Integrand size = 35, antiderivative size = 416 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx=\frac {\left (26 a^2 A b^2-15 A b^4-a^4 (3 A-8 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3 a^3 \left (a^2-b^2\right )^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (5 A b^2-a^2 (3 A-2 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{3 a^2 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}-\frac {5 A b \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{a^3 d \sqrt {a+b \cos (c+d x)}}-\frac {b \left (5 A b^2-a^2 (3 A-2 C)\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac {b \left (26 a^2 A b^2-15 A b^4-a^4 (3 A-8 C)\right ) \sin (c+d x)}{3 a^3 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}+\frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}} \]
-1/3*b*(5*A*b^2-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-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-a^4*(3*A-8*C))*(co s(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*co s(d*x+c))/(a+b))^(1/2)-1/3*(5*A*b^2-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*(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*cos(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 = 8.32 (sec) , antiderivative size = 786, normalized size of antiderivative = 1.89 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx=\frac {\cos ^2(c+d x) \left (C+A \sec ^2(c+d x)\right ) \left (\frac {2 \left (36 a^3 A b^2-20 a A b^4+12 a^5 C+4 a^3 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}+\frac {2 \left (-33 a^4 A b+86 a^2 A b^3-45 A b^5+8 a^4 b C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}-\frac {2 i \left (-3 a^4 A b+26 a^2 A b^3-15 A b^5+8 a^4 b C\right ) \sqrt {\frac {b-b \cos (c+d x)}{a+b}} \sqrt {-\frac {b+b \cos (c+d x)}{a-b}} \cos (2 (c+d x)) \left (2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )-b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sin (c+d x)}{a \sqrt {-\frac {1}{a+b}} \sqrt {1-\cos ^2(c+d x)} \sqrt {-\frac {a^2-b^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2}{b^2}} \left (2 a^2-b^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2\right )}\right )}{6 a^3 (-a+b)^2 (a+b)^2 d (2 A+C+C \cos (2 c+2 d x))}+\frac {\cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \left (C+A \sec ^2(c+d x)\right ) \left (-\frac {4 \left (A b^3 \sin (c+d x)+a^2 b C \sin (c+d x)\right )}{3 a^2 \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {8 \left (5 a^2 A b^3 \sin (c+d x)-3 A b^5 \sin (c+d x)+2 a^4 b C \sin (c+d x)\right )}{3 a^3 \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}+\frac {2 A \tan (c+d x)}{a^3}\right )}{d (2 A+C+C \cos (2 c+2 d x))} \]
(Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2)*((2*(36*a^3*A*b^2 - 20*a*A*b^4 + 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 + 8*a^4*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticP i[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 + 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*EllipticPi[(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 + C*Cos[2*c + 2*d*x])) + (Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x] ]*(C + A*Sec[c + d*x]^2)*((-4*(A*b^3*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) - (8*(5*a^2*A*b^3*Sin[c + d*x] - 3*A*b^5*Sin[c + d*x] + 2*a^4*b*C*Sin[c + d*x]))/(3*a^3*(a^2 - b^2)^2*(a + b*Cos[c + d*x])) + (2*A*Tan[c + d*x])/a^3))/(d*(2*A + C + C*Cos[2*c + 2 *d*x]))
Time = 3.62 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.08, number of steps used = 24, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.686, Rules used = {3042, 3535, 27, 3042, 3534, 27, 3042, 3534, 27, 3042, 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.
