Integrand size = 43, antiderivative size = 372 \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=-\frac {\left (12 a^2 B-24 b^2 B+a b (27 A-56 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{12 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\left (12 a^3 B+48 a b^2 B+8 b^3 (3 A+C)+a^2 b (33 A+16 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{12 d \sqrt {a+b \cos (c+d x)}}+\frac {a \left (15 A b^2+20 a b B+4 a^2 (A+2 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{4 d \sqrt {a+b \cos (c+d x)}}-\frac {b (21 A b+12 a B-8 b C) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{12 d}+\frac {(5 A b+4 a B) (a+b \cos (c+d x))^{3/2} \tan (c+d x)}{4 d}+\frac {A (a+b \cos (c+d x))^{5/2} \sec (c+d x) \tan (c+d x)}{2 d} \] Output:
-1/12*(12*B*a^2-24*B*b^2+a*b*(27*A-56*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))/d/((a+b*cos(d*x+c))/(a+b))^(1 /2)+1/12*(12*B*a^3+48*B*a*b^2+8*b^3*(3*A+C)+a^2*b*(33*A+16*C))*((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)) /d/(a+b*cos(d*x+c))^(1/2)+1/4*a*(15*A*b^2+20*B*a*b+4*a^2*(A+2*C))*((a+b*co s(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))/d/(a+b*cos(d*x+c))^(1/2)-1/12*b*(21*A*b+12*B*a-8*C*b)*(a+b*cos(d*x+c ))^(1/2)*sin(d*x+c)/d+1/4*(5*A*b+4*B*a)*(a+b*cos(d*x+c))^(3/2)*tan(d*x+c)/ d+1/2*A*(a+b*cos(d*x+c))^(5/2)*sec(d*x+c)*tan(d*x+c)/d
Result contains complex when optimal does not.
Time = 6.18 (sec) , antiderivative size = 492, normalized size of antiderivative = 1.32 \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\frac {\frac {8 b \left (36 a b B+4 b^2 (3 A+C)+3 a^2 (A+12 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}+\frac {2 \left (108 a^2 b B+24 b^3 B+24 a^3 (A+2 C)+7 a b^2 (9 A+8 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}+\frac {2 i \left (-27 a A b-12 a^2 B+24 b^2 B+56 a b C\right ) \sqrt {-\frac {b (-1+\cos (c+d x))}{a+b}} \sqrt {\frac {b (1+\cos (c+d x))}{-a+b}} \csc (c+d x) \left (-2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (-2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )+b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right )}{a b \sqrt {-\frac {1}{a+b}}}+4 \sqrt {a+b \cos (c+d x)} \sec (c+d x) \left (3 a (9 A b+4 a B) \sin (c+d x)+4 b^2 C \sin (2 (c+d x))+6 a^2 A \tan (c+d x)\right )}{48 d} \] Input:
Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^ 2)*Sec[c + d*x]^3,x]
Output:
((8*b*(36*a*b*B + 4*b^2*(3*A + C) + 3*a^2*(A + 12*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*(108*a^2*b*B + 24*b^3*B + 24*a^3*(A + 2*C) + 7*a*b^2*(9*A + 8*C) )*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + ((2*I)*(-27*a*A*b - 12*a^2*B + 24*b^2*B + 56*a*b*C)*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d *x]))/(-a + b)]*Csc[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)])))/(a*b*Sqrt[-(a + b)^(-1)]) + 4*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*(3*a*(9*A*b + 4*a*B)*Sin[c + d*x] + 4*b^2*C*Sin[2*(c + d*x)] + 6*a^2*A*Tan[c + d*x]))/(48*d)
Time = 3.43 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.03, number of steps used = 24, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.558, Rules used = {3042, 3526, 27, 3042, 3526, 27, 3042, 3528, 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.
| \(\displaystyle \int \sec ^3(c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{2} (a+b \cos (c+d x))^{3/2} \left (-b (3 A-4 C) \cos ^2(c+d x)+2 (2 b B+a (A+2 C)) \cos (c+d x)+5 A b+4 a B\right ) \sec ^2(c+d x)dx+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{5/2}}{2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \int (a+b \cos (c+d x))^{3/2} \left (-b (3 A-4 C) \cos ^2(c+d x)+2 (2 b B+a (A+2 C)) \cos (c+d x)+5 A b+4 a B\right ) \sec ^2(c+d x)dx+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{5/2}}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (-b (3 A-4 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 (2 b B+a (A+2 C)) \sin \left (c+d x+\frac {\pi }{2}\right )+5 A b+4 a B\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{5/2}}{2 d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{4} \left (\int \frac {1}{2} \sqrt {a+b \cos (c+d x)} \left (4 (A+2 C) a^2+20 b B a+15 A b^2-b (21 A b-8 C b+12 a B) \cos ^2(c+d x)+2 b (4 b B-a (A-8 C)) \cos (c+d x)\right ) \sec (c+d x)dx+\frac {(4 a B+5 A b) \tan (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{5/2}}{2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \int \sqrt {a+b \cos (c+d x)} \left (4 (A+2 C) a^2+20 b B a+15 A b^2-b (21 A b-8 C b+12 a B) \cos ^2(c+d x)-2 b (a A-4 b B-8 a C) \cos (c+d x)\right ) \sec (c+d x)dx+\frac {(4 a B+5 A b) \tan (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{5/2}}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \int \frac {\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (4 (A+2 C) a^2+20 b B a+15 A b^2-b (21 A b-8 C b+12 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2-2 b (a A-4 b B-8 a C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {(4 a B+5 A b) \tan (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{5/2}}{2 d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\frac {2}{3} \int \frac {\left (-b \left (12 B a^2+b (27 A-56 C) a-24 b^2 B\right ) \cos ^2(c+d x)+2 b \left (3 (A+12 C) a^2+36 b B a+4 b^2 (3 A+C)\right ) \cos (c+d x)+3 a \left (4 (A+2 C) a^2+20 b B a+15 A b^2\right )\right ) \sec (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx-\frac {2 b \sin (c+d x) (12 a B+21 A b-8 b C) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {(4 a B+5 A b) \tan (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{5/2}}{2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\frac {1}{3} \int \frac {\left (-b \left (12 B a^2+b (27 A-56 C) a-24 b^2 B\right ) \cos ^2(c+d x)+2 b \left (3 (A+12 C) a^2+36 b B a+4 b^2 (3 A+C)\right ) \cos (c+d x)+3 a \left (4 (A+2 C) a^2+20 b B a+15 A b^2\right )\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx-\frac {2 b \sin (c+d x) (12 a B+21 A b-8 b C) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {(4 a B+5 A b) \tan (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{5/2}}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\frac {1}{3} \int \frac {-b \left (12 B a^2+b (27 A-56 C) a-24 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 b \left (3 (A+12 C) a^2+36 b B a+4 b^2 (3 A+C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 a \left (4 (A+2 C) a^2+20 b B a+15 A b^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 b \sin (c+d x) (12 a B+21 A b-8 b C) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {(4 a B+5 A b) \tan (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{5/2}}{2 d}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\frac {1}{3} \left (-\left (\left (12 a^2 B+a b (27 A-56 C)-24 b^2 B\right ) \int \sqrt {a+b \cos (c+d x)}dx\right )-\frac {\int -\frac {\left (3 a b \left (4 (A+2 C) a^2+20 b B a+15 A b^2\right )+b \left (12 B a^3+b (33 A+16 C) a^2+48 b^2 B a+8 b^3 (3 A+C)\right ) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}\right )-\frac {2 b \sin (c+d x) (12 a B+21 A b-8 b C) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {(4 a B+5 A b) \tan (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{5/2}}{2 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\frac {1}{3} \left (\frac {\int \frac {\left (3 a b \left (4 (A+2 C) a^2+20 b B a+15 A b^2\right )+b \left (12 B a^3+b (33 A+16 C) a^2+48 b^2 B a+8 b^3 (3 A+C)\right ) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\left (12 a^2 B+a b (27 A-56 C)-24 b^2 B\right ) \int \sqrt {a+b \cos (c+d x)}dx\right )-\frac {2 b \sin (c+d x) (12 a B+21 A b-8 b C) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {(4 a B+5 A b) \tan (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{5/2}}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\frac {1}{3} \left (\frac {\int \frac {3 a b \left (4 (A+2 C) a^2+20 b B a+15 A b^2\right )+b \left (12 B a^3+b (33 A+16 C) a^2+48 b^2 B a+8 b^3 (3 A+C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\left (12 a^2 B+a b (27 A-56 C)-24 b^2 B\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )-\frac {2 b \sin (c+d x) (12 a B+21 A b-8 b C) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {(4 a B+5 A b) \tan (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{5/2}}{2 d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\frac {1}{3} \left (\frac {\int \frac {3 a b \left (4 (A+2 C) a^2+20 b B a+15 A b^2\right )+b \left (12 B a^3+b (33 A+16 C) a^2+48 b^2 B a+8 