Integrand size = 45, antiderivative size = 545 \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}} \, dx=-\frac {2 \left (48 A b^4+25 a^3 b B-40 a b^3 B-6 a^2 b^2 (4 A-5 C)-3 a^4 (3 A+5 C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{15 a^5 \sqrt {a+b} d}-\frac {2 \left (48 A b^3+4 a b^2 (9 A-10 B)+6 a^2 b (2 A-5 B+5 C)+a^3 (9 A-5 B+15 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{15 a^4 \sqrt {a+b} d}+\frac {2 \left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{a \left (a^2-b^2\right ) d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (6 A b^2-5 a b B-a^2 (A-5 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{5 a^2 \left (a^2-b^2\right ) d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (24 A b^3+5 a^3 B-20 a b^2 B-a^2 (9 A b-15 b C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x)} \] Output:
-2/15*(48*A*b^4+25*B*a^3*b-40*B*a*b^3-6*a^2*b^2*(4*A-5*C)-3*a^4*(3*A+5*C)) *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*(1-sec(d*x+c))/(a+b))^(1/2)*(a*(1+sec(d*x+c))/(a- b))^(1/2)/a^5/(a+b)^(1/2)/d-2/15*(48*A*b^3+4*a*b^2*(9*A-10*B)+6*a^2*b*(2*A -5*B+5*C)+a^3*(9*A-5*B+15*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*(1-sec(d*x+c))/(a+b) )^(1/2)*(a*(1+sec(d*x+c))/(a-b))^(1/2)/a^4/(a+b)^(1/2)/d+2*(A*b^2-a*(B*b-C *a))*sin(d*x+c)/a/(a^2-b^2)/d/cos(d*x+c)^(5/2)/(a+b*cos(d*x+c))^(1/2)-2/5* (6*A*b^2-5*B*a*b-a^2*(A-5*C))*(a+b*cos(d*x+c))^(1/2)*sin(d*x+c)/a^2/(a^2-b ^2)/d/cos(d*x+c)^(5/2)+2/15*(24*A*b^3+5*B*a^3-20*B*a*b^2-a^2*(9*A*b-15*C*b ))*(a+b*cos(d*x+c))^(1/2)*sin(d*x+c)/a^3/(a^2-b^2)/d/cos(d*x+c)^(3/2)
Result contains complex when optimal does not.
Time = 7.81 (sec) , antiderivative size = 1511, normalized size of antiderivative = 2.77 \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}} \, dx =\text {Too large to display} \] Input:
Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + b*Cos[c + d*x])^(3/2)),x]
Output:
((-4*a*(12*a^4*A*b + 36*a^2*A*b^3 - 48*A*b^5 - 5*a^5*B - 35*a^3*b^2*B + 40 *a*b^4*B + 30*a^4*b*C - 30*a^2*b^3*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]]) - 4* a*(9*a^5*A + 24*a^3*A*b^2 - 48*a*A*b^4 - 25*a^4*b*B + 40*a^2*b^3*B + 15*a^ 5*C - 30*a^3*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]]) - (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]*Ellip ticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqr t[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*(9*a^4*A*b + 24*a^2*A*b^3 - 48*A*b^5 - 25*a^3*b^2*B + 40*a*b^4*B + 15*a^4*b*C - 30*a^2*b^3*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]]*...
Time = 2.68 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.03, 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.
| \(\displaystyle \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {2 \int -\frac {-\left ((A-5 C) a^2\right )-5 b B a+(A b+C b-a B) \cos (c+d x) a+6 A b^2-4 \left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x)}{2 \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{a \left (a^2-b^2\right )}+\frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}-\frac {\int \frac {-\left ((A-5 C) a^2\right )-5 b B a+(A b+C b-a B) \cos (c+d x) a+6 A b^2-4 \left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}-\frac {\int \frac {-\left ((A-5 C) a^2\right )-5 b B a+(A b+C b-a B) \sin \left (c+d x+\frac {\pi }{2}\right ) a+6 A b^2-4 \left (A b^2-a (b B-a C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {2 \int -\frac {5 B a^3-3 b (3 A-5 C) a^2-20 b^2 B a+\left ((3 A+5 C) a^2-5 b B a+2 A b^2\right ) \cos (c+d x) a+24 A b^3-2 b \left (-\left ((A-5 C) a^2\right )-5 b B a+6 A b^2\right ) \cos ^2(c+d x)}{2 \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{5 a}+\frac {2 \sin (c+d x) \left (-\left (a^2 (A-5 C)\right )-5 a b B+6 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (-\left (a^2 (A-5 C)\right )-5 a b B+6 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\int \frac {5 B a^3-(9 A b-15 b C) a^2-20 b^2 B a+\left ((3 A+5 C) a^2-5 b B a+2 A b^2\right ) \cos (c+d x) a+24 A b^3-2 b \left (-\left ((A-5 C) a^2\right )-5 b B a+6 A b^2\right ) \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{5 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (-\left (a^2 (A-5 C)\right )-5 a b B+6 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\int \frac {5 B a^3-(9 A b-15 b C) a^2-20 b^2 B a+\left ((3 A+5 C) a^2-5 b B a+2 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+24 A b^3-2 b \left (-\left ((A-5 C) a^2\right )-5 b B a+6 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (-\left (a^2 (A-5 C)\right )-5 a b B+6 