| \(\displaystyle \int \frac {\sec ^2(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 3535 |
| \(\displaystyle \frac {\int -\frac {\left (-3 A b \cos ^2(c+d x)-2 a C \cos (c+d x)+5 A b\right ) \sec (c+d x)}{2 (a+b \cos (c+d x))^{5/2}}dx}{a}+\frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\int \frac {\left (-3 A b \cos ^2(c+d x)-2 a C \cos (c+d x)+5 A b\right ) \sec (c+d x)}{(a+b \cos (c+d x))^{5/2}}dx}{2 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\int \frac {-3 A b \sin \left (c+d x+\frac {\pi }{2}\right )^2-2 a C \sin \left (c+d x+\frac {\pi }{2}\right )+5 A b}{\sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx}{2 a}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {2 \int \frac {\left (b \left (5 A b^2-a^2 (3 A-2 C)\right ) \cos ^2(c+d x)-6 a \left (C a^2+A b^2\right ) \cos (c+d x)+15 A b \left (a^2-b^2\right )\right ) \sec (c+d x)}{2 (a+b \cos (c+d x))^{3/2}}dx}{3 a \left (a^2-b^2\right )}+\frac {2 b \left (5 A b^2-a^2 (3 A-2 C)\right ) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\int \frac {\left (b \left (5 A b^2-a^2 (3 A-2 C)\right ) \cos ^2(c+d x)-6 a \left (C a^2+A b^2\right ) \cos (c+d x)+15 A b \left (a^2-b^2\right )\right ) \sec (c+d x)}{(a+b \cos (c+d x))^{3/2}}dx}{3 a \left (a^2-b^2\right )}+\frac {2 b \left (5 A b^2-a^2 (3 A-2 C)\right ) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\int \frac {b \left (5 A b^2-a^2 (3 A-2 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-6 a \left (C a^2+A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+15 A b \left (a^2-b^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{3 a \left (a^2-b^2\right )}+\frac {2 b \left (5 A b^2-a^2 (3 A-2 C)\right ) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {2 \int \frac {\left (15 A b \left (a^2-b^2\right )^2-b \left (-\left ((3 A-8 C) a^4\right )+26 A b^2 a^2-15 A b^4\right ) \cos ^2(c+d x)+2 a \left (-3 C a^4-b^2 (9 A+C) a^2+5 A b^4\right ) \cos (c+d x)\right ) \sec (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx}{a \left (a^2-b^2\right )}+\frac {2 b \left (-\left (a^4 (3 A-8 C)\right )+26 a^2 A b^2-15 A b^4\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \left (5 A b^2-a^2 (3 A-2 C)\right ) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\int \frac {\left (15 A b \left (a^2-b^2\right )^2-b \left (-\left ((3 A-8 C) a^4\right )+26 A b^2 a^2-15 A b^4\right ) \cos ^2(c+d x)+2 a \left (-3 C a^4-b^2 (9 A+C) a^2+5 A b^4\right ) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{a \left (a^2-b^2\right )}+\frac {2 b \left (-\left (a^4 (3 A-8 C)\right )+26 a^2 A b^2-15 A b^4\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \left (5 A b^2-a^2 (3 A-2 C)\right ) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\int \frac {15 A b \left (a^2-b^2\right )^2-b \left (-\left ((3 A-8 C) a^4\right )+26 A b^2 a^2-15 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 a \left (-3 C a^4-b^2 (9 A+C) a^2+5 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \left (a^2-b^2\right )}+\frac {2 b \left (-\left (a^4 (3 A-8 C)\right )+26 a^2 A b^2-15 A b^4\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \left (5 A b^2-a^2 (3 A-2 C)\right ) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {-\frac {\int -\frac {\left (15 A b^2 \left (a^2-b^2\right )^2+a b \left (5 A b^2-a^2 (3 A-2 C)\right ) \cos (c+d x) \left (a^2-b^2\right )\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\left (\left (-\left (a^4 (3 A-8 C)\right )+26 a^2 A b^2-15 A b^4\right ) \int \sqrt {a+b \cos (c+d x)}dx\right )}{a \left (a^2-b^2\right )}+\frac {2 b \left (-\left (a^4 (3 A-8 C)\right )+26 a^2 A b^2-15 A b^4\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \left (5 A b^2-a^2 (3 A-2 C)\right ) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\frac {\int \frac {\left (15 A b^2 \left (a^2-b^2\right )^2+a b \left (5 A b^2-a^2 (3 A-2 C)\right ) \cos (c+d x) \left (a^2-b^2\right )\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\left (-\left (a^4 (3 A-8 C)\right )+26 a^2 A b^2-15 A b^4\right ) \int \sqrt {a+b \cos (c+d x)}dx}{a \left (a^2-b^2\right )}+\frac {2 b \left (-\left (a^4 (3 A-8 C)\right )+26 a^2 A b^2-15 A b^4\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \left (5 A b^2-a^2 (3 A-2 C)\right ) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\frac {\int \frac {15 A b^2 \left (a^2-b^2\right )^2+a b \left (5 A b^2-a^2 (3 A-2 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a^2-b^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\left (-\left (a^4 (3 A-8 C)\right )+26 a^2 A b^2-15 A b^4\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a \left (a^2-b^2\right )}+\frac {2 b \left (-\left (a^4 (3 A-8 C)\right )+26 a^2 A b^2-15 A