Integrand size = 45, antiderivative size = 466 \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx=-\frac {2 (a-b) \sqrt {a+b} \left (48 A b^3-63 a^3 B-56 a b^2 B+a^2 (44 A b+70 b C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{105 a^5 d}+\frac {2 \sqrt {a+b} \left (48 A b^3-4 a b^2 (3 A+14 B)+a^3 (25 A-63 B+35 C)+2 a^2 b (22 A+7 (B+5 C))\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{105 a^4 d}+\frac {2 A \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {2 (6 A b-7 a B) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{35 a^2 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (24 A b^2-28 a b B+5 a^2 (5 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 a^3 d \cos ^{\frac {3}{2}}(c+d x)} \]
2/7*A*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/a/d/cos(d*x+c)^(7/2)-2/35*(6*A*b-7 *B*a)*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/a^2/d/cos(d*x+c)^(5/2)+2/105*(24*A *b^2-28*B*a*b+5*a^2*(5*A+7*C))*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/a^3/d/cos (d*x+c)^(3/2)-2/105*(a-b)*(48*A*b^3-63*B*a^3-56*B*a*b^2+a^2*(44*A*b+70*C*b ))*cot(d*x+c)*EllipticE((a+b*cos(d*x+c))^(1/2)/(a+b)^(1/2)/cos(d*x+c)^(1/2 ),((-a-b)/(a-b))^(1/2))*(a+b)^(1/2)*(a*(1-sec(d*x+c))/(a+b))^(1/2)*(a*(1+s ec(d*x+c))/(a-b))^(1/2)/a^5/d+2/105*(48*A*b^3-4*a*b^2*(3*A+14*B)+a^3*(25*A -63*B+35*C)+2*a^2*b*(22*A+7*B+35*C))*cot(d*x+c)*EllipticF((a+b*cos(d*x+c)) ^(1/2)/(a+b)^(1/2)/cos(d*x+c)^(1/2),((-a-b)/(a-b))^(1/2))*(a+b)^(1/2)*(a*( 1-sec(d*x+c))/(a+b))^(1/2)*(a*(1+sec(d*x+c))/(a-b))^(1/2)/a^4/d
Result contains complex when optimal does not.
Time = 7.46 (sec) , antiderivative size = 1468, normalized size of antiderivative = 3.15 \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx =\text {Too large to display} \]
Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(9/2)*Sqrt [a + b*Cos[c + d*x]]),x]
((-4*a*(25*a^4*A + 32*a^2*A*b^2 + 48*A*b^4 - 49*a^3*b*B - 56*a*b^3*B + 35* a^4*C + 70*a^2*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(( (a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Cs c[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d* x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/ ((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(44*a^3*A*b + 48*a*A*b^3 - 63*a^4*B - 56*a^2*b^2*B + 70*a^3*b*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a) ]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF [ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a) /(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[ c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)* Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d *x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d *x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4) /(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(44*a^2*A*b^2 + 48*A *b^4 - 63*a^3*b*B - 56*a*b^3*B + 70*a^2*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]] ], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]] *Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a...
Time = 2.21 (sec) , antiderivative size = 486, normalized size of antiderivative = 1.04, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.311, Rules used = {3042, 3534, 27, 3042, 3534, 27, 3042, 3534, 27, 3042, 3477, 3042, 3295, 3473}
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.
