Integrand size = 35, antiderivative size = 365 \[ \int \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\frac {\left (3 A b^2-8 a^2 (2 A+3 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{24 a^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (A b^2-8 a^2 (2 A+3 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{24 a d \sqrt {a+b \cos (c+d x)}}+\frac {b \left (A b^2+4 a^2 (A+2 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{8 a^2 d \sqrt {a+b \cos (c+d x)}}-\frac {\left (3 A b^2-8 a^2 (2 A+3 C)\right ) \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{24 a^2 d}+\frac {A b \sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{12 a d}+\frac {A \sqrt {a+b \cos (c+d x)} \sec ^2(c+d x) \tan (c+d x)}{3 d} \]
1/24*(3*A*b^2-8*a^2*(2*A+3*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^2/d/((a+b*cos(d*x+c))/(a+b))^(1/2)-1/24*(A*b^2-8*a^2*(2*A+3*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/d/(a+b*cos(d* x+c))^(1/2)+1/8*b*(A*b^2+4*a^2*(A+2*C))*(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^2/d/(a+b*cos(d*x+c))^(1/2)-1/24*(3*A*b^2-8*a ^2*(2*A+3*C))*(a+b*cos(d*x+c))^(1/2)*tan(d*x+c)/a^2/d+1/12*A*b*sec(d*x+c)* (a+b*cos(d*x+c))^(1/2)*tan(d*x+c)/a/d+1/3*A*sec(d*x+c)^2*(a+b*cos(d*x+c))^ (1/2)*tan(d*x+c)/d
Result contains complex when optimal does not.
Time = 6.95 (sec) , antiderivative size = 601, normalized size of antiderivative = 1.65 \[ \int \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=-\frac {b \left (-\frac {8 a A b \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}+\frac {2 \left (-8 a^2 A-9 A b^2-24 a^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}-\frac {2 i \left (16 a^2 A-3 A b^2+24 a^2 C\right ) \sqrt {\frac {b-b \cos (c+d x)}{a+b}} \sqrt {-\frac {b+b \cos (c+d x)}{a-b}} \cos (2 (c+d x)) \left (2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )-b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sin (c+d x)}{a \sqrt {-\frac {1}{a+b}} \sqrt {1-\cos ^2(c+d x)} \sqrt {-\frac {a^2-b^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2}{b^2}} \left (2 a^2-b^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2\right )}\right )}{96 a^2 d}+\frac {\sqrt {a+b \cos (c+d x)} \left (\frac {\sec (c+d x) \left (16 a^2 A \sin (c+d x)-3 A b^2 \sin (c+d x)+24 a^2 C \sin (c+d x)\right )}{24 a^2}+\frac {A b \sec (c+d x) \tan (c+d x)}{12 a}+\frac {1}{3} A \sec ^2(c+d x) \tan (c+d x)\right )}{d} \]
-1/96*(b*((-8*a*A*b*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*(-8*a^2*A - 9*A*b^2 - 24 *a^2*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)*(16*a^2*A - 3*A*b^2 + 24*a^2 *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)]*S qrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sq rt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] - b*Elliptic Pi[(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) )))/(a^2*d) + (Sqrt[a + b*Cos[c + d*x]]*((Sec[c + d*x]*(16*a^2*A*Sin[c + d *x] - 3*A*b^2*Sin[c + d*x] + 24*a^2*C*Sin[c + d*x]))/(24*a^2) + (A*b*Sec[c + d*x]*Tan[c + d*x])/(12*a) + (A*Sec[c + d*x]^2*Tan[c + d*x])/3))/d
Time = 3.20 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.03, number of steps used = 24, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.686, Rules used = {3042, 3527, 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.
