Integrand size = 43, antiderivative size = 416 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx=-\frac {\left (15 A b^3+4 a^3 B-12 a b^2 B-a^2 (7 A b-8 b C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{4 a^3 \left (a^2-b^2\right ) d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {(5 A b-4 a B) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{4 a^2 d \sqrt {a+b \cos (c+d x)}}+\frac {\left (15 A b^2-12 a b B+4 a^2 (A+2 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{4 a^3 d \sqrt {a+b \cos (c+d x)}}+\frac {b \left (15 A b^3+4 a^3 B-12 a b^2 B-a^2 (7 A b-8 b C)\right ) \sin (c+d x)}{4 a^3 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}-\frac {(5 A b-4 a B) \tan (c+d x)}{4 a^2 d \sqrt {a+b \cos (c+d x)}}+\frac {A \sec (c+d x) \tan (c+d x)}{2 a d \sqrt {a+b \cos (c+d x)}} \] Output:
-1/4*(15*A*b^3+4*B*a^3-12*B*a*b^2-a^2*(7*A*b-8*C*b))*(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^3/(a^2-b^2)/d/(( a+b*cos(d*x+c))/(a+b))^(1/2)-1/4*(5*A*b-4*B*a)*((a+b*cos(d*x+c))/(a+b))^(1 /2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2)*(b/(a+b))^(1/2))/a^2/d/(a+b*cos( d*x+c))^(1/2)+1/4*(15*A*b^2-12*B*a*b+4*a^2*(A+2*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^3/d/(a +b*cos(d*x+c))^(1/2)+1/4*b*(15*A*b^3+4*B*a^3-12*B*a*b^2-a^2*(7*A*b-8*C*b)) *sin(d*x+c)/a^3/(a^2-b^2)/d/(a+b*cos(d*x+c))^(1/2)-1/4*(5*A*b-4*B*a)*tan(d *x+c)/a^2/d/(a+b*cos(d*x+c))^(1/2)+1/2*A*sec(d*x+c)*tan(d*x+c)/a/d/(a+b*co s(d*x+c))^(1/2)
Result contains complex when optimal does not.
Time = 7.57 (sec) , antiderivative size = 723, normalized size of antiderivative = 1.74 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx=-\frac {\frac {2 \left (4 a^3 A b-20 a A b^3+16 a^2 b^2 B-16 a^3 b C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}+\frac {2 \left (8 a^4 A+29 a^2 A b^2-45 A b^4-28 a^3 b B+36 a b^3 B+16 a^4 C-24 a^2 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}-\frac {2 i \left (7 a^2 A b^2-15 A b^4-4 a^3 b B+12 a b^3 B-8 a^2 b^2 C\right ) \sqrt {\frac {b-b \cos (c+d x)}{a+b}} \sqrt {-\frac {b+b \cos (c+d x)}{a-b}} \cos (2 (c+d x)) \left (2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )-b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sin (c+d x)}{a \sqrt {-\frac {1}{a+b}} \sqrt {1-\cos ^2(c+d x)} \sqrt {-\frac {a^2-b^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2}{b^2}} \left (2 a^2-b^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2\right )}}{16 a^3 (-a+b) (a+b) d}+\frac {\sqrt {a+b \cos (c+d x)} \left (\frac {\sec (c+d x) (-7 A b \sin (c+d x)+4 a B \sin (c+d x))}{4 a^3}+\frac {2 \left (A b^4 \sin (c+d x)-a b^3 B \sin (c+d x)+a^2 b^2 C \sin (c+d x)\right )}{a^3 \left (a^2-b^2\right ) (a+b \cos (c+d x))}+\frac {A \sec (c+d x) \tan (c+d x)}{2 a^2}\right )}{d} \] Input:
Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b* Cos[c + d*x])^(3/2),x]
Output:
-1/16*((2*(4*a^3*A*b - 20*a*A*b^3 + 16*a^2*b^2*B - 16*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*(8*a^4*A + 29*a^2*A*b^2 - 45*A*b^4 - 28*a^3*b*B + 36*a* b^3*B + 16*a^4*C - 24*a^2*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*Ellipt icPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(7* a^2*A*b^2 - 15*A*b^4 - 4*a^3*b*B + 12*a*b^3*B - 8*a^2*b^2*C)*Sqrt[(b - b*C os[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*A rcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)]))*S in[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^3*(-a + b)* (a + b)*d) + (Sqrt[a + b*Cos[c + d*x]]*((Sec[c + d*x]*(-7*A*b*Sin[c + d*x] + 4*a*B*Sin[c + d*x]))/(4*a^3) + (2*(A*b^4*Sin[c + d*x] - a*b^3*B*Sin[c + d*x] + a^2*b^2*C*Sin[c + d*x]))/(a^3*(a^2 - b^2)*(a + b*Cos[c + d*x])) + (A*Sec[c + d*x]*Tan[c + d*x])/(2*a^2)))/d
