Integrand size = 35, antiderivative size = 579 \[ \int \frac {A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)} \, dx=-\frac {\left (a^2 b^4 (145 A-192 C)-3 a^4 b^2 (25 A-187 C)-315 a^6 C-8 b^6 (5 A+3 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{20 b^5 \left (a^2-b^2\right )^2 d}+\frac {a \left (a^2 b^4 (33 A-64 C)-3 a^4 b^2 (5 A-43 C)-63 a^6 C-8 b^6 (3 A+C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{4 b^6 \left (a^2-b^2\right )^2 d}+\frac {a^2 \left (35 A b^6-a^2 b^4 (38 A-99 C)+15 a^4 b^2 (A-10 C)+63 a^6 C\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{4 (a-b)^2 b^6 (a+b)^3 d}-\frac {\left (A b^2+a^2 C\right ) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2 \sec ^{\frac {7}{2}}(c+d x)}+\frac {\left (7 A b^4-a^2 b^2 (A-15 C)-9 a^4 C\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x)) \sec ^{\frac {5}{2}}(c+d x)}+\frac {\left (a^2 b^2 (15 A-101 C)-b^4 (45 A-8 C)+63 a^4 C\right ) \sin (c+d x)}{20 b^3 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {a \left (b^4 (11 A-8 C)-5 a^2 b^2 (A-7 C)-21 a^4 C\right ) \sin (c+d x)}{4 b^4 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}} \] Output:
-1/20*(a^2*b^4*(145*A-192*C)-3*a^4*b^2*(25*A-187*C)-315*a^6*C-8*b^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)/b^5/(a^2-b^2)^2/d+1/4*a*(a^2*b^4*(33*A-64*C)-3*a^4*b^2*(5*A-43*C)-63*a^ 6*C-8*b^6*(3*A+C))*cos(d*x+c)^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2)) *sec(d*x+c)^(1/2)/b^6/(a^2-b^2)^2/d+1/4*a^2*(35*A*b^6-a^2*b^4*(38*A-99*C)+ 15*a^4*b^2*(A-10*C)+63*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-b)^2/b^6/(a+b)^3/d-1/2*(A*b^2+C* a^2)*sin(d*x+c)/b/(a^2-b^2)/d/(a+b*cos(d*x+c))^2/sec(d*x+c)^(7/2)+1/4*(7*A *b^4-a^2*b^2*(A-15*C)-9*a^4*C)*sin(d*x+c)/b^2/(a^2-b^2)^2/d/(a+b*cos(d*x+c ))/sec(d*x+c)^(5/2)+1/20*(a^2*b^2*(15*A-101*C)-b^4*(45*A-8*C)+63*a^4*C)*si n(d*x+c)/b^3/(a^2-b^2)^2/d/sec(d*x+c)^(3/2)+1/4*a*(b^4*(11*A-8*C)-5*a^2*b^ 2*(A-7*C)-21*a^4*C)*sin(d*x+c)/b^4/(a^2-b^2)^2/d/sec(d*x+c)^(1/2)
Time = 7.51 (sec) , antiderivative size = 925, normalized size of antiderivative = 1.60 \[ \int \frac {A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)} \, dx =\text {Too large to display} \] Input:
Integrate[(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^3*Sec[c + d*x]^(7/2 )),x]
Output:
((2*(25*a^4*A*b^2 - 35*a^2*A*b^4 + 40*A*b^6 + 105*a^6*C - 211*a^4*b^2*C + 112*a^2*b^4*C + 24*b^6*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*(40*a^3*A*b^3 - 160*a*A*b^5 + 168*a^5*b*C - 256 *a^3*b^3*C - 32*a*b^5*C)*Cos[c + d*x]^2*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 ])/(b*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((75*a^4*A*b^2 - 145*a^ 2*A*b^4 + 40*A*b^6 + 315*a^6*C - 561*a^4*b^2*C + 192*a^2*b^4*C + 24*b^6*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[ArcSin[Sqrt[Sec[c + d*x]]], -1]* Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), Arc Sin[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)))/(80*(a - b)^2* b^4*(a + b)^2*d) + (Sqrt[Sec[c + d*x]]*(-1/20*((35*a^4*A*b^2 - 65*a^2*A*b^ 4 + 75*a^6*C - 107*a^4*b^2*C + 4*a^2*b^4*C - 2*b^6*C)*Sin[c + d*x])/(b^5*( a^2 - b^2)^2) - (-(a^4*A*b^2*Sin[c + d*x]) - a^6*C*Sin[c + d*x])/(2*b^5...
