Integrand size = 43, antiderivative size = 503 \[ \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {\left (9 A b^3-128 a^3 B-24 a b^2 B-12 a^2 b (13 A+20 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{192 a^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (3 A b^3-128 a^3 B-136 a b^2 B-12 a^2 b (19 A+28 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{192 a d \sqrt {a+b \cos (c+d x)}}+\frac {\left (3 A b^4+96 a^3 b B-8 a b^3 B+24 a^2 b^2 (A+2 C)+16 a^4 (3 A+4 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{64 a^2 d \sqrt {a+b \cos (c+d x)}}-\frac {\left (9 A b^3-128 a^3 B-24 a b^2 B-12 a^2 b (13 A+20 C)\right ) \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{192 a^2 d}+\frac {\left (3 A b^2+56 a b B+12 a^2 (3 A+4 C)\right ) \sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{96 a d}+\frac {(3 A b+8 a B) \sqrt {a+b \cos (c+d x)} \sec ^2(c+d x) \tan (c+d x)}{24 d}+\frac {A (a+b \cos (c+d x))^{3/2} \sec ^3(c+d x) \tan (c+d x)}{4 d} \]
1/192*(9*A*b^3-128*B*a^3-24*B*a*b^2-12*a^2*b*(13*A+20*C))*(cos(1/2*d*x+1/2 *c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2)*(b/(a +b))^(1/2))*(a+b*cos(d*x+c))^(1/2)/a^2/d/((a+b*cos(d*x+c))/(a+b))^(1/2)-1/ 192*(3*A*b^3-128*B*a^3-136*B*a*b^2-12*a^2*b*(19*A+28*C))*(cos(1/2*d*x+1/2* c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2)*(b/(a+ b))^(1/2))*((a+b*cos(d*x+c))/(a+b))^(1/2)/a/d/(a+b*cos(d*x+c))^(1/2)+1/64* (3*A*b^4+96*B*a^3*b-8*B*a*b^3+24*a^2*b^2*(A+2*C)+16*a^4*(3*A+4*C))*(cos(1/ 2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticPi(sin(1/2*d*x+1/2*c),2,2 ^(1/2)*(b/(a+b))^(1/2))*((a+b*cos(d*x+c))/(a+b))^(1/2)/a^2/d/(a+b*cos(d*x+ c))^(1/2)+1/4*A*(a+b*cos(d*x+c))^(3/2)*sec(d*x+c)^3*tan(d*x+c)/d-1/192*(9* A*b^3-128*B*a^3-24*B*a*b^2-12*a^2*b*(13*A+20*C))*(a+b*cos(d*x+c))^(1/2)*ta n(d*x+c)/a^2/d+1/96*(3*A*b^2+56*B*a*b+12*a^2*(3*A+4*C))*sec(d*x+c)*(a+b*co s(d*x+c))^(1/2)*tan(d*x+c)/a/d+1/24*(3*A*b+8*B*a)*sec(d*x+c)^2*(a+b*cos(d* x+c))^(1/2)*tan(d*x+c)/d
Result contains complex when optimal does not.