b^4\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \left (5 A b^2-a^2 (3 A-2 C)\right ) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\frac {\int \frac {15 A b^2 \left (a^2-b^2\right )^2+a b \left (5 A b^2-a^2 (3 A-2 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a^2-b^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (-\left (a^4 (3 A-8 C)\right )+26 a^2 A b^2-15 A b^4\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 b \left (-\left (a^4 (3 A-8 C)\right )+26 a^2 A b^2-15 A b^4\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \left (5 A b^2-a^2 (3 A-2 C)\right ) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\frac {\int \frac {15 A b^2 \left (a^2-b^2\right )^2+a b \left (5 A b^2-a^2 (3 A-2 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a^2-b^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (-\left (a^4 (3 A-8 C)\right )+26 a^2 A b^2-15 A b^4\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 b \left (-\left (a^4 (3 A-8 C)\right )+26 a^2 A b^2-15 A b^4\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \left (5 A b^2-a^2 (3 A-2 C)\right ) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\frac {\int \frac {15 A b^2 \left (a^2-b^2\right )^2+a b \left (5 A b^2-a^2 (3 A-2 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a^2-b^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \left (-\left (a^4 (3 A-8 C)\right )+26 a^2 A b^2-15 A b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 b \left (-\left (a^4 (3 A-8 C)\right )+26 a^2 A b^2-15 A b^4\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \left (5 A b^2-a^2 (3 A-2 C)\right ) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\frac {a b \left (a^2-b^2\right ) \left (5 A b^2-a^2 (3 A-2 C)\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx+15 A b^2 \left (a^2-b^2\right )^2 \int \frac {\sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\frac {2 \left (-\left (a^4 (3 A-8 C)\right )+26 a^2 A b^2-15 A b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 b \left (-\left (a^4 (3 A-8 C)\right )+26 a^2 A b^2-15 A b^4\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \left (5 A b^2-a^2 (3 A-2 C)\right ) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\frac {a b \left (a^2-b^2\right ) \left (5 A b^2-a^2 (3 A-2 C)\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+15 A b^2 \left (a^2-b^2\right )^2 \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \left (-\left (a^4 (3 A-8 C)\right )+26 a^2 A b^2-15 A b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 b \left (-\left (a^4 (3 A-8 C)\right )+26 a^2 A b^2-15 A b^4\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \left (5 A b^2-a^2 (3 A-2 C)\right ) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\frac {\frac {a b \left (a^2-b^2\right ) \left (5 A b^2-a^2 (3 A-2 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}+15 A b^2 \left (a^2-b^2\right )^2 \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \left (-\left (a^4 (3 A-8 C)\right )+26 a^2 A b^2-15 A b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 b \left (-\left (a^4 (3 A-8 C)\right )+26 a^2 A b^2-15 A b^4\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \left (5 A b^2-a^2 (3 A-2 C)\right ) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\frac {\frac {a b \left (a^2-b^2\right ) \left (5 A b^2-a^2 (3 A-2 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}+15 A b^2 \left (a^2-b^2\right )^2 \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \left (-\left (a^4 (3 A-8 C)\right )+26 a^2 A b^2-15 A b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 b \left (-\left (a^4 (3 A-8 C)\right )+26 a^2 A b^2-15 A b^4\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \left (5 A b^2-a^2 (3 A-2 C)\right ) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\frac {15 A b^2 \left (a^2-b^2\right )^2 \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a b \left (a^2-b^2\right ) \left (5 A b^2-a^2 (3 A-2 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 \left (-\left (a^4 (3 A-8 C)\right )+26 a^2 A b^2-15 A b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 b \left (-\left (a^4 (3 A-8 C)\right )+26 a^2 A b^2-15 A b^4\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \left (5 A b^2-a^2 (3 A-2 C)\right ) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\frac {\frac {15 A b^2 \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {\sec (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}+\frac {2 a b \left (a^2-b^2\right ) \left (5 A b^2-a^2 (3 A-2 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 \left (-\left (a^4 (3 A-8 C)\right )+26 a^2 A b^2-15 A b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 b \left (-\left (a^4 (3 A-8 C)\right )+26 a^2 A b^2-15 A