Integrand size = 43, antiderivative size = 669 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\frac {\left (35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B+3 a^4 b (8 A-3 C)-a^2 b^3 (65 A-3 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{4 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (35 A b^4+33 a^3 b B-15 a b^3 B+a^4 (8 A-21 C)-a^2 b^2 (61 A-3 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{12 a^3 \left (a^2-b^2\right )^2 d}+\frac {\left (35 A b^6-35 a^5 b B+38 a^3 b^3 B-15 a b^5 B-a^2 b^4 (86 A-3 C)+3 a^4 b^2 (21 A-2 C)+15 a^6 C\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{4 a^4 (a-b)^2 (a+b)^3 d}-\frac {\left (35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B+3 a^4 b (8 A-3 C)-a^2 b^3 (65 A-3 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (35 A b^4+33 a^3 b B-15 a b^3 B+a^4 (8 A-21 C)-a^2 b^2 (61 A-3 C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d}+\frac {\left (A b^2-a (b B-a C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac {\left (7 A b^4+9 a^3 b B-3 a b^3 B-5 a^4 C-a^2 b^2 (13 A+C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))} \] Output:
1/4*(35*A*b^5-8*a^5*B+29*a^3*b^2*B-15*a*b^4*B+3*a^4*b*(8*A-3*C)-a^2*b^3*(6 5*A-3*C))*cos(d*x+c)^(1/2)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))*sec(d*x+c )^(1/2)/a^4/(a^2-b^2)^2/d+1/12*(35*A*b^4+33*B*a^3*b-15*B*a*b^3+a^4*(8*A-21 *C)-a^2*b^2*(61*A-3*C))*cos(d*x+c)^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^( 1/2))*sec(d*x+c)^(1/2)/a^3/(a^2-b^2)^2/d+1/4*(35*A*b^6-35*B*a^5*b+38*B*a^3 *b^3-15*B*a*b^5-a^2*b^4*(86*A-3*C)+3*a^4*b^2*(21*A-2*C)+15*a^6*C)*cos(d*x+ c)^(1/2)*EllipticPi(sin(1/2*d*x+1/2*c),2*b/(a+b),2^(1/2))*sec(d*x+c)^(1/2) /a^4/(a-b)^2/(a+b)^3/d-1/4*(35*A*b^5-8*a^5*B+29*a^3*b^2*B-15*a*b^4*B+3*a^4 *b*(8*A-3*C)-a^2*b^3*(65*A-3*C))*sec(d*x+c)^(1/2)*sin(d*x+c)/a^4/(a^2-b^2) ^2/d+1/12*(35*A*b^4+33*B*a^3*b-15*B*a*b^3+a^4*(8*A-21*C)-a^2*b^2*(61*A-3*C ))*sec(d*x+c)^(3/2)*sin(d*x+c)/a^3/(a^2-b^2)^2/d+1/2*(A*b^2-a*(B*b-C*a))*s ec(d*x+c)^(3/2)*sin(d*x+c)/a/(a^2-b^2)/d/(a+b*cos(d*x+c))^2-1/4*(7*A*b^4+9 *B*a^3*b-3*B*a*b^3-5*a^4*C-a^2*b^2*(13*A+C))*sec(d*x+c)^(3/2)*sin(d*x+c)/a ^2/(a^2-b^2)^2/d/(a+b*cos(d*x+c))
Time = 7.57 (sec) , antiderivative size = 1013, normalized size of antiderivative = 1.51 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx =\text {Too large to display} \] Input:
Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x])^3,x]
Output:
((2*(16*a^6*A + 328*a^4*A*b^2 - 641*a^2*A*b^4 + 315*A*b^6 - 168*a^5*b*B + 285*a^3*b^3*B - 135*a*b^5*B + 48*a^6*C - 57*a^4*b^2*C + 27*a^2*b^4*C)*Cos[ c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d* x]^2]*Sin[c + d*x])/(a*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(16 0*a^5*A*b - 512*a^3*A*b^3 + 280*a*A*b^5 - 48*a^6*B + 240*a^4*b^2*B - 120*a ^2*b^4*B - 96*a^5*b*C + 24*a^3*b^3*C)*Cos[c + d*x]^2*EllipticPi[-(a/b), Ar cSin[Sqrt[Sec[c + d*x]]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2 ]*Sin[c + d*x])/(b*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((72*a^4*A *b^2 - 195*a^2*A*b^4 + 105*A*b^6 - 24*a^5*b*B + 87*a^3*b^3*B - 45*a*b^5*B - 27*a^4*b^2*C + 9*a^2*b^4*C)*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a* b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1] *Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - b)*b*EllipticF[Arc Sin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x ]]*Sqrt[1 - Sec[c + d*x]^2] + 2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a *b^2*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec [c + d*x]^2)))/(48*a^4*(a - b)^2*(a + b)^2*d) + (Sqrt[Sec[c + d*x]]*(((-24 *a^4*A*b + 65*a^2*A*b^3 - 35*A*b^5 + 8*a^5*B - 29*a^3*b^2*B + 15*a*b^4*...