Time = 3.83 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.06, number of steps used = 24, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.558, Rules used = {3042, 3534, 27, 3042, 3534, 27, 3042, 3534, 27, 3042, 3538, 25, 3042, 3134, 3042, 3132, 3481, 3042, 3142, 3042, 3140, 3286, 3042, 3284}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sec ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\int -\frac {\left (-3 A b \cos ^2(c+d x)-2 a (A+2 C) \cos (c+d x)+5 A b-4 a B\right ) \sec ^2(c+d x)}{2 (a+b \cos (c+d x))^{3/2}}dx}{2 a}+\frac {A \tan (c+d x) \sec (c+d x)}{2 a d \sqrt {a+b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d \sqrt {a+b \cos (c+d x)}}-\frac {\int \frac {\left (-3 A b \cos ^2(c+d x)-2 a (A+2 C) \cos (c+d x)+5 A b-4 a B\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^{3/2}}dx}{4 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d \sqrt {a+b \cos (c+d x)}}-\frac {\int \frac {-3 A b \sin \left (c+d x+\frac {\pi }{2}\right )^2-2 a (A+2 C) \sin \left (c+d x+\frac {\pi }{2}\right )+5 A b-4 a B}{\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{4 a}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d \sqrt {a+b \cos (c+d x)}}-\frac {\frac {\int -\frac {\left (4 (A+2 C) a^2-12 b B a+6 A b \cos (c+d x) a+15 A b^2-b (5 A b-4 a B) \cos ^2(c+d x)\right ) \sec (c+d x)}{2 (a+b \cos (c+d x))^{3/2}}dx}{a}+\frac {(5 A b-4 a B) \tan (c+d x)}{a d \sqrt {a+b \cos (c+d x)}}}{4 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d \sqrt {a+b \cos (c+d x)}}-\frac {\frac {(5 A b-4 a B) \tan (c+d x)}{a d \sqrt {a+b \cos (c+d x)}}-\frac {\int \frac {\left (4 (A+2 C) a^2-12 b B a+6 A b \cos (c+d x) a+15 A b^2-b (5 A b-4 a B) \cos ^2(c+d x)\right ) \sec (c+d x)}{(a+b \cos (c+d x))^{3/2}}dx}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d \sqrt {a+b \cos (c+d x)}}-\frac {\frac {(5 A b-4 a B) \tan (c+d x)}{a d \sqrt {a+b \cos (c+d x)}}-\frac {\int \frac {4 (A+2 C) a^2-12 b B a+6 A b \sin \left (c+d x+\frac {\pi }{2}\right ) a+15 A b^2-b (5 A b-4 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d \sqrt {a+b \cos (c+d x)}}-\frac {\frac {(5 A b-4 a B) \tan (c+d x)}{a d \sqrt {a+b \cos (c+d x)}}-\frac {\frac {2 \int \frac {\left (-b \left (4 B a^3-(7 A b-8 b C) a^2-12 b^2 B a+15 A b^3\right ) \cos ^2(c+d x)-2 a b \left (-\left ((A-4 C) a^2\right )-4 b B a+5 A b^2\right ) \cos (c+d x)+\left (a^2-b^2\right ) \left (4 (A+2 C) a^2-12 b B a+15 A b^2\right )\right ) \sec (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx}{a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (4 a^3 B-a^2 (7 A b-8 b C)-12 a b^2 B+15 A b^3\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d \sqrt {a+b \cos (c+d x)}}-\frac {\frac {(5 A b-4 a B) \tan (c+d x)}{a d \sqrt {a+b \cos (c+d x)}}-\frac {\frac {\int \frac {\left (-b \left (4 B a^3-(7 A b-8 b C) a^2-12 b^2 B a+15 A b^3\right ) \cos ^2(c+d x)-2 a b \left (-\left ((A-4 C) a^2\right )-4 b B a+5 A b^2\right ) \cos (c+d x)+\left (a^2-b^2\right ) \left (4 (A+2 C) a^2-12 b B a+15 A b^2\right )\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (4 a^3 B-a^2 (7 A b-8 b C)-12 a b^2 B+15 A b^3\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d \sqrt {a+b \cos (c+d x)}}-\frac {\frac {(5 A b-4 a B) \tan (c+d x)}{a d \sqrt {a+b \cos (c+d x)}}-\frac {\frac {\int \frac {-b \left (4 B a^3-(7 A b-8 b C) a^2-12 b^2 B a+15 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-2 a b \left (-\left ((A-4 C) a^2\right )-4 b B a+5 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+\left (a^2-b^2\right ) \left (4 (A+2 C) a^2-12 b B a+15 A b^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (4 a^3 B-a^2 (7 A b-8 b C)-12 a b^2 B+15 A b^3\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d \sqrt {a+b \cos (c+d x)}}-\frac {\frac {(5 A b-4 a B) \tan (c+d x)}{a d \sqrt {a+b \cos (c+d x)}}-\frac {\frac {-\frac {\int -\frac {\left (b \left (a^2-b^2\right ) \left (4 (A+2 C) a^2-12 b B a+15 A b^2\right )-a b \left (a^2-b^2\right ) (5 A b-4 a B) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\left (\left (4 a^3 B-a^2 b (7 A-8 C)-12 a b^2 B+15 A b^3\right ) \int \sqrt {a+b \cos (c+d x)}dx\right )}{a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (4 a^3 B-a^2 (7 A b-8 b C)-12 a b^2 B+15 A b^3\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d \sqrt {a+b \cos (c+d x)}}-\frac {\frac {(5 A b-4 a B) \tan (c+d x)}{a d \sqrt {a+b \cos (c+d x)}}-\frac {\frac {\frac {\int \frac {\left (b \left (a^2-b^2\right ) \left (4 (A+2 C) a^2-12 b B a+15 A b^2\right )-a b \left (a^2-b^2\right ) (5 A b-4 a B) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\left (4 a^3 B-a^2 b (7 A-8 C)-12 a b^2 B+15 A b^3\right ) \int \sqrt {a+b \cos (c+d x)}dx}{a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (4 a^3 B-a^2 (7 A b-8 b C)-12 a b^2 B+15 A b^3\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d \sqrt {a+b \cos (c+d x)}}-\frac {\frac {(5 A b-4 a B) \tan (c+d x)}{a d \sqrt {a+b \cos (c+d x)}}-\frac {\frac {\frac {\int \frac {b \left (a^2-b^2\right ) \left (4 (A+2 C) a^2-12 b B a+15 A b^2\right )-a b \left (a^2-b^2\right ) (5 A b-4 a B) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\left (4 a^3 B-a^2 b (7 A-8 C)-12 a b^2 B+15 A b^3\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (4 a^3 B-a^2 (7 A b-8 b C)-12 a b^2 B+15 A b^3\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d \sqrt {a+b \cos (c+d x)}}-\frac {\frac {(5 A b-4 a B) \tan (c+d x)}{a d \sqrt {a+b \cos (c+d x)}}-\frac {\frac {\frac {\int \frac {b \left (a^2-b^2\right ) \left (4 (A+2 C) a^2-12 b B a+15 A b^2\right )-a b \left (a^2-b^2\right ) (5 A b-4 a B) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (4 a^3 B-a^2 b (7 A-8 C)-12 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (4 a^3 B-a^2 (7 A b-8 b C)-12 a b^2 B+15 A b^3\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d \sqrt {a+b \cos (c+d x)}}-\frac {\frac {(5 A b-4 a B) \tan (c+d x)}{a d \sqrt {a+b \cos (c+d x)}}-\frac {\frac {\frac {\int \frac {b \left (a^2-b^2\right ) \left (4 (A+2 C) a^2-12 b B a+15 A b^2\right )-a b \left (a^2-b^2\right ) (5 A b-4 a B) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (4 a^3 B-a^2 b (7 A-8 C)-12 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (4 a^3 B-a^2 (7 A b-8 b C)-12 a b^2 B+15 A b^3\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d \sqrt {a+b \cos (c+d x)}}-\frac {\frac {(5 A b-4 a B) \tan (c+d x)}{a d \sqrt {a+b \cos (c+d x)}}-\frac {\frac {\frac {\int \frac {b \left (a^2-b^2\right ) \left (4 (A+2 C) a^2-12 b B a+15 A b^2\right )-a b \left (a^2-b^2\right ) (5 A b-4 a B) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \left (4 a^3 B-a^2 b (7 A-8 C)-12 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (4 a^3 B-a^2 (7 A b-8 b C)-12 a b^2 B+15 A b^3\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d \sqrt {a+b \cos (c+d x)}}-\frac {\frac {(5 A b-4 a B) \tan (c+d x)}{a d \sqrt {a+b \cos (c+d x)}}-\frac {\frac {\frac {b \left (a^2-b^2\right ) \left (4 a^2 (A+2 C)-12 a b B+15 A b^2\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx-a b \left (a^2-b^2\right ) (5 A b-4 a B) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\frac {2 \left (4 a^3 B-a^2 b (7 A-8 C)-12 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (4 a^3 B-a^2 (7 A b-8 b C)-12 a b^2 B+15 A b^3\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d \sqrt {a+b \cos (c+d x)}}-\frac {\frac {(5 A b-4 a B) \tan (c+d x)}{a d \sqrt {a+b \cos (c+d x)}}-\frac {\frac {\frac {b \left (a^2-b^2\right ) \left (4 a^2 (A+2 C)-12 a b B+15 A b^2\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-a b \left (a^2-b^2\right ) (5 A b-4 a B) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \left (4 a^3 B-a^2 b (7 A-8 C)-12 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (4 a^3 B-a^2 (7 A b-8 b C)-12 a b^2 B+15 A b^3\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d \sqrt {a+b \cos (c+d x)}}-\frac {\frac {(5 A b-4 a B) \tan (c+d x)}{a d \sqrt {a+b \cos (c+d x)}}-\frac {\frac {\frac {b \left (a^2-b^2\right ) \left (4 a^2 (A+2 C)-12 a b B+15 A b^2\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {a b \left (a^2-b^2\right ) (5 A b-4 a B) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 \left (4 a^3 B-a^2 b (7 A-8 C)-12 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (4 a^3 B-a^2 (7 A b-8 b C)-12 a b^2 B+15 A b^3\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d \sqrt {a+b \cos (c+d x)}}-\frac {\frac {(5 A b-4 a B) \tan (c+d x)}{a d \sqrt {a+b \cos (c+d x)}}-\frac {\frac {\frac {b \left (a^2-b^2\right ) \left (4 a^2 (A+2 C)-12 a b B+15 A b^2\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {a b \left (a^2-b^2\right ) (5 A b-4 a B) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 \left (4 a^3 B-a^2 b (7 A-8 C)-12 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (4 a^3 B-a^2 (7 A b-8 b C)-12 a b^2 B+15 A b^3\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d \sqrt {a+b \cos (c+d x)}}-\frac {\frac {(5 A b-4 a B) \tan (c+d x)}{a d \sqrt {a+b \cos (c+d x)}}-\frac {\frac {\frac {b \left (a^2-b^2\right ) \left (4 a^2 (A+2 C)-12 a b B+15 A b^2\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 a b \left (a^2-b^2\right ) (5 A b-4 a B) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 \left (4 a^3 B-a^2 b (7 A-8 C)-12 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (4 a^3 B-a^2 (7 A b-8 b C)-12 a b^2 B+15 A b^3\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d \sqrt {a+b \cos (c+d x)}}-\frac {\frac {(5 A b-4 a B) \tan (c+d x)}{a d \sqrt {a+b \cos (c+d x)}}-\frac {\frac {\frac {\frac {b \left (a^2-b^2\right ) \left (4 a^2 (A+2 C)-12 a b B+15 A b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {\sec (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}-\frac {2 a b \left (a^2-b^2\right ) (5 A b-4 a B) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 \left (4 a^3 B-a^2 b (7 A-8 C)-12 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (4 a^3 B-a^2 (7 A b-8 b C)-12 a b^2 B+15 A b^3\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d \sqrt {a+b \cos (c+d x)}}-\frac {\frac {(5 A b-4 a B) \tan (c+d x)}{a d \sqrt {a+b \cos (c+d x)}}-\frac {\frac {\frac {\frac {b \left (a^2-b^2\right ) \left (4 a^2 (A+2 C)-12 a b B+15 A b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}-\frac {2 a b \left (a^2-b^2\right ) (5 A b-4 a B) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 \left (4 a^3 B-a^2 b (7 A-8 C)-12 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (4 a^3 B-a^2 (7 A b-8 b C)-12 a b^2 B+15 A b^3\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d \sqrt {a+b \cos (c+d x)}}-\frac {\frac {(5 A b-4 a B) \tan (c+d x)}{a d \sqrt {a+b \cos (c+d x)}}-\frac {\frac {2 b \sin (c+d x) \left (4 a^3 B-a^2 (7 A b-8 b C)-12 a b^2 B+15 A b^3\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}+\frac {\frac {\frac {2 b \left (a^2-b^2\right ) \left (4 a^2 (A+2 C)-12 a b B+15 A b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}-\frac {2 a b \left (a^2-b^2\right ) (5 A b-4 a B) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 \left (4 a^3 B-a^2 b (7 A-8 C)-12 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}}{2 a}}{4 a}\) |
Input:
Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^(3/2),x]
Output:
(A*Sec[c + d*x]*Tan[c + d*x])/(2*a*d*Sqrt[a + b*Cos[c + d*x]]) - (-1/2*((( -2*(15*A*b^3 + 4*a^3*B - 12*a*b^2*B - a^2*b*(7*A - 8*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x] )/(a + b)]) + ((-2*a*b*(a^2 - b^2)*(5*A*b - 4*a*B)*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*(a^2 - b^2)*(15*A*b^2 - 12*a*b*B + 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*Sq rt[a + b*Cos[c + d*x]]))/b)/(a*(a^2 - b^2)) + (2*b*(15*A*b^3 + 4*a^3*B - 1 2*a*b^2*B - a^2*(7*A*b - 8*b*C))*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b *Cos[c + d*x]]))/a + ((5*A*b - 4*a*B)*Tan[c + d*x])/(a*d*Sqrt[a + b*Cos[c + d*x]]))/(4*a)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 0]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[ 2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a, b, c , d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt [c + d*Sin[e + f*x]] Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/(c + d))*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a* d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !GtQ[c + d, 0]
Int[(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]))/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[ B/d Int[(a + b*Sin[e + f*x])^m, x], x] - Simp[(B*c - A*d)/d Int[(a + b* Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int [(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A *b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ [n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) | | EqQ[a, 0])))
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[C/(b*d) Int[Sqrt[a + b*Sin[e + f*x]], x] , x] - Simp[1/(b*d) Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[ e + f*x], x]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; Fre eQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0 ] && NeQ[c^2 - d^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1580\) vs. \(2(400)=800\).
