Integrand size = 43, antiderivative size = 407 \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=-\frac {\left (54 a b B+3 b^2 (11 A-16 C)+8 a^2 (2 A+3 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{24 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\left (66 a^2 b B+48 b^3 B+8 a^3 (2 A+3 C)+a b^2 (59 A+96 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{24 d \sqrt {a+b \cos (c+d x)}}+\frac {\left (5 A b^3+8 a^3 B+30 a b^2 B+20 a^2 b (A+2 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{8 d \sqrt {a+b \cos (c+d x)}}+\frac {\left (15 A b^2+42 a b B+8 a^2 (2 A+3 C)\right ) \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{24 d}+\frac {(5 A b+6 a B) (a+b \cos (c+d x))^{3/2} \sec (c+d x) \tan (c+d x)}{12 d}+\frac {A (a+b \cos (c+d x))^{5/2} \sec ^2(c+d x) \tan (c+d x)}{3 d} \] Output:
-1/24*(54*B*a*b+3*b^2*(11*A-16*C)+8*a^2*(2*A+3*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)+1/24*(66*B*a^2*b+48*B*b^3+8*a^3*(2*A+3*C)+a*b^2*(59*A+96*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)+1/8*(5*A*b^3+8*B*a^3+30*B*a*b^2+20*a^ 2*b*(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))/d/(a+b*cos(d*x+c))^(1/2)+1/24*(15*A*b^2+42*B*a* b+8*a^2*(2*A+3*C))*(a+b*cos(d*x+c))^(1/2)*tan(d*x+c)/d+1/12*(5*A*b+6*B*a)* (a+b*cos(d*x+c))^(3/2)*sec(d*x+c)*tan(d*x+c)/d+1/3*A*(a+b*cos(d*x+c))^(5/2 )*sec(d*x+c)^2*tan(d*x+c)/d
Result contains complex when optimal does not.
Time = 7.37 (sec) , antiderivative size = 519, normalized size of antiderivative = 1.28 \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\frac {\frac {8 b \left (6 a^2 B+24 b^2 B+a b (13 A+72 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}+\frac {2 \left (48 a^3 B+126 a b^2 B-3 b^3 (A-16 C)+8 a^2 b (13 A+27 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}-\frac {2 i \left (54 a b B+3 b^2 (11 A-16 C)+8 a^2 (2 A+3 C)\right ) \sqrt {-\frac {b (-1+\cos (c+d x))}{a+b}} \sqrt {\frac {b (1+\cos (c+d x))}{-a+b}} \csc (c+d x) \left (-2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (-2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )+b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right )}{a b \sqrt {-\frac {1}{a+b}}}+4 \sqrt {a+b \cos (c+d x)} \sec ^2(c+d x) \left (2 a (13 A b+6 a B) \sin (c+d x)+\left (\frac {33 A b^2}{2}+27 a b B+4 a^2 (2 A+3 C)\right ) \sin (2 (c+d x))+8 a^2 A \tan (c+d x)\right )}{96 d} \] 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]^4,x]
Output:
((8*b*(6*a^2*B + 24*b^2*B + a*b*(13*A + 72*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*(48*a^3*B + 126*a*b^2*B - 3*b^3*(A - 16*C) + 8*a^2*b*(13*A + 27*C))*Sqr t[(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)*(54*a*b*B + 3*b^2*(11*A - 16*C) + 8*a^2 *(2*A + 3*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*Elliptic F[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*Co s[c + d*x]]*Sec[c + d*x]^2*(2*a*(13*A*b + 6*a*B)*Sin[c + d*x] + ((33*A*b^2 )/2 + 27*a*b*B + 4*a^2*(2*A + 3*C))*Sin[2*(c + d*x)] + 8*a^2*A*Tan[c + d*x ]))/(96*d)
Time = 3.76 (sec) , antiderivative size = 420, 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, 3526, 27, 3042, 3538, 25, 3042, 3134, 3042, 3132, 3481, 3042, 3142, 3042, 3140, 3286, 3042, 3284}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \sec ^4(c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{3} \int \frac {1}{2} (a+b \cos (c+d x))^{3/2} \left (-b (A-6 C) \cos ^2(c+d x)+2 (2 a A+3 b B+3 a C) \cos (c+d x)+5 A b+6 a B\right ) \sec ^3(c+d x)dx+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \int (a+b \cos (c+d x))^{3/2} \left (-b (A-6 C) \cos ^2(c+d x)+2 (2 a A+3 b B+3 a C) \cos (c+d x)+5 A b+6 a B\right ) \sec ^3(c+d x)dx+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (-b (A-6 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 (2 a A+3 b B+3 a C) \sin \left (c+d x+\frac {\pi }{2}\right )+5 A b+6 a B\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{2} \int \frac {1}{2} \sqrt {a+b \cos (c+d x)} \left (8 (2 A+3 C) a^2+42 b B a+15 A b^2-3 b (3 A b-8 C b+2 a B) \cos ^2(c+d x)+2 \left (6 B a^2+b (11 A+24 C) a+12 b^2 B\right ) \cos (c+d x)\right ) \sec ^2(c+d x)dx+\frac {(6 a B+5 A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \int \sqrt {a+b \cos (c+d x)} \left (8 (2 A+3 C) a^2+42 b B a+15 A b^2-3 b (3 A b-8 C b+2 a B) \cos ^2(c+d x)+2 \left (6 B a^2+b (11 A+24 C) a+12 