a+b*cos(d*x+c)),x)
Output:
int(sqrt(cos(c + d*x))/(cos(c + d*x)**7*b + cos(c + d*x)**6*a),x)*a + int( sqrt(cos(c + d*x))/(cos(c + d*x)**5*b + cos(c + d*x)**4*a),x)*c