Time = 5.16 (sec) , antiderivative size = 619, normalized size of antiderivative = 0.93, number of steps used = 23, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.535, Rules used = {3042, 4709, 3042, 3534, 27, 3042, 3534, 27, 3042, 3534, 27, 3042, 3534, 27, 3042, 3538, 25, 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.
| \(\displaystyle \int \frac {\sec ^{\frac {5}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sec (c+d x)^{5/2} \left (A+B \cos (c+d x)+C \cos (c+d x)^2\right )}{(a+b \cos (c+d x))^3}dx\) |
| \(\Big \downarrow \) 4709 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {C \cos ^2(c+d x)+B \cos (c+d x)+A}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {C \sin \left (c+d x+\frac {\pi }{2}\right )^2+B \sin \left (c+d x+\frac {\pi }{2}\right )+A}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int -\frac {-\left ((4 A-3 C) a^2\right )-3 b B a+4 (A b+C b-a B) \cos (c+d x) a+7 A b^2-5 \left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x)}{2 \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\int \frac {-\left ((4 A-3 C) a^2\right )-3 b B a+4 (A b+C b-a B) \cos (c+d x) a+7 A b^2-5 \left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}dx}{4 a \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\int \frac {-\left ((4 A-3 C) a^2\right )-3 b B a+4 (A b+C b-a B) \sin \left (c+d x+\frac {\pi }{2}\right ) a+7 A b^2-5 \left (A b^2-a (b B-a C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{4 a \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\frac {\int -\frac {(8 A-21 C) a^4+33 b B a^3-b^2 (61 A-3 C) a^2-15 b^3 B a+4 \left (2 B a^3-b (4 A+3 C) a^2+b^2 B a+A b^3\right ) \cos (c+d x) a+35 A b^4-3 \left (-5 C a^4+9 b B a^3-b^2 (13 A+C) a^2-3 b^3 B a+7 A b^4\right ) \cos ^2(c+d x)}{2 \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}dx}{a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}}{4 a \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\int \frac {(8 A-21 C) a^4+33 b B a^3-b^2 (61 A-3 C) a^2-15 b^3 B a+4 \left (2 B a^3-b (4 A+3 C) a^2+b^2 B a+A b^3\right ) \cos (c+d x) a+35 A b^4-3 \left (-5 C a^4+9 b B a^3-b^2 (13 A+C) a^2-3 b^3 B a+7 A b^4\right ) \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}dx}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\int \frac {(8 A-21 C) a^4+33 b B a^3-b^2 (61 A-3 C) a^2-15 b^3 B a+4 \left (2 B a^3-b (4 A+3 C) a^2+b^2 B a+A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+35 A b^4-3 \left (-5 C a^4+9 b B a^3-b^2 (13 A+C) a^2-3 b^3 B a+7 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \int -\frac {-b \left ((8 A-21 C) a^4+33 b B a^3-b^2 (61 A-3 C) a^2-15 b^3 B a+35 A b^4\right ) \cos ^2(c+d x)+4 a \left (-2 (A+3 C) a^4+12 b B a^3-b^2 (14 A+3 C) a^2-3 b^3 B a+7 A b^4\right ) \cos (c+d x)+3 \left (-8 B a^5+(24 A b-9 b C) a^4+29 b^2 B a^3-b^3 (65 A-3 C) a^2-15 b^4 B a+35 A b^5\right )}{2 \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}dx}{3 a}+\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-b \left ((8 A-21 C) a^4+33 b B a^3-b^2 (61 A-3 C) a^2-15 b^3 B a+35 A b^4\right ) \cos ^2(c+d x)+4 a \left (-2 (A+3 C) a^4+12 b B a^3-b^2 (14 A+3 C) a^2-3 b^3 B a+7 A b^4\right ) \cos (c+d x)+3 \left (-8 B a^5+3 b (8 A-3 C) a^4+29 b^2 B a^3-b^3 (65 A-3 C) a^2-15 b^4 B a+35 A b^5\right )}{\cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}dx}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-b \left ((8 A-21 C) a^4+33 b B a^3-b^2 (61 A-3 C) a^2-15 b^3 B a+35 