| \(\displaystyle \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {2 \int -\frac {-4 A b \cos ^2(c+d x)-a (5 A+7 C) \cos (c+d x)+6 A b-7 a B}{2 \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{7 a}+\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\int \frac {-4 A b \cos ^2(c+d x)-a (5 A+7 C) \cos (c+d x)+6 A b-7 a B}{\cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{7 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\int \frac {-4 A b \sin \left (c+d x+\frac {\pi }{2}\right )^2-a (5 A+7 C) \sin \left (c+d x+\frac {\pi }{2}\right )+6 A b-7 a B}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{7 a}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 \int -\frac {5 (5 A+7 C) a^2-28 b B a+(2 A b+21 a B) \cos (c+d x) a+24 A b^2-2 b (6 A b-7 a B) \cos ^2(c+d x)}{2 \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{5 a}+\frac {2 (6 A b-7 a B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 (6 A b-7 a B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\int \frac {5 (5 A+7 C) a^2-28 b B a+(2 A b+21 a B) \cos (c+d x) a+24 A b^2-2 b (6 A b-7 a B) \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{5 a}}{7 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 (6 A b-7 a B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\int \frac {5 (5 A+7 C) a^2-28 b B a+(2 A b+21 a B) \sin \left (c+d x+\frac {\pi }{2}\right ) a+24 A b^2-2 b (6 A b-7 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 a}}{7 a}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 (6 A b-7 a B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \int -\frac {-63 B a^3+2 b (22 A+35 C) a^2-56 b^2 B a+\left (-5 (5 A+7 C) a^2-14 b B a+12 A b^2\right ) \cos (c+d x) a+48 A b^3}{2 \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{3 a}+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)-28 a b B+24 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{3 a d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}}{7 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 (6 A b-7 a B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)-28 a b B+24 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-63 B a^3+(44 A b+70 C b) a^2-56 b^2 B a+\left (-5 (5 A+7 C) a^2-14 b B a+12 A b^2\right ) \cos (c+d x) a+48 A b^3}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{3 a}}{5 a}}{7 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 (6 A b-7 a B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)-28 a b B+24 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-63 B a^3+(44 A b+70 C b) a^2-56 b^2 B a+\left (-5 (5 A+7 C) a^2-14 b B a+12 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+48 A b^3}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 a}}{5 a}}{7 a}\) |
| \(\Big \downarrow \) 3477 |
| \(\displaystyle \frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 (6 A b-7 a B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)-28 a b B+24 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\left (-63 a^3 B+2 a^2 b (22 A+35 C)-56 a b^2 B+48 A b^3\right ) \int \frac {\cos (c+d x)+1}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx-\left (a^3 (25 A-63 B+35 C)+2 a^2 b (22 A+7 (B+5 C))-4 a b^2 (3 A+14 B)+48 A b^3\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx}{3 a}}{5 a}}{7 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 (6 A b-7 a B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)-28 a b B+24 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\left (-63 a^3 B+2 a^2 b (22 A+35 C)-56 a b^2 B+48 A b^3\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\left (a^3 (25 A-63 B+35 C)+2 a^2 b (22 A+7 (B+5 C))-4 a b^2 (3 A+14 B)+48 A b^3\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 a}}{5 a}}{7 a}\) |
| \(\Big \downarrow \) 3295 |
| \(\displaystyle \frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 (6 A b-7 a B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)-28 a b B+24 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\left (-63 a^3 B+2 a^2 b (22 A+35 C)-56 a b^2 B+48 A b^3\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 \sqrt {a+b} \cot (c+d x) \left (a^3 (25 A-63 B+35 C)+2 a^2 b (22 A+7 (B+5 C))-4 a b^2 (3 A+14 B)+48 A b^3\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{a d}}{3 a}}{5 a}}{7 a}\) |
| \(\Big \downarrow \) 3473 |
| \(\displaystyle \frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 (6 A b-7 a B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)-28 a b B+24 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (-63 a^3 B+2 a^2 b (22 A+35 C)-56 a b^2 B+48 A b^3\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{a^2 d}-\frac {2 \sqrt {a+b} \cot (c+d x) \left (a^3 (25 A-63 B+35 C)+2 a^2 b (22 A+7 (B+5 C))-4 a b^2 (3 A+14 B)+48 A b^3\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{a d}}{3 a}}{5 a}}{7 a}\) |
