Integrand size = 35, antiderivative size = 436 \[ \int (a+b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{64 a^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {b \left (A b^2-4 a^2 (19 A+28 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{64 a d \sqrt {a+b \cos (c+d x)}}+\frac {\left (3 A b^4+24 a^2 b^2 (A+2 C)+16 a^4 (3 A+4 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{64 a^2 d \sqrt {a+b \cos (c+d x)}}-\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{64 a^2 d}+\frac {\left (A b^2+4 a^2 (3 A+4 C)\right ) \sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{32 a d}+\frac {A b \sqrt {a+b \cos (c+d x)} \sec ^2(c+d x) \tan (c+d x)}{8 d}+\frac {A (a+b \cos (c+d x))^{3/2} \sec ^3(c+d x) \tan (c+d x)}{4 d} \] Output:
1/64*b*(3*A*b^2-4*a^2*(13*A+20*C))*(a+b*cos(d*x+c))^(1/2)*EllipticE(sin(1/ 2*d*x+1/2*c),2^(1/2)*(b/(a+b))^(1/2))/a^2/d/((a+b*cos(d*x+c))/(a+b))^(1/2) -1/64*b*(A*b^2-4*a^2*(19*A+28*C))*((a+b*cos(d*x+c))/(a+b))^(1/2)*InverseJa cobiAM(1/2*d*x+1/2*c,2^(1/2)*(b/(a+b))^(1/2))/a/d/(a+b*cos(d*x+c))^(1/2)+1 /64*(3*A*b^4+24*a^2*b^2*(A+2*C)+16*a^4*(3*A+4*C))*((a+b*cos(d*x+c))/(a+b)) ^(1/2)*EllipticPi(sin(1/2*d*x+1/2*c),2,2^(1/2)*(b/(a+b))^(1/2))/a^2/d/(a+b *cos(d*x+c))^(1/2)-1/64*b*(3*A*b^2-4*a^2*(13*A+20*C))*(a+b*cos(d*x+c))^(1/ 2)*tan(d*x+c)/a^2/d+1/32*(A*b^2+4*a^2*(3*A+4*C))*(a+b*cos(d*x+c))^(1/2)*se c(d*x+c)*tan(d*x+c)/a/d+1/8*A*b*(a+b*cos(d*x+c))^(1/2)*sec(d*x+c)^2*tan(d* x+c)/d+1/4*A*(a+b*cos(d*x+c))^(3/2)*sec(d*x+c)^3*tan(d*x+c)/d
Result contains complex when optimal does not.
Time = 7.29 (sec) , antiderivative size = 696, normalized size of antiderivative = 1.60 \[ \int (a+b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {\frac {2 \left (48 a^3 A b+4 a A b^3+64 a^3 b C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}+\frac {2 \left (96 a^4 A-4 a^2 A b^2+9 A b^4+128 a^4 C+16 a^2 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}-\frac {2 i \left (-52 a^2 A b^2+3 A b^4-80 a^2 b^2 C\right ) \sqrt {\frac {b-b \cos (c+d x)}{a+b}} \sqrt {-\frac {b+b \cos (c+d x)}{a-b}} \cos (2 (c+d x)) \left (2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )-b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sin (c+d x)}{a \sqrt {-\frac {1}{a+b}} \sqrt {1-\cos ^2(c+d x)} \sqrt {-\frac {a^2-b^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2}{b^2}} \left (2 a^2-b^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2\right )}}{256 a^2 d}+\frac {\sqrt {a+b \cos (c+d x)} \left (\frac {\sec ^2(c+d x) \left (12 a^2 A \sin (c+d x)+A b^2 \sin (c+d x)+16 a^2 C \sin (c+d x)\right )}{32 a}+\frac {\sec (c+d x) \left (52 a^2 A b \sin (c+d x)-3 A b^3 \sin (c+d x)+80 a^2 b C \sin (c+d x)\right )}{64 a^2}+\frac {3}{8} A b \sec ^2(c+d x) \tan (c+d x)+\frac {1}{4} a A \sec ^3(c+d x) \tan (c+d x)\right )}{d} \] Input:
Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5 ,x]
Output:
((2*(48*a^3*A*b + 4*a*A*b^3 + 64*a^3*b*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*(9 6*a^4*A - 4*a^2*A*b^2 + 9*A*b^4 + 128*a^4*C + 16*a^2*b^2*C)*Sqrt[(a + b*Co s[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b *Cos[c + d*x]] - ((2*I)*(-52*a^2*A*b^2 + 3*A*b^4 - 80*a^2*b^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)]*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)))/(256*a^2*d ) + (Sqrt[a + b*Cos[c + d*x]]*((Sec[c + d*x]^2*(12*a^2*A*Sin[c + d*x] + A* b^2*Sin[c + d*x] + 16*a^2*C*Sin[c + d*x]))/(32*a) + (Sec[c + d*x]*(52*a^2* A*b*Sin[c + d*x] - 3*A*b^3*Sin[c + d*x] + 80*a^2*b*C*Sin[c + d*x]))/(64*a^ 2) + (3*A*b*Sec[c + d*x]^2*Tan[c + d*x])/8 + (a*A*Sec[c + d*x]^3*Tan[c + d *x])/4))/d
Time = 4.01 (sec) , antiderivative size = 450, normalized size of antiderivative = 1.03, number of steps used = 27, number of rules used = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.771, Rules used = {3042, 3527, 27, 3042, 3526, 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.
