Integrand size = 41, antiderivative size = 359 \[ \int \frac {\cos (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx=-\frac {2 \left (2 a^3 b B-6 a b^3 B+3 b^4 (A-C)-8 a^4 C+a^2 b^2 (A+15 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3 b^3 \left (a^2-b^2\right )^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 \left (2 a^2 b B-3 b^3 B-8 a^3 C+a b^2 (A+9 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{3 b^3 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {2 a \left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}+\frac {2 \left (3 A b^4+2 a^3 b B-6 a b^3 B-5 a^4 C+a^2 b^2 (A+9 C)\right ) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}} \]
2/3*a*(A*b^2-a*(B*b-C*a))*sin(d*x+c)/b^2/(a^2-b^2)/d/(a+b*cos(d*x+c))^(3/2 )+2/3*(3*A*b^4+2*B*a^3*b-6*B*a*b^3-5*a^4*C+a^2*b^2*(A+9*C))*sin(d*x+c)/b^2 /(a^2-b^2)^2/d/(a+b*cos(d*x+c))^(1/2)-2/3*(2*B*a^3*b-6*B*a*b^3+3*b^4*(A-C) -8*a^4*C+a^2*b^2*(A+15*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c) *EllipticE(sin(1/2*d*x+1/2*c),2^(1/2)*(b/(a+b))^(1/2))*(a+b*cos(d*x+c))^(1 /2)/b^3/(a^2-b^2)^2/d/((a+b*cos(d*x+c))/(a+b))^(1/2)+2/3*(2*B*a^2*b-3*B*b^ 3-8*a^3*C+a*b^2*(A+9*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*E llipticF(sin(1/2*d*x+1/2*c),2^(1/2)*(b/(a+b))^(1/2))*((a+b*cos(d*x+c))/(a+ b))^(1/2)/b^3/(a^2-b^2)/d/(a+b*cos(d*x+c))^(1/2)
Time = 4.32 (sec) , antiderivative size = 323, normalized size of antiderivative = 0.90 \[ \int \frac {\cos (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx=\frac {2 \left (\frac {(-a-b \cos (c+d x)) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (-b^2 \left (a^2 b B+3 b^3 B+2 a^3 C-2 a b^2 (2 A+3 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )+\left (2 a^3 b B-6 a b^3 B+3 b^4 (A-C)-8 a^4 C+a^2 b^2 (A+15 C)\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )\right )}{(a-b)^2 (a+b)^2}+\frac {b \left (a \left (2 A b^4+a^3 b B-5 a b^3 B-4 a^4 C+2 a^2 b^2 (A+4 C)\right )+b \left (3 A b^4+2 a^3 b B-6 a b^3 B-5 a^4 C+a^2 b^2 (A+9 C)\right ) \cos (c+d x)\right ) \sin (c+d x)}{\left (a^2-b^2\right )^2}\right )}{3 b^3 d (a+b \cos (c+d x))^{3/2}} \]
Integrate[(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Co s[c + d*x])^(5/2),x]
(2*(((-a - b*Cos[c + d*x])*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(-(b^2*(a^2* b*B + 3*b^3*B + 2*a^3*C - 2*a*b^2*(2*A + 3*C))*EllipticF[(c + d*x)/2, (2*b )/(a + b)]) + (2*a^3*b*B - 6*a*b^3*B + 3*b^4*(A - C) - 8*a^4*C + a^2*b^2*( A + 15*C))*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])))/((a - b)^2*(a + b)^2) + (b*(a*(2*A*b^4 + a^3* b*B - 5*a*b^3*B - 4*a^4*C + 2*a^2*b^2*(A + 4*C)) + b*(3*A*b^4 + 2*a^3*b*B - 6*a*b^3*B - 5*a^4*C + a^2*b^2*(A + 9*C))*Cos[c + d*x])*Sin[c + d*x])/(a^ 2 - b^2)^2))/(3*b^3*d*(a + b*Cos[c + d*x])^(3/2))
Time = 1.75 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.02, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.366, Rules used = {3042, 3510, 27, 3042, 3500, 27, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}
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.
