Integrand size = 42, antiderivative size = 420 \[ \int \frac {B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx=-\frac {\left (8 a^4 B-29 a^2 b^2 B+15 b^4 B+9 a^3 b C-3 a b^3 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 a^3 \left (a^2-b^2\right )^2 d}+\frac {\left (11 a^2 b B-5 b^3 B-7 a^3 C+a b^2 C\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{4 a^2 \left (a^2-b^2\right )^2 d}-\frac {\left (35 a^4 b B-38 a^2 b^3 B+15 b^5 B-15 a^5 C+6 a^3 b^2 C-3 a b^4 C\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{4 a^3 (a-b)^2 (a+b)^3 d}+\frac {\left (8 a^4 B-29 a^2 b^2 B+15 b^4 B+9 a^3 b C-3 a b^3 C\right ) \sin (c+d x)}{4 a^3 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)}}+\frac {b (b B-a C) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}+\frac {b \left (11 a^2 b B-5 b^3 B-7 a^3 C+a b^2 C\right ) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \]
-1/4*(8*B*a^4-29*B*a^2*b^2+15*B*b^4+9*C*a^3*b-3*C*a*b^3)*(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))/a^3/( a^2-b^2)^2/d+1/4*(11*B*a^2*b-5*B*b^3-7*C*a^3+C*a*b^2)*(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))/a^2/(a^2 -b^2)^2/d-1/4*(35*B*a^4*b-38*B*a^2*b^3+15*B*b^5-15*C*a^5+6*C*a^3*b^2-3*C*a *b^4)*(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*b/(a+b),2^(1/2))/a^3/(a-b)^2/(a+b)^3/d+1/4*(8*B*a^4-29*B*a^2*b ^2+15*B*b^4+9*C*a^3*b-3*C*a*b^3)*sin(d*x+c)/a^3/(a^2-b^2)^2/d/cos(d*x+c)^( 1/2)+1/2*b*(B*b-C*a)*sin(d*x+c)/a/(a^2-b^2)/d/(a+b*cos(d*x+c))^2/cos(d*x+c )^(1/2)+1/4*b*(11*B*a^2*b-5*B*b^3-7*C*a^3+C*a*b^2)*sin(d*x+c)/a^2/(a^2-b^2 )^2/d/(a+b*cos(d*x+c))/cos(d*x+c)^(1/2)
Time = 5.92 (sec) , antiderivative size = 458, normalized size of antiderivative = 1.09 \[ \int \frac {B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx=\frac {-\frac {\frac {\left (56 a^4 b B-95 a^2 b^3 B+45 b^5 B-16 a^5 C+19 a^3 b^2 C-9 a b^4 C\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{a+b}+\frac {8 a \left (2 a^4 B-10 a^2 b^2 B+5 b^4 B+4 a^3 b C-a b^3 C\right ) \left ((a+b) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-a \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )\right )}{b (a+b)}+\frac {\left (8 a^4 B-29 a^2 b^2 B+15 b^4 B+9 a^3 b C-3 a b^3 C\right ) \left (-2 a b E\left (\left .\arcsin \left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 a (a+b) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )+\left (-2 a^2+b^2\right ) \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )\right ) \sin (c+d x)}{a b \sqrt {\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}+\frac {\sqrt {\cos (c+d x)} \left (2 a b \left (16 a^4 B-47 a^2 b^2 B+25 b^4 B+11 a^3 b C-5 a b^3 C\right ) \sin (c+d x)+b^2 \left (8 a^4 B-29 a^2 b^2 B+15 b^4 B+9 a^3 b C-3 a b^3 C\right ) \sin (2 (c+d x))+16 \left (a^3-a b^2\right )^2 B \tan (c+d x)\right )}{\left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}}{8 a^3 d} \]
(-((((56*a^4*b*B - 95*a^2*b^3*B + 45*b^5*B - 16*a^5*C + 19*a^3*b^2*C - 9*a *b^4*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*a*(2*a^4*B - 10*a^2*b^2*B + 5*b^4*B + 4*a^3*b*C - a*b^3*C)*((a + b)*EllipticF[(c + d *x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(b*(a + b)) + (( 8*a^4*B - 29*a^2*b^2*B + 15*b^4*B + 9*a^3*b*C - 3*a*b^3*C)*(-2*a*b*Ellipti cE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos [c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d *x]]], -1])*Sin[c + d*x])/(a*b*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2 )) + (Sqrt[Cos[c + d*x]]*(2*a*b*(16*a^4*B - 47*a^2*b^2*B + 25*b^4*B + 11*a ^3*b*C - 5*a*b^3*C)*Sin[c + d*x] + b^2*(8*a^4*B - 29*a^2*b^2*B + 15*b^4*B + 9*a^3*b*C - 3*a*b^3*C)*Sin[2*(c + d*x)] + 16*(a^3 - a*b^2)^2*B*Tan[c + d *x]))/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2))/(8*a^3*d)
Time = 3.22 (sec) , antiderivative size = 412, normalized size of antiderivative = 0.98, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {3042, 3508, 3042, 3479, 27, 3042, 3534, 27, 3042, 3534, 27, 3042, 3538, 25, 3042, 3119, 3481, 3042, 3120, 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.
| \(\displaystyle \int \frac {B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 3508 |
| \(\displaystyle \int \frac {B+C \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {B+C \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 3479 |
| \(\displaystyle \frac {\int \frac {4 B a^2+b C a-4 (b B-a C) \cos (c+d x) a+3 b (b B-a C) \cos ^2(c+d x)-5 b^2 B}{2 \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}+\frac {b (b B-a C) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {4 B a^2+b C a-4 (b B-a C) \cos (c+d x) a+3 b (b B-a C) \cos ^2(c+d x)-5 b^2 B}{\cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}dx}{4 a \left (a^2-b^2\right )}+\frac {b (b B-a C) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {4 B a^2+b C a-4 (b B-a C) \sin \left (c+d x+\frac {\pi }{2}\right ) a+3 b (b B-a C) \sin \left (c+d x+\frac {\pi }{2}\right )^2-5 b^2 B}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{4 a \left (a^2-b^2\right )}+\frac {b (b B-a C) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\int \frac {8 B a^4+9 b C a^3-29 b^2 B a^2-3 b^3 C a-4 \left (-2 C a^3+4 b B a^2-b^2 C a-b^3 B\right ) \cos (c+d x) a+b \left (-7 C a^3+11 b B a^2+b^2 C a-5 b^3 B\right ) \cos ^2(c+d x)+15 b^4 B}{2 \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}dx}{a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 C+11 a^2 b B+a b^2 C-5 b^3 B\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b (b B-a C) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {8 B a^4+9 b C a^3-29 b^2 B a^2-3 b^3 C a-4 \left (-2 C a^3+4 b B a^2-b^2 C a-b^3 B\right ) \cos (c+d x) a+b \left (-7 C a^3+11 b B a^2+b^2 C a-5 b^3 B\right ) \cos ^2(c+d x)+15 b^4 B}{\cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}dx}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 C+11 a^2 b B+a b^2 C-5 b^3 B\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b (b B-a C) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {8 B a^4+9 b C a^3-29 b^2 B a^2-3 b^3 C a-4 \left (-2 C a^3+4 b B a^2-b^2 C a-b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+b \left (-7 C a^3+11 b B a^2+b^2 C a-5 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+15 b^4 B}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 C+11 a^2 b B+a b^2 C-5 b^3 B\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b (b B-a C) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\frac {2 \int -\frac {-8 C a^5+24 b B a^4+5 b^2 C a^3-33 b^3 B a^2-3 b^4 C a+4 \left (2 B a^4+4 b C a^3-10 b^2 B a^2-b^3 C a+5 b^4 B\right ) \cos (c+d x) a+b \left (8 B a^4+9 b C a^3-29 b^2 B a^2-3 b^3 C a+15 b^4 B\right ) \cos ^2(c+d x)+15 b^5 B}{2 \sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{a}+\frac {2 \left (8 a^4 B+9 a^3 b C-29 a^2 b^2 B-3 a b^3 C+15 b^4 B\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 C+11 a^2 b B+a b^2 C-5 b^3 B\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b (b B-a C) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {2 \left (8 a^4 B+9 a^3 b C-29 a^2 b^2 B-3 a b^3 C+15 b^4 B\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {\int \frac {-8 C a^5+24 b B a^4+5 b^2 C a^3-33 b^3 B a^2-3 b^4 C a+4 \left (2 B a^4+4 b C a^3-10 b^2 B a^2-b^3 C a+5 b^4 B\right ) \cos (c+d x) a+b \left (8 B a^4+9 b C a^3-29 b^2 B a^2-3 b^3 C a+15 b^4 B\right ) \cos ^2(c+d x)+15 b^5 B}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{a}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 C+11 a^2 b B+a b^2 C-5 b^3 B\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b (b B-a C) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (8 a^4 B+9 a^3 b C-29 a^2 b^2 B-3 a b^3 C+15 b^4 B\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {\int \frac {-8 C a^5+24 b B a^4+5 b^2 C a^3-33 b^3 B a^2-3 b^4 C a+4 \left (2 B a^4+4 b C a^3-10 b^2 B a^2-b^3 C a+5 b^4 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+b \left (8 B a^4+9 b C a^3-29 b^2 B a^2-3 b^3 C a+15 b^4 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+15 b^5 B}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 C+11 a^2 b B+a b^2 C-5 b^3 B\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b (b B-a C) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle \frac {\frac {\frac {2 \left (8 a^4 B+9 a^3 b C-29 a^2 b^2 B-3 a b^3 C+15 b^4 B\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {\left (8 a^4 B+9 a^3 b C-29 a^2 b^2 B-3 a b^3 C+15 b^4 B\right ) \int \sqrt {\cos (c+d x)}dx-\frac {\int -\frac {b \left (-8 C a^5+24 b B a^4+5 b^2 C a^3-33 b^3 B a^2-3 b^4 C a+15 b^5 B\right )-a b^2 \left (-7 C a^3+11 b B a^2+b^2 C a-5 b^3 B\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}}{a}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 C+11 