| \(\displaystyle \int \sec ^5(c+d x) (a+b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^5}dx\) |
| \(\Big \downarrow \) 3527 |
| \(\displaystyle \frac {1}{4} \int \frac {1}{2} \sqrt {a+b \cos (c+d x)} \left (b (3 A+8 C) \cos ^2(c+d x)+2 a (3 A+4 C) \cos (c+d x)+3 A b\right ) \sec ^4(c+d x)dx+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \int \sqrt {a+b \cos (c+d x)} \left (b (3 A+8 C) \cos ^2(c+d x)+2 a (3 A+4 C) \cos (c+d x)+3 A b\right ) \sec ^4(c+d x)dx+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \int \frac {\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (b (3 A+8 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 a (3 A+4 C) \sin \left (c+d x+\frac {\pi }{2}\right )+3 A b\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{3} \int \frac {3 \left (4 (3 A+4 C) a^2+2 b (11 A+16 C) \cos (c+d x) a+A b^2+b^2 (9 A+16 C) \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx+\frac {A b \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \int \frac {\left (4 (3 A+4 C) a^2+2 b (11 A+16 C) \cos (c+d x) a+A b^2+b^2 (9 A+16 C) \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{\sqrt {a+b \cos (c+d x)}}dx+\frac {A b \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \int \frac {4 (3 A+4 C) a^2+2 b (11 A+16 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a+A b^2+b^2 (9 A+16 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {A b \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {\int -\frac {\left (3 A b^3-\left (4 (3 A+4 C) a^2+A b^2\right ) \cos ^2(c+d x) b-2 a^2 (26 A b+40 C b)-2 a \left (4 (3 A+4 C) a^2+b^2 (19 A+32 C)\right ) \cos (c+d x)\right ) \sec ^2(c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx}{2 a}+\frac {\left (4 a^2 (3 A+4 C)+A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}\right )+\frac {A b \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {\left (4 a^2 (3 A+4 C)+A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\int \frac {\left (-b \left (4 (3 A+4 C) a^2+A b^2\right ) \cos ^2(c+d x)-2 a \left (4 (3 A+4 C) a^2+b^2 (19 A+32 C)\right ) \cos (c+d x)+b \left (3 A b^2-4 a^2 (13 A+20 C)\right )\right ) \sec ^2(c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{4 a}\right )+\frac {A b \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {\left (4 a^2 (3 A+4 C)+A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\int \frac {-b \left (4 (3 A+4 C) a^2+A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-2 a \left (4 (3 A+4 C) a^2+b^2 (19 A+32 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+b \left (3 A b^2-4 a^2 (13 A+20 C)\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^2 \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{4 a}\right )+\frac {A b \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {\left (4 a^2 (3 A+4 C)+A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\int -\frac {\left (16 (3 A+4 C) a^4+24 b^2 (A+2 C) a^2+2 b \left (4 (3 A+4 C) a^2+A b^2\right ) \cos (c+d x) a+3 A b^4+b^2 \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \cos ^2(c+d x)\right ) \sec (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx}{a}+\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}}{4 a}\right )+\frac {A b \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {\left (4 a^2 (3 A+4 C)+A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\int \frac {\left (16 (3 A+4 C) a^4+24 b^2 (A+2 C) a^2+2 b \left (4 (3 A+4 C) a^2+A b^2\right ) \cos (c+d x) a+3 A b^4+b^2 \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \cos ^2(c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{2 a}}{4 a}\right )+\frac {A b \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {\left (4 a^2 (3 A+4 C)+A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\int \frac {16 (3 A+4 C) a^4+24 b^2 (A+2 C) a^2+2 b \left (4 (3 A+4 C) a^2+A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+3 A b^4+b^2 \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}}{4 a}\right )+\frac {A b \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {\left (4 a^2 (3 A+4 C)+A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \int \sqrt {a+b \cos (c+d x)}dx-\frac {\int -\frac {\left (b \left (16 (3 A+4 C) a^4+24 b^2 (A+2 C) a^2+3 A b^4\right )-a b^2 \left (A b^2-4 a^2 (19 A+28 C)\right ) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}}{2 a}}{4 a}\right )+\frac {A b \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {\left (4 a^2 (3 A+4 C)+A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \int \sqrt {a+b \cos (c+d x)}dx+\frac {\int \frac {\left (b \left (16 (3 A+4 C) a^4+24 b^2 (A+2 C) a^2+3 A b^4\right )-a b^2 \left (A b^2-4 a^2 (19 A+28 C)\right ) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}}{2 a}}{4 a}\right )+\frac {A b \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {\left (4 a^2 (3 A+4 C)+A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {\int \frac {b \left (16 (3 A+4 C) a^4+24 b^2 (A+2 C) a^2+3 A b^4\right )-a b^2 \left (A b^2-4 a^2 (19 A+28 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{2 a}}{4 a}\right )+\frac {A b \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {\left (4 a^2 (3 A+4 C)+A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\int \frac {b \left (16 (3 A+4 C) a^4+24 b^2 (A+2 C) a^2+3 A b^4\right )-a b^2 \left (A b^2-4 a^2 (19 A+28 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{2 a}}{4 a}\right )+\frac {A b \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {\left (4 a^2 (3 A+4 C)+A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\int \frac {b \left (16 (3 A+4 C) a^4+24 b^2 (A+2 C) a^2+3 A b^4\right )-a b^2 \left (A b^2-4 a^2 (19 A+28 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{2 a}}{4 a}\right )+\frac {A b \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {\left (4 a^2 (3 A+4 C)+A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {\int \frac {b \left (16 (3 A+4 C) a^4+24 b^2 (A+2 C) a^2+3 A b^4\right )-a b^2 \left (A b^2-4 a^2 (19 A+28 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}+\frac {2 b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}}{4 a}\right )+\frac {A b \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {\left (4 a^2 (3 A+4 C)+A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {b \left (16 a^4 (3 A+4 C)+24 a^2 b^2 (A+2 C)+3 A b^4\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx-a b^2 \left (A b^2-4 a^2 (19 A+28 C)\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}+\frac {2 b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}}{4 a}\right )+\frac {A b \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {\left (4 a^2 (3 A+4 C)+A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {b \left (16 a^4 (3 A+4 C)+24 a^2 b^2 (A+2 C)+3 A b^4\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-a b^2 \left (A b^2-4 a^2 (19 A+28 C)\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}+\frac {2 b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}}{4 a}\right )+\frac {A b \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {\left (4 a^2 (3 A+4 C)+A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {b \left (16 a^4 (3 A+4 C)+24 a^2 b^2 (A+2 C)+3 A b^4\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {a b^2 \left (A b^2-4 a^2 (19 A+28 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}}{4 a}\right )+\frac {A b \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {\left (4 a^2 (3 A+4 C)+A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {b \left (16 a^4 (3 A+4 C)+24 a^2 b^2 (A+2 C)+3 A b^4\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {a b^2 \left (A b^2-4 a^2 (19 A+28 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}}{4 a}\right )+\frac {A b \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {\left (4 a^2 (3 A+4 C)+A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {b \left (16 a^4 (3 A+4 C)+24 a^2 b^2 (A+2 C)+3 A b^4\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 a b^2 \left (A b^2-4 a^2 (19 A+28 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}}{4 a}\right )+\frac {A b \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {\left (4 a^2 (3 A+4 C)+A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {\frac {b \left (16 a^4 (3 A+4 C)+24 a^2 b^2 (A+2 C)+3 A b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {\sec (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}-\frac {2 a b^2 \left (A b^2-4 a^2 (19 A+28 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}}{4 a}\right )+\frac {A b \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {\left (4 a^2 (3 A+4 C)+A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {\frac {b \left (16 a^4 (3 A+4 C)+24 a^2 b^2 (A+2 C)+3 A b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}-\frac {2 a b^2 \left (A b^2-4 a^2 (19 A+28 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}}{4 a}\right )+\frac {A b \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {\left (4 a^2 (3 A+4 C)+A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {2 b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\frac {2 b \left (16 a^4 (3 A+4 C)+24 a^2 b^2 (A+2 C)+3 A b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}-\frac {2 a b^2 \left (A b^2-4 a^2 (19 A+28 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}}{2 a}}{4 a}\right )+\frac {A b \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
Input:
Int[(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]
Output:
(A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + ((A*b*S qrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/d + (((A*b^2 + 4*a^2* (3*A + 4*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) - (-1/2*((2*b*(3*A*b^2 - 4*a^2*(13*A + 20*C))*Sqrt[a + b*Cos[c + d*x]]*Elli pticE[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + ((-2*a*b^2*(A*b^2 - 4*a^2*(19*A + 28*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]]) + (2*b*(3*A*b^4 + 24*a^2*b^2*(A + 2*C) + 16*a^4*(3*A + 4*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 + (b*(3*A*b^2 - 4*a^2*(13*A + 20*C))*Sqrt[a + b*Co s[c + d*x]]*Tan[c + d*x])/(a*d))/(4*a))/2)/8
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[(-(c^2*C - B*c*d + A*d^2))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x ] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f *x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d , 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C + A*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^ 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A* d*(b*d*m + a*c*(n + 1)) + c*C*(b*c*m + a*d*(n + 1)) - (A*d*(a*d*(n + 2) - b *c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*( A*d^2*(m + n + 2) + C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(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. \(3301\) vs. \(2(416)=832\).