\(\displaystyle \int \frac {\cos (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx\) |
\(\Big \downarrow \) 3510 |
\(\displaystyle \frac {2 \int -\frac {-3 b \left (a^2-b^2\right ) C \cos ^2(c+d x)-\left (-2 C a^3+2 b B a^2+b^2 (A+3 C) a-3 b^3 B\right ) \cos (c+d x)+3 b \left (A b^2-a (b B-a C)\right )}{2 (a+b \cos (c+d x))^{3/2}}dx}{3 b^2 \left (a^2-b^2\right )}+\frac {2 a \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 a \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}-\frac {\int \frac {-3 b \left (a^2-b^2\right ) C \cos ^2(c+d x)-\left (-2 C a^3+2 b B a^2+b^2 (A+3 C) a-3 b^3 B\right ) \cos (c+d x)+3 b \left (A b^2-a (b B-a C)\right )}{(a+b \cos (c+d x))^{3/2}}dx}{3 b^2 \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 a \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}-\frac {\int \frac {-3 b \left (a^2-b^2\right ) C \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (2 C a^3-2 b B a^2-b^2 (A+3 C) a+3 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 b \left (A b^2-a (b B-a C)\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{3 b^2 \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3500 |
\(\displaystyle \frac {2 a \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}-\frac {-\frac {2 \int \frac {b^2 \left (2 C a^3+b B a^2-2 b^2 (2 A+3 C) a+3 b^3 B\right )-b \left (-8 C a^4+2 b B a^3+b^2 (A+15 C) a^2-6 b^3 B a+3 b^4 (A-C)\right ) \cos (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx}{b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \left (-5 a^4 C+2 a^3 b B+a^2 b^2 (A+9 C)-6 a b^3 B+3 A b^4\right )}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b^2 \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 a \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}-\frac {-\frac {\int \frac {b^2 \left (2 C a^3+b B a^2-2 b^2 (2 A+3 C) a+3 b^3 B\right )-b \left (-8 C a^4+2 b B a^3+b^2 (A+15 C) a^2-6 b^3 B a+3 b^4 (A-C)\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \left (-5 a^4 C+2 a^3 b B+a^2 b^2 (A+9 C)-6 a b^3 B+3 A b^4\right )}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b^2 \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 a \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}-\frac {-\frac {\int \frac {b^2 \left (2 C a^3+b B a^2-2 b^2 (2 A+3 C) a+3 b^3 B\right )-b \left (-8 C a^4+2 b B a^3+b^2 (A+15 C) a^2-6 b^3 B a+3 b^4 (A-C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \left (-5 a^4 C+2 a^3 b B+a^2 b^2 (A+9 C)-6 a b^3 B+3 A b^4\right )}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b^2 \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3231 |
\(\displaystyle \frac {2 a \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}-\frac {-\frac {\left (a^2-b^2\right ) \left (-8 a^3 C+2 a^2 b B+a b^2 (A+9 C)-3 b^3 B\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx-\left (-8 a^4 C+2 a^3 b B+a^2 b^2 (A+15 C)-6 a b^3 B+3 b^4 (A-C)\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \left (-5 a^4 C+2 a^3 b B+a^2 b^2 (A+9 C)-6 a b^3 B+3 A b^4\right )}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b^2 \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 a \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}-\frac {-\frac {\left (a^2-b^2\right ) \left (-8 a^3 C+2 a^2 b B+a b^2 (A+9 C)-3 b^3 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\left (-8 a^4 C+2 a^3 b B+a^2 b^2 (A+15 C)-6 a b^3 B+3 b^4 (A-C)\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \left (-5 a^4 C+2 a^3 b B+a^2 b^2 (A+9 C)-6 a b^3 B+3 A b^4\right )}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b^2 \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3134 |
