Integrand size = 45, antiderivative size = 506 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=-\frac {2 (a-b) \sqrt {a+b} \left (48 A b^3-63 a^3 B-56 a b^2 B+a^2 (44 A b+70 b C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{105 a^5 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {a+b} \left (48 A b^3-4 a b^2 (3 A+14 B)+a^3 (25 A-63 B+35 C)+2 a^2 b (22 A+7 (B+5 C))\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{105 a^4 d \sqrt {\sec (c+d x)}}+\frac {2 \left (24 A b^2-28 a b B+5 a^2 (5 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 a^3 d}-\frac {2 (6 A b-7 a B) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 a^2 d}+\frac {2 A \sqrt {a+b \cos (c+d x)} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 a d} \] Output:
-2/105*(a-b)*(a+b)^(1/2)*(48*A*b^3-63*B*a^3-56*B*a*b^2+a^2*(44*A*b+70*C*b) )*cos(d*x+c)^(1/2)*csc(d*x+c)*EllipticE((a+b*cos(d*x+c))^(1/2)/(a+b)^(1/2) /cos(d*x+c)^(1/2),(-(a+b)/(a-b))^(1/2))*(a*(1-sec(d*x+c))/(a+b))^(1/2)*(a* (1+sec(d*x+c))/(a-b))^(1/2)/a^5/d/sec(d*x+c)^(1/2)+2/105*(a+b)^(1/2)*(48*A *b^3-4*a*b^2*(3*A+14*B)+a^3*(25*A-63*B+35*C)+2*a^2*b*(22*A+7*B+35*C))*cos( d*x+c)^(1/2)*csc(d*x+c)*EllipticF((a+b*cos(d*x+c))^(1/2)/(a+b)^(1/2)/cos(d *x+c)^(1/2),(-(a+b)/(a-b))^(1/2))*(a*(1-sec(d*x+c))/(a+b))^(1/2)*(a*(1+sec (d*x+c))/(a-b))^(1/2)/a^4/d/sec(d*x+c)^(1/2)+2/105*(24*A*b^2-28*B*a*b+5*a^ 2*(5*A+7*C))*(a+b*cos(d*x+c))^(1/2)*sec(d*x+c)^(3/2)*sin(d*x+c)/a^3/d-2/35 *(6*A*b-7*B*a)*(a+b*cos(d*x+c))^(1/2)*sec(d*x+c)^(5/2)*sin(d*x+c)/a^2/d+2/ 7*A*(a+b*cos(d*x+c))^(1/2)*sec(d*x+c)^(7/2)*sin(d*x+c)/a/d
Leaf count is larger than twice the leaf count of optimal. \(3704\) vs. \(2(506)=1012\).
Time = 24.48 (sec) , antiderivative size = 3704, normalized size of antiderivative = 7.32 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=\text {Result too large to show} \] Input:
Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2))/Sqr t[a + b*Cos[c + d*x]],x]
Output:
(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*(-44*a^2*A*b - 48*A*b^3 + 63*a^3*B + 56*a*b^2*B - 70*a^2*b*C)*Sin[c + d*x])/(105*a^4) + (2*Sec[c + d*x]^2*(-6*A*b*Sin[c + d*x] + 7*a*B*Sin[c + d*x]))/(35*a^2) + (2*Sec[c + d *x]*(25*a^2*A*Sin[c + d*x] + 24*A*b^2*Sin[c + d*x] - 28*a*b*B*Sin[c + d*x] + 35*a^2*C*Sin[c + d*x]))/(105*a^3) + (2*A*Sec[c + d*x]^2*Tan[c + d*x])/( 7*a)))/d + (2*((44*A*b)/(105*a*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]] ) + (16*A*b^3)/(35*a^3*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (3*B )/(5*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*b^2*B)/(15*a^2*Sqrt [a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*b*C)/(3*a*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (5*A*Sqrt[Sec[c + d*x]])/(21*Sqrt[a + b*Cos[c + d*x]]) + (32*A*b^2*Sqrt[Sec[c + d*x]])/(105*a^2*Sqrt[a + b*Cos[c + d*x] ]) + (16*A*b^4*Sqrt[Sec[c + d*x]])/(35*a^4*Sqrt[a + b*Cos[c + d*x]]) - (7* b*B*Sqrt[Sec[c + d*x]])/(15*a*Sqrt[a + b*Cos[c + d*x]]) - (8*b^3*B*Sqrt[Se c[c + d*x]])/(15*a^3*Sqrt[a + b*Cos[c + d*x]]) + (C*Sqrt[Sec[c + d*x]])/(3 *Sqrt[a + b*Cos[c + d*x]]) + (2*b^2*C*Sqrt[Sec[c + d*x]])/(3*a^2*Sqrt[a + b*Cos[c + d*x]]) + (44*A*b^2*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*a^2 *Sqrt[a + b*Cos[c + d*x]]) + (16*A*b^4*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]] )/(35*a^4*Sqrt[a + b*Cos[c + d*x]]) - (3*b*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*a*Sqrt[a + b*Cos[c + d*x]]) - (8*b^3*B*Cos[2*(c + d*x)]*Sqrt[Se c[c + d*x]])/(15*a^3*Sqrt[a + b*Cos[c + d*x]]) + (2*b^2*C*Cos[2*(c + d*...