b^2 B\right ) \cos (c+d x)\right ) \sec ^2(c+d x)dx+\frac {(6 a B+5 A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \int \frac {\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (8 (2 A+3 C) a^2+42 b B a+15 A b^2-3 b (3 A b-8 C b+2 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (6 B a^2+b (11 A+24 C) a+12 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {(6 a B+5 A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\int \frac {\left (-b \left (8 (2 A+3 C) a^2+54 b B a+3 b^2 (11 A-16 C)\right ) \cos ^2(c+d x)+2 b \left (6 B a^2+b (13 A+72 C) a+24 b^2 B\right ) \cos (c+d x)+3 \left (8 B a^3+20 b (A+2 C) a^2+30 b^2 B a+5 A b^3\right )\right ) \sec (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx+\frac {\tan (c+d x) \left (8 a^2 (2 A+3 C)+42 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {(6 a B+5 A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \int \frac {\left (-b \left (8 (2 A+3 C) a^2+54 b B a+3 b^2 (11 A-16 C)\right ) \cos ^2(c+d x)+2 b \left (6 B a^2+b (13 A+72 C) a+24 b^2 B\right ) \cos (c+d x)+3 \left (8 B a^3+20 b (A+2 C) a^2+30 b^2 B a+5 A b^3\right )\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx+\frac {\tan (c+d x) \left (8 a^2 (2 A+3 C)+42 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {(6 a B+5 A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \int \frac {-b \left (8 (2 A+3 C) a^2+54 b B a+3 b^2 (11 A-16 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 b \left (6 B a^2+b (13 A+72 C) a+24 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 \left (8 B a^3+20 b (A+2 C) a^2+30 b^2 B a+5 A b^3\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {\tan (c+d x) \left (8 a^2 (2 A+3 C)+42 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {(6 a B+5 A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (-\left (\left (8 a^2 (2 A+3 C)+54 a b B+3 b^2 (11 A-16 C)\right ) \int \sqrt {a+b \cos (c+d x)}dx\right )-\frac {\int -\frac {\left (3 b \left (8 B a^3+20 b (A+2 C) a^2+30 b^2 B a+5 A b^3\right )+b \left (8 (2 A+3 C) a^3+66 b B a^2+b^2 (59 A+96 C) a+48 b^3 B\right ) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}\right )+\frac {\tan (c+d x) \left (8 a^2 (2 A+3 C)+42 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {(6 a B+5 A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {\int \frac {\left (3 b \left (8 B a^3+20 b (A+2 C) a^2+30 b^2 B a+5 A b^3\right )+b \left (8 (2 A+3 C) a^3+66 b B a^2+b^2 (59 A+96 C) a+48 b^3 B\right ) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\left (8 a^2 (2 A+3 C)+54 a b B+3 b^2 (11 A-16 C)\right ) \int \sqrt {a+b \cos (c+d x)}dx\right )+\frac {\tan (c+d x) \left (8 a^2 (2 A+3 C)+42 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {(6 a B+5 A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {\int \frac {3 b \left (8 B a^3+20 b (A+2 C) a^2+30 b^2 B a+5 A b^3\right )+b \left (8 (2 A+3 C) a^3+66 b B a^2+b^2 (59 A+96 C) a+48 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\left (8 a^2 (2 A+3 C)+54 a b B+3 b^2 (11 A-16 C)\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )+\frac {\tan (c+d x) \left (8 a^2 (2 A+3 C)+42 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {(6 a B+5 A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {\int \frac {3 b \left (8 B a^3+20 b (A+2 C) a^2+30 b^2 B a+5 A b^3\right )+b \left (8 (2 A+3 C) a^3+66 b B a^2+b^2 (59 A+96 C) a+48 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (8 a^2 (2 A+3 C)+54 a b B+3 b^2 (11 A-16 C)\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {\tan (c+d x) \left (8 a^2 (2 A+3 C)+42 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {(6 a B+5 A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {\int \frac {3 b \left (8 B a^3+20 b (A+2 C) a^2+30 b^2 B a+5 A b^3\right )+b \left (8 (2 A+3 C) a^3+66 b B a^2+b^2 (59 A+96 C) a+48 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (8 a^2 (2 A+3 C)+54 a b B+3 b^2 (11 A-16 C)\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {\tan (c+d x) \left (8 a^2 (2 A+3 C)+42 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {(6 a B+5 A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {\int \frac {3 b \left (8 B a^3+20 b (A+2 C) a^2+30 b^2 B a+5 A b^3\right )+b \left (8 (2 A+3 C) a^3+66 b B a^2+b^2 (59 A+96 C) a+48 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \left (8 a^2 (2 A+3 C)+54 a b B+3 