\(\displaystyle \frac {2 a \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}-\frac {-\frac {\left (a^2-b^2\right ) \left (-8 a^3 C+2 a^2 b B+a b^2 (A+9 C)-3 b^3 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {\left (-8 a^4 C+2 a^3 b B+a^2 b^2 (A+15 C)-6 a b^3 B+3 b^4 (A-C)\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \left (-5 a^4 C+2 a^3 b B+a^2 b^2 (A+9 C)-6 a b^3 B+3 A b^4\right )}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b^2 \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 a \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}-\frac {-\frac {\left (a^2-b^2\right ) \left (-8 a^3 C+2 a^2 b B+a b^2 (A+9 C)-3 b^3 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {\left (-8 a^4 C+2 a^3 b B+a^2 b^2 (A+15 C)-6 a b^3 B+3 b^4 (A-C)\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \left (-5 a^4 C+2 a^3 b B+a^2 b^2 (A+9 C)-6 a b^3 B+3 A b^4\right )}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b^2 \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3132 |
\(\displaystyle \frac {2 a \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}-\frac {-\frac {\left (a^2-b^2\right ) \left (-8 a^3 C+2 a^2 b B+a b^2 (A+9 C)-3 b^3 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 \left (-8 a^4 C+2 a^3 b B+a^2 b^2 (A+15 C)-6 a b^3 B+3 b^4 (A-C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \left (-5 a^4 C+2 a^3 b B+a^2 b^2 (A+9 C)-6 a b^3 B+3 A b^4\right )}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b^2 \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3142 |
\(\displaystyle \frac {2 a \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}-\frac {-\frac {\frac {\left (a^2-b^2\right ) \left (-8 a^3 C+2 a^2 b B+a b^2 (A+9 C)-3 b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-8 a^4 C+2 a^3 b B+a^2 b^2 (A+15 C)-6 a b^3 B+3 b^4 (A-C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \left (-5 a^4 C+2 a^3 b B+a^2 b^2 (A+9 C)-6 a b^3 B+3 A b^4\right )}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b^2 \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 a \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}-\frac {-\frac {\frac {\left (a^2-b^2\right ) \left (-8 a^3 C+2 a^2 b B+a b^2 (A+9 C)-3 b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-8 a^4 C+2 a^3 b B+a^2 b^2 (A+15 C)-6 a b^3 B+3 b^4 (A-C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \left (-5 a^4 C+2 a^3 b B+a^2 b^2 (A+9 C)-6 a b^3 B+3 A b^4\right )}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b^2 \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3140 |
\(\displaystyle \frac {2 a \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}-\frac {-\frac {2 \sin (c+d x) \left (-5 a^4 C+2 a^3 b B+a^2 b^2 (A+9 C)-6 a b^3 B+3 A b^4\right )}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {2 \left (a^2-b^2\right ) \left (-8 a^3 C+2 a^2 b B+a b^2 (A+9 C)-3 b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-8 a^4 C+2 a^3 b B+a^2 b^2 (A+15 C)-6 a b^3 B+3 b^4 (A-C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{b \left (a^2-b^2\right )}}{3 b^2 \left (a^2-b^2\right )}\) |
(2*a*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d*(a + b*Cos [c + d*x])^(3/2)) - (-(((-2*(2*a^3*b*B - 6*a*b^3*B + 3*b^4*(A - C) - 8*a^4 *C + a^2*b^2*(A + 15*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^2 - b^2)*(2* a^2*b*B - 3*b^3*B - 8*a^3*C + a*b^2*(A + 9*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]] ))/(b*(a^2 - b^2))) - (2*(3*A*b^4 + 2*a^3*b*B - 6*a*b^3*B - 5*a^4*C + a^2* b^2*(A + 9*C))*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]))/(3* b^2*(a^2 - b^2))
3.11.55.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 0]
Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + ( f_.)*(x_)]], x_Symbol] :> Simp[(b*c - a*d)/b Int[1/Sqrt[a + b*Sin[e + f*x ]], x], x] + Simp[d/b Int[Sqrt[a + b*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]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((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)/(b*f*(m + 1)* (a^2 - b^2))), x] + Simp[1/(b*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x ])^(m + 1)*Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A *b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[(-(b*c - a*d))*(A*b^2 - a*b*B + a^2*C)*Cos[ e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b^2*f*(m + 1)*(a^2 - b^2))), x] - S imp[1/(b^2*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*( m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1)) ))*Sin[e + f*x] - b*C*d*(m + 1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; F reeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(966\) vs. \(2(397)=794\).