(2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*a*d*Cos[c + d*x]^(7/2)) - ( (2*(6*A*b - 7*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*a*d*Cos[c + d *x]^(5/2)) - (-1/3*((2*(a - b)*Sqrt[a + b]*(48*A*b^3 - 63*a^3*B - 56*a*b^2 *B + 2*a^2*b*(22*A + 35*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*d) - (2*Sqrt[a + b]*(48*A*b^3 - 4*a*b^2*(3*A + 14*B) + a^3*(25*A - 63*B + 35*C ) + 2*a^2*b*(22*A + 7*(B + 5*C)))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b *Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt [(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a* d))/a + (2*(24*A*b^2 - 28*a*b*B + 5*a^2*(5*A + 7*C))*Sqrt[a + b*Cos[c + d* x]]*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)))/(5*a))/(7*a)
3.12.42.3.1 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[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f _.)*(x_)]]), x_Symbol] :> Simp[-2*(Tan[e + f*x]/(a*f))*Rt[(a + b)/d, 2]*Sqr t[a*((1 - Csc[e + f*x])/(a + b))]*Sqrt[a*((1 + Csc[e + f*x])/(a - b))]*Elli pticF[ArcSin[Sqrt[a + b*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]/Rt[(a + b)/d, 2] ], -(a + b)/(a - b)], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && PosQ[(a + b)/d]
Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((b_.)*sin[(e_.) + (f_.)*(x_)]) ^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*A* (c - d)*(Tan[e + f*x]/(f*b*c^2))*Rt[(c + d)/b, 2]*Sqrt[c*((1 + Csc[e + f*x] )/(c - d))]*Sqrt[c*((1 - Csc[e + f*x])/(c + d))]*EllipticE[ArcSin[Sqrt[c + d*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]]/Rt[(c + d)/b, 2]], -(c + d)/(c - d)], x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && PosQ[(c + d)/b]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ .)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> S imp[(A - B)/(a - b) Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f* x]]), x], x] - Simp[(A*b - a*B)/(a - b) Int[(1 + Sin[e + f*x])/((a + b*Si n[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e , f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && NeQ[A, B]
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])))
Leaf count of result is larger than twice the leaf count of optimal. \(5790\) vs. \(2(427)=854\).
Time = 38.61 (sec) , antiderivative size = 5791, normalized size of antiderivative = 12.43
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(5791\) |
| default | \(\text {Expression too large to display}\) | \(5875\) |
int((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(9/2)/(a+b*cos(d*x+c))^(1/2 ),x,method=_RETURNVERBOSE)
\[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx=\int { \frac {C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A}{\sqrt {b \cos \left (d x + c\right ) + a} \cos \left (d x + c\right )^{\frac {9}{2}}} \,d x } \]
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(9/2)/(a+b*cos(d*x+c) )^(1/2),x, algorithm="fricas")
integral((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*sqrt(b*cos(d*x + c) + a)* sqrt(cos(d*x + c))/(b*cos(d*x + c)^6 + a*cos(d*x + c)^5), x)
Timed out. \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx=\text {Timed out} \]
\[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx=\int { \frac {C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A}{\sqrt {b \cos \left (d x + c\right ) + a} \cos \left (d x + c\right )^{\frac {9}{2}}} \,d x } \]
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(9/2)/(a+b*cos(d*x+c) )^(1/2),x, algorithm="maxima")
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)/(sqrt(b*cos(d*x + c) + a )*cos(d*x + c)^(9/2)), x)
\[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx=\int { \frac {C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A}{\sqrt {b \cos \left (d x + c\right ) + a} \cos \left (d x + c\right )^{\frac {9}{2}}} \,d x } \]
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(9/2)/(a+b*cos(d*x+c) )^(1/2),x, algorithm="giac")
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)/(sqrt(b*cos(d*x + c) + a )*cos(d*x + c)^(9/2)), x)
Timed out. \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx=\int \frac {C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A}{{\cos \left (c+d\,x\right )}^{9/2}\,\sqrt {a+b\,\cos \left (c+d\,x\right )}} \,d x \]
int((A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^(9/2)*(a + b*cos (c + d*x))^(1/2)),x)