a^2 b B+a b^2 C-5 b^3 B\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b (b B-a C) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {2 \left (8 a^4 B+9 a^3 b C-29 a^2 b^2 B-3 a b^3 C+15 b^4 B\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {\left (8 a^4 B+9 a^3 b C-29 a^2 b^2 B-3 a b^3 C+15 b^4 B\right ) \int \sqrt {\cos (c+d x)}dx+\frac {\int \frac {b \left (-8 C a^5+24 b B a^4+5 b^2 C a^3-33 b^3 B a^2-3 b^4 C a+15 b^5 B\right )-a b^2 \left (-7 C a^3+11 b B a^2+b^2 C a-5 b^3 B\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}}{a}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 C+11 a^2 b B+a b^2 C-5 b^3 B\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b (b B-a C) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (8 a^4 B+9 a^3 b C-29 a^2 b^2 B-3 a b^3 C+15 b^4 B\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {\left (8 a^4 B+9 a^3 b C-29 a^2 b^2 B-3 a b^3 C+15 b^4 B\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {\int \frac {b \left (-8 C a^5+24 b B a^4+5 b^2 C a^3-33 b^3 B a^2-3 b^4 C a+15 b^5 B\right )-a b^2 \left (-7 C a^3+11 b B a^2+b^2 C a-5 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}}{a}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 C+11 a^2 b B+a b^2 C-5 b^3 B\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b (b B-a C) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\frac {\frac {2 \left (8 a^4 B+9 a^3 b C-29 a^2 b^2 B-3 a b^3 C+15 b^4 B\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {\int \frac {b \left (-8 C a^5+24 b B a^4+5 b^2 C a^3-33 b^3 B a^2-3 b^4 C a+15 b^5 B\right )-a b^2 \left (-7 C a^3+11 b B a^2+b^2 C a-5 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}+\frac {2 \left (8 a^4 B+9 a^3 b C-29 a^2 b^2 B-3 a b^3 C+15 b^4 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{a}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 C+11 a^2 b B+a b^2 C-5 b^3 B\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b (b B-a C) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle \frac {\frac {\frac {2 \left (8 a^4 B+9 a^3 b C-29 a^2 b^2 B-3 a b^3 C+15 b^4 B\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {b \left (-15 a^5 C+35 a^4 b B+6 a^3 b^2 C-38 a^2 b^3 B-3 a b^4 C+15 b^5 B\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx-a b \left (-7 a^3 C+11 a^2 b B+a b^2 C-5 b^3 B\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{b}+\frac {2 \left (8 a^4 B+9 a^3 b C-29 a^2 b^2 B-3 a b^3 C+15 b^4 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{a}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 C+11 a^2 b B+a b^2 C-5 b^3 B\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b (b B-a C) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (8 a^4 B+9 a^3 b C-29 a^2 b^2 B-3 a b^3 C+15 b^4 B\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {b \left (-15 a^5 C+35 a^4 b B+6 a^3 b^2 C-38 a^2 b^3 B-3 a b^4 C+15 b^5 B\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx-a b \left (-7 a^3 C+11 a^2 b B+a b^2 C-5 b^3 B\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}+\frac {2 \left (8 a^4 B+9 a^3 b C-29 a^2 b^2 B-3 a b^3 C+15 b^4 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{a}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 C+11 a^2 b B+a b^2 C-5 b^3 B\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b (b B-a C) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {\frac {\frac {2 \left (8 a^4 B+9 a^3 b C-29 a^2 b^2 B-3 a b^3 C+15 b^4 B\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {b \left (-15 a^5 C+35 a^4 b B+6 a^3 b^2 C-38 a^2 b^3 B-3 a b^4 C+15 b^5 B\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx-\frac {2 a b \left (-7 a^3 C+11 a^2 b B+a b^2 C-5 b^3 B\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}}{b}+\frac {2 \left (8 a^4 B+9 a^3 b C-29 a^2 b^2 B-3 a b^3 C+15 b^4 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{a}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 C+11 a^2 b B+a b^2 C-5 b^3 B\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b (b B-a C) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {b (b B-a C) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}+\frac {\frac {b \left (-7 a^3 C+11 a^2 b B+a b^2 C-5 b^3 B\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))}+\frac {\frac {2 \left (8 a^4 B+9 a^3 b C-29 a^2 b^2 B-3 a b^3 C+15 b^4 B\right ) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {2 \left (8 a^4 B+9 a^3 b C-29 a^2 b^2 B-3 a b^3 C+15 b^4 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {\frac {2 b \left (-15 a^5 C+35 a^4 b B+6 a^3 b^2 C-38 a^2 b^3 B-3 a b^4 C+15 b^5 B\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{d (a+b)}-\frac {2 a b \left (-7 a^3 C+11 a^2 b B+a b^2 C-5 b^3 B\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}}{b}}{a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\) |