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \int -\frac {-3 (3 A+5 C) a^4+25 b B a^3-6 b^2 (4 A-5 C) a^2-40 b^3 B a+\left (-5 B a^3+3 b (A+5 C) a^2-10 b^2 B a+12 A b^3\right ) \cos (c+d x) a+48 A b^4}{2 \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{3 a}+\frac {2 \sin (c+d x) \left (5 a^3 B-3 a^2 b (3 A-5 C)-20 a b^2 B+24 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{3 a d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (-\left (a^2 (A-5 C)\right )-5 a b B+6 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (5 a^3 B-3 a^2 b (3 A-5 C)-20 a b^2 B+24 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-3 (3 A+5 C) a^4+25 b B a^3-6 b^2 (4 A-5 C) a^2-40 b^3 B a+\left (-5 B a^3+3 b (A+5 C) a^2-10 b^2 B a+12 A b^3\right ) \cos (c+d x) a+48 A b^4}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{3 a}}{5 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (-\left (a^2 (A-5 C)\right )-5 a b B+6 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (5 a^3 B-3 a^2 b (3 A-5 C)-20 a b^2 B+24 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-3 (3 A+5 C) a^4+25 b B a^3-6 b^2 (4 A-5 C) a^2-40 b^3 B a+\left (-5 B a^3+3 b (A+5 C) a^2-10 b^2 B a+12 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+48 A b^4}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 a}}{5 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3477 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (-\left (a^2 (A-5 C)\right )-5 a b B+6 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (5 a^3 B-3 a^2 b (3 A-5 C)-20 a b^2 B+24 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {(a-b) \left (a^3 (9 A-5 B+15 C)+6 a^2 b (2 A-5 B+5 C)+4 a b^2 (9 A-10 B)+48 A b^3\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx+\left (-3 a^4 (3 A+5 C)+25 a^3 b B-6 a^2 b^2 (4 A-5 C)-40 a b^3 B+48 A b^4\right ) \int \frac {\cos (c+d x)+1}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{3 a}}{5 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (-\left (a^2 (A-5 C)\right )-5 a b B+6 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (5 a^3 B-3 a^2 b (3 A-5 C)-20 a b^2 B+24 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {(a-b) \left (a^3 (9 A-5 B+15 C)+6 a^2 b (2 A-5 B+5 C)+4 a b^2 (9 A-10 B)+48 A b^3\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\left (-3 a^4 (3 A+5 C)+25 a^3 b B-6 a^2 b^2 (4 A-5 C)-40 a b^3 B+48 A b^4\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 a}}{5 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3295 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (-\left (a^2 (A-5 C)\right )-5 a b B+6 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (5 a^3 B-3 a^2 b (3 A-5 C)-20 a b^2 B+24 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\left (-3 a^4 (3 A+5 C)+25 a^3 b B-6 a^2 b^2 (4 A-5 C)-40 a b^3 B+48 A b^4\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (a^3 (9 A-5 B+15 C)+6 a^2 b (2 A-5 B+5 C)+4 a b^2 (9 A-10 B)+48 A b^3\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{a d}}{3 a}}{5 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3473 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (-\left (a^2 (A-5 C)\right )-5 a b B+6 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (5 a^3 B-3 a^2 b (3 A-5 C)-20 a b^2 B+24 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (a^3 (9 A-5 B+15 C)+6 a^2 b (2 A-5 B+5 C)+4 a b^2 (9 A-10 B)+48 A b^3\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{a d}+\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (-3 a^4 (3 A+5 C)+25 a^3 b B-6 a^2 b^2 (4 A-5 C)-40 a b^3 B+48 A b^4\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{a^2 d}}{3 a}}{5 a}}{a \left (a^2-b^2\right )}\) |
Input:
Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + b*Cos [c + d*x])^(3/2)),x]
Output:
(2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(a*(a^2 - b^2)*d*Cos[c + d*x]^(5/ 2)*Sqrt[a + b*Cos[c + d*x]]) - ((2*(6*A*b^2 - 5*a*b*B - a^2*(A - 5*C))*Sqr t[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^4 + 25*a^3*b*B - 40*a*b^3*B - 6*a^2*b^2*(4*A - 5*C) - 3*a^4*(3*A + 5*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*(a - b)*Sqrt[a + b]*(48*A*b^3 + 4*a*b^2*(9*A - 10*B) + 6*a^2*b*(2*A - 5*B + 5*C) + a^3*(9*A - 5*B + 15*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))]*S qrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/ (a*d))/a + (2*(24*A*b^3 + 5*a^3*B - 20*a*b^2*B - 3*a^2*b*(3*A - 5*C))*Sqrt [a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)))/(5*a))/(a*( a^2 - b^2))
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. \(3434\) vs. \(2(503)=1006\).