Time = 4.04 (sec) , antiderivative size = 536, normalized size of antiderivative = 0.93, number of steps used = 23, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.657, Rules used = {3042, 4709, 3042, 3527, 27, 3042, 3526, 27, 3042, 3528, 27, 3042, 3528, 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.
| \(\displaystyle \int \frac {A+C \cos ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+C \cos (c+d x)^2}{\sec (c+d x)^{7/2} (a+b \cos (c+d x))^3}dx\) |
| \(\Big \downarrow \) 4709 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (C \cos ^2(c+d x)+A\right )}{(a+b \cos (c+d x))^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2} \left (C \sin \left (c+d x+\frac {\pi }{2}\right )^2+A\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 3527 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {\int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (-\left (\left (9 C a^2+5 A b^2-4 b^2 C\right ) \cos ^2(c+d x)\right )-4 a b (A+C) \cos (c+d x)+7 \left (C a^2+A b^2\right )\right )}{2 (a+b \cos (c+d x))^2}dx}{2 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {\int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (-\left (\left (9 C a^2+5 A b^2-4 b^2 C\right ) \cos ^2(c+d x)\right )-4 a b (A+C) \cos (c+d x)+7 \left (C a^2+A b^2\right )\right )}{(a+b \cos (c+d x))^2}dx}{4 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (\left (-9 C a^2-5 A b^2+4 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-4 a b (A+C) \sin \left (c+d x+\frac {\pi }{2}\right )+7 \left (C a^2+A b^2\right )\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{4 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\right )\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {-\frac {\int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (\left (63 C a^4+b^2 (15 A-101 C) a^2-b^4 (45 A-8 C)\right ) \cos ^2(c+d x)-4 a b \left (-C a^2+3 A b^2+4 b^2 C\right ) \cos (c+d x)+5 \left (-9 C a^4-b^2 (A-15 C) a^2+7 A b^4\right )\right )}{2 (a+b \cos (c+d x))}dx}{b \left (a^2-b^2\right )}-\frac {\left (-9 a^4 C-a^2 b^2 (A-15 C)+7 A b^4\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {-\frac {\int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (\left (63 C a^4+b^2 (15 A-101 C) a^2-b^4 (45 A-8 C)\right ) \cos ^2(c+d x)-4 a b \left (-C a^2+3 A b^2+4 b^2 C\right ) \cos (c+d x)+5 \left (-9 C a^4-b^2 (A-15 C) a^2+7 A b^4\right )\right )}{a+b \cos (c+d x)}dx}{2 b \left (a^2-b^2\right )}-\frac {\left (-9 a^4 C-a^2 b^2 (A-15 C)+7 A b^4\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {-\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (\left (63 C a^4+b^2 (15 A-101 C) a^2-b^4 (45 A-8 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-4 a b \left (-C a^2+3 A b^2+4 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+5 \left (-9 C a^4-b^2 (A-15 C) a^2+7 A b^4\right )\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{2 b \left (a^2-b^2\right )}-\frac {\left (-9 a^4 C-a^2 b^2 (A-15 C)+7 A b^4\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\right )\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {-\frac {\frac {2 \int \frac {\sqrt {\cos (c+d x)} \left (15 a \left (-21 C a^4-5 b^2 (A-7 C) a^2+b^4 (11 A-8 C)\right ) \cos ^2(c+d x)-4 b \left (9 C a^4-b^2 (5 A+18 C) a^2-2 b^4 (5 A+3 C)\right ) \cos (c+d x)+3 a \left (63 C a^4+b^2 (15 A-101 C) a^2-b^4 (45 A-8 C)\right )\right )}{2 (a+b \cos (c+d x))}dx}{5 b}+\frac {2 \left (63 a^4 C+a^2 b^2 (15 A-101 C)-b^4 (45 A-8 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{2 b \left (a^2-b^2\right )}-\frac {\left (-9 a^4 C-a^2 b^2 (A-15 C)+7 A b^4\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {-\frac {\frac {\int \frac {\sqrt {\cos (c+d x)} \left (15 a \left (-21 C a^4-5 b^2 (A-7 C) a^2+b^4 (11 A-8 C)\right ) \cos ^2(c+d x)-4 b \left (9 C a^4-b^2 (5 A+18 C) a^2-2 b^4 (5 A+3 C)\right ) \cos (c+d x)+3 a \left (63 C a^4+b^2 (15 A-101 C) a^2-b^4 (45 A-8 C)\right )\right )}{a+b \cos (c+d x)}dx}{5 