b^2 (11 A-16 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {\tan (c+d x) \left (8 a^2 (2 A+3 C)+42 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {(6 a B+5 A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {b \left (8 a^3 (2 A+3 C)+66 a^2 b B+a b^2 (59 A+96 C)+48 b^3 B\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx+3 b \left (8 a^3 B+20 a^2 b (A+2 C)+30 a b^2 B+5 A b^3\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\frac {2 \left (8 a^2 (2 A+3 C)+54 a b B+3 b^2 (11 A-16 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {\tan (c+d x) \left (8 a^2 (2 A+3 C)+42 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {(6 a B+5 A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {b \left (8 a^3 (2 A+3 C)+66 a^2 b B+a b^2 (59 A+96 C)+48 b^3 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+3 b \left (8 a^3 B+20 a^2 b (A+2 C)+30 a b^2 B+5 A b^3\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \left (8 a^2 (2 A+3 C)+54 a b B+3 b^2 (11 A-16 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {\tan (c+d x) \left (8 a^2 (2 A+3 C)+42 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {(6 a B+5 A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {3 b \left (8 a^3 B+20 a^2 b (A+2 C)+30 a b^2 B+5 A b^3\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {b \left (8 a^3 (2 A+3 C)+66 a^2 b B+a b^2 (59 A+96 C)+48 b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 \left (8 a^2 (2 A+3 C)+54 a b B+3 b^2 (11 A-16 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {\tan (c+d x) \left (8 a^2 (2 A+3 C)+42 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {(6 a B+5 A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {3 b \left (8 a^3 B+20 a^2 b (A+2 C)+30 a b^2 B+5 A b^3\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {b \left (8 a^3 (2 A+3 C)+66 a^2 b B+a b^2 (59 A+96 C)+48 b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 \left (8 a^2 (2 A+3 C)+54 a b B+3 b^2 (11 A-16 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {\tan (c+d x) \left (8 a^2 (2 A+3 C)+42 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {(6 a B+5 A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {3 b \left (8 a^3 B+20 a^2 b (A+2 C)+30 a b^2 B+5 A b^3\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 b \left (8 a^3 (2 A+3 C)+66 a^2 b B+a b^2 (59 A+96 C)+48 b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 \left (8 a^2 (2 A+3 C)+54 a b B+3 b^2 (11 A-16 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {\tan (c+d x) \left (8 a^2 (2 A+3 C)+42 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {(6 a B+5 A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {\frac {3 b \left (8 a^3 B+20 a^2 b (A+2 C)+30 a b^2 B+5 A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {\sec (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}+\frac {2 b \left (8 a^3 (2 A+3 C)+66 a^2 b B+a b^2 (59 A+96 C)+48 b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 \left (8 a^2 (2 A+3 C)+54 a b B+3 b^2 (11 A-16 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {\tan (c+d x) \left (8 a^2 (2 A+3 C)+42 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {(6 a B+5 A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {\frac {3 b \left (8 a^3 B+20 a^2 b (A+2 C)+30 a b^2 B+5 A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}+\frac {2 b \left (8 a^3 (2 A+3 C)+66 a^2 b B+a b^2 (59 A+96 C)+48 b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 \left (8 a^2 (2 A+3 C)+54 a b B+3 b^2 (11 A-16 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {\tan (c+d x) \left (8 a^2 (2 A+3 C)+42 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {(6 a B+5 A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {\tan (c+d x) \left (8 a^2 (2 A+3 C)+42 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{d}+\frac {1}{2} \left (\frac {\frac {2 b \left (8 a^3 (2 A+3 C)+66 a^2 b B+a b^2 (59 A+96 C)+48 b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}+\frac {6 b \left (8 a^3 B+20 a^2 b (A+2 C)+30 a b^2 B+5 A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 \left (8 a^2 (2 A+3 C)+54 a b B+3 b^2 (11 A-16 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )\right )+\frac {(6 a B+5 A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{3 d}\) |
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]^4,x]
Output:
