Integrand size = 43, antiderivative size = 461 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx=\frac {\left (26 a^2 A b^2-15 A b^4-14 a^3 b B+6 a b^3 B-a^4 (3 A-8 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3 a^3 \left (a^2-b^2\right )^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (5 A b^2-2 a b B-a^2 (3 A-2 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{3 a^2 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}-\frac {(5 A b-2 a B) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{a^3 d \sqrt {a+b \cos (c+d x)}}-\frac {b \left (5 A b^2-2 a b B-a^2 (3 A-2 C)\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac {b \left (26 a^2 A b^2-15 A b^4-14 a^3 b B+6 a b^3 B-a^4 (3 A-8 C)\right ) \sin (c+d x)}{3 a^3 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}+\frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}} \]
-1/3*b*(5*A*b^2-2*B*a*b-a^2*(3*A-2*C))*sin(d*x+c)/a^2/(a^2-b^2)/d/(a+b*cos (d*x+c))^(3/2)-1/3*b*(26*A*a^2*b^2-15*A*b^4-14*B*a^3*b+6*B*a*b^3-a^4*(3*A- 8*C))*sin(d*x+c)/a^3/(a^2-b^2)^2/d/(a+b*cos(d*x+c))^(1/2)+1/3*(26*A*a^2*b^ 2-15*A*b^4-14*B*a^3*b+6*B*a*b^3-a^4*(3*A-8*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2 )/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2)*(b/(a+b))^(1/2)) *(a+b*cos(d*x+c))^(1/2)/a^3/(a^2-b^2)^2/d/((a+b*cos(d*x+c))/(a+b))^(1/2)-1 /3*(5*A*b^2-2*B*a*b-a^2*(3*A-2*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d* x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2)*(b/(a+b))^(1/2))*((a+b*cos(d *x+c))/(a+b))^(1/2)/a^2/(a^2-b^2)/d/(a+b*cos(d*x+c))^(1/2)-(5*A*b-2*B*a)*( cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticPi(sin(1/2*d*x+1/2* c),2,2^(1/2)*(b/(a+b))^(1/2))*((a+b*cos(d*x+c))/(a+b))^(1/2)/a^3/d/(a+b*co s(d*x+c))^(1/2)+A*tan(d*x+c)/a/d/(a+b*cos(d*x+c))^(3/2)
Result contains complex when optimal does not.
Time = 9.88 (sec) , antiderivative size = 915, normalized size of antiderivative = 1.98 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx=\frac {\cos ^2(c+d x) \left (C+B \sec (c+d x)+A \sec ^2(c+d x)\right ) \left (\frac {2 \left (36 a^3 A b^2-20 a A b^4-24 a^4 b B+8 a^2 b^3 B+12 a^5 C+4 a^3 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}+\frac {2 \left (-33 a^4 A b+86 a^2 A b^3-45 A b^5+12 a^5 B-38 a^3 b^2 B+18 a b^4 B+8 a^4 b C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}-\frac {2 i \left (-3 a^4 A b+26 a^2 A b^3-15 A b^5-14 a^3 b^2 B+6 a b^4 B+8 a^4 b C\right ) \sqrt {\frac {b-b \cos (c+d x)}{a+b}} \sqrt {-\frac {b+b \cos (c+d x)}{a-b}} \cos (2 (c+d x)) \left (2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )-b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sin (c+d x)}{a \sqrt {-\frac {1}{a+b}} \sqrt {1-\cos ^2(c+d x)} \sqrt {-\frac {a^2-b^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2}{b^2}} \left (2 a^2-b^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2\right )}\right )}{6 a^3 (-a+b)^2 (a+b)^2 d (2 A+C+2 B \cos (c+d x)+C \cos (2 c+2 d x))}+\frac {\cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \left (C+B \sec (c+d x)+A \sec ^2(c+d x)\right ) \left (-\frac {4 \left (A b^3 \sin (c+d x)-a b^2 B \sin (c+d x)+a^2 b C \sin (c+d x)\right )}{3 a^2 \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {4 \left (10 a^2 A b^3 \sin (c+d x)-6 A b^5 \sin (c+d x)-7 a^3 b^2 B \sin (c+d x)+3 a b^4 B \sin (c+d x)+4 a^4 b C \sin (c+d x)\right )}{3 a^3 \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}+\frac {2 A \tan (c+d x)}{a^3}\right )}{d (2 A+C+2 B \cos (c+d x)+C \cos (2 c+2 d x))} \]
Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b* Cos[c + d*x])^(5/2),x]
(Cos[c + d*x]^2*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*((2*(36*a^3*A*b^2 - 20*a*A*b^4 - 24*a^4*b*B + 8*a^2*b^3*B + 12*a^5*C + 4*a^3*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(-33*a^4*A*b + 86*a^2*A*b^3 - 45*A*b^5 + 12*a^5*B - 38*a^3*b^2*B + 18*a*b^4*B + 8*a^4*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]* EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2* I)*(-3*a^4*A*b + 26*a^2*A*b^3 - 15*A*b^5 - 14*a^3*b^2*B + 6*a*b^4*B + 8*a^ 4*b*C)*Sqrt[(b - b*Cos[c + d*x])/(a + b)]*Sqrt[-((b + b*Cos[c + d*x])/(a - b))]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1) ]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh [Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] - b*Ellip ticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)]))*Sin[c + d*x])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Cos[c + d *x]^2]*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Cos[c + d*x]) + (a + b*Cos[c + d*x]) ^2)/b^2)]*(2*a^2 - b^2 - 4*a*(a + b*Cos[c + d*x]) + 2*(a + b*Cos[c + d*x]) ^2))))/(6*a^3*(-a + b)^2*(a + b)^2*d*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2 *c + 2*d*x])) + (Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*(C + B*Sec[c + d* x] + A*Sec[c + d*x]^2)*((-4*(A*b^3*Sin[c + d*x] - a*b^2*B*Sin[c + d*x] + a ^2*b*C*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) - (4*(10* a^2*A*b^3*Sin[c + d*x] - 6*A*b^5*Sin[c + d*x] - 7*a^3*b^2*B*Sin[c + d*x...