(b*(b*B - a*C)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b* Cos[c + d*x])^2) + ((b*(11*a^2*b*B - 5*b^3*B - 7*a^3*C + a*b^2*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])) + (-(((2*( 8*a^4*B - 29*a^2*b^2*B + 15*b^4*B + 9*a^3*b*C - 3*a*b^3*C)*EllipticE[(c + d*x)/2, 2])/d + ((-2*a*b*(11*a^2*b*B - 5*b^3*B - 7*a^3*C + a*b^2*C)*Ellipt icF[(c + d*x)/2, 2])/d + (2*b*(35*a^4*b*B - 38*a^2*b^3*B + 15*b^5*B - 15*a ^5*C + 6*a^3*b^2*C - 3*a*b^4*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]) /((a + b)*d))/b)/a) + (2*(8*a^4*B - 29*a^2*b^2*B + 15*b^4*B + 9*a^3*b*C - 3*a*b^3*C)*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]))/(2*a*(a^2 - b^2)))/(4*a *(a^2 - b^2))
3.9.96.3.1 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[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
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[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[(-(A*b^2 - a*b*B))*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin [e + f*x])^(1 + n)/(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[(a*A - b*B)*(b*c - a*d)*(m + 1) + b*d*(A*b - a*B)*(m + n + 2) + (A*b - a*B)*(a*d*(m + 1) - b*c*(m + 2))*Sin[e + f*x] - b*d*(A*b - a*B) *(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n }, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && Rat ionalQ[m] && m < -1 && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ[n]) || !(I ntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) || EqQ[a, 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[1/b^2 Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*(b*B - a*C + b*C*Sin[e + f*x]), x], x] /; FreeQ [{a, b, c, d, e, f, A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[A*b^2 - a*b*B + a^2*C, 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. \(1974\) vs. \(2(480)=960\).
Time = 18.11 (sec) , antiderivative size = 1975, normalized size of antiderivative = 4.70
int((B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(5/2)/(a+cos(d*x+c)*b)^3,x,me thod=_RETURNVERBOSE)
-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*B/a^3/sin(1/ 2*d*x+1/2*c)^2/(2*sin(1/2*d*x+1/2*c)^2-1)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2 *d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-(sin(1/2*d *x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+ 1/2*c),2^(1/2)))-2*B/a^2*b*(-1/a*b^2/(a^2-b^2)*cos(1/2*d*x+1/2*c)*(-2*sin( 1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(2*b*cos(1/2*d*x+1/2*c)^2+a-b )-1/2/a/(a+b)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/ 2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2* d*x+1/2*c),2^(1/2))-1/2/(a^2-b^2)*b/a*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos (1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^ (1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+1/2/(a^2-b^2)*b/a*(sin(1/2*d*x +1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^ 4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))-3*a/(a ^2-b^2)/(-2*a*b+2*b^2)*b*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2* c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*Ellipti cPi(cos(1/2*d*x+1/2*c),-2*b/(a-b),2^(1/2))+1/a/(a^2-b^2)/(-2*a*b+2*b^2)*b^ 3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1 /2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c), -2*b/(a-b),2^(1/2)))+4*B/a^3*b^2/(-2*a*b+2*b^2)*(sin(1/2*d*x+1/2*c)^2)^(1/ 2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d...
Timed out. \[ \int \frac {B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx=\text {Timed out} \]
integrate((B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(5/2)/(a+b*cos(d*x+c))^ 3,x, algorithm="fricas")
Timed out. \[ \int \frac {B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx=\text {Timed out} \]
integrate((B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(5/2)/(a+b*cos(d*x+c))^ 3,x, algorithm="maxima")
\[ \int \frac {B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx=\int { \frac {C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3} \cos \left (d x + c\right )^{\frac {5}{2}}} \,d x } \]
integrate((B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(5/2)/(a+b*cos(d*x+c))^ 3,x, algorithm="giac")
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c))/((b*cos(d*x + c) + a)^3*cos( d*x + c)^(5/2)), x)
Timed out. \[ \int \frac {B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx=\int \frac {C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{5/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3} \,d x \]