Time = 2.48 (sec) , antiderivative size = 507, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.356, Rules used = {3042, 4709, 3042, 3534, 27, 3042, 3534, 27, 3042, 3534, 27, 3042, 3477, 3042, 3295, 3473}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sec ^{\frac {9}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {a+b \cos (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sec (c+d x)^{9/2} \left (A+B \cos (c+d x)+C \cos (c+d x)^2\right )}{\sqrt {a+b \cos (c+d x)}}dx\) |
| \(\Big \downarrow \) 4709 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {C \cos ^2(c+d x)+B \cos (c+d x)+A}{\cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {C \sin \left (c+d x+\frac {\pi }{2}\right )^2+B \sin \left (c+d x+\frac {\pi }{2}\right )+A}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 \int -\frac {-4 A b \cos ^2(c+d x)-a (5 A+7 C) \cos (c+d x)+6 A b-7 a B}{2 \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{7 a}+\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\int \frac {-4 A b \cos ^2(c+d x)-a (5 A+7 C) \cos (c+d x)+6 A b-7 a B}{\cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{7 a}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\int \frac {-4 A b \sin \left (c+d x+\frac {\pi }{2}\right )^2-a (5 A+7 C) \sin \left (c+d x+\frac {\pi }{2}\right )+6 A b-7 a B}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{7 a}\right )\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 \int -\frac {5 (5 A+7 C) a^2-28 b B a+(2 A b+21 a B) \cos (c+d x) a+24 A b^2-2 b (6 A b-7 a B) \cos ^2(c+d x)}{2 \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{5 a}+\frac {2 (6 A b-7 a B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 (6 A b-7 a B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\int \frac {5 (5 A+7 C) a^2-28 b B a+(2 A b+21 a B) \cos (c+d x) a+24 A b^2-2 b (6 A b-7 a B) \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{5 a}}{7 a}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 (6 A b-7 a B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\int \frac {5 (5 A+7 C) a^2-28 b B a+(2 A b+21 a B) \sin \left (c+d x+\frac {\pi }{2}\right ) a+24 A b^2-2 b (6 A b-7 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 a}}{7 a}\right )\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 (6 A b-7 a B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \int -\frac {-63 B a^3+2 b (22 A+35 C) a^2-56 b^2 B a+\left (-5 (5 A+7 C) a^2-14 b B a+12 A b^2\right ) \cos (c+d x) a+48 A b^3}{2 \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{3 a}+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)-28 a b B+24 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{3 a d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}}{7 a}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 (6 A b-7 a B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)-28 a b B+24 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-63 B a^3+(44 A b+70 C b) a^2-56 b^2 B a+\left (-5 (5 A+7 C) a^2-14 b B a+12 A b^2\right ) \cos (c+d x) a+48 A b^3}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{3 a}}{5 a}}{7 a}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 (6 A b-7 a B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)-28 a b B+24 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-63 B a^3+(44 A b+70 C b) a^2-56 b^2 B a+\left (-5 (5 A+7 C) a^2-14 b B a+12 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+48 A b^3}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 a}}{5 a}}{7 a}\right )\) |
| \(\Big \downarrow \) 3477 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 (6 A b-7 a B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)-28 a b B+24 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\left (-63 a^3 B+2 a^2 b (22 A+35 C)-56 a b^2 B+48 A b^3\right ) \int \frac {\cos (c+d x)+1}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx-\left (a^3 (25 A-63 B+35 C)+2 a^2 b (22 A+7 (B+5 C))-4 a b^2 (3 A+14 B)+48 A b^3\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx}{3 a}}{5 a}}{7 a}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 (6 A b-7 a B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)-28 a b B+24 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\left (-63 a^3 B+2 a^2 b (22 A+35 C)-56 a b^2 B+48 A b^3\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\left (a^3 (25 A-63 B+35 C)+2 a^2 b (22 A+7 (B+5 C))-4 a b^2 (3 A+14 B)+48 A b^3\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 a}}{5 a}}{7 a}\right )\) |