\(\displaystyle \int \sec ^4(c+d x) \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx\) |
\(\Big \downarrow \) 3527 |
\(\displaystyle \frac {1}{3} \int \frac {\left (3 b (A+2 C) \cos ^2(c+d x)+2 a (2 A+3 C) \cos (c+d x)+A b\right ) \sec ^3(c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{6} \int \frac {\left (3 b (A+2 C) \cos ^2(c+d x)+2 a (2 A+3 C) \cos (c+d x)+A b\right ) \sec ^3(c+d x)}{\sqrt {a+b \cos (c+d x)}}dx+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} \int \frac {3 b (A+2 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 a (2 A+3 C) \sin \left (c+d x+\frac {\pi }{2}\right )+A b}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3534 |
\(\displaystyle \frac {1}{6} \left (\frac {\int -\frac {\left (-8 (2 A+3 C) a^2-2 b (7 A+12 C) \cos (c+d x) a+3 A b^2-A b^2 \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx}{2 a}+\frac {A b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{6} \left (\frac {A b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\int \frac {\left (-8 (2 A+3 C) a^2-2 b (7 A+12 C) \cos (c+d x) a+3 A b^2-A b^2 \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{4 a}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} \left (\frac {A b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\int \frac {-8 (2 A+3 C) a^2-2 b (7 A+12 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a+3 A b^2-A b^2 \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^2 \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{4 a}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3534 |
\(\displaystyle \frac {1}{6} \left (\frac {A b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\int -\frac {\left (2 a A \cos (c+d x) b^2+\left (3 A b^2-8 a^2 (2 A+3 C)\right ) \cos ^2(c+d x) b+3 \left (4 (A+2 C) a^2+A b^2\right ) b\right ) \sec (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx}{a}+\frac {\left (3 A b^2-8 a^2 (2 A+3 C)\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}}{4 a}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{6} \left (\frac {A b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\left (3 A b^2-8 a^2 (2 A+3 C)\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\int \frac {\left (2 a A \cos (c+d x) b^2+\left (3 A b^2-8 a^2 (2 A+3 C)\right ) \cos ^2(c+d x) b+3 \left (4 (A+2 C) a^2+A b^2\right ) b\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{2 a}}{4 a}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} \left (\frac {A b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\left (3 A b^2-8 a^2 (2 A+3 C)\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\int \frac {2 a A \sin \left (c+d x+\frac {\pi }{2}\right ) b^2+\left (3 A b^2-8 a^2 (2 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2 b+3 \left (4 (A+2 C) a^2+A b^2\right ) b}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}}{4 a}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3538 |
\(\displaystyle \frac {1}{6} \left (\frac {A b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\left (3 A b^2-8 a^2 (2 A+3 C)\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\left (3 A b^2-8 a^2 (2 A+3 C)\right ) \int \sqrt {a+b \cos (c+d x)}dx-\frac {\int -\frac {\left (3 b^2 \left (4 (A+2 C) a^2+A b^2\right )-a b \left (A b^2-8 a^2 (2 A+3 C)\right ) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}}{2 a}}{4 a}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{6} \left (\frac {A b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\left (3 A b^2-8 a^2 (2 A+3 C)\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\left (3 A b^2-8 a^2 (2 A+3 C)\right ) \int \sqrt {a+b \cos (c+d x)}dx+\frac {\int \frac {\left (3 b^2 \left (4 (A+2 C) a^2+A b^2\right )-a b \left (A b^2-8 a^2 (2 A+3 C)\right ) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}}{2 a}}{4 a}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} \left (\frac {A b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\left (3 A b^2-8 a^2 (2 A+3 C)\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\left (3 A b^2-8 a^2 (2 A+3 C)\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {\int \frac {3 b^2 \left (4 (A+2 C) a^2+A b^2\right )-a b \left (A b^2-8 a^2 (2 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{2 a}}{4 a}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3134 |
\(\displaystyle \frac {1}{6} \left (\frac {A b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\left (3 A b^2-8 a^2 (2 A+3 C)\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {\int \frac {3 b^2 \left (4 (A+2 C) a^2+A b^2\right )-a b \left (A b^2-8 a^2 (2 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}+\frac {\left (3 A b^2-8 a^2 (2 A+3 C)\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}}{4 a}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} \left (\frac {A b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\left (3 A b^2-8 a^2 (2 A+3 C)\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {\int \frac {3 b^2 \left (4 (A+2 C) a^2+A b^2\right )-a b \left (A b^2-8 a^2 (2 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}+\frac {\left (3 A b^2-8 a^2 (2 A+3 C)\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}}{4 a}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3132 |
\(\displaystyle \frac {1}{6} \left (\frac {A b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\left (3 A b^2-8 a^2 (2 A+3 C)\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {\int \frac {3 b^2 \left (4 (A+2 C) a^2+A b^2\right )-a b \left (A b^2-8 a^2 (2 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}+\frac {2 \left (3 A b^2-8 a^2 (2 A+3 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}}{4 a}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3481 |