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+4 a \left (-2 (A+3 C) a^4+12 b B a^3-b^2 (14 A+3 C) a^2-3 b^3 B a+7 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 \left (-8 B a^5+3 b (8 A-3 C) a^4+29 b^2 B a^3-b^3 (65 A-3 C) a^2-15 b^4 B a+35 A b^5\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \int -\frac {8 (A+3 C) a^6-72 b B a^5+b^2 (128 A-15 C) a^4+99 b^3 B a^3-b^4 (223 A-9 C) a^2-45 b^5 B a+4 \left (-6 B a^5+4 b (5 A-3 C) a^4+30 b^2 B a^3-b^3 (64 A-3 C) a^2-15 b^4 B a+35 A b^5\right ) \cos (c+d x) a+105 A b^6+3 b \left (-8 B a^5+3 b (8 A-3 C) a^4+29 b^2 B a^3-b^3 (65 A-3 C) a^2-15 b^4 B a+35 A b^5\right ) \cos ^2(c+d x)}{2 \sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{a}+\frac {6 \sin (c+d x) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{a d \sqrt {\cos (c+d x)}}}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \sin (c+d x) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{a d \sqrt {\cos (c+d x)}}-\frac {\int \frac {8 (A+3 C) a^6-72 b B a^5+b^2 (128 A-15 C) a^4+99 b^3 B a^3-b^4 (223 A-9 C) a^2-45 b^5 B a+4 \left (-6 B a^5+4 b (5 A-3 C) a^4+30 b^2 B a^3-b^3 (64 A-3 C) a^2-15 b^4 B a+35 A b^5\right ) \cos (c+d x) a+105 A b^6+3 b \left (-8 B a^5+3 b (8 A-3 C) a^4+29 b^2 B a^3-b^3 (65 A-3 C) a^2-15 b^4 B a+35 A b^5\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{a}}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \sin (c+d x) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{a d \sqrt {\cos (c+d x)}}-\frac {\int \frac {8 (A+3 C) a^6-72 b B a^5+b^2 (128 A-15 C) a^4+99 b^3 B a^3-b^4 (223 A-9 C) a^2-45 b^5 B a+4 \left (-6 B a^5+4 b (5 A-3 C) a^4+30 b^2 B a^3-b^3 (64 A-3 C) a^2-15 b^4 B a+35 A b^5\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+105 A b^6+3 b \left (-8 B a^5+3 b (8 A-3 C) a^4+29 b^2 B a^3-b^3 (65 A-3 C) a^2-15 b^4 B a+35 A b^5\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \sin (c+d x) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{a d \sqrt {\cos (c+d x)}}-\frac {3 \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right ) \int \sqrt {\cos (c+d x)}dx-\frac {\int -\frac {a \left ((8 A-21 C) a^4+33 b B a^3-b^2 (61 A-3 C) a^2-15 b^3 B a+35 A b^4\right ) \cos (c+d x) b^2+\left (8 (A+3 C) a^6-72 b B a^5+b^2 (128 A-15 C) a^4+99 b^3 B a^3-b^4 (223 A-9 C) a^2-45 b^5 B a+105 A b^6\right ) b}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}}{a}}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \sin (c+d x) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{a d \sqrt {\cos (c+d x)}}-\frac {3 \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right ) \int \sqrt {\cos (c+d x)}dx+\frac {\int \frac {a \left ((8 A-21 C) a^4+33 b B a^3-b^2 (61 A-3 C) a^2-15 b^3 B a+35 A b^4\right ) \cos (c+d x) b^2+\left (8 (A+3 C) a^6-72 b B a^5+b^2 (128 A-15 C) a^4+99 b^3 B a^3-b^4 (223 A-9 C) a^2-45 b^5 B a+105 A b^6\right ) b}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}}{a}}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \sin (c+d x) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{a d \sqrt {\cos (c+d x)}}-\frac {3 \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {\int \frac {a \left ((8 A-21 C) a^4+33 b B a^3-b^2 (61 A-3 C) a^2-15 b^3 B a+35 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) b^2+\left (8 (A+3 C) a^6-72 b B a^5+b^2 (128 A-15 C) a^4+99 b^3 B a^3-b^4 (223 A-9 C) a^2-45 b^5 B a+105 A b^6\right ) b}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}}{a}}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \sin (c+d x) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {\int \frac {a \left ((8 A-21 C) a^4+33 b B a^3-b^2 (61 A-3 C) a^2-15 b^3 B a+35 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) b^2+\left (8 (A+3 C) a^6-72 b B a^5+b^2 (128 A-15 C) a^4+99 b^3 B a^3-b^4 (223 A-9 C) a^2-45 b^5 B a+105 A b^6\right ) b}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}+\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{d}}{a}}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \sin (c+d x) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {a b \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx+3 b \left (15 a^6 C-35 a^5 b B+3 a^4 b^2 (21 A-2 C)+38 a^3 b^3 B-a^2 b^4 (86 A-3 C)-15 a b^5 B+35 A b^6\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}+\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{d}}{a}}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \sin (c+d x) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {a b \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+3 b \left (15 a^6 C-35 a^5 b B+3 a^4 b^2 (21 A-2 C)+38 a^3 b^3 B-a^2 b^4 (86 A-3 C)-15 a b^5 B+35 A b^6\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}+\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{d}}{a}}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \sin (c+d x) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {3 b \left (15 a^6 C-35 a^5 b B+3 a^4 b^2 (21 A-2 C)+38 a^3 b^3 B-a^2 b^4 (86 A-3 C)-15 a b^5 B+35 A b^6\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx+\frac {2 a b \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{d}}{b}+\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{d}}{a}}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \sin (c+d x) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{d}+\frac {\frac {2 a b \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{d}+\frac {6 b \left (15 a^6 C-35 a^5 b B+3 a^4 b^2 (21 A-2 C)+38 a^3 b^3 B-a^2 b^4 (86 A-3 C)-15 a b^5 B+35 A b^6\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{d (a+b)}}{b}}{a}}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\right )\) |
Input:
Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + b*Co s[c + d*x])^3,x]
Output:
Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(((A*b^2 - a*(b*B - a*C))*Sin[c + d* x])/(2*a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2) - (((7*A *b^4 + 9*a^3*b*B - 3*a*b^3*B - 5*a^4*C - a^2*b^2*(13*A + C))*Sin[c + d*x]) /(a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])) - ((2*(35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B + a^4*(8*A - 21*C) - a^2*b^2*(61*A - 3*C))*Sin[ c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) - (-(((6*(35*A*b^5 - 8*a^5*B + 29*a^3 *b^2*B - 15*a*b^4*B + 3*a^4*b*(8*A - 3*C) - a^2*b^3*(65*A - 3*C))*Elliptic E[(c + d*x)/2, 2])/d + ((2*a*b*(35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B + a^4*( 8*A - 21*C) - a^2*b^2*(61*A - 3*C))*EllipticF[(c + d*x)/2, 2])/d + (6*b*(3 5*A*b^6 - 35*a^5*b*B + 38*a^3*b^3*B - 15*a*b^5*B - a^2*b^4*(86*A - 3*C) + 3*a^4*b^2*(21*A - 2*C) + 15*a^6*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/((a + b)*d))/b)/a) + (6*(35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4* B + 3*a^4*b*(8*A - 3*C) - a^2*b^3*(65*A - 3*C))*Sin[c + d*x])/(a*d*Sqrt[Co s[c + d*x]]))/(3*a))/(2*a*(a^2 - b^2)))/(4*a*(a^2 - b^2)))
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]
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. \(2138\) vs. \(2(640)=1280\).