b}+\frac {2 \left (63 a^4 C+a^2 b^2 (15 A-101 C)-b^4 (45 A-8 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{2 b \left (a^2-b^2\right )}-\frac {\left (-9 a^4 C-a^2 b^2 (A-15 C)+7 A b^4\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {-\frac {\frac {\int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (15 a \left (-21 C a^4-5 b^2 (A-7 C) a^2+b^4 (11 A-8 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-4 b \left (9 C a^4-b^2 (5 A+18 C) a^2-2 b^4 (5 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 a \left (63 C a^4+b^2 (15 A-101 C) a^2-b^4 (45 A-8 C)\right )\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{5 b}+\frac {2 \left (63 a^4 C+a^2 b^2 (15 A-101 C)-b^4 (45 A-8 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{2 b \left (a^2-b^2\right )}-\frac {\left (-9 a^4 C-a^2 b^2 (A-15 C)+7 A b^4\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\right )\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {-\frac {\frac {\frac {2 \int \frac {3 \left (5 \left (-21 C a^4-5 b^2 (A-7 C) a^2+b^4 (11 A-8 C)\right ) a^2+4 b \left (21 C a^4+b^2 (5 A-32 C) a^2-4 b^4 (5 A+C)\right ) \cos (c+d x) a-\left (-315 C a^6-3 b^2 (25 A-187 C) a^4+b^4 (145 A-192 C) a^2-8 b^6 (5 A+3 C)\right ) \cos ^2(c+d x)\right )}{2 \sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{3 b}+\frac {10 a \left (-21 a^4 C-5 a^2 b^2 (A-7 C)+b^4 (11 A-8 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{b d}}{5 b}+\frac {2 \left (63 a^4 C+a^2 b^2 (15 A-101 C)-b^4 (45 A-8 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{2 b \left (a^2-b^2\right )}-\frac {\left (-9 a^4 C-a^2 b^2 (A-15 C)+7 A b^4\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {-\frac {\frac {\frac {\int \frac {5 \left (-21 C a^4-5 b^2 (A-7 C) a^2+b^4 (11 A-8 C)\right ) a^2+4 b \left (21 C a^4+b^2 (5 A-32 C) a^2-4 b^4 (5 A+C)\right ) \cos (c+d x) a-\left (-315 C a^6-3 b^2 (25 A-187 C) a^4+b^4 (145 A-192 C) a^2-8 b^6 (5 A+3 C)\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}+\frac {10 a \left (-21 a^4 C-5 a^2 b^2 (A-7 C)+b^4 (11 A-8 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{b d}}{5 b}+\frac {2 \left (63 a^4 C+a^2 b^2 (15 A-101 C)-b^4 (45 A-8 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{2 b \left (a^2-b^2\right )}-\frac {\left (-9 a^4 C-a^2 b^2 (A-15 C)+7 A b^4\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {-\frac {\frac {\frac {\int \frac {5 \left (-21 C a^4-5 b^2 (A-7 C) a^2+b^4 (11 A-8 C)\right ) a^2+4 b \left (21 C a^4+b^2 (5 A-32 C) a^2-4 b^4 (5 A+C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+\left (315 C a^6+3 b^2 (25 A-187 C) a^4-b^4 (145 A-192 C) a^2+8 b^6 (5 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}+\frac {10 a \left (-21 a^4 C-5 a^2 b^2 (A-7 C)+b^4 (11 A-8 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{b d}}{5 b}+\frac {2 \left (63 a^4 C+a^2 b^2 (15 A-101 C)-b^4 (45 A-8 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{2 b \left (a^2-b^2\right )}-\frac {\left (-9 a^4 C-a^2 b^2 (A-15 C)+7 A b^4\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\right )\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {-\frac {\frac {\frac {-\frac {\left (-315 a^6 C-3 a^4 b^2 (25 A-187 C)+a^2 b^4 (145 A-192 C)-8 b^6 (5 A+3 C)\right ) \int \sqrt {\cos (c+d x)}dx}{b}-\frac {\int -\frac {5 \left (b \left (-21 C a^4-5 b^2 (A-7 C) a^2+b^4 (11 A-8 C)\right ) a^2+\left (-63 C a^6-3 b^2 (5 A-43 C) a^4+b^4 (33 A-64 C) a^2-8 b^6 (3 A+C)\right ) \cos (c+d x) a\right )}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}}{b}+\frac {10 a \left (-21 a^4 C-5 a^2 b^2 (A-7 C)+b^4 (11 A-8 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{b d}}{5 b}+\frac {2 \left (63 a^4 C+a^2 b^2 (15 A-101 C)-b^4 (45 A-8 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{2 b \left (a^2-b^2\right )}-\frac {\left (-9 a^4 C-a^2 b^2 (A-15 C)+7 A b^4\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {-\frac {\frac {\frac {\frac {5 \int \frac {b \left (-21 C a^4-5 b^2 (A-7 C) a^2+b^4 (11 A-8 C)\right ) a^2+\left (-63 C a^6-3 b^2 (5 A-43 C) a^4+b^4 (33 A-64 C) a^2-8 b^6 (3 A+C)\right ) \cos (c+d x) a}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}-\frac {\left (-315 a^6 C-3 a^4 b^2 (25 A-187 C)+a^2 b^4 (145 A-192 C)-8 b^6 (5 A+3 C)\right ) \int \sqrt {\cos (c+d x)}dx}{b}}{b}+\frac {10 a \left (-21 a^4 C-5 a^2 b^2 (A-7 C)+b^4 (11 A-8 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{b d}}{5 b}+\frac {2 \left (63 a^4 C+a^2 b^2 (15 A-101 C)-b^4 (45 A-8 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{2 b \left (a^2-b^2\right )}-\frac {\left (-9 a^4 C-a^2 b^2 (A-15 C)+7 A b^4\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {-\frac {\frac {\frac {\frac {5 \int \frac {b \left (-21 C a^4-5 b^2 (A-7 C) a^2+b^4 (11 A-8 C)\right ) a^2+\left (-63 C a^6-3 b^2 (5 A-43 C) a^4+b^4 (33 A-64 C) a^2-8 b^6 (3 A+C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}-\frac {\left (-315 a^6 C-3 a^4 b^2 (25 A-187 C)+a^2 b^4 (145 A-192 C)-8 b^6 (5 A+3 C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}}{b}+\frac {10 a \left (-21 a^4 C-5 a^2 b^2 (A-7 C)+b^4 (11 A-8 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{b d}}{5 b}+\frac {2 \left (63 a^4 C+a^2 b^2 (15 A-101 C)-b^4 (45 A-8 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{2 b \left (a^2-b^2\right )}-\frac {\left (-9 a^4 C-a^2 b^2 (A-15 C)+7 A b^4\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\right )\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {-\frac {\frac {\frac {\frac {5 \int \frac {b \left (-21 C a^4-5 b^2 (A-7 C) a^2+b^4 (11 A-8 C)\right ) a^2+\left (-63 C a^6-3 b^2 (5 A-43 C) a^4+b^4 (33 A-64 C) a^2-8 b^6 (3 A+C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}-\frac {2 \left (-315 a^6 C-3 a^4 b^2 (25 A-187 C)+a^2 b^4 (145 A-192 C)-8 b^6 (5 A+3 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b d}}{b}+\frac {10 a \left (-21 a^4 C-5 a^2 b^2 (A-7 C)+b^4 (11 A-8 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{b d}}{5 b}+\frac {2 \left (63 a^4 C+a^2 b^2 (15 A-101 C)-b^4 (45 A-8 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{2 b \left (a^2-b^2\right )}-\frac {\left (-9 a^4 C-a^2 b^2 (A-15 C)+7 A b^4\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\right )\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {-\frac {\frac {\frac {\frac {5 \left (\frac {a^2 \left (63 a^6 C+15 a^4 b^2 (A-10 C)-a^2 b^4 (38 A-99 C)+35 A b^6\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}+\frac {a \left (-63 a^6 C-3 a^4 b^2 (5 A-43 C)+a^2 b^4 (33 A-64 C)-8 b^6 (3 A+C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{b}\right )}{b}-\frac {2 \left (-315 a^6 C-3 a^4 b^2 (25 A-187 C)+a^2 b^4 (145 A-192 C)-8 b^6 (5 A+3 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b d}}{b}+\frac {10 a \left (-21 a^4 C-5 a^2 b^2 (A-7 C)+b^4 (11 A-8 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{b d}}{5 b}+\frac {2 \left (63 a^4 C+a^2 b^2 (15 A-101 C)-b^4 (45 A-8 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{2 b \left (a^2-b^2\right )}-\frac {\left (-9 a^4 C-a^2 b^2 (A-15 C)+7 A b^4\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {-\frac {\frac {\frac {\frac {5 \left (\frac {a^2 \left (63 a^6 C+15 a^4 b^2 (A-10 C)-a^2 b^4 (38 A-99 C)+35 A b^6\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}+\frac {a \left (-63 a^6 C-3 a^4 b^2 (5 A-43 C)+a^2 b^4 (33 A-64 C)-8 b^6 (3 A+C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}\right )}{b}-\frac {2 \left (-315 a^6 C-3 a^4 b^2 (25 A-187 C)+a^2 b^4 (145 A-192 C)-8 b^6 (5 A+3 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b d}}{b}+\frac {10 a \left (-21 a^4 C-5 a^2 b^2 (A-7 C)+b^4 (11 A-8 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{b d}}{5 b}+\frac {2 \left (63 a^4 C+a^2 b^2 (15 A-101 C)-b^4 (45 A-8 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{2 b \left (a^2-b^2\right )}-\frac {\left (-9 a^4 C-a^2 b^2 (A-15 C)+7 A