b^3 (3 A+C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (12 a^2 B+a b (27 A-56 C)-24 b^2 B\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 b \sin (c+d x) (12 a B+21 A b-8 b C) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {(4 a B+5 A b) \tan (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{5/2}}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\frac {1}{3} \left (\frac {\int \frac {3 a b \left (4 (A+2 C) a^2+20 b B a+15 A b^2\right )+b \left (12 B a^3+b (33 A+16 C) a^2+48 b^2 B a+8 b^3 (3 A+C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (12 a^2 B+a b (27 A-56 C)-24 b^2 B\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 b \sin (c+d x) (12 a B+21 A b-8 b C) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {(4 a B+5 A b) \tan (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{5/2}}{2 d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\frac {1}{3} \left (\frac {\int \frac {3 a b \left (4 (A+2 C) a^2+20 b B a+15 A b^2\right )+b \left (12 B a^3+b (33 A+16 C) a^2+48 b^2 B a+8 b^3 (3 A+C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \left (12 a^2 B+a b (27 A-56 C)-24 b^2 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 b \sin (c+d x) (12 a B+21 A b-8 b C) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {(4 a B+5 A b) \tan (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{5/2}}{2 d}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\frac {1}{3} \left (\frac {3 a b \left (4 a^2 (A+2 C)+20 a b B+15 A b^2\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx+b \left (12 a^3 B+a^2 b (33 A+16 C)+48 a b^2 B+8 b^3 (3 A+C)\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\frac {2 \left (12 a^2 B+a b (27 A-56 C)-24 b^2 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 b \sin (c+d x) (12 a B+21 A b-8 b C) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {(4 a B+5 A b) \tan (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{5/2}}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\frac {1}{3} \left (\frac {3 a b \left (4 a^2 (A+2 C)+20 a b B+15 A b^2\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+b \left (12 a^3 B+a^2 b (33 A+16 C)+48 a b^2 B+8 b^3 (3 A+C)\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \left (12 a^2 B+a b (27 A-56 C)-24 b^2 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 b \sin (c+d x) (12 a B+21 A b-8 b C) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {(4 a B+5 A b) \tan (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{5/2}}{2 d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\frac {1}{3} \left (\frac {3 a b \left (4 a^2 (A+2 C)+20 a b B+15 A b^2\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {b \left (12 a^3 B+a^2 b (33 A+16 C)+48 a b^2 B+8 b^3 (3 A+C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 \left (12 a^2 B+a b (27 A-56 C)-24 b^2 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 b \sin (c+d x) (12 a B+21 A b-8 b C) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {(4 a B+5 A b) \tan (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{5/2}}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\frac {1}{3} \left (\frac {3 a b \left (4 a^2 (A+2 C)+20 a b B+15 A b^2\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {b \left (12 a^3 B+a^2 b (33 A+16 C)+48 a b^2 B+8 b^3 (3 A+C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 \left (12 a^2 B+a b (27 A-56 C)-24 b^2 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 b \sin (c+d x) (12 a B+21 A b-8 b C) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {(4 a B+5 A b) \tan (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{5/2}}{2 d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\frac {1}{3} \left (\frac {3 a b \left (4 a^2 (A+2 C)+20 a b B+15 A b^2\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 b \left (12 a^3 B+a^2 b (33 A+16 C)+48 a b^2 B+8 b^3 (3 A+C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 \left (12 a^2 B+a b (27 A-56 C)-24 b^2 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 b \sin (c+d x) (12 a B+21 A b-8 b C) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {(4 a B+5 A b) \tan (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{5/2}}{2 d}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\frac {1}{3} \left (\frac {\frac {3 a b \left (4 a^2 (A+2 C)+20 a b B+15 A b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {\sec (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}+\frac {2 b \left (12 a^3 B+a^2 b (33 A+16 C)+48 a b^2 B+8 b^3 (3 A+C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 \left (12 a^2 B+a b (27 A-56 C)-24 b^2 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 b \sin (c+d x) (12 a B+21 A b-8 b C) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {(4 a B+5 A b) \tan (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{5/2}}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\frac {1}{3} \left (\frac {\frac {3 a b \left (4 a^2 (A+2 C)+20 a b B+15 A b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}+\frac {2 b \left (12 a^3 B+a^2 b (33 A+16 C)+48 a b^2 B+8 b^3 (3 A+C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 \left (12 a^2 B+a b (27 A-56 C)-24 b^2 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 b \sin (c+d x) (12 a B+21 A b-8 b C) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {(4 a B+5 A b) \tan (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{5/2}}{2 d}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\frac {1}{3} \left (\frac {\frac {6 a b \left (4 a^2 (A+2 C)+20 a b B+15 A b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}+\frac {2 b \left (12 a^3 B+a^2 b (33 A+16 C)+48 a b^2 B+8 b^3 (3 A+C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 \left (12 a^2 B+a b (27 A-56 C)-24 b^2 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 b \sin (c+d x) (12 a B+21 A b-8 b C) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {(4 a B+5 A b) \tan (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{5/2}}{2 d}\) |