Time = 7.69 (sec) , antiderivative size = 783, normalized size of antiderivative = 1.56 \[ \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {\frac {2 \left (144 a^3 A b+12 a A b^3+224 a^2 b^2 B+192 a^3 b C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}+\frac {2 \left (288 a^4 A-12 a^2 A b^2+27 A b^4+448 a^3 b B-72 a b^3 B+384 a^4 C+48 a^2 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}-\frac {2 i \left (-156 a^2 A b^2+9 A b^4-128 a^3 b B-24 a b^3 B-240 a^2 b^2 C\right ) \sqrt {\frac {b-b \cos (c+d x)}{a+b}} \sqrt {-\frac {b+b \cos (c+d x)}{a-b}} \cos (2 (c+d x)) \left (2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )-b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sin (c+d x)}{a \sqrt {-\frac {1}{a+b}} \sqrt {1-\cos ^2(c+d x)} \sqrt {-\frac {a^2-b^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2}{b^2}} \left (2 a^2-b^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2\right )}}{768 a^2 d}+\frac {\sqrt {a+b \cos (c+d x)} \left (\frac {1}{24} \sec ^3(c+d x) (9 A b \sin (c+d x)+8 a B \sin (c+d x))+\frac {\sec ^2(c+d x) \left (36 a^2 A \sin (c+d x)+3 A b^2 \sin (c+d x)+56 a b B \sin (c+d x)+48 a^2 C \sin (c+d x)\right )}{96 a}+\frac {\sec (c+d x) \left (156 a^2 A b \sin (c+d x)-9 A b^3 \sin (c+d x)+128 a^3 B \sin (c+d x)+24 a b^2 B \sin (c+d x)+240 a^2 b C \sin (c+d x)\right )}{192 a^2}+\frac {1}{4} a A \sec ^3(c+d x) \tan (c+d x)\right )}{d} \]
Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^ 2)*Sec[c + d*x]^5,x]
((2*(144*a^3*A*b + 12*a*A*b^3 + 224*a^2*b^2*B + 192*a^3*b*C)*Sqrt[(a + b*C os[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Co s[c + d*x]] + (2*(288*a^4*A - 12*a^2*A*b^2 + 27*A*b^4 + 448*a^3*b*B - 72*a *b^3*B + 384*a^4*C + 48*a^2*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*Elli pticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*( -156*a^2*A*b^2 + 9*A*b^4 - 128*a^3*b*B - 24*a*b^3*B - 240*a^2*b^2*C)*Sqrt[ (b - b*Cos[c + d*x])/(a + b)]*Sqrt[-((b + b*Cos[c + d*x])/(a - b))]*Cos[2* (c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b *Cos[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b )/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)]))*Sin[c + d*x])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[ -((a^2 - b^2 - 2*a*(a + b*Cos[c + d*x]) + (a + b*Cos[c + d*x])^2)/b^2)]*(2 *a^2 - b^2 - 4*a*(a + b*Cos[c + d*x]) + 2*(a + b*Cos[c + d*x])^2)))/(768*a ^2*d) + (Sqrt[a + b*Cos[c + d*x]]*((Sec[c + d*x]^3*(9*A*b*Sin[c + d*x] + 8 *a*B*Sin[c + d*x]))/24 + (Sec[c + d*x]^2*(36*a^2*A*Sin[c + d*x] + 3*A*b^2* Sin[c + d*x] + 56*a*b*B*Sin[c + d*x] + 48*a^2*C*Sin[c + d*x]))/(96*a) + (S ec[c + d*x]*(156*a^2*A*b*Sin[c + d*x] - 9*A*b^3*Sin[c + d*x] + 128*a^3*B*S in[c + d*x] + 24*a*b^2*B*Sin[c + d*x] + 240*a^2*b*C*Sin[c + d*x]))/(192*a^ 2) + (a*A*Sec[c + d*x]^3*Tan[c + d*x])/4))/d
Time = 4.53 (sec) , antiderivative size = 519, normalized size of antiderivative = 1.03, number of steps used = 27, number of rules used = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.628, Rules used = {3042, 3526, 27, 3042, 3526, 27, 3042, 3534, 27, 3042, 3534, 27, 3042, 3538, 25, 3042, 3134, 3042, 3132, 3481, 3042, 3142, 3042, 3140, 3286, 3042, 3284}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \sec ^5(c+d x) (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^5}dx\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{4} \int \frac {1}{2} \sqrt {a+b \cos (c+d x)} \left (b (3 A+8 C) \cos ^2(c+d x)+2 (3 a A+4 b B+4 a C) \cos (c+d x)+3 A b+8 a B\right ) \sec ^4(c+d x)dx+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \int \sqrt {a+b \cos (c+d x)} \left (b (3 A+8 C) \cos ^2(c+d x)+2 (3 a A+4 b B+4 a C) \cos (c+d x)+3 A b+8 a B\right ) \sec ^4(c+d x)dx+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \int \frac {\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (b (3 A+8 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 (3 a A+4 b B+4 a C) \sin \left (c+d x+\frac {\pi }{2}\right )+3 A b+8 a B\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{3} \int \frac {\left (12 (3 A+4 C) a^2+56 b B a+3 A b^2+3 b (9 A b+16 C b+8 a B) \cos ^2(c+d x)+2 \left (16 B a^2+33 A b a+48 b C a+24 b^2 B\right ) \cos (c+d x)\right ) \sec ^3(c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx+\frac {(8 a B+3 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \int \frac {\left (12 (3 A+4 C) a^2+56 b B a+3 A b^2+3 b (9 A b+16 C b+8 a B) \cos ^2(c+d x)+2 \left (16 B a^2+33 A b a+48 b C a+24 b^2 B\right ) \cos (c+d x)\right ) \sec ^3(c+d x)}{\sqrt {a+b \cos (c+d x)}}dx+\frac {(8 a B+3 