Time = 3.96 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.07, number of steps used = 24, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.558, Rules used = {3042, 3534, 27, 3042, 3534, 27, 3042, 3534, 27, 3042, 3538, 25, 3042, 3134, 3042, 3132, 3481, 3042, 3142, 3042, 3140, 3286, 3042, 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.
| \(\displaystyle \int \frac {\sec ^2(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\int -\frac {\left (-3 A b \cos ^2(c+d x)-2 a C \cos (c+d x)+5 A b-2 a B\right ) \sec (c+d x)}{2 (a+b \cos (c+d x))^{5/2}}dx}{a}+\frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\int \frac {\left (-3 A b \cos ^2(c+d x)-2 a C \cos (c+d x)+5 A b-2 a B\right ) \sec (c+d x)}{(a+b \cos (c+d x))^{5/2}}dx}{2 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\int \frac {-3 A b \sin \left (c+d x+\frac {\pi }{2}\right )^2-2 a C \sin \left (c+d x+\frac {\pi }{2}\right )+5 A b-2 a B}{\sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx}{2 a}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {2 \int \frac {\left (b \left (-\left ((3 A-2 C) a^2\right )-2 b B a+5 A b^2\right ) \cos ^2(c+d x)-6 a \left (A b^2-a (b B-a C)\right ) \cos (c+d x)+3 \left (a^2-b^2\right ) (5 A b-2 a B)\right ) \sec (c+d x)}{2 (a+b \cos (c+d x))^{3/2}}dx}{3 a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (-\left (a^2 (3 A-2 C)\right )-2 a b B+5 A b^2\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\int \frac {\left (b \left (-\left ((3 A-2 C) a^2\right )-2 b B a+5 A b^2\right ) \cos ^2(c+d x)-6 a \left (A b^2-a (b B-a C)\right ) \cos (c+d x)+3 \left (a^2-b^2\right ) (5 A b-2 a B)\right ) \sec (c+d x)}{(a+b \cos (c+d x))^{3/2}}dx}{3 a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (-\left (a^2 (3 A-2 C)\right )-2 a b B+5 A b^2\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\int \frac {b \left (-\left ((3 A-2 C) a^2\right )-2 b B a+5 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-6 a \left (A b^2-a (b B-a C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 \left (a^2-b^2\right ) (5 A b-2 a B)}{\sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{3 a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (-\left (a^2 (3 A-2 C)\right )-2 a b B+5 A b^2\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {2 \int \frac {\left (3 (5 A b-2 a B) \left (a^2-b^2\right )^2-b \left (-\left ((3 A-8 C) a^4\right )-14 b B a^3+26 A b^2 a^2+6 b^3 B a-15 A b^4\right ) \cos ^2(c+d x)+2 a \left (-3 C a^4+6 b B a^3-b^2 (9 A+C) a^2-2 b^3 B a+5 A b^4\right ) \cos (c+d x)\right ) \sec (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx}{a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (-\left (a^2 (3 A-2 C)\right )-2 a b B+5 A b^2\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\int \frac {\left (3 (5 A b-2 a B) \left (a^2-b^2\right )^2-b \left (-\left ((3 A-8 C) a^4\right )-14 b B a^3+26 A b^2 a^2+6 b^3 B a-15 A b^4\right ) \cos ^2(c+d x)+2 a \left (-3 C a^4+6 b B a^3-b^2 (9 A+C) a^2-2 b^3 B a+5 A b^4\right ) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (-\left (a^2 (3 A-2 C)\right )-2 a b B+5 A b^2\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\int \frac {3 (5 A b-2 a B) \left (a^2-b^2\right )^2-b \left (-\left ((3 A-8 C) a^4\right )-14 b B a^3+26 A b^2 a^2+6 b^3 B a-15 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 a \left (-3 C a^4+6 b B a^3-b^2 (9 A+C) a^2-2 b^3 B a+5 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (-\left (a^2 (3 A-2 C)\right )-2 a b B+5 A b^2\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {-\frac {\int -\frac {\left (3 b (5 A b-2 a B) \left (a^2-b^2\right )^2+a b \left (-\left ((3 A-2 C) a^2\right )-2 b B a+5 A b^2\right ) \cos (c+d x) \left (a^2-b^2\right )\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\left (\left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right ) \int \sqrt {a+b \cos (c+d x)}dx\right )}{a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (-\left (a^2 (3 A-2 C)\right )-2 a b B+5 A b^2\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\frac {\int \frac {\left (3 b (5 A b-2 a B) \left (a^2-b^2\right )^2+a b \left (-\left ((3 A-2 C) a^2\right )-2 b B a+5 A b^2\right ) \cos (c+d x) \left (a^2-b^2\right )\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right ) \int \sqrt {a+b \cos (c+d x)}dx}{a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (-\left (a^2 (3 A-2 C)\right )-2 a b B+5 A b^2\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\frac {\int \frac {3 b (5 A b-2 a B) \left (a^2-b^2\right )^2+a b \left (-\left ((3 A-2 C) a^2\right )-2 b B a+5 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a^2-b^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (-\left (a^2 (3 A-2 C)\right )-2 a b B+5 A b^2\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\frac {\int \frac {3 b (5 A b-2 a B) \left (a^2-b^2\right )^2+a b \left (-\left ((3 A-2 C) a^2\right )-2 b B a+5 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a^2-b^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (-\left (a^2 (3 A-2 C)\right )-2 a b B+5 A b^2\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\frac {\int \frac {3 b (5 A b-2 a B) \left (a^2-b^2\right )^2+a b \left (-\left ((3 A-2 C) a^2\right )-2 b B a+5 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a^2-b^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (-\left (a^2 (3 A-2 C)\right )-2 a b B+5 A b^2\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\frac {\int \frac {3 b (5 A b-2 a B) \left (a^2-b^2\right )^2+a b \left (-\left ((3 A-2 C) a^2\right )-2 b B a+5 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a^2-b^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (-\left (a^2 (3 A-2 C)\right )-2 a b B+5 A b^2\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\frac {a b \left (a^2-b^2\right ) \left (-\left (a^2 (3 A-2 C)\right )-2 a b B+5 A b^2\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx+3 b \left (a^2-b^2\right )^2 (5 A b-2 a B) \int \frac {\sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\frac {2 \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (-\left (a^2 (3 A-2 C)\right )-2 a b B+5 A b^2\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\frac {a b \left (a^2-b^2\right ) \left (-\left (a^2 (3 A-2 C)\right )-2 a b B+5 A b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+3 b \left (a^2-b^2\right )^2 (5 A b-2 a B) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (-\left (a^2 (3 A-2 C)\right )-2 a b B+5 A b^2\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\frac {\frac {a b \left (a^2-b^2\right ) \left (-\left (a^2 (3 A-2 C)\right )-2 a b B+5 A b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}+3 b \left (a^2-b^2\right )^2 (5 A b-2 a B) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (-\left (a^2 (3 A-2 C)\right )-2 a b B+5 A b^2\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\frac {\frac {a b \left (a^2-b^2\right ) \left (-\left (a^2 (3 A-2 C)\right )-2 a b B+5 A b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}+3 b \left (a^2-b^2\right )^2 (5 A b-2 a B) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (-\left (a^2 (3 A-2 C)\right )-2 a b B+5 A b^2\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\frac {3 b \left (a^2-b^2\right )^2 (5 A b-2 a B) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a b \left (a^2-b^2\right ) \left (-\left (a^2 (3 A-2 C)\right )-2 a b B+5 A b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (-\left (a^2 (3 A-2 C)\right )-2 a b B+5 A b^2\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\frac {\frac {3 b \left (a^2-b^2\right )^2 (5 A b-2 a B) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {\sec (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}+\frac {2 a b \left (a^2-b^2\right ) \left (-\left (a^2 (3 A-2 C)\right )-2 a b B+5 A b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (-\left (a^2 (3 A-2 C)\right )-2 a b B+5 A b^2\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\frac {\frac {3 b \left (a^2-b^2\right )^2 (5 A b-2 a B) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}+\frac {2 a b \left (a^2-b^2\right ) \left (-\left (a^2 (3 A-2 C)\right )-2 a b B+5 A b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (-\left (a^2 (3 A-2 C)\right )-2 a b B+5 A b^2\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {2 b \sin (c+d x) \left (-\left (a^2 (3 A-2 C)\right )-2 a b B+5 A b^2\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}+\frac {\frac {2 b \sin (c+d x) \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}+\frac {\frac {\frac {2 a b \left (a^2-b^2\right ) \left (-\left (a^2 (3 A-2 C)\right )-2 a b B+5 A b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}+\frac {6 b \left (a^2-b^2\right )^2 (5 A b-2 a B) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 \left (-\left (a^4 (3 A-8 C)\right )-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}}{2 a}\) |