b^4\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\right )\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {-\frac {\frac {\frac {\frac {5 \left (\frac {a^2 \left (63 a^6 C+15 a^4 b^2 (A-10 C)-a^2 b^4 (38 A-99 C)+35 A b^6\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}+\frac {2 a \left (-63 a^6 C-3 a^4 b^2 (5 A-43 C)+a^2 b^4 (33 A-64 C)-8 b^6 (3 A+C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{b d}\right )}{b}-\frac {2 \left (-315 a^6 C-3 a^4 b^2 (25 A-187 C)+a^2 b^4 (145 A-192 C)-8 b^6 (5 A+3 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b d}}{b}+\frac {10 a \left (-21 a^4 C-5 a^2 b^2 (A-7 C)+b^4 (11 A-8 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{b d}}{5 b}+\frac {2 \left (63 a^4 C+a^2 b^2 (15 A-101 C)-b^4 (45 A-8 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{2 b \left (a^2-b^2\right )}-\frac {\left (-9 a^4 C-a^2 b^2 (A-15 C)+7 A b^4\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\right )\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {\left (-9 a^4 C-a^2 b^2 (A-15 C)+7 A b^4\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {2 \left (63 a^4 C+a^2 b^2 (15 A-101 C)-b^4 (45 A-8 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}+\frac {\frac {10 a \left (-21 a^4 C-5 a^2 b^2 (A-7 C)+b^4 (11 A-8 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{b d}+\frac {\frac {5 \left (\frac {2 a \left (-63 a^6 C-3 a^4 b^2 (5 A-43 C)+a^2 b^4 (33 A-64 C)-8 b^6 (3 A+C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{b d}+\frac {2 a^2 \left (63 a^6 C+15 a^4 b^2 (A-10 C)-a^2 b^4 (38 A-99 C)+35 A b^6\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{b d (a+b)}\right )}{b}-\frac {2 \left (-315 a^6 C-3 a^4 b^2 (25 A-187 C)+a^2 b^4 (145 A-192 C)-8 b^6 (5 A+3 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b d}}{b}}{5 b}}{2 b \left (a^2-b^2\right )}}{4 b \left (a^2-b^2\right )}\right )\) |
Input:
Int[(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)),x]
Output:
Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(-1/2*((A*b^2 + a^2*C)*Cos[c + d*x]^ (7/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (-(((7*A*b^ 4 - a^2*b^2*(A - 15*C) - 9*a^4*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))) - ((2*(a^2*b^2*(15*A - 101*C) - b^4*(45*A - 8*C) + 63*a^4*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*b*d) + (((-2*(a^2* b^4*(145*A - 192*C) - 3*a^4*b^2*(25*A - 187*C) - 315*a^6*C - 8*b^6*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2])/(b*d) + (5*((2*a*(a^2*b^4*(33*A - 64*C) - 3*a^4*b^2*(5*A - 43*C) - 63*a^6*C - 8*b^6*(3*A + C))*EllipticF[(c + d*x)/ 2, 2])/(b*d) + (2*a^2*(35*A*b^6 - a^2*b^4*(38*A - 99*C) + 15*a^4*b^2*(A - 10*C) + 63*a^6*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(b*(a + b)*d) ))/b)/b + (10*a*(b^4*(11*A - 8*C) - 5*a^2*b^2*(A - 7*C) - 21*a^4*C)*Sqrt[C os[c + d*x]]*Sin[c + d*x])/(b*d))/(5*b))/(2*b*(a^2 - b^2)))/(4*b*(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[(-(c^2*C - B*c*d + A*d^2))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x ] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f *x]^2, x], x], x] /; FreeQ[{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] && GtQ[m, 0] && LtQ[n, -1]
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[(-(c^2*C + A*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^ 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A* d*(b*d*m + a*c*(n + 1)) + c*C*(b*c*m + a*d*(n + 1)) - (A*d*(a*d*(n + 2) - b *c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*( A*d^2*(m + n + 2) + C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
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[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a *c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*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] && GtQ[ m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 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. \(2465\) vs. \(2(550)=1100\).