| \(\Big \downarrow \) 3295 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 (6 A b-7 a B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)-28 a b B+24 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\left (-63 a^3 B+2 a^2 b (22 A+35 C)-56 a b^2 B+48 A b^3\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 \sqrt {a+b} \cot (c+d x) \left (a^3 (25 A-63 B+35 C)+2 a^2 b (22 A+7 (B+5 C))-4 a b^2 (3 A+14 B)+48 A b^3\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{a d}}{3 a}}{5 a}}{7 a}\right )\) |
| \(\Big \downarrow \) 3473 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 (6 A b-7 a B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)-28 a b B+24 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (-63 a^3 B+2 a^2 b (22 A+35 C)-56 a b^2 B+48 A b^3\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{a^2 d}-\frac {2 \sqrt {a+b} \cot (c+d x) \left (a^3 (25 A-63 B+35 C)+2 a^2 b (22 A+7 (B+5 C))-4 a b^2 (3 A+14 B)+48 A b^3\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{a d}}{3 a}}{5 a}}{7 a}\right )\) |
Input:
Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2))/Sqrt[a + b*Cos[c + d*x]],x]
Output:
Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*a*d*Cos[c + d*x]^(7/2)) - ((2*(6*A*b - 7*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*a*d*Cos[c + d*x]^(5/2)) - (-1/3*((2*(a - b)*Sqrt [a + b]*(48*A*b^3 - 63*a^3*B - 56*a*b^2*B + 2*a^2*b*(22*A + 35*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a* (1 + Sec[c + d*x]))/(a - b)])/(a^2*d) - (2*Sqrt[a + b]*(48*A*b^3 - 4*a*b^2 *(3*A + 14*B) + a^3*(25*A - 63*B + 35*C) + 2*a^2*b*(22*A + 7*(B + 5*C)))*C ot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Co s[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sq rt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d))/a + (2*(24*A*b^2 - 28*a*b*B + 5 *a^2*(5*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a*d*Cos[c + d* x]^(3/2)))/(5*a))/(7*a))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f _.)*(x_)]]), x_Symbol] :> Simp[-2*(Tan[e + f*x]/(a*f))*Rt[(a + b)/d, 2]*Sqr t[a*((1 - Csc[e + f*x])/(a + b))]*Sqrt[a*((1 + Csc[e + f*x])/(a - b))]*Elli pticF[ArcSin[Sqrt[a + b*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]/Rt[(a + b)/d, 2] ], -(a + b)/(a - b)], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && PosQ[(a + b)/d]
Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((b_.)*sin[(e_.) + (f_.)*(x_)]) ^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*A* (c - d)*(Tan[e + f*x]/(f*b*c^2))*Rt[(c + d)/b, 2]*Sqrt[c*((1 + Csc[e + f*x] )/(c - d))]*Sqrt[c*((1 - Csc[e + f*x])/(c + d))]*EllipticE[ArcSin[Sqrt[c + d*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]]/Rt[(c + d)/b, 2]], -(c + d)/(c - d)], x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && PosQ[(c + d)/b]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ .)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> S imp[(A - B)/(a - b) Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f* x]]), x], x] - Simp[(A*b - a*B)/(a - b) Int[(1 + Sin[e + f*x])/((a + b*Si n[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e , f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && NeQ[A, B]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(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[(u_)*((c_.)*sec[(a_.) + (b_.)*(x_)])^(m_.), x_Symbol] :> Simp[(c*Sec[a + b*x])^m*(c*Cos[a + b*x])^m Int[ActivateTrig[u]/(c*Cos[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownSineIntegrandQ[u, x]
Leaf count of result is larger than twice the leaf count of optimal. \(2633\) vs. \(2(453)=906\).