(A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (((5*A* b + 6*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + ( ((-2*(54*a*b*B + 3*b^2*(11*A - 16*C) + 8*a^2*(2*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x ])/(a + b)]) + ((2*b*(66*a^2*b*B + 48*b^3*B + 8*a^3*(2*A + 3*C) + a*b^2*(5 9*A + 96*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2* b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (6*b*(5*A*b^3 + 8*a^3*B + 30*a *b^2*B + 20*a^2*b*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi [2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]))/b)/2 + ((15 *A*b^2 + 42*a*b*B + 8*a^2*(2*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d* x])/d)/4)/6
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_)] + (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. \(2790\) vs. \(2(391)=782\).
Time = 437.79 (sec) , antiderivative size = 2791, normalized size of antiderivative = 6.86
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2791\) |
| parts | \(\text {Expression too large to display}\) | \(3838\) |
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)^4,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*B*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*A*a^3*(-1/3*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)^3+5/12*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)^2-1/24*(16*a^2+15 *b^2)/a^3*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)+5/48*b^2/a^2*(sin(1/2*d*x+1/2* c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2)/(-2*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))+1/3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2* c)^2+a-b)/(a-b))^(1/2)/(-2*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))-1/3*(sin(1/2*d* x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2)/(-2*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))+1/3/a*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2* d*x+1/2*c)^2+a-b)/(a-b))^(1/2)/(-2*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))-5/16* b^2/a^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a...
Timed out. \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\text {Timed out} \] 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 )^4,x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\text {Timed out} \] 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)**4,x)
Output:
Timed out
\[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{4} \,d x } \] 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 )^4,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)^4, x)
\[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{4} \,d x } \] 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 )^4,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)^4, x)
Timed out. \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\int \frac {{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\cos \left (c+d\,x\right )}^4} \,d x \] 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)^4,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)^4, x)
\[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=3 \left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right ) \sec \left (d x +c \right )^{4}d x \right ) a^{2} b +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )^{4}d x \right ) b^{2} c +2 \left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{4}d x \right ) a b c +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{4}d x \right ) b^{3}+\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{4}d x \right ) a^{2} c +3 \left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{4}d x \right ) a \,b^{2}+\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{4}d x \right ) a^{3} \] 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)^4,x)
Output:
3*int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)*sec(c + d*x)**4,x)*a**2*b + in t(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**4*sec(c + d*x)**4,x)*b**2*c + 2*i nt(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**3*sec(c + d*x)**4,x)*a*b*c + int (sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**3*sec(c + d*x)**4,x)*b**3 + int(sq rt(cos(c + d*x)*b + a)*cos(c + d*x)**2*sec(c + d*x)**4,x)*a**2*c + 3*int(s qrt(cos(c + d*x)*b + a)*cos(c + d*x)**2*sec(c + d*x)**4,x)*a*b**2 + int(sq rt(cos(c + d*x)*b + a)*sec(c + d*x)**4,x)*a**3