Time = 8.55 (sec) , antiderivative size = 3302, normalized size of antiderivative = 7.57
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(3302\) |
| default | \(\text {Expression too large to display}\) | \(3534\) |
Input:
int((a+b*cos(d*x+c))^(3/2)*(A+C*cos(d*x+c)^2)*sec(d*x+c)^5,x,method=_RETUR NVERBOSE)
Output:
-1/64*A*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*((-166 4*a^2*b^2+96*b^4)*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^10+(1216*a^3*b+332 8*a^2*b^2-16*a*b^3-192*b^4)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-192* a^4-1824*a^3*b-2704*a^2*b^2+24*a*b^3+144*b^4)*sin(1/2*d*x+1/2*c)^6*cos(1/2 *d*x+1/2*c)+(192*a^4+1072*a^3*b+1040*a^2*b^2-12*a*b^3-48*b^4)*sin(1/2*d*x+ 1/2*c)^4*cos(1/2*d*x+1/2*c)+(-80*a^4-232*a^3*b-156*a^2*b^2+2*a*b^3+6*b^4)* sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+16*(-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)*(76*b*EllipticF(cos(1/2* d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3-b^3*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/ (a-b))^(1/2))*a-52*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3*b+ 52*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^2*b^2+3*EllipticE(co s(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a*b^3-3*b^4*EllipticE(cos(1/2*d*x+1/2 *c),(-2*b/(a-b))^(1/2))-48*EllipticPi(cos(1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1 /2))*a^4-24*EllipticPi(cos(1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1/2))*b^2*a^2-3* EllipticPi(cos(1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1/2))*b^4)*sin(1/2*d*x+1/2*c )^8-32*(-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)*(76*b*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3-b ^3*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a-52*EllipticE(cos(1/2 *d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3*b+52*EllipticE(cos(1/2*d*x+1/2*c),(-2* b/(a-b))^(1/2))*a^2*b^2+3*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/...
Timed out. \[ \int (a+b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))^(3/2)*(A+C*cos(d*x+c)^2)*sec(d*x+c)^5,x, algori thm="fricas")
Output:
Timed out
Timed out. \[ \int (a+b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))**(3/2)*(A+C*cos(d*x+c)**2)*sec(d*x+c)**5,x)
Output:
Timed out
\[ \int (a+b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{5} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(3/2)*(A+C*cos(d*x+c)^2)*sec(d*x+c)^5,x, algori thm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c) + a)^(3/2)*sec(d*x + c)^5 , x)
\[ \int (a+b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{5} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(3/2)*(A+C*cos(d*x+c)^2)*sec(d*x+c)^5,x, algori thm="giac")
Output:
integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c) + a)^(3/2)*sec(d*x + c)^5 , x)
Timed out. \[ \int (a+b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\int \frac {\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2}}{{\cos \left (c+d\,x\right )}^5} \,d x \] Input:
int(((A + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(3/2))/cos(c + d*x)^5,x)
Output:
int(((A + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(3/2))/cos(c + d*x)^5, x)
\[ \int (a+b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right ) \sec \left (d x +c \right )^{5}d x \right ) a b +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{5}d x \right ) b c +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{5}d x \right ) a c +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{5}d x \right ) a^{2} \] Input:
int((a+b*cos(d*x+c))^(3/2)*(A+C*cos(d*x+c)^2)*sec(d*x+c)^5,x)
Output:
int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)*sec(c + d*x)**5,x)*a*b + int(sqr t(cos(c + d*x)*b + a)*cos(c + d*x)**3*sec(c + d*x)**5,x)*b*c + int(sqrt(co s(c + d*x)*b + a)*cos(c + d*x)**2*sec(c + d*x)**5,x)*a*c + int(sqrt(cos(c + d*x)*b + a)*sec(c + d*x)**5,x)*a**2