-1/2*((2*b*(5*A*b^2 - 2*a*b*B - a^2*(3*A - 2*C))*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (((-2*(26*a^2*A*b^2 - 15*A*b^4 - 14* a^3*b*B + 6*a*b^3*B - a^4*(3*A - 8*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[ (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + ((2* a*b*(a^2 - b^2)*(5*A*b^2 - 2*a*b*B - a^2*(3*A - 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (6*b*(a^2 - b^2)^2*(5*A*b - 2*a*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d* x]]))/b)/(a*(a^2 - b^2)) + (2*b*(26*a^2*A*b^2 - 15*A*b^4 - 14*a^3*b*B + 6* a*b^3*B - a^4*(3*A - 8*C))*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]))/(3*a*(a^2 - b^2)))/a + (A*Tan[c + d*x])/(a*d*(a + b*Cos[c + d*x ])^(3/2))
3.11.58.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 0]
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[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt [c + d*Sin[e + f*x]] Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/(c + d))*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a* d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !GtQ[c + d, 0]
Int[(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]))/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[ B/d Int[(a + b*Sin[e + f*x])^m, x], x] - Simp[(B*c - A*d)/d Int[(a + b* Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int [(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A *b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ [n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) | | EqQ[a, 0])))
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[C/(b*d) Int[Sqrt[a + b*Sin[e + f*x]], x] , x] - Simp[1/(b*d) Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[ e + f*x], x]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; Fre eQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0 ] && NeQ[c^2 - d^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1351\) vs. \(2(522)=1044\).
Time = 12.43 (sec) , antiderivative size = 1352, normalized size of antiderivative = 2.93
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1352\) |
| parts | \(\text {Expression too large to display}\) | \(2664\) |
int((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^2/(a+b*cos(d*x+c))^(5/2),x, method=_RETURNVERBOSE)
-(-(-2*cos(1/2*d*x+1/2*c)^2*b-a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*A/a^2*(- cos(1/2*d*x+1/2*c)/a*(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2 )^(1/2)/(-1+2*cos(1/2*d*x+1/2*c)^2)+1/2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*c os(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*s in(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2) )-1/2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^ (1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*Ellipti cE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))+1/2/a*(sin(1/2*d*x+1/2*c)^2)^(1/ 2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4 +(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*b*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a -b))^(1/2))+1/2/a*b*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2* b+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2) ^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1/2)))-2*(-2*A*b+B*a) /a^3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^( 1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*Elliptic Pi(cos(1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1/2))-2*b*(2*A*b-B*a)/a^3/sin(1/2*d* x+1/2*c)^2/(2*b*sin(1/2*d*x+1/2*c)^2-a-b)/(a^2-b^2)*(-2*b*sin(1/2*d*x+1/2* c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*b*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*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a +b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a-(si...
Timed out. \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx=\text {Timed out} \]
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^2/(a+b*cos(d*x+c))^(5 /2),x, algorithm="fricas")
Timed out. \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{2}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^2/(a+b*cos(d*x+c))^(5 /2),x, algorithm="maxima")
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*sec(d*x + c)^2/(b*cos(d* x + c) + a)^(5/2), x)
\[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{2}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^2/(a+b*cos(d*x+c))^(5 /2),x, algorithm="giac")
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*sec(d*x + c)^2/(b*cos(d* x + c) + a)^(5/2), x)
Timed out. \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx=\int \frac {C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A}{{\cos \left (c+d\,x\right )}^2\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \]