Integrand size = 45, antiderivative size = 764 \[ \int \frac {(a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=-\frac {(a-b) \sqrt {a+b} \left (24 a^2 b B+128 b^3 B-9 a^3 C+12 a b^2 (20 A+13 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{192 a b^2 d \sqrt {\sec (c+d x)}}-\frac {\sqrt {a+b} \left (9 a^3 C-6 a^2 b (4 B+C)-8 b^3 (12 A+16 B+9 C)-4 a b^2 (60 A+28 B+39 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{192 b^2 d \sqrt {\sec (c+d x)}}+\frac {\sqrt {a+b} \left (8 a^3 b B-96 a b^3 B-3 a^4 C-24 a^2 b^2 (2 A+C)-16 b^4 (4 A+3 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{64 b^3 d \sqrt {\sec (c+d x)}}+\frac {\left (4 b^2 (4 A+3 C)+a (8 b B-3 a C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{32 b d \sqrt {\sec (c+d x)}}+\frac {(8 b B-3 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 b d \sqrt {\sec (c+d x)}}+\frac {C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{4 b d \sqrt {\sec (c+d x)}}+\frac {\left (24 a^2 b B+128 b^3 B-9 a^3 C+12 a b^2 (20 A+13 C)\right ) \sqrt {a+b \cos (c+d x)} \sqrt {\sec (c+d x)} \sin (c+d x)}{192 b^2 d} \]
1/24*(8*B*b-3*C*a)*(a+b*cos(d*x+c))^(3/2)*sin(d*x+c)/b/d/sec(d*x+c)^(1/2)+ 1/4*C*(a+b*cos(d*x+c))^(5/2)*sin(d*x+c)/b/d/sec(d*x+c)^(1/2)+1/32*(4*b^2*( 4*A+3*C)+a*(8*B*b-3*C*a))*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/b/d/sec(d*x+c) ^(1/2)+1/192*(24*B*a^2*b+128*B*b^3-9*a^3*C+12*a*b^2*(20*A+13*C))*sin(d*x+c )*(a+b*cos(d*x+c))^(1/2)*sec(d*x+c)^(1/2)/b^2/d-1/192*(a-b)*(24*B*a^2*b+12 8*B*b^3-9*a^3*C+12*a*b^2*(20*A+13*C))*csc(d*x+c)*EllipticE((a+b*cos(d*x+c) )^(1/2)/(a+b)^(1/2)/cos(d*x+c)^(1/2),((-a-b)/(a-b))^(1/2))*(a+b)^(1/2)*cos (d*x+c)^(1/2)*(a*(1-sec(d*x+c))/(a+b))^(1/2)*(a*(1+sec(d*x+c))/(a-b))^(1/2 )/a/b^2/d/sec(d*x+c)^(1/2)-1/192*(9*a^3*C-6*a^2*b*(4*B+C)-8*b^3*(12*A+16*B +9*C)-4*a*b^2*(60*A+28*B+39*C))*csc(d*x+c)*EllipticF((a+b*cos(d*x+c))^(1/2 )/(a+b)^(1/2)/cos(d*x+c)^(1/2),((-a-b)/(a-b))^(1/2))*(a+b)^(1/2)*cos(d*x+c )^(1/2)*(a*(1-sec(d*x+c))/(a+b))^(1/2)*(a*(1+sec(d*x+c))/(a-b))^(1/2)/b^2/ d/sec(d*x+c)^(1/2)+1/64*(8*B*a^3*b-96*B*a*b^3-3*a^4*C-24*a^2*b^2*(2*A+C)-1 6*b^4*(4*A+3*C))*csc(d*x+c)*EllipticPi((a+b*cos(d*x+c))^(1/2)/(a+b)^(1/2)/ cos(d*x+c)^(1/2),(a+b)/b,((-a-b)/(a-b))^(1/2))*(a+b)^(1/2)*cos(d*x+c)^(1/2 )*(a*(1-sec(d*x+c))/(a+b))^(1/2)*(a*(1+sec(d*x+c))/(a-b))^(1/2)/b^3/d/sec( d*x+c)^(1/2)
Time = 11.86 (sec) , antiderivative size = 601, normalized size of antiderivative = 0.79 \[ \int \frac {(a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\frac {-\frac {-b (a+b) \left (24 a^2 b B+128 b^3 B-9 a^3 C+12 a b^2 (20 A+13 C)\right ) E\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {-a+b}{a+b}\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {(a+b \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+a (a+b) \left (9 a^3 C-6 a^2 b (4 B+3 C)+12 a b^2 (12 A+4 B+7 C)+8 b^3 (12 A+16 B+9 C)\right ) \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {(a+b \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+3 \left (8 a^3 b B-96 a b^3 B-3 a^4 C-24 a^2 b^2 (2 A+C)-16 b^4 (4 A+3 C)\right ) \left ((a-b) \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right )+2 b \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right )\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {(a+b \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}-b \left (24 a^2 b B+128 b^3 B-9 a^3 C+12 a b^2 (20 A+13 C)\right ) (a+b \cos (c+d x)) \left (\cos (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )\right )^{3/2} \sec (c+d x) \tan \left (\frac {1}{2} (c+d x)\right )}{b^3 \left (\cos (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )\right )^{3/2}}+\frac {2 (a+b \cos (c+d x)) \left (48 A b^2+56 a b B+3 a^2 C+48 b^2 C+4 b (8 b B+9 a C) \cos (c+d x)+12 b^2 C \cos (2 (c+d x))\right ) \tan (c+d x)}{b}}{192 d \sqrt {a+b \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x)} \]
Integrate[((a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x] ^2))/Sqrt[Sec[c + d*x]],x]
(-((-(b*(a + b)*(24*a^2*b*B + 128*b^3*B - 9*a^3*C + 12*a*b^2*(20*A + 13*C) )*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2 *Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]) + a*(a + b)*(9*a ^3*C - 6*a^2*b*(4*B + 3*C) + 12*a*b^2*(12*A + 4*B + 7*C) + 8*b^3*(12*A + 1 6*B + 9*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + 3*(8* a^3*b*B - 96*a*b^3*B - 3*a^4*C - 24*a^2*b^2*(2*A + C) - 16*b^4*(4*A + 3*C) )*((a - b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*b*Ell ipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)])*Sec[(c + d*x)/2]^ 2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - b*(24*a^2*b*B + 128*b^3*B - 9*a^3*C + 12*a*b^2*(20*A + 13*C))*(a + b*Cos[c + d*x])*(Cos[ c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Tan[(c + d*x)/2])/(b^3*(Co s[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2))) + (2*(a + b*Cos[c + d*x])*(48*A*b^2 + 56*a*b*B + 3*a^2*C + 48*b^2*C + 4*b*(8*b*B + 9*a*C)*Cos[c + d*x] + 12*b ^2*C*Cos[2*(c + d*x)])*Tan[c + d*x])/b)/(192*d*Sqrt[a + b*Cos[c + d*x]]*Se c[c + d*x]^(3/2))
Time = 4.09 (sec) , antiderivative size = 730, normalized size of antiderivative = 0.96, number of steps used = 22, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.489, Rules used = {3042, 4709, 3042, 3528, 27, 3042, 3528, 27, 3042, 3528, 27, 3042, 3540, 25, 3042, 3532, 3042, 3288, 3477, 3042, 3295, 3473}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos (c+d x)^2\right )}{\sqrt {\sec (c+d x)}}dx\) |
| \(\Big \downarrow \) 4709 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \left (C \cos ^2(c+d x)+B \cos (c+d x)+A\right )dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (C \sin \left (c+d x+\frac {\pi }{2}\right )^2+B \sin \left (c+d x+\frac {\pi }{2}\right )+A\right )dx\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \frac {(a+b \cos (c+d x))^{3/2} \left ((8 b B-3 a C) \cos ^2(c+d x)+2 b (4 A+3 C) \cos (c+d x)+a C\right )}{2 \sqrt {\cos (c+d x)}}dx}{4 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \frac {(a+b \cos (c+d x))^{3/2} \left ((8 b B-3 a C) \cos ^2(c+d x)+2 b (4 A+3 C) \cos (c+d x)+a C\right )}{\sqrt {\cos (c+d x)}}dx}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left ((8 b B-3 a C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 b (4 A+3 C) \sin \left (c+d x+\frac {\pi }{2}\right )+a C\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\right )\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{3} \int \frac {\sqrt {a+b \cos (c+d x)} \left (3 \left (4 (4 A+3 C) b^2+a (8 b B-3 a C)\right ) \cos ^2(c+d x)+2 b (24 a A+16 b B+15 a C) \cos (c+d x)+a (8 b B+3 a C)\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {(8 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{6} \int \frac {\sqrt {a+b \cos (c+d x)} \left (3 \left (4 (4 A+3 C) b^2+a (8 b B-3 a C)\right ) \cos ^2(c+d x)+2 b (24 a A+16 b B+15 a C) \cos (c+d x)+a (8 b B+3 a C)\right )}{\sqrt {\cos (c+d x)}}dx+\frac {(8 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{6} \int \frac {\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (3 \left (4 (4 A+3 C) b^2+a (8 b B-3 a C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 b (24 a A+16 b B+15 a C) \sin \left (c+d x+\frac {\pi }{2}\right )+a (8 b B+3 a C)\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {(8 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\right )\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{6} \left (\frac {1}{2} \int \frac {\left (-9 C a^3+24 b B a^2+12 b^2 (20 A+13 C) a+128 b^3 B\right ) \cos ^2(c+d x)+2 b \left ((96 A+57 C) a^2+104 b B a+12 b^2 (4 A+3 C)\right ) \cos (c+d x)+a \left (3 C a^2+56 b B a+48 A b^2+36 b^2 C\right )}{2 \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a (8 b B-3 a C)+4 b^2 (4 A+3 C)\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {(8 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{6} \left (\frac {1}{4} \int \frac {\left (-9 C a^3+24 b B a^2+12 b^2 (20 A+13 C) a+128 b^3 B\right ) \cos ^2(c+d x)+2 b \left ((96 A+57 C) a^2+104 b B a+12 b^2 (4 A+3 C)\right ) \cos (c+d x)+a \left (3 C a^2+56 b B a+48 A b^2+36 b^2 C\right )}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a (8 b B-3 a C)+4 b^2 (4 A+3 C)\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {(8 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{6} \left (\frac {1}{4} \int \frac {\left (-9 C a^3+24 b B a^2+12 b^2 (20 A+13 C) a+128 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 b \left ((96 A+57 C) a^2+104 b B a+12 b^2 (4 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left (3 C a^2+56 b B a+48 A b^2+36 b^2 C\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a (8 b B-3 a C)+4 b^2 (4 A+3 C)\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {(8 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\right )\) |
| \(\Big \downarrow \) 3540 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{6} \left (\frac {1}{4} \left (\frac {\int -\frac {3 \left (-3 C a^4+8 b B a^3-24 b^2 (2 A+C) a^2-96 b^3 B a-16 b^4 (4 A+3 C)\right ) \cos ^2(c+d x)-2 a b \left (3 C a^2+56 b B a+48 A b^2+36 b^2 C\right ) \cos (c+d x)+a \left (-9 C a^3+24 b B a^2+12 b^2 (20 A+13 C) a+128 b^3 B\right )}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{2 b}+\frac {\sin (c+d x) \left (-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a (8 b B-3 a C)+4 b^2 (4 A+3 C)\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {(8 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {\int \frac {3 \left (-3 C a^4+8 b B a^3-24 b^2 (2 A+C) a^2-96 b^3 B a-16 b^4 (4 A+3 C)\right ) \cos ^2(c+d x)-2 a b \left (3 C a^2+56 b B a+48 A b^2+36 b^2 C\right ) \cos (c+d x)+a \left (-9 C a^3+24 b B a^2+12 b^2 (20 A+13 C) a+128 b^3 B\right )}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a (8 b B-3 a C)+4 b^2 (4 A+3 C)\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {(8 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {\int \frac {3 \left (-3 C a^4+8 b B a^3-24 b^2 (2 A+C) a^2-96 b^3 B a-16 b^4 (4 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-2 a b \left (3 C a^2+56 b B a+48 A b^2+36 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left (-9 C a^3+24 b B a^2+12 b^2 (20 A+13 C) a+128 b^3 B\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a (8 b B-3 a C)+4 b^2 (4 A+3 C)\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {(8 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\right )\) |
| \(\Big \downarrow \) 3532 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {\int \frac {a \left (-9 C a^3+24 b B a^2+12 b^2 (20 A+13 C) a+128 b^3 B\right )-2 a b \left (3 C a^2+56 b B a+48 A b^2+36 b^2 C\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx+3 \left (-3 a^4 C+8 a^3 b B-24 a^2 b^2 (2 A+C)-96 a b^3 B-16 b^4 (4 A+3 C)\right ) \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {a+b \cos (c+d x)}}dx}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a (8 b B-3 a C)+4 b^2 (4 A+3 C)\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {(8 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {\int \frac {a \left (-9 C a^3+24 b B a^2+12 b^2 (20 A+13 C) a+128 b^3 B\right )-2 a b \left (3 C a^2+56 b B a+48 A b^2+36 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+3 \left (-3 a^4 C+8 a^3 b B-24 a^2 b^2 (2 A+C)-96 a b^3 B-16 b^4 (4 A+3 C)\right ) \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a (8 b B-3 a C)+4 b^2 (4 A+3 C)\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {(8 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\right )\) |
| \(\Big \downarrow \) 3288 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {\int \frac {a \left (-9 C a^3+24 b B a^2+12 b^2 (20 A+13 C) a+128 b^3 B\right )-2 a b \left (3 C a^2+56 b B a+48 A b^2+36 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 \sqrt {a+b} \cot (c+d x) \left (-3 a^4 C+8 a^3 b B-24 a^2 b^2 (2 A+C)-96 a b^3 B-16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b d}}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a (8 b B-3 a C)+4 b^2 (4 A+3 C)\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {(8 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\right )\) |