Time = 7.59 (sec) , antiderivative size = 1581, normalized size of antiderivative = 3.80
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1581\) |
| parts | \(\text {Expression too large to display}\) | \(2822\) |
Input:
int((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+b*cos(d*x+c))^(3/2),x, method=_RETURNVERBOSE)
Output:
-(-(-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/ 2*cos(1/2*d*x+1/2*c)/a*(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c) ^2)^(1/2)/(-1+2*cos(1/2*d*x+1/2*c)^2)^2+3/4*b/a^2*cos(1/2*d*x+1/2*c)*(-2*b *sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)/(-1+2*cos(1/2*d*x+ 1/2*c)^2)-1/8*b/a*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+ a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(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))+3/8/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*b*sin(1/2* d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*b*EllipticE(cos(1/2*d*x+1/2 *c),(-2*b/(a-b))^(1/2))-3/8*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*b*sin(1/2*d*x+1/2*c)^4+(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/ 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*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi( cos(1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1/2))-3/8/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*b*sin(1/2*d*x+1/2*c)^ 4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),2,(-2*b/ (a-b))^(1/2))*b^2)-2*(A*b-B*a)/a^2*(-cos(1/2*d*x+1/2*c)/a*(-2*b*sin(1/2*d* x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)/(-1+2*cos(1/2*d*x+1/2*c)^2)+1 /2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^...
Timed out. \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+b*cos(d*x+c))^(3 /2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx=\int \frac {\left (A + B \cos {\left (c + d x \right )} + C \cos ^{2}{\left (c + d x \right )}\right ) \sec ^{3}{\left (c + d x \right )}}{\left (a + b \cos {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x+c)**3/(a+b*cos(d*x+c))* *(3/2),x)
Output:
Integral((A + B*cos(c + d*x) + C*cos(c + d*x)**2)*sec(c + d*x)**3/(a + b*c os(c + d*x))**(3/2), x)
Timed out. \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+b*cos(d*x+c))^(3 /2),x, algorithm="maxima")
Output:
Timed out
\[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{3}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+b*cos(d*x+c))^(3 /2),x, algorithm="giac")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*sec(d*x + c)^3/(b*cos(d* x + c) + a)^(3/2), x)
Timed out. \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx=\int \frac {C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A}{{\cos \left (c+d\,x\right )}^3\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \] Input:
int((A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^3*(a + b*cos(c + d*x))^(3/2)),x)
Output:
int((A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^3*(a + b*cos(c + d*x))^(3/2)), x)
\[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx=\left (\int \frac {\sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right ) \sec \left (d x +c \right )^{3}}{\cos \left (d x +c \right )^{2} b^{2}+2 \cos \left (d x +c \right ) a b +a^{2}}d x \right ) b +\left (\int \frac {\sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{3}}{\cos \left (d x +c \right )^{2} b^{2}+2 \cos \left (d x +c \right ) a b +a^{2}}d x \right ) c +\left (\int \frac {\sqrt {\cos \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{3}}{\cos \left (d x +c \right )^{2} b^{2}+2 \cos \left (d x +c \right ) a b +a^{2}}d x \right ) a \] Input:
int((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+b*cos(d*x+c))^(3/2),x)
Output:
int((sqrt(cos(c + d*x)*b + a)*cos(c + d*x)*sec(c + d*x)**3)/(cos(c + d*x)* *2*b**2 + 2*cos(c + d*x)*a*b + a**2),x)*b + int((sqrt(cos(c + d*x)*b + a)* cos(c + d*x)**2*sec(c + d*x)**3)/(cos(c + d*x)**2*b**2 + 2*cos(c + d*x)*a* b + a**2),x)*c + int((sqrt(cos(c + d*x)*b + a)*sec(c + d*x)**3)/(cos(c + d *x)**2*b**2 + 2*cos(c + d*x)*a*b + a**2),x)*a