Time = 23.61 (sec) , antiderivative size = 3435, normalized size of antiderivative = 6.30
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(3435\) |
| default | \(\text {Expression too large to display}\) | \(3500\) |
Input:
int((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(7/2)/(a+b*cos(d*x+c))^(3/2 ),x,method=_RETURNVERBOSE)
Output:
2/5*A/d/(a+b)/(a-b)/a^4*((cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*c os(d*x+c))/(1+cos(d*x+c)))^(1/2)*a^5*EllipticE(-csc(d*x+c)+cot(d*x+c),(-(a -b)/(a+b))^(1/2))*(3*cos(d*x+c)^4+6*cos(d*x+c)^3+3*cos(d*x+c)^2)+(cos(d*x+ c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a ^4*b*EllipticE(-csc(d*x+c)+cot(d*x+c),(-(a-b)/(a+b))^(1/2))*(3*cos(d*x+c)^ 4+6*cos(d*x+c)^3+3*cos(d*x+c)^2)+(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b )*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a^3*b^2*EllipticE(-csc(d*x+c)+cot (d*x+c),(-(a-b)/(a+b))^(1/2))*(8*cos(d*x+c)^4+16*cos(d*x+c)^3+8*cos(d*x+c) ^2)+(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x +c)))^(1/2)*a^2*b^3*EllipticE(-csc(d*x+c)+cot(d*x+c),(-(a-b)/(a+b))^(1/2)) *(8*cos(d*x+c)^4+16*cos(d*x+c)^3+8*cos(d*x+c)^2)+(cos(d*x+c)/(1+cos(d*x+c) ))^(1/2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a*b^4*EllipticE(- csc(d*x+c)+cot(d*x+c),(-(a-b)/(a+b))^(1/2))*(-16*cos(d*x+c)^4-32*cos(d*x+c )^3-16*cos(d*x+c)^2)+(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*cos(d *x+c))/(1+cos(d*x+c)))^(1/2)*b^5*EllipticE(-csc(d*x+c)+cot(d*x+c),(-(a-b)/ (a+b))^(1/2))*(-16*cos(d*x+c)^4-32*cos(d*x+c)^3-16*cos(d*x+c)^2)+(cos(d*x+ c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a ^5*EllipticF(-csc(d*x+c)+cot(d*x+c),(-(a-b)/(a+b))^(1/2))*(-3*cos(d*x+c)^4 -6*cos(d*x+c)^3-3*cos(d*x+c)^2)+(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b) *(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a^4*b*EllipticF(-csc(d*x+c)+cot...
\[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}} \, dx=\int { \frac {C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(7/2)/(a+b*cos(d*x+c) )^(3/2),x, algorithm="fricas")
Output:
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^2*cos(d*x + c)^6 + 2*a*b*cos(d*x + c)^5 + a^2*cos(d* x + c)^4), x)
Timed out. \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)**2)/cos(d*x+c)**(7/2)/(a+b*cos(d*x+ c))**(3/2),x)
Output:
Timed out
\[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}} \, dx=\int { \frac {C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(7/2)/(a+b*cos(d*x+c) )^(3/2),x, algorithm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)/((b*cos(d*x + c) + a)^(3 /2)*cos(d*x + c)^(7/2)), x)
\[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}} \, dx=\int { \frac {C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(7/2)/(a+b*cos(d*x+c) )^(3/2),x, algorithm="giac")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)/((b*cos(d*x + c) + a)^(3 /2)*cos(d*x + c)^(7/2)), x)
Timed out. \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}} \, dx=\int \frac {C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A}{{\cos \left (c+d\,x\right )}^{7/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \] Input:
int((A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^(7/2)*(a + b*cos (c + d*x))^(3/2)),x)
Output:
int((A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^(7/2)*(a + b*cos (c + d*x))^(3/2)), x)
\[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}} \, dx=\left (\int \frac {\sqrt {\cos \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{6} b^{2}+2 \cos \left (d x +c \right )^{5} a b +\cos \left (d x +c \right )^{4} a^{2}}d x \right ) a +\left (\int \frac {\sqrt {\cos \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{5} b^{2}+2 \cos \left (d x +c \right )^{4} a b +\cos \left (d x +c \right )^{3} a^{2}}d x \right ) b +\left (\int \frac {\sqrt {\cos \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{4} b^{2}+2 \cos \left (d x +c \right )^{3} a b +\cos \left (d x +c \right )^{2} a^{2}}d x \right ) c \] Input:
int((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(7/2)/(a+b*cos(d*x+c))^(3/2 ),x)
Output:
int((sqrt(cos(c + d*x)*b + a)*sqrt(cos(c + d*x)))/(cos(c + d*x)**6*b**2 + 2*cos(c + d*x)**5*a*b + cos(c + d*x)**4*a**2),x)*a + int((sqrt(cos(c + d*x )*b + a)*sqrt(cos(c + d*x)))/(cos(c + d*x)**5*b**2 + 2*cos(c + d*x)**4*a*b + cos(c + d*x)**3*a**2),x)*b + int((sqrt(cos(c + d*x)*b + a)*sqrt(cos(c + d*x)))/(cos(c + d*x)**4*b**2 + 2*cos(c + d*x)**3*a*b + cos(c + d*x)**2*a* *2),x)*c