Time = 73.03 (sec) , antiderivative size = 2634, normalized size of antiderivative = 5.21
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2634\) |
| parts | \(\text {Expression too large to display}\) | \(2691\) |
Input:
int((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(9/2)/(a+b*cos(d*x+c))^(1/2 ),x,method=_RETURNVERBOSE)
Output:
2/105/d/a^4*(a+b*cos(d*x+c))^(1/2)*sec(d*x+c)^(9/2)/(b*cos(d*x+c)^2+a*cos( d*x+c)+b*cos(d*x+c)+a)*(B*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b* cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a^4*EllipticF(-csc(d*x+c)+cot(d*x+c),(-( a-b)/(a+b))^(1/2))*(-63*cos(d*x+c)^6-126*cos(d*x+c)^5-63*cos(d*x+c)^4)+C*( cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c))) ^(1/2)*a^4*EllipticF(-csc(d*x+c)+cot(d*x+c),(-(a-b)/(a+b))^(1/2))*(-35*cos (d*x+c)^6-70*cos(d*x+c)^5-35*cos(d*x+c)^4)+B*(cos(d*x+c)/(1+cos(d*x+c)))^( 1/2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a^4*EllipticE(-csc(d* x+c)+cot(d*x+c),(-(a-b)/(a+b))^(1/2))*(63*cos(d*x+c)^6+126*cos(d*x+c)^5+63 *cos(d*x+c)^4)+A*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*cos(d*x+c ))/(1+cos(d*x+c)))^(1/2)*a^4*EllipticF(-csc(d*x+c)+cot(d*x+c),(-(a-b)/(a+b ))^(1/2))*(-25*cos(d*x+c)^6-50*cos(d*x+c)^5-25*cos(d*x+c)^4)+sin(d*x+c)*co s(d*x+c)^2*(63*cos(d*x+c)^2+21*cos(d*x+c)+21)*B*a^4+sin(d*x+c)*cos(d*x+c)^ 3*(35*cos(d*x+c)+35)*a^4*C+(25*cos(d*x+c)^3+25*cos(d*x+c)^2+15*cos(d*x+c)+ 15)*sin(d*x+c)*A*a^4*cos(d*x+c)+A*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+ b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*b^4*EllipticE(-csc(d*x+c)+cot(d* x+c),(-(a-b)/(a+b))^(1/2))*(-48*cos(d*x+c)^6-96*cos(d*x+c)^5-48*cos(d*x+c) ^4)-48*A*b^4*cos(d*x+c)^5*sin(d*x+c)+B*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*( 1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a^3*b*EllipticE(-csc(d*x+c) +cot(d*x+c),(-(a-b)/(a+b))^(1/2))*(63*cos(d*x+c)^6+126*cos(d*x+c)^5+63*...
\[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{\frac {9}{2}}}{\sqrt {b \cos \left (d x + c\right ) + a}} \,d x } \] Input:
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(9/2)/(a+b*cos(d*x+c) )^(1/2),x, algorithm="fricas")
Output:
integral((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*sec(d*x + c)^(9/2)/sqrt(b *cos(d*x + c) + a), x)
Timed out. \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=\text {Timed out} \] Input:
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x+c)**(9/2)/(a+b*cos(d*x+ c))**(1/2),x)
Output:
Timed out
\[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{\frac {9}{2}}}{\sqrt {b \cos \left (d x + c\right ) + a}} \,d x } \] Input:
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(9/2)/(a+b*cos(d*x+c) )^(1/2),x, algorithm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*sec(d*x + c)^(9/2)/sqrt( b*cos(d*x + c) + a), x)
\[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{\frac {9}{2}}}{\sqrt {b \cos \left (d x + c\right ) + a}} \,d x } \] Input:
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(9/2)/(a+b*cos(d*x+c) )^(1/2),x, algorithm="giac")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*sec(d*x + c)^(9/2)/sqrt( b*cos(d*x + c) + a), x)
Timed out. \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=\int \frac {{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{\sqrt {a+b\,\cos \left (c+d\,x\right )}} \,d x \] Input:
int(((1/cos(c + d*x))^(9/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + b*cos(c + d*x))^(1/2),x)
Output:
int(((1/cos(c + d*x))^(9/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + b*cos(c + d*x))^(1/2), x)
\[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right ) \sec \left (d x +c \right )^{4}}{\cos \left (d x +c \right ) b +a}d x \right ) b +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{4}}{\cos \left (d x +c \right ) b +a}d x \right ) c +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{4}}{\cos \left (d x +c \right ) b +a}d x \right ) a \] Input:
int((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(9/2)/(a+b*cos(d*x+c))^(1/2 ),x)
Output:
int((sqrt(sec(c + d*x))*sqrt(cos(c + d*x)*b + a)*cos(c + d*x)*sec(c + d*x) **4)/(cos(c + d*x)*b + a),x)*b + int((sqrt(sec(c + d*x))*sqrt(cos(c + d*x) *b + a)*cos(c + d*x)**2*sec(c + d*x)**4)/(cos(c + d*x)*b + a),x)*c + int(( sqrt(sec(c + d*x))*sqrt(cos(c + d*x)*b + a)*sec(c + d*x)**4)/(cos(c + d*x) *b + a),x)*a