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \int \frac {12 (3 A+4 C) a^2+56 b B a+3 A b^2+3 b (9 A b+16 C b+8 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (16 B a^2+33 A b a+48 b C a+24 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {(8 a B+3 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\int -\frac {\left (-128 B a^3-12 b (13 A+20 C) a^2-24 b^2 B a-2 \left (12 (3 A+4 C) a^2+104 b B a+3 b^2 (19 A+32 C)\right ) \cos (c+d x) a+9 A b^3-b \left (12 (3 A+4 C) a^2+56 b B a+3 A b^2\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx}{2 a}+\frac {\tan (c+d x) \sec (c+d x) \left (12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}\right )+\frac {(8 a B+3 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\tan (c+d x) \sec (c+d x) \left (12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\int \frac {\left (-128 B a^3-12 b (13 A+20 C) a^2-24 b^2 B a-2 \left (12 (3 A+4 C) a^2+104 b B a+3 b^2 (19 A+32 C)\right ) \cos (c+d x) a+9 A b^3-b \left (12 (3 A+4 C) a^2+56 b B a+3 A b^2\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{4 a}\right )+\frac {(8 a B+3 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\tan (c+d x) \sec (c+d x) \left (12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\int \frac {-128 B a^3-12 b (13 A+20 C) a^2-24 b^2 B a-2 \left (12 (3 A+4 C) a^2+104 b B a+3 b^2 (19 A+32 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+9 A b^3-b \left (12 (3 A+4 C) a^2+56 b B a+3 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^2 \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{4 a}\right )+\frac {(8 a B+3 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\tan (c+d x) \sec (c+d x) \left (12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\int -\frac {\left (b \left (-128 B a^3-12 b (13 A+20 C) a^2-24 b^2 B a+9 A b^3\right ) \cos ^2(c+d x)+2 a b \left (12 (3 A+4 C) a^2+56 b B a+3 A b^2\right ) \cos (c+d x)+3 \left (16 (3 A+4 C) a^4+96 b B a^3+24 b^2 (A+2 C) a^2-8 b^3 B a+3 A b^4\right )\right ) \sec (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx}{a}+\frac {\tan (c+d x) \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{a d}}{4 a}\right )+\frac {(8 a B+3 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\tan (c+d x) \sec (c+d x) \left (12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\tan (c+d x) \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\int \frac {\left (b \left (-128 B a^3-12 b (13 A+20 C) a^2-24 b^2 B a+9 A b^3\right ) \cos ^2(c+d x)+2 a b \left (12 (3 A+4 C) a^2+56 b B a+3 A b^2\right ) \cos (c+d x)+3 \left (16 (3 A+4 C) a^4+96 b B a^3+24 b^2 (A+2 C) a^2-8 b^3 B a+3 A b^4\right )\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{2 a}}{4 a}\right )+\frac {(8 a B+3 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\tan (c+d x) \sec (c+d x) \left (12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\tan (c+d x) \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\int \frac {b \left (-128 B a^3-12 b (13 A+20 C) a^2-24 b^2 B a+9 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 a b \left (12 (3 A+4 C) a^2+56 b B a+3 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 \left (16 (3 A+4 C) a^4+96 b B a^3+24 b^2 (A+2 C) a^2-8 b^3 B a+3 A b^4\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}}{4 a}\right )+\frac {(8 a B+3 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\tan (c+d x) \sec (c+d x) \left (12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\tan (c+d x) \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \int \sqrt {a+b \cos (c+d x)}dx-\frac {\int -\frac {\left (3 b \left (16 (3 A+4 C) a^4+96 b B a^3+24 b^2 (A+2 C) a^2-8 b^3 B a+3 A b^4\right )-a b \left (-128 B a^3-12 b (19 A+28 C) a^2-136 b^2 B a+3 A b^3\right ) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}}{2 a}}{4 a}\right )+\frac {(8 a B+3 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\tan (c+d x) \sec (c+d x) \left (12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\tan (c+d x) \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \int \sqrt {a+b \cos (c+d x)}dx+\frac {\int \frac {\left (3 b \left (16 (3 A+4 C) a^4+96 b B a^3+24 b^2 (A+2 C) a^2-8 b^3 B a+3 A b^4\right )-a b \left (-128 B a^3-12 b (19 A+28 C) a^2-136 b^2 B a+3 A b^3\right ) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}}{2 a}}{4 a}\right )+\frac {(8 a B+3 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\tan (c+d x) \sec (c+d x) \left (12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\tan (c+d x) \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {\int \frac {3 b \left (16 (3 A+4 