Input:
Int[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec [c + d*x]^3,x]
Output:
(A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + ((((-2*(1 2*a^2*B - 24*b^2*B + a*b*(27*A - 56*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 *b*(12*a^3*B + 48*a*b^2*B + 8*b^3*(3*A + C) + a^2*b*(33*A + 16*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt [a + b*Cos[c + d*x]]) + (6*a*b*(15*A*b^2 + 20*a*b*B + 4*a^2*(A + 2*C))*Sqr t[(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)/3 - (2*b*(21*A*b + 12*a*B - 8*b*C)*Sqrt[ a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d))/2 + ((5*A*b + 4*a*B)*(a + b*Cos[c + d*x])^(3/2)*Tan[c + d*x])/d)/4
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_.) + (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)/(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. \(2364\) vs. \(2(356)=712\).
Time = 157.84 (sec) , antiderivative size = 2365, normalized size of antiderivative = 6.36
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2365\) |
| parts | \(\text {Expression too large to display}\) | \(2624\) |
Input:
int((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3,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*b^3*(s in(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))+2*C*b^3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2 *b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2)/(-2*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))-2*b*(B*b+3*C*a-2*C*b)*(a-b)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2 *d*x+1/2*c)^2+a-b)/(a-b))^(1/2)/(-2*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))-Ellip ticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2)))+2*A*a^3*(-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*c os(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(co s(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)^...
\[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{3} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c )^3,x, algorithm="fricas")
Output:
integral((C*b^2*cos(d*x + c)^4 + (2*C*a*b + B*b^2)*cos(d*x + c)^3 + A*a^2 + (C*a^2 + 2*B*a*b + A*b^2)*cos(d*x + c)^2 + (B*a^2 + 2*A*a*b)*cos(d*x + c ))*sqrt(b*cos(d*x + c) + a)*sec(d*x + c)^3, x)
Timed out. \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))**(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x +c)**3,x)
Output:
Timed out
\[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{3} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c )^3,x, algorithm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(5/ 2)*sec(d*x + c)^3, x)
\[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{3} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c )^3,x, algorithm="giac")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(5/ 2)*sec(d*x + c)^3, x)
Timed out. \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\int \frac {{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\cos \left (c+d\,x\right )}^3} \,d x \] Input:
int(((a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/c os(c + d*x)^3,x)
Output:
int(((a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/c os(c + d*x)^3, x)
\[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=3 \left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right ) \sec \left (d x +c \right )^{3}d x \right ) a^{2} b +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )^{3}d x \right ) b^{2} c +2 \left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{3}d x \right ) a b c +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{3}d x \right ) b^{3}+\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{3}d x \right ) a^{2} c +3 \left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{3}d x \right ) a \,b^{2}+\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{3}d x \right ) a^{3} \] Input:
int((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3,x)
Output:
3*int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)*sec(c + d*x)**3,x)*a**2*b + in t(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**4*sec(c + d*x)**3,x)*b**2*c + 2*i nt(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**3*sec(c + d*x)**3,x)*a*b*c + int (sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**3*sec(c + d*x)**3,x)*b**3 + int(sq rt(cos(c + d*x)*b + a)*cos(c + d*x)**2*sec(c + d*x)**3,x)*a**2*c + 3*int(s qrt(cos(c + d*x)*b + a)*cos(c + d*x)**2*sec(c + d*x)**3,x)*a*b**2 + int(sq rt(cos(c + d*x)*b + a)*sec(c + d*x)**3,x)*a**3