Time = 297.80 (sec) , antiderivative size = 2139, normalized size of antiderivative = 3.20
Input:
int((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(5/2)/(a+b*cos(d*x+c))^3,x, method=_RETURNVERBOSE)
Output:
-(-(-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^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)/(c os(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*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*E llipticF(cos(1/2*d*x+1/2*c),2^(1/2)))-2*(3*A*b-B*a)/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)))+2*(A*b^2-B*a*b+C*a^2)/a^2*(-1/2/a*b^2/(a^2-b^2)*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)/(2*cos(1/2*d*x+1/2*c)^2 *b+a-b)^2-3/4*b^2*(3*a^2-b^2)/a^2/(a^2-b^2)^2*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)/(2*cos(1/2*d*x+1/2*c)^2*b+a-b) -7/8/(a+b)/(a^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)*EllipticF(c os(1/2*d*x+1/2*c),2^(1/2))+1/4/(a+b)/(a^2-b^2)/a*(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))*b+3/8/(a+b)/(a^2-b^ 2)/a^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)*EllipticF(cos(1/2*d*x+1/2 *c),2^(1/2))*b^2-9/8*b/(a^2-b^2)^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos...
Timed out. \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\text {Timed out} \] Input:
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(5/2)/(a+b*cos(d*x+c) )^3,x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\text {Timed out} \] Input:
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x+c)**(5/2)/(a+b*cos(d*x+ c))**3,x)
Output:
Timed out
Timed out. \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\text {Timed out} \] Input:
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(5/2)/(a+b*cos(d*x+c) )^3,x, algorithm="maxima")
Output:
Timed out
\[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{\frac {5}{2}}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3}} \,d x } \] Input:
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(5/2)/(a+b*cos(d*x+c) )^3,x, algorithm="giac")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*sec(d*x + c)^(5/2)/(b*co s(d*x + c) + a)^3, x)
Timed out. \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\int \frac {{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3} \,d x \] Input:
int(((1/cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + b*cos(c + d*x))^3,x)
Output:
int(((1/cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + b*cos(c + d*x))^3, x)
\[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right ) \sec \left (d x +c \right )^{2}}{\cos \left (d x +c \right )^{3} b^{3}+3 \cos \left (d x +c \right )^{2} a \,b^{2}+3 \cos \left (d x +c \right ) a^{2} b +a^{3}}d x \right ) b +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{2}}{\cos \left (d x +c \right )^{3} b^{3}+3 \cos \left (d x +c \right )^{2} a \,b^{2}+3 \cos \left (d x +c \right ) a^{2} b +a^{3}}d x \right ) c +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )^{2}}{\cos \left (d x +c \right )^{3} b^{3}+3 \cos \left (d x +c \right )^{2} a \,b^{2}+3 \cos \left (d x +c \right ) a^{2} b +a^{3}}d x \right ) a \] Input:
int((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(5/2)/(a+b*cos(d*x+c))^3,x)
Output:
int((sqrt(sec(c + d*x))*cos(c + d*x)*sec(c + d*x)**2)/(cos(c + d*x)**3*b** 3 + 3*cos(c + d*x)**2*a*b**2 + 3*cos(c + d*x)*a**2*b + a**3),x)*b + int((s qrt(sec(c + d*x))*cos(c + d*x)**2*sec(c + d*x)**2)/(cos(c + d*x)**3*b**3 + 3*cos(c + d*x)**2*a*b**2 + 3*cos(c + d*x)*a**2*b + a**3),x)*c + int((sqrt (sec(c + d*x))*sec(c + d*x)**2)/(cos(c + d*x)**3*b**3 + 3*cos(c + d*x)**2* a*b**2 + 3*cos(c + d*x)*a**2*b + a**3),x)*a