| \(\Big \downarrow \) 3477 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {a \left (-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right ) \int \frac {\cos (c+d x)+1}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx+a \left (9 a^3 C-6 a^2 b (4 B+C)-4 a b^2 (60 A+28 B+39 C)-8 b^3 (12 A+16 B+9 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx-\frac {6 \sqrt {a+b} \cot (c+d x) \left (-3 a^4 C+8 a^3 b B-24 a^2 b^2 (2 A+C)-96 a b^3 B-16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b d}}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a (8 b B-3 a C)+4 b^2 (4 A+3 C)\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {(8 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {a \left (9 a^3 C-6 a^2 b (4 B+C)-4 a b^2 (60 A+28 B+39 C)-8 b^3 (12 A+16 B+9 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+a \left (-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 \sqrt {a+b} \cot (c+d x) \left (-3 a^4 C+8 a^3 b B-24 a^2 b^2 (2 A+C)-96 a b^3 B-16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b d}}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a (8 b B-3 a C)+4 b^2 (4 A+3 C)\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {(8 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\right )\) |
| \(\Big \downarrow \) 3295 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {a \left (-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sqrt {a+b} \cot (c+d x) \left (9 a^3 C-6 a^2 b (4 B+C)-4 a b^2 (60 A+28 B+39 C)-8 b^3 (12 A+16 B+9 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}-\frac {6 \sqrt {a+b} \cot (c+d x) \left (-3 a^4 C+8 a^3 b B-24 a^2 b^2 (2 A+C)-96 a b^3 B-16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b d}}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a (8 b B-3 a C)+4 b^2 (4 A+3 C)\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {(8 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\right )\) |
| \(\Big \downarrow \) 3473 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {\frac {2 \sqrt {a+b} \cot (c+d x) \left (9 a^3 C-6 a^2 b (4 B+C)-4 a b^2 (60 A+28 B+39 C)-8 b^3 (12 A+16 B+9 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}+\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{a d}-\frac {6 \sqrt {a+b} \cot (c+d x) \left (-3 a^4 C+8 a^3 b B-24 a^2 b^2 (2 A+C)-96 a b^3 B-16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b d}}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a (8 b B-3 a C)+4 b^2 (4 A+3 C)\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {(8 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\right )\) |
Int[((a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/S qrt[Sec[c + d*x]],x]
Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*b*d) + (((8*b*B - 3*a*C)*Sqrt[Cos[c + d*x]] *(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d) + ((3*(4*b^2*(4*A + 3*C) + a*(8*b*B - 3*a*C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d* x])/(2*d) + (-1/2*((2*(a - b)*Sqrt[a + b]*(24*a^2*b*B + 128*b^3*B - 9*a^3* C + 12*a*b^2*(20*A + 13*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d) + ( 2*Sqrt[a + b]*(9*a^3*C - 6*a^2*b*(4*B + C) - 8*b^3*(12*A + 16*B + 9*C) - 4 *a*b^2*(60*A + 28*B + 39*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[ c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*( 1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d - (6*S qrt[a + b]*(8*a^3*b*B - 96*a*b^3*B - 3*a^4*C - 24*a^2*b^2*(2*A + C) - 16*b ^4*(4*A + 3*C))*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d))/b + ((24*a^2*b*B + 128*b^3*B - 9*a^3*C + 12*a*b^2*(20*A + 13*C))*Sqrt[a + b* Cos[c + d*x]]*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]]))/4)/6)/(8*b))
3.16.16.3.1 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[(b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.) *(x_)]], x_Symbol] :> Simp[2*b*(Tan[e + f*x]/(d*f))*Rt[(c + d)/b, 2]*Sqrt[c *((1 + Csc[e + f*x])/(c - d))]*Sqrt[c*((1 - Csc[e + f*x])/(c + d))]*Ellipti cPi[(c + d)/d, ArcSin[Sqrt[c + d*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]]/Rt[(c + d)/b, 2]], -(c + d)/(c - d)], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] && PosQ[(c + d)/b]
Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f _.)*(x_)]]), x_Symbol] :> Simp[-2*(Tan[e + f*x]/(a*f))*Rt[(a + b)/d, 2]*Sqr t[a*((1 - Csc[e + f*x])/(a + b))]*Sqrt[a*((1 + Csc[e + f*x])/(a - b))]*Elli pticF[ArcSin[Sqrt[a + b*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]/Rt[(a + b)/d, 2] ], -(a + b)/(a - b)], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && PosQ[(a + b)/d]
Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((b_.)*sin[(e_.) + (f_.)*(x_)]) ^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*A* (c - d)*(Tan[e + f*x]/(f*b*c^2))*Rt[(c + d)/b, 2]*Sqrt[c*((1 + Csc[e + f*x] )/(c - d))]*Sqrt[c*((1 - Csc[e + f*x])/(c + d))]*EllipticE[ArcSin[Sqrt[c + d*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]]/Rt[(c + d)/b, 2]], -(c + d)/(c - d)], x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && PosQ[(c + d)/b]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ .)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> S imp[(A - B)/(a - b) Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f* x]]), x], x] - Simp[(A*b - a*B)/(a - b) Int[(1 + Sin[e + f*x])/((a + b*Si n[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e , f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && NeQ[A, B]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_ .) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a *c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n} , x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[ m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_.) + (d_.)*sin[(e _.) + (f_.)*(x_)]]), x_Symbol] :> Simp[C/b^2 Int[Sqrt[a + b*Sin[e + f*x]] /Sqrt[c + d*Sin[e + f*x]], x], x] + Simp[1/b^2 Int[(A*b^2 - a^2*C + b*(b* B - 2*a*C)*Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x ]]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] & & NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(Sqrt[c + d*Sin[e + f *x]]/(d*f*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[1/(2*d) Int[(1/((a + b*Si n[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]))*Simp[2*a*A*d - C*(b*c - a*d) - 2*(a*c*C - d*(A*b + a*B))*Sin[e + f*x] + (2*b*B*d - C*(b*c + a*d))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a *d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[(u_)*((c_.)*sec[(a_.) + (b_.)*(x_)])^(m_.), x_Symbol] :> Simp[(c*Sec[a + b*x])^m*(c*Cos[a + b*x])^m Int[ActivateTrig[u]/(c*Cos[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownSineIntegrandQ[u, x]
Leaf count of result is larger than twice the leaf count of optimal. \(7355\) vs. \(2(698)=1396\).
Time = 13.84 (sec) , antiderivative size = 7356, normalized size of antiderivative = 9.63
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(7356\) |
| default | \(\text {Expression too large to display}\) | \(7452\) |
int((a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/sec(d*x+c)^(1/2 ),x,method=_RETURNVERBOSE)
\[ \int \frac {(a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{\sqrt {\sec \left (d x + c\right )}} \,d x } \]
integrate((a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/sec(d*x+c )^(1/2),x, algorithm="fricas")
integral((C*b*cos(d*x + c)^3 + (C*a + B*b)*cos(d*x + c)^2 + A*a + (B*a + A *b)*cos(d*x + c))*sqrt(b*cos(d*x + c) + a)/sqrt(sec(d*x + c)), x)
Timed out. \[ \int \frac {(a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\text {Timed out} \]
\[ \int \frac {(a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{\sqrt {\sec \left (d x + c\right )}} \,d x } \]
integrate((a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/sec(d*x+c )^(1/2),x, algorithm="maxima")
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(3/ 2)/sqrt(sec(d*x + c)), x)
\[ \int \frac {(a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{\sqrt {\sec \left (d x + c\right )}} \,d x } \]
integrate((a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/sec(d*x+c )^(1/2),x, algorithm="giac")
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(3/ 2)/sqrt(sec(d*x + c)), x)
Timed out. \[ \int \frac {(a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\int \frac {{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \,d x \]
int(((a + b*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/( 1/cos(c + d*x))^(1/2),x)