C) a^4+96 b B a^3+24 b^2 (A+2 C) a^2-8 b^3 B a+3 A b^4\right )-a b \left (-128 B a^3-12 b (19 A+28 C) a^2-136 b^2 B a+3 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{2 a}}{4 a}\right )+\frac {(8 a B+3 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\tan (c+d x) \sec (c+d x) \left (12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\tan (c+d x) \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {\left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\int \frac {3 b \left (16 (3 A+4 C) a^4+96 b B a^3+24 b^2 (A+2 C) a^2-8 b^3 B a+3 A b^4\right )-a b \left (-128 B a^3-12 b (19 A+28 C) a^2-136 b^2 B a+3 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{2 a}}{4 a}\right )+\frac {(8 a B+3 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\tan (c+d x) \sec (c+d x) \left (12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\tan (c+d x) \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {\left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\int \frac {3 b \left (16 (3 A+4 C) a^4+96 b B a^3+24 b^2 (A+2 C) a^2-8 b^3 B a+3 A b^4\right )-a b \left (-128 B a^3-12 b (19 A+28 C) a^2-136 b^2 B a+3 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{2 a}}{4 a}\right )+\frac {(8 a B+3 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\tan (c+d x) \sec (c+d x) \left (12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\tan (c+d x) \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {\int \frac {3 b \left (16 (3 A+4 C) a^4+96 b B a^3+24 b^2 (A+2 C) a^2-8 b^3 B a+3 A b^4\right )-a b \left (-128 B a^3-12 b (19 A+28 C) a^2-136 b^2 B a+3 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}+\frac {2 \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}}{4 a}\right )+\frac {(8 a B+3 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\tan (c+d x) \sec (c+d x) \left (12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\tan (c+d x) \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {3 b \left (16 a^4 (3 A+4 C)+96 a^3 b B+24 a^2 b^2 (A+2 C)-8 a b^3 B+3 A b^4\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx-a b \left (-128 a^3 B-12 a^2 b (19 A+28 C)-136 a b^2 B+3 A b^3\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}+\frac {2 \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}}{4 a}\right )+\frac {(8 a B+3 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\tan (c+d x) \sec (c+d x) \left (12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\tan (c+d x) \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {3 b \left (16 a^4 (3 A+4 C)+96 a^3 b B+24 a^2 b^2 (A+2 C)-8 a b^3 B+3 A b^4\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-a b \left (-128 a^3 B-12 a^2 b (19 A+28 C)-136 a b^2 B+3 A b^3\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}+\frac {2 \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}}{4 a}\right )+\frac {(8 a B+3 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\tan (c+d x) \sec (c+d x) \left (12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\tan (c+d x) \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {3 b \left (16 a^4 (3 A+4 C)+96 a^3 b B+24 a^2 b^2 (A+2 C)-8 a b^3 B+3 A b^4\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {a b \left (-128 a^3 B-12 a^2 b (19 A+28 C)-136 a b^2 B+3 A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}}{4 a}\right )+\frac {(8 a B+3 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\tan (c+d x) \sec (c+d x) \left (12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\tan (c+d x) \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {3 b \left (16 a^4 (3 A+4 C)+96 a^3 b B+24 a^2 b^2 (A+2 C)-8 a b^3 B+3 A b^4\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {a b \left (-128 a^3 B-12 a^2 b (19 A+28 C)-136 a b^2 B+3 A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}}{4 a}\right )+\frac {(8 a B+3 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\tan (c+d x) \sec (c+d x) \left (12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\tan (c+d x) \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {3 b \left (16 a^4 (3 A+4 C)+96 a^3 b B+24 a^2 b^2 (A+2 C)-8 a b^3 B+3 A b^4\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 a b \left (-128 a^3 B-12 a^2 b (19 A+28 C)-136 a b^2 B+3 A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}}{4 a}\right )+\frac {(8 a B+3 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\tan (c+d x) \sec (c+d x) \left (12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\tan (c+d x) \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {\frac {3 b \left (16 a^4 (3 A+4 C)+96 a^3 b B+24 a^2 b^2 (A+2 C)-8 a b^3 B+3 A b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {\sec (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}-\frac {2 a b \left (-128 a^3 B-12 a^2 b (19 A+28 C)-136 a b^2 B+3 A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}}{4 a}\right )+\frac {(8 a B+3 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\tan (c+d x) \sec (c+d x) \left (12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\tan (c+d x) \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {\frac {3 b \left (16 a^4 (3 A+4 C)+96 a^3 b B+24 a^2 b^2 (A+2 C)-8 a b^3 B+3 A b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}-\frac {2 a b \left (-128 a^3 B-12 a^2 b (19 A+28 C)-136 a b^2 B+3 A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}}{4 a}\right )+\frac {(8 a B+3 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\tan (c+d x) \sec (c+d x) \left (12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\tan (c+d x) \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {2 \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\frac {6 b \left (16 a^4 (3 A+4 C)+96 a^3 b B+24 a^2 b^2 (A+2 C)-8 a b^3 B+3 A b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}-\frac {2 a b \left (-128 a^3 B-12 a^2 b (19 A+28 C)-136 a b^2 B+3 A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}}{2 a}}{4 a}\right )+\frac {(8 a B+3 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
(A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (((3*A* b + 8*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + ( ((3*A*b^2 + 56*a*b*B + 12*a^2*(3*A + 4*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) - (-1/2*((2*(9*A*b^3 - 128*a^3*B - 24*a*b^2*B - 12*a^2*b*(13*A + 20*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*(3*A*b^ 3 - 128*a^3*B - 136*a*b^2*B - 12*a^2*b*(19*A + 28*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (6*b*(3*A*b^4 + 96*a^3*b*B - 8*a*b^3*B + 24*a^2*b^2*(A + 2*C) + 16*a^4*(3*A + 4*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]))/b)/a + ((9*A*b^3 - 128*a^3*B - 24*a*b^2*B - 12*a^2*b*(13*A + 20*C))*Sqrt[a + b*Cos[c + d*x]]* Tan[c + d*x])/(a*d))/(4*a))/6)/8
3.11.28.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(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[(-(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. \(3550\) vs. \(2(556)=1112\).
Time = 16.65 (sec) , antiderivative size = 3551, normalized size of antiderivative = 7.06
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(3551\) |
| parts | \(\text {Expression too large to display}\) | \(5047\) |
int((a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^5,x, method=_RETURNVERBOSE)
-(-(-2*cos(1/2*d*x+1/2*c)^2*b-a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*C*b^2*( sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+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))+2*A*a^2*(-1/4*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)^4+7/24*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)^3-1/96*(36*a^ 2+35*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)^2+5/192*b*(20*a^2+21*b^2)/ a^4*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)-7/96*b/a*(sin(1/2*d*x+1/2*c)^2)^(1/2 )*((2*cos(1/2*d*x+1/2*c)^2*b+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))-35/384*b^3/a^3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c )^2*b+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))+25/96/a*(sin(1 /2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+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))-25/96*b^2/a^2*(sin(1/2*d*x+1/2*c)^2)^(1/2) *((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^...
Timed out. \[ \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\text {Timed out} \]
integrate((a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c )^5,x, algorithm="fricas")
Timed out. \[ \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\text {Timed out} \]
\[ \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{5} \,d x } \]
integrate((a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c )^5,x, algorithm="maxima")
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(3/ 2)*sec(d*x + c)^5, x)
\[ \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{5} \,d x } \]
integrate((a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c )^5,x, algorithm="giac")
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(3/ 2)*sec(d*x + c)^5, x)
Timed out. \[ \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\int \frac {{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\cos \left (c+d\,x\right )}^5} \,d x \]