Time = 8.91 (sec) , antiderivative size = 2466, normalized size of antiderivative = 4.26
Input:
int((A+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^3/sec(d*x+c)^(7/2),x,method=_RETUR NVERBOSE)
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^4*(A*b^2+C *a^2)/b^6*(-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(cos(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)*Elli pticF(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(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))+3/8*b^3/a^2/(a^2-b^2)^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))+9/8*b/(a^2-b^2)^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)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))- 3/8*b^3/a^2/(a^2-b^2)^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/...
Timed out. \[ \int \frac {A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:
integrate((A+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^3/sec(d*x+c)^(7/2),x, algori thm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:
integrate((A+C*cos(d*x+c)**2)/(a+b*cos(d*x+c))**3/sec(d*x+c)**(7/2),x)
Output:
Timed out
\[ \int \frac {A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)} \, dx=\int { \frac {C \cos \left (d x + c\right )^{2} + A}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate((A+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^3/sec(d*x+c)^(7/2),x, algori thm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + A)/((b*cos(d*x + c) + a)^3*sec(d*x + c)^(7/2 )), x)
\[ \int \frac {A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)} \, dx=\int { \frac {C \cos \left (d x + c\right )^{2} + A}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate((A+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^3/sec(d*x+c)^(7/2),x, algori thm="giac")
Output:
integrate((C*cos(d*x + c)^2 + A)/((b*cos(d*x + c) + a)^3*sec(d*x + c)^(7/2 )), x)
Timed out. \[ \int \frac {A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)} \, dx=\int \frac {C\,{\cos \left (c+d\,x\right )}^2+A}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3} \,d x \] Input:
int((A + C*cos(c + d*x)^2)/((1/cos(c + d*x))^(7/2)*(a + b*cos(c + d*x))^3) ,x)
Output:
int((A + C*cos(c + d*x)^2)/((1/cos(c + d*x))^(7/2)*(a + b*cos(c + d*x))^3) , x)
\[ \int \frac {A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)} \, dx=\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{4} b^{3}+3 \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{4} a \,b^{2}+3 \cos \left (d x +c \right ) \sec \left (d x +c \right )^{4} a^{2} b +\sec \left (d x +c \right )^{4} a^{3}}d x \right ) a +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}}{\cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{4} b^{3}+3 \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{4} a \,b^{2}+3 \cos \left (d x +c \right ) \sec \left (d x +c \right )^{4} a^{2} b +\sec \left (d x +c \right )^{4} a^{3}}d x \right ) c \] Input:
int((A+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^3/sec(d*x+c)^(7/2),x)
Output:
int(sqrt(sec(c + d*x))/(cos(c + d*x)**3*sec(c + d*x)**4*b**3 + 3*cos(c + d *x)**2*sec(c + d*x)**4*a*b**2 + 3*cos(c + d*x)*sec(c + d*x)**4*a**2*b + se c(c + d*x)**4*a**3),x)*a + int((sqrt(sec(c + d*x))*cos(c + d*x)**2)/(cos(c + d*x)**3*sec(c + d*x)**4*b**3 + 3*cos(c + d*x)**2*sec(c + d*x)**4*a*b**2 + 3*cos(c + d*x)*sec(c + d*x)**4*a**2*b + sec(c + d*x)**4*a**3),x)*c