b^4\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \left (5 A b^2-a^2 (3 A-2 C)\right ) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\frac {\frac {15 A b^2 \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}+\frac {2 a b \left (a^2-b^2\right ) \left (5 A b^2-a^2 (3 A-2 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 \left (-\left (a^4 (3 A-8 C)\right )+26 a^2 A b^2-15 A b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 b \left (-\left (a^4 (3 A-8 C)\right )+26 a^2 A b^2-15 A b^4\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \left (5 A b^2-a^2 (3 A-2 C)\right ) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {2 b \left (5 A b^2-a^2 (3 A-2 C)\right ) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}+\frac {\frac {2 b \left (-\left (a^4 (3 A-8 C)\right )+26 a^2 A b^2-15 A b^4\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}+\frac {\frac {\frac {2 a b \left (a^2-b^2\right ) \left (5 A b^2-a^2 (3 A-2 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}+\frac {30 A b^2 \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 \left (-\left (a^4 (3 A-8 C)\right )+26 a^2 A b^2-15 A b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}}{2 a}\) |
-1/2*((2*b*(5*A*b^2 - 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 - a^4*(3*A - 8* C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqr t[(a + b*Cos[c + d*x])/(a + b)]) + ((2*a*b*(a^2 - b^2)*(5*A*b^2 - 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]]) + (30*A*b^2*(a^2 - b^2)^2*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqr t[a + b*Cos[c + d*x]]))/b)/(a*(a^2 - b^2)) + (2*b*(26*a^2*A*b^2 - 15*A*b^4 - 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 ))
3.7.68.3.1 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_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 + a^2*C))*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*S in[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin [e + f*x])^n*Simp[a*(m + 1)*(b*c - a*d)*(A + C) + d*(A*b^2 + a^2*C)*(m + n + 2) - (c*(A*b^2 + a^2*C) + b*(m + 1)*(b*c - a*d)*(A + C))*Sin[e + f*x] - d *(A*b^2 + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ[n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) || EqQ[a, 0])))
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[C/(b*d) Int[Sqrt[a + b*Sin[e + f*x]], x] , x] - Simp[1/(b*d) Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[ e + f*x], x]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; Fre eQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0 ] && NeQ[c^2 - d^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1334\) vs. \(2(477)=954\).
Time = 42.12 (sec) , antiderivative size = 1335, normalized size of antiderivative = 3.21
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1335\) |
| parts | \(\text {Expression too large to display}\) | \(1815\) |
-(-(-2*b*cos(1/2*d*x+1/2*c)^2-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*sin(1/2*d*x+1/2*c)^4*b+(a+b)*sin(1/2*d*x+1/2*c)^2 )^(1/2)/(2*cos(1/2*d*x+1/2*c)^2-1)+1/2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b* cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4*b+(a+b)*si n(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*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^( 1/2)/(-2*sin(1/2*d*x+1/2*c)^4*b+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*Elliptic E(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*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4*b+ (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*b*cos(1/2*d*x+1/2*c)^2 +a-b)/(a-b))^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4*b+(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)))+4/a^3*b*A*(sin( 1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2)/(-2*s in(1/2*d*x+1/2*c)^4*b+(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))-4*A*b^2/a^3/sin(1/2*d*x+1/2*c)^2/(2*b*si n(1/2*d*x+1/2*c)^2-a-b)/(a^2-b^2)*(-2*sin(1/2*d*x+1/2*c)^4*b+(a+b)*sin(1/2 *d*x+1/2*c)^2)^(1/2)*(2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2*b+Elliptic E(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+ (a+b)/(a-b))^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*a-EllipticE(cos(1/2*d*x...
Timed out. \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx=\int \frac {\left (A + C \cos ^{2}{\left (c + d x \right )}\right ) \sec ^{2}{\left (c + d x \right )}}{\left (a + b \cos {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \sec \left (d x + c\right )^{2}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \sec \left (d x + c\right )^{2}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx=\int \frac {C\,{\cos \left (c+d\,x\right )}^2+A}{{\cos \left (c+d\,x\right )}^2\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \]