Integrand size = 43, antiderivative size = 572 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx=\frac {\left (105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B+a^4 b (33 A-56 C)-2 a^2 b^3 (85 A-12 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{12 a^4 \left (a^2-b^2\right )^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\left (35 A b^3+12 a^3 B-20 a b^2 B-a^2 (27 A b-8 b C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{12 a^3 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {\left (35 A b^2-20 a b B+4 a^2 (A+2 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{4 a^4 d \sqrt {a+b \cos (c+d x)}}+\frac {b \left (35 A b^3+12 a^3 B-20 a b^2 B-a^2 (27 A b-8 b C)\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac {b \left (105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B+a^4 b (33 A-56 C)-2 a^2 b^3 (85 A-12 C)\right ) \sin (c+d x)}{12 a^4 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}-\frac {(7 A b-4 a B) \tan (c+d x)}{4 a^2 d (a+b \cos (c+d x))^{3/2}}+\frac {A \sec (c+d x) \tan (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}} \] Output:
1/12*(105*A*b^5-12*a^5*B+104*a^3*b^2*B-60*a*b^4*B+a^4*b*(33*A-56*C)-2*a^2* b^3*(85*A-12*C))*(a+b*cos(d*x+c))^(1/2)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/ 2)*(b/(a+b))^(1/2))/a^4/(a^2-b^2)^2/d/((a+b*cos(d*x+c))/(a+b))^(1/2)+1/12* (35*A*b^3+12*B*a^3-20*B*a*b^2-a^2*(27*A*b-8*C*b))*((a+b*cos(d*x+c))/(a+b)) ^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2)*(b/(a+b))^(1/2))/a^3/(a^2-b^2 )/d/(a+b*cos(d*x+c))^(1/2)+1/4*(35*A*b^2-20*B*a*b+4*a^2*(A+2*C))*((a+b*cos (d*x+c))/(a+b))^(1/2)*EllipticPi(sin(1/2*d*x+1/2*c),2,2^(1/2)*(b/(a+b))^(1 /2))/a^4/d/(a+b*cos(d*x+c))^(1/2)+1/12*b*(35*A*b^3+12*B*a^3-20*B*a*b^2-a^2 *(27*A*b-8*C*b))*sin(d*x+c)/a^3/(a^2-b^2)/d/(a+b*cos(d*x+c))^(3/2)-1/12*b* (105*A*b^5-12*a^5*B+104*a^3*b^2*B-60*a*b^4*B+a^4*b*(33*A-56*C)-2*a^2*b^3*( 85*A-12*C))*sin(d*x+c)/a^4/(a^2-b^2)^2/d/(a+b*cos(d*x+c))^(1/2)-1/4*(7*A*b -4*B*a)*tan(d*x+c)/a^2/d/(a+b*cos(d*x+c))^(3/2)+1/2*A*sec(d*x+c)*tan(d*x+c )/a/d/(a+b*cos(d*x+c))^(3/2)
Result contains complex when optimal does not.
Time = 8.14 (sec) , antiderivative size = 922, normalized size of antiderivative = 1.61 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx=\frac {\frac {2 \left (12 a^5 A b-216 a^3 A b^3+140 a A b^5+144 a^4 b^2 B-80 a^2 b^4 B-96 a^5 b C+32 a^3 b^3 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 (24 a^6 A+195 a^4 A b^2-566 a^2 A b^4+315 A b^6-132 a^5 b B+344 a^3 b^3 B-180 a b^5 B+48 a^6 C-152 a^4 b^2 C+72 a^2 b^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 )}{\sqrt {a+b \cos (c+d x)}}-\frac {2 i \left (33 a^4 A b^2-170 a^2 A b^4+105 A b^6-12 a^5 b B+104 a^3 b^3 B-60 a b^5 B-56 a^4 b^2 C+24 a^2 b^4 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 )}}{48 a^4 (a-b)^2 (a+b)^2 d}+\frac {\sqrt {a+b \cos (c+d x)} \left (\frac {\sec (c+d x) (-11 A b \sin (c+d x)+4 a B \sin (c+d x))}{4 a^4}+\frac {2 \left (A b^4 \sin (c+d x)-a b^3 B \sin (c+d x)+a^2 b^2 C \sin (c+d x)\right )}{3 a^3 \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}+\frac {2 \left (13 a^2 A b^4 \sin (c+d x)-9 A b^6 \sin (c+d x)-10 a^3 b^3 B \sin (c+d x)+6 a b^5 B \sin (c+d x)+7 a^4 b^2 C \sin (c+d x)-3 a^2 b^4 C \sin (c+d x)\right )}{3 a^4 \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}+\frac {A \sec (c+d x) \tan (c+d x)}{2 a^3}\right )}{d} \] Input:
Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b* Cos[c + d*x])^(5/2),x]
Output:
((2*(12*a^5*A*b - 216*a^3*A*b^3 + 140*a*A*b^5 + 144*a^4*b^2*B - 80*a^2*b^4 *B - 96*a^5*b*C + 32*a^3*b^3*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*Ellipti cF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(24*a^6*A + 195*a^4*A*b^2 - 566*a^2*A*b^4 + 315*A*b^6 - 132*a^5*b*B + 344*a^3*b^3*B - 180*a*b^5*B + 48*a^6*C - 152*a^4*b^2*C + 72*a^2*b^4*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)*(33*a^4*A*b^2 - 170*a^2*A*b^4 + 105*A*b^6 - 12*a^5*b*B + 104*a^3*b^3*B - 60*a*b^5*B - 56*a^4*b^2*C + 24*a^2*b^4*C)*Sqrt[(b - b*Co s[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*Ar cSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)]))*Si n[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)))/(48*a^4*(a - b) ^2*(a + b)^2*d) + (Sqrt[a + b*Cos[c + d*x]]*((Sec[c + d*x]*(-11*A*b*Sin[c + d*x] + 4*a*B*Sin[c + d*x]))/(4*a^4) + (2*(A*b^4*Sin[c + d*x] - a*b^3*B*S in[c + d*x] + a^2*b^2*C*Sin[c + d*x]))/(3*a^3*(a^2 - b^2)*(a + b*Cos[c + d *x])^2) + (2*(13*a^2*A*b^4*Sin[c + d*x] - 9*A*b^6*Sin[c + d*x] - 10*a^3...
Time = 5.18 (sec) , antiderivative size = 610, normalized size of antiderivative = 1.07, number of steps used = 27, number of rules used = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.628, Rules used = {3042, 3534, 27, 3042, 3534, 27, 3042, 3534, 27, 3042, 3534, 27, 3042, 3538, 25, 3042, 3134, 3042, 3132, 3481, 3042, 3142, 3042, 3140, 3286, 3042, 3284}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sec ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\int -\frac {\left (-5 A b \cos ^2(c+d x)-2 a (A+2 C) \cos (c+d x)+7 A b-4 a B\right ) \sec ^2(c+d x)}{2 (a+b \cos (c+d x))^{5/2}}dx}{2 a}+\frac {A \tan (c+d x) \sec (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}-\frac {\int \frac {\left (-5 A b \cos ^2(c+d x)-2 a (A+2 C) \cos (c+d x)+7 A b-4 a B\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}}dx}{4 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}-\frac {\int \frac {-5 A b \sin \left (c+d x+\frac {\pi }{2}\right )^2-2 a (A+2 C) \sin \left (c+d x+\frac {\pi }{2}\right )+7 A b-4 a B}{\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx}{4 a}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\int -\frac {\left (4 (A+2 C) a^2-20 b B a+10 A b \cos (c+d x) a+35 A b^2-3 b (7 A b-4 a B) \cos ^2(c+d x)\right ) \sec (c+d x)}{2 (a+b \cos (c+d x))^{5/2}}dx}{a}+\frac {(7 A b-4 a B) \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}}{4 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {(7 A b-4 a B) \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\int \frac {\left (4 (A+2 C) a^2-20 b B a+10 A b \cos (c+d x) a+35 A b^2-3 b (7 A b-4 a B) \cos ^2(c+d x)\right ) \sec (c+d x)}{(a+b \cos (c+d x))^{5/2}}dx}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {(7 A b-4 a B) \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\int \frac {4 (A+2 C) a^2-20 b B a+10 A b \sin \left (c+d x+\frac {\pi }{2}\right ) a+35 A b^2-3 b (7 A b-4 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {(7 A b-4 a B) \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {2 \int \frac {\left (b \left (12 B a^3-(27 A b-8 b C) a^2-20 b^2 B a+35 A b^3\right ) \cos ^2(c+d x)-6 a b \left (-\left ((3 A-4 C) a^2\right )-4 b B a+7 A b^2\right ) \cos (c+d x)+3 \left (a^2-b^2\right ) \left (4 (A+2 C) a^2-20 b B a+35 A b^2\right )\right ) \sec (c+d x)}{2 (a+b \cos (c+d x))^{3/2}}dx}{3 a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (12 a^3 B-a^2 (27 A b-8 b C)-20 a b^2 B+35 A b^3\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {(7 A b-4 a B) \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\int \frac {\left (b \left (12 B a^3-(27 A b-8 b C) a^2-20 b^2 B a+35 A b^3\right ) \cos ^2(c+d x)-6 a b \left (-\left ((3 A-4 C) a^2\right )-4 b B a+7 A b^2\right ) \cos (c+d x)+3 \left (a^2-b^2\right ) \left (4 (A+2 C) a^2-20 b B a+35 A b^2\right )\right ) \sec (c+d x)}{(a+b \cos (c+d x))^{3/2}}dx}{3 a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (12 a^3 B-a^2 (27 A b-8 b C)-20 a b^2 B+35 A b^3\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {(7 A b-4 a B) \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\int \frac {b \left (12 B a^3-(27 A b-8 b C) a^2-20 b^2 B a+35 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-6 a b \left (-\left ((3 A-4 C) a^2\right )-4 b B a+7 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 \left (a^2-b^2\right ) \left (4 (A+2 C) a^2-20 b B a+35 A b^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{3 a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (12 a^3 B-a^2 (27 A b-8 b C)-20 a b^2 B+35 A b^3\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {(7 A b-4 a B) \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {2 \int \frac {\left (3 \left (4 (A+2 C) a^2-20 b B a+35 A b^2\right ) \left (a^2-b^2\right )^2+b \left (-12 B a^5+b (33 A-56 C) a^4+104 b^2 B a^3-2 b^3 (85 A-12 C) a^2-60 b^4 B a+105 A b^5\right ) \cos ^2(c+d x)+2 a b \left (3 (A-8 C) a^4+36 b B a^3-2 b^2 (27 A-4 C) a^2-20 b^3 B a+35 A b^4\right ) \cos (c+d x)\right ) \sec (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx}{a \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x) \left (-12 a^5 B+a^4 (33 A b-56 b C)+104 a^3 b^2 B-2 a^2 b^3 (85 A-12 C)-60 a b^4 B+105 A b^5\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (12 a^3 B-a^2 (27 A b-8 b C)-20 a b^2 B+35 A b^3\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {(7 A b-4 a B) \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\int \frac {\left (3 \left (4 (A+2 C) a^2-20 b B a+35 A b^2\right ) \left (a^2-b^2\right )^2+b \left (-12 B a^5+b (33 A-56 C) a^4+104 b^2 B a^3-2 b^3 (85 A-12 C) a^2-60 b^4 B a+105 A b^5\right ) \cos ^2(c+d x)+2 a b \left (3 (A-8 C) a^4+36 b B a^3-2 b^2 (27 A-4 C) a^2-20 b^3 B a+35 A b^4\right ) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{a \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x) \left (-12 a^5 B+a^4 (33 A b-56 b C)+104 a^3 b^2 B-2 a^2 b^3 (85 A-12 C)-60 a b^4 B+105 A b^5\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (12 a^3 B-a^2 (27 A b-8 b C)-20 a b^2 B+35 A b^3\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {(7 A b-4 a B) \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\int \frac {3 \left (4 (A+2 C) a^2-20 b B a+35 A b^2\right ) \left (a^2-b^2\right )^2+b \left (-12 B a^5+b (33 A-56 C) a^4+104 b^2 B a^3-2 b^3 (85 A-12 C) a^2-60 b^4 B a+105 A b^5\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 a b \left (3 (A-8 C) a^4+36 b B a^3-2 b^2 (27 A-4 C) a^2-20 b^3 B a+35 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x) \left (-12 a^5 B+a^4 (33 A b-56 b C)+104 a^3 b^2 B-2 a^2 b^3 (85 A-12 C)-60 a b^4 B+105 A b^5\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (12 a^3 B-a^2 (27 A b-8 b C)-20 a b^2 B+35 A b^3\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {(7 A b-4 a B) \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\left (-12 a^5 B+a^4 b (33 A-56 C)+104 a^3 b^2 B-2 a^2 b^3 (85 A-12 C)-60 a b^4 B+105 A b^5\right ) \int \sqrt {a+b \cos (c+d x)}dx-\frac {\int -\frac {\left (3 b \left (4 (A+2 C) a^2-20 b B a+35 A b^2\right ) \left (a^2-b^2\right )^2+a b \left (12 B a^3-(27 A b-8 b C) a^2-20 b^2 B a+35 A b^3\right ) \cos (c+d x) \left (a^2-b^2\right )\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}}{a \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x) \left (-12 a^5 B+a^4 (33 A b-56 b C)+104 a^3 b^2 B-2 a^2 b^3 (85 A-12 C)-60 a b^4 B+105 A b^5\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (12 a^3 B-a^2 (27 A b-8 b C)-20 a b^2 B+35 A b^3\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {(7 A b-4 a B) \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\frac {\int \frac {\left (3 b \left (4 (A+2 C) a^2-20 b B a+35 A b^2\right ) \left (a^2-b^2\right )^2+a b \left (12 B a^3-(27 A b-8 b C) a^2-20 b^2 B a+35 A b^3\right ) \cos (c+d x) \left (a^2-b^2\right )\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}+\left (-12 a^5 B+a^4 b (33 A-56 C)+104 a^3 b^2 B-2 a^2 b^3 (85 A-12 C)-60 a b^4 B+105 A b^5\right ) \int \sqrt {a+b \cos (c+d x)}dx}{a \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x) \left (-12 a^5 B+a^4 (33 A b-56 b C)+104 a^3 b^2 B-2 a^2 b^3 (85 A-12 C)-60 a b^4 B+105 A b^5\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (12 a^3 B-a^2 (27 A b-8 b C)-20 a b^2 B+35 A b^3\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {(7 A b-4 a B) \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\frac {\int \frac {3 b \left (4 (A+2 C) a^2-20 b B a+35 A b^2\right ) \left (a^2-b^2\right )^2+a b \left (12 B a^3-(27 A b-8 b C) a^2-20 b^2 B a+35 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a^2-b^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}+\left (-12 a^5 B+a^4 b (33 A-56 C)+104 a^3 b^2 B-2 a^2 b^3 (85 A-12 C)-60 a b^4 B+105 A b^5\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x) \left (-12 a^5 B+a^4 (33 A b-56 b C)+104 a^3 b^2 B-2 a^2 b^3 (85 A-12 C)-60 a b^4 B+105 A b^5\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (12 a^3 B-a^2 (27 A b-8 b C)-20 a b^2 B+35 A b^3\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {(7 A b-4 a B) \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\frac {\int \frac {3 b \left (4 (A+2 C) a^2-20 b B a+35 A b^2\right ) \left (a^2-b^2\right )^2+a b \left (12 B a^3-(27 A b-8 b C) a^2-20 b^2 B a+35 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a^2-b^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}+\frac {\left (-12 a^5 B+a^4 b (33 A-56 C)+104 a^3 b^2 B-2 a^2 b^3 (85 A-12 C)-60 a b^4 B+105 A b^5\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x) \left (-12 a^5 B+a^4 (33 A b-56 b C)+104 a^3 b^2 B-2 a^2 b^3 (85 A-12 C)-60 a b^4 B+105 A b^5\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (12 a^3 B-a^2 (27 A b-8 b C)-20 a b^2 B+35 A b^3\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {(7 A b-4 a B) \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\frac {\int \frac {3 b \left (4 (A+2 C) a^2-20 b B a+35 A b^2\right ) \left (a^2-b^2\right )^2+a b \left (12 B a^3-(27 A b-8 b C) a^2-20 b^2 B a+35 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a^2-b^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}+\frac {\left (-12 a^5 B+a^4 b (33 A-56 C)+104 a^3 b^2 B-2 a^2 b^3 (85 A-12 C)-60 a b^4 B+105 A b^5\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x) \left (-12 a^5 B+a^4 (33 A b-56 b C)+104 a^3 b^2 B-2 a^2 b^3 (85 A-12 C)-60 a b^4 B+105 A b^5\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (12 a^3 B-a^2 (27 A b-8 b C)-20 a b^2 B+35 A b^3\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {(7 A b-4 a B) \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\frac {\int \frac {3 b \left (4 (A+2 C) a^2-20 b B a+35 A b^2\right ) \left (a^2-b^2\right )^2+a b \left (12 B a^3-(27 A b-8 b C) a^2-20 b^2 B a+35 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a^2-b^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}+\frac {2 \left (-12 a^5 B+a^4 b (33 A-56 C)+104 a^3 b^2 B-2 a^2 b^3 (85 A-12 C)-60 a b^4 B+105 A b^5\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x) \left (-12 a^5 B+a^4 (33 A b-56 b C)+104 a^3 b^2 B-2 a^2 b^3 (85 A-12 C)-60 a b^4 B+105 A b^5\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (12 a^3 B-a^2 (27 A b-8 b C)-20 a b^2 B+35 A b^3\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {(7 A b-4 a B) \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\frac {3 b \left (a^2-b^2\right )^2 \left (4 a^2 (A+2 C)-20 a b B+35 A b^2\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx+a b \left (a^2-b^2\right ) \left (12 a^3 B-a^2 (27 A b-8 b C)-20 a b^2 B+35 A b^3\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}+\frac {2 \left (-12 a^5 B+a^4 b (33 A-56 C)+104 a^3 b^2 B-2 a^2 b^3 (85 A-12 C)-60 a b^4 B+105 A b^5\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x) \left (-12 a^5 B+a^4 (33 A b-56 b C)+104 a^3 b^2 B-2 a^2 b^3 (85 A-12 C)-60 a b^4 B+105 A b^5\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (12 a^3 B-a^2 (27 A b-8 b C)-20 a b^2 B+35 A b^3\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {(7 A b-4 a B) \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\frac {3 b \left (a^2-b^2\right )^2 \left (4 a^2 (A+2 C)-20 a b B+35 A b^2\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+a b \left (a^2-b^2\right ) \left (12 a^3 B-a^2 (27 A b-8 b C)-20 a b^2 B+35 A b^3\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}+\frac {2 \left (-12 a^5 B+a^4 b (33 A-56 C)+104 a^3 b^2 B-2 a^2 b^3 (85 A-12 C)-60 a b^4 B+105 A b^5\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x) \left (-12 a^5 B+a^4 (33 A b-56 b C)+104 a^3 b^2 B-2 a^2 b^3 (85 A-12 C)-60 a b^4 B+105 A b^5\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (12 a^3 B-a^2 (27 A b-8 b C)-20 a b^2 B+35 A b^3\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {(7 A b-4 a B) \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\frac {3 b \left (a^2-b^2\right )^2 \left (4 a^2 (A+2 C)-20 a b B+35 A b^2\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {a b \left (a^2-b^2\right ) \left (12 a^3 B-a^2 (27 A b-8 b C)-20 a b^2 B+35 A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 \left (-12 a^5 B+a^4 b (33 A-56 C)+104 a^3 b^2 B-2 a^2 b^3 (85 A-12 C)-60 a b^4 B+105 A b^5\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x) \left (-12 a^5 B+a^4 (33 A b-56 b C)+104 a^3 b^2 B-2 a^2 b^3 (85 A-12 C)-60 a b^4 B+105 A b^5\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (12 a^3 B-a^2 (27 A b-8 b C)-20 a b^2 B+35 A b^3\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {(7 A b-4 a B) \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\frac {3 b \left (a^2-b^2\right )^2 \left (4 a^2 (A+2 C)-20 a b B+35 A b^2\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {a b \left (a^2-b^2\right ) \left (12 a^3 B-a^2 (27 A b-8 b C)-20 a b^2 B+35 A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 \left (-12 a^5 B+a^4 b (33 A-56 C)+104 a^3 b^2 B-2 a^2 b^3 (85 A-12 C)-60 a b^4 B+105 A b^5\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x) \left (-12 a^5 B+a^4 (33 A b-56 b C)+104 a^3 b^2 B-2 a^2 b^3 (85 A-12 C)-60 a b^4 B+105 A b^5\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (12 a^3 B-a^2 (27 A b-8 b C)-20 a b^2 B+35 A b^3\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {(7 A b-4 a B) \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\frac {3 b \left (a^2-b^2\right )^2 \left (4 a^2 (A+2 C)-20 a b B+35 A b^2\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a b \left (a^2-b^2\right ) \left (12 a^3 B-a^2 (27 A b-8 b C)-20 a b^2 B+35 A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 \left (-12 a^5 B+a^4 b (33 A-56 C)+104 a^3 b^2 B-2 a^2 b^3 (85 A-12 C)-60 a b^4 B+105 A b^5\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x) \left (-12 a^5 B+a^4 (33 A b-56 b C)+104 a^3 b^2 B-2 a^2 b^3 (85 A-12 C)-60 a b^4 B+105 A b^5\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (12 a^3 B-a^2 (27 A b-8 b C)-20 a b^2 B+35 A b^3\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {(7 A b-4 a B) \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\frac {\frac {3 b \left (a^2-b^2\right )^2 \left (4 a^2 (A+2 C)-20 a b B+35 A b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {\sec (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}+\frac {2 a b \left (a^2-b^2\right ) \left (12 a^3 B-a^2 (27 A b-8 b C)-20 a b^2 B+35 A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 \left (-12 a^5 B+a^4 b (33 A-56 C)+104 a^3 b^2 B-2 a^2 b^3 (85 A-12 C)-60 a b^4 B+105 A b^5\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x) \left (-12 a^5 B+a^4 (33 A b-56 b C)+104 a^3 b^2 B-2 a^2 b^3 (85 A-12 C)-60 a b^4 B+105 A b^5\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (12 a^3 B-a^2 (27 A b-8 b C)-20 a b^2 B+35 A b^3\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {(7 A b-4 a B) \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {\frac {\frac {\frac {3 b \left (a^2-b^2\right )^2 \left (4 a^2 (A+2 C)-20 a b B+35 A b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}+\frac {2 a b \left (a^2-b^2\right ) \left (12 a^3 B-a^2 (27 A b-8 b C)-20 a b^2 B+35 A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 \left (-12 a^5 B+a^4 b (33 A-56 C)+104 a^3 b^2 B-2 a^2 b^3 (85 A-12 C)-60 a b^4 B+105 A b^5\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x) \left (-12 a^5 B+a^4 (33 A b-56 b C)+104 a^3 b^2 B-2 a^2 b^3 (85 A-12 C)-60 a b^4 B+105 A b^5\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b \sin (c+d x) \left (12 a^3 B-a^2 (27 A b-8 b C)-20 a b^2 B+35 A b^3\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {A \tan (c+d x) \sec (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {(7 A b-4 a B) \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\frac {2 b \sin (c+d x) \left (12 a^3 B-a^2 (27 A b-8 b C)-20 a b^2 B+35 A b^3\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}+\frac {\frac {\frac {\frac {6 b \left (a^2-b^2\right )^2 \left (4 a^2 (A+2 C)-20 a b B+35 A b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}+\frac {2 a b \left (a^2-b^2\right ) \left (12 a^3 B-a^2 (27 A b-8 b C)-20 a b^2 B+35 A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 \left (-12 a^5 B+a^4 b (33 A-56 C)+104 a^3 b^2 B-2 a^2 b^3 (85 A-12 C)-60 a b^4 B+105 A b^5\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x) \left (-12 a^5 B+a^4 (33 A b-56 b C)+104 a^3 b^2 B-2 a^2 b^3 (85 A-12 C)-60 a b^4 B+105 A b^5\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 a \left (a^2-b^2\right )}}{2 a}}{4 a}\) |
Input:
Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^(5/2),x]
Output:
(A*Sec[c + d*x]*Tan[c + d*x])/(2*a*d*(a + b*Cos[c + d*x])^(3/2)) - (-1/2*( (2*b*(35*A*b^3 + 12*a^3*B - 20*a*b^2*B - a^2*(27*A*b - 8*b*C))*Sin[c + d*x ])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (((2*(105*A*b^5 - 12*a ^5*B + 104*a^3*b^2*B - 60*a*b^4*B + a^4*b*(33*A - 56*C) - 2*a^2*b^3*(85*A - 12*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/( d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + ((2*a*b*(a^2 - b^2)*(35*A*b^3 + 12 *a^3*B - 20*a*b^2*B - a^2*(27*A*b - 8*b*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*(a^2 - b^2)^2*(35*A*b^2 - 20*a*b*B + 4*a^2*(A + 2*C))*Sqrt[(a + b*Co s[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*(a^2 - b^2)) - (2*b*(105*A*b^5 - 12*a^5*B + 104* a^3*b^2*B - 60*a*b^4*B - 2*a^2*b^3*(85*A - 12*C) + a^4*(33*A*b - 56*b*C))* Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]))/(3*a*(a^2 - b^2) ))/a + ((7*A*b - 4*a*B)*Tan[c + d*x])/(a*d*(a + b*Cos[c + d*x])^(3/2)))/(4 *a)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 0]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[ 2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a, b, c , d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt [c + d*Sin[e + f*x]] Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/(c + d))*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a* d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !GtQ[c + d, 0]
Int[(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]))/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[ B/d Int[(a + b*Sin[e + f*x])^m, x], x] - Simp[(B*c - A*d)/d Int[(a + b* Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int [(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A *b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ [n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) | | EqQ[a, 0])))
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[C/(b*d) Int[Sqrt[a + b*Sin[e + f*x]], x] , x] - Simp[1/(b*d) Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[ e + f*x], x]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; Fre eQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0 ] && NeQ[c^2 - d^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(2023\) vs. \(2(552)=1104\).
Time = 8.12 (sec) , antiderivative size = 2024, normalized size of antiderivative = 3.54
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2024\) |
| parts | \(\text {Expression too large to display}\) | \(4147\) |
Input:
int((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+b*cos(d*x+c))^(5/2),x, method=_RETURNVERBOSE)
Output:
-(-(-2*b*cos(1/2*d*x+1/2*c)^2-a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*A/a^2*(- 1/2*cos(1/2*d*x+1/2*c)/a*(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2* c)^2)^(1/2)/(-1+2*cos(1/2*d*x+1/2*c)^2)^2+3/4*b/a^2*cos(1/2*d*x+1/2*c)*(-2 *b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)/(-1+2*cos(1/2*d* x+1/2*c)^2)-1/8*b/a*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^ 2+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2) ^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))+3/8/a*(sin(1/2*d*x +1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2)/(-2*b*sin(1/ 2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*b*EllipticE(cos(1/2*d*x+1 /2*c),(-2*b/(a-b))^(1/2))-3/8*b^2/a^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*c os(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin (1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))- 1/2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1 /2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticP i(cos(1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1/2))-3/8/a^2*(sin(1/2*d*x+1/2*c)^2)^ (1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c )^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),2,(-2* b/(a-b))^(1/2))*b^2)-2*(2*A*b-B*a)/a^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)+1/2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-...
Timed out. \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+b*cos(d*x+c))^(5 /2),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x+c)**3/(a+b*cos(d*x+c))* *(5/2),x)
Output:
Timed out
Timed out. \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+b*cos(d*x+c))^(5 /2),x, algorithm="maxima")
Output:
Timed out
\[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{3}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+b*cos(d*x+c))^(5 /2),x, algorithm="giac")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*sec(d*x + c)^3/(b*cos(d* x + c) + a)^(5/2), x)
Timed out. \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx=\int \frac {C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A}{{\cos \left (c+d\,x\right )}^3\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \] Input:
int((A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^3*(a + b*cos(c + d*x))^(5/2)),x)
Output:
int((A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^3*(a + b*cos(c + d*x))^(5/2)), x)
\[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx=\left (\int \frac {\sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right ) \sec \left (d x +c \right )^{3}}{\cos \left (d x +c \right )^{3} b^{3}+3 \cos \left (d x +c \right )^{2} a \,b^{2}+3 \cos \left (d x +c \right ) a^{2} b +a^{3}}d x \right ) b +\left (\int \frac {\sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{3}}{\cos \left (d x +c \right )^{3} b^{3}+3 \cos \left (d x +c \right )^{2} a \,b^{2}+3 \cos \left (d x +c \right ) a^{2} b +a^{3}}d x \right ) c +\left (\int \frac {\sqrt {\cos \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{3}}{\cos \left (d x +c \right )^{3} b^{3}+3 \cos \left (d x +c \right )^{2} a \,b^{2}+3 \cos \left (d x +c \right ) a^{2} b +a^{3}}d x \right ) a \] Input:
int((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+b*cos(d*x+c))^(5/2),x)
Output:
int((sqrt(cos(c + d*x)*b + a)*cos(c + d*x)*sec(c + d*x)**3)/(cos(c + d*x)* *3*b**3 + 3*cos(c + d*x)**2*a*b**2 + 3*cos(c + d*x)*a**2*b + a**3),x)*b + int((sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**2*sec(c + d*x)**3)/(cos(c + d* x)**3*b**3 + 3*cos(c + d*x)**2*a*b**2 + 3*cos(c + d*x)*a**2*b + a**3),x)*c + int((sqrt(cos(c + d*x)*b + a)*sec(c + d*x)**3)/(cos(c + d*x)**3*b**3 + 3*cos(c + d*x)**2*a*b**2 + 3*cos(c + d*x)*a**2*b + a**3),x)*a