\(\displaystyle \frac {1}{6} \left (\frac {A b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\left (3 A b^2-8 a^2 (2 A+3 C)\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {3 b^2 \left (4 a^2 (A+2 C)+A b^2\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx-a b \left (A b^2-8 a^2 (2 A+3 C)\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}+\frac {2 \left (3 A b^2-8 a^2 (2 A+3 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}}{4 a}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} \left (\frac {A b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\left (3 A b^2-8 a^2 (2 A+3 C)\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {3 b^2 \left (4 a^2 (A+2 C)+A b^2\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-a b \left (A b^2-8 a^2 (2 A+3 C)\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}+\frac {2 \left (3 A b^2-8 a^2 (2 A+3 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}}{4 a}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3142 |
\(\displaystyle \frac {1}{6} \left (\frac {A b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\left (3 A b^2-8 a^2 (2 A+3 C)\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {3 b^2 \left (4 a^2 (A+2 C)+A b^2\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {a b \left (A b^2-8 a^2 (2 A+3 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 \left (3 A b^2-8 a^2 (2 A+3 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}}{4 a}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} \left (\frac {A b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\left (3 A b^2-8 a^2 (2 A+3 C)\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {3 b^2 \left (4 a^2 (A+2 C)+A b^2\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {a b \left (A b^2-8 a^2 (2 A+3 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 \left (3 A b^2-8 a^2 (2 A+3 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}}{4 a}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3140 |
\(\displaystyle \frac {1}{6} \left (\frac {A b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\left (3 A b^2-8 a^2 (2 A+3 C)\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {3 b^2 \left (4 a^2 (A+2 C)+A b^2\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 a b \left (A b^2-8 a^2 (2 A+3 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 \left (3 A b^2-8 a^2 (2 A+3 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}}{4 a}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3286 |
\(\displaystyle \frac {1}{6} \left (\frac {A b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\left (3 A b^2-8 a^2 (2 A+3 C)\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {\frac {3 b^2 \left (4 a^2 (A+2 C)+A b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {\sec (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}-\frac {2 a b \left (A b^2-8 a^2 (2 A+3 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 \left (3 A b^2-8 a^2 (2 A+3 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}}{4 a}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} \left (\frac {A b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\left (3 A b^2-8 a^2 (2 A+3 C)\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {\frac {3 b^2 \left (4 a^2 (A+2 C)+A b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}-\frac {2 a b \left (A b^2-8 a^2 (2 A+3 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 \left (3 A b^2-8 a^2 (2 A+3 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}}{4 a}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3284 |
\(\displaystyle \frac {1}{6} \left (\frac {A b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\left (3 A b^2-8 a^2 (2 A+3 C)\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {2 \left (3 A b^2-8 a^2 (2 A+3 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\frac {6 b^2 \left (4 a^2 (A+2 C)+A b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}-\frac {2 a b \left (A b^2-8 a^2 (2 A+3 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}}{2 a}}{4 a}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
(A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + ((A*b*Sqr t[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) - (-1/2*((2*(3*A* b^2 - 8*a^2*(2*A + 3*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*b^2 - 8*a^2*(2*A + 3*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^2*(A*b^2 + 4*a^2*(A + 2*C))*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 + ((3*A*b^2 - 8*a^2*(2*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(a*d))/(4*a))/6
3.7.29.3.1 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_.) + (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[(-(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. \(2308\) vs. \(2(422)=844\).
Time = 27.23 (sec) , antiderivative size = 2309, normalized size of antiderivative = 6.33
method | result | size |
default | \(\text {Expression too large to display}\) | \(2309\) |
parts | \(\text {Expression too large to display}\) | \(2361\) |
-(-(-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*(-1/ 3*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)^3+5/12*b/a^2*cos(1/2*d*x+1/2*c)*(-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)/(2*cos(1/2*d*x+1/2 *c)^2-1)^2-1/24*(16*a^2+15*b^2)/a^3*cos(1/2*d*x+1/2*c)*(-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)+5/48* b^2/a^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)*Ellip ticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))+1/3*(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)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b ))^(1/2))-1/3*(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) *EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))+1/3/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))-5/16*b^2/a^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)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))+5/16/ a^3*(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))...
Timed out. \[ \int \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\text {Timed out} \]
Timed out. \[ \int \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\text {Timed out} \]
\[ \int \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sec \left (d x + c\right )^{4} \,d x } \]
\[ \int \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sec \left (d x + c\right )^{4} \,d x } \]
Timed out. \[ \int \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\int \frac {\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,\sqrt {a+b\,\cos \left (c+d\,x\right )}}{{\cos \left (c+d\,x\right )}^4} \,d x \]