Time = 11.08 (sec) , antiderivative size = 967, normalized size of antiderivative = 2.69
method | result | size |
default | \(\text {Expression too large to display}\) | \(967\) |
parts | \(\text {Expression too large to display}\) | \(2509\) |
int(cos(d*x+c)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(5/2),x,me thod=_RETURNVERBOSE)
-(-(-2*cos(1/2*d*x+1/2*c)^2*b-a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2/b^3/(-2* b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(sin(1/2*d*x+1/2* c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*(B*Ellipti cF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*b-3*C*EllipticF(cos(1/2*d*x+1/2* c),(-2*b/(a-b))^(1/2))*a+C*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2) )*a-C*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*b)-2/b^3*(A*b^2-2*B *a*b+3*C*a^2)/sin(1/2*d*x+1/2*c)^2/(2*b*sin(1/2*d*x+1/2*c)^2-a-b)/(a^2-b^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)*(2*b*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*b/(a-b) *sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2* b/(a-b))^(1/2))*a-(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2 *c)^2+(a+b)/(a-b))^(1/2)*b*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2) ))-2*a*(A*b^2-B*a*b+C*a^2)/b^3*(1/6/b/(a-b)/(a+b)*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)/(cos(1/2*d*x+1/2*c )^2+1/2/b*(a-b))^2+8/3*b*sin(1/2*d*x+1/2*c)^2/(a-b)^2/(a+b)^2*cos(1/2*d*x+ 1/2*c)*a/(-(-2*cos(1/2*d*x+1/2*c)^2*b-a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)+(3* a-b)/(3*a^3+3*a^2*b-3*a*b^2-3*b^3)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/ 2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/ 2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))-4/3 *a/(a-b)/(a+b)^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.20 (sec) , antiderivative size = 1310, normalized size of antiderivative = 3.65 \[ \int \frac {\cos (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx=\text {Too large to display} \]
integrate(cos(d*x+c)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(5/2 ),x, algorithm="fricas")
-1/9*(6*(4*C*a^5*b^2 - B*a^4*b^3 - 2*(A + 4*C)*a^3*b^4 + 5*B*a^2*b^5 - 2*A *a*b^6 + (5*C*a^4*b^3 - 2*B*a^3*b^4 - (A + 9*C)*a^2*b^5 + 6*B*a*b^6 - 3*A* b^7)*cos(d*x + c))*sqrt(b*cos(d*x + c) + a)*sin(d*x + c) - (sqrt(2)*(16*I* C*a^5*b^2 - 4*I*B*a^4*b^3 - 2*I*(A + 18*C)*a^3*b^4 + 9*I*B*a^2*b^5 + 6*I*( A + 4*C)*a*b^6 - 9*I*B*b^7)*cos(d*x + c)^2 - 2*sqrt(2)*(-16*I*C*a^6*b + 4* I*B*a^5*b^2 + 2*I*(A + 18*C)*a^4*b^3 - 9*I*B*a^3*b^4 - 6*I*(A + 4*C)*a^2*b ^5 + 9*I*B*a*b^6)*cos(d*x + c) + sqrt(2)*(16*I*C*a^7 - 4*I*B*a^6*b - 2*I*( A + 18*C)*a^5*b^2 + 9*I*B*a^4*b^3 + 6*I*(A + 4*C)*a^3*b^4 - 9*I*B*a^2*b^5) )*sqrt(b)*weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a* b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*a)/b) - (sqrt(2)* (-16*I*C*a^5*b^2 + 4*I*B*a^4*b^3 + 2*I*(A + 18*C)*a^3*b^4 - 9*I*B*a^2*b^5 - 6*I*(A + 4*C)*a*b^6 + 9*I*B*b^7)*cos(d*x + c)^2 - 2*sqrt(2)*(16*I*C*a^6* b - 4*I*B*a^5*b^2 - 2*I*(A + 18*C)*a^4*b^3 + 9*I*B*a^3*b^4 + 6*I*(A + 4*C) *a^2*b^5 - 9*I*B*a*b^6)*cos(d*x + c) + sqrt(2)*(-16*I*C*a^7 + 4*I*B*a^6*b + 2*I*(A + 18*C)*a^5*b^2 - 9*I*B*a^4*b^3 - 6*I*(A + 4*C)*a^3*b^4 + 9*I*B*a ^2*b^5))*sqrt(b)*weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) - 3*I*b*sin(d*x + c) + 2*a)/b) + 3* (sqrt(2)*(-8*I*C*a^4*b^3 + 2*I*B*a^3*b^4 + I*(A + 15*C)*a^2*b^5 - 6*I*B*a* b^6 + 3*I*(A - C)*b^7)*cos(d*x + c)^2 + 2*sqrt(2)*(-8*I*C*a^5*b^2 + 2*I*B* a^4*b^3 + I*(A + 15*C)*a^3*b^4 - 6*I*B*a^2*b^5 + 3*I*(A - C)*a*b^6)*cos...
Timed out. \[ \int \frac {\cos (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\cos (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
integrate(cos(d*x+c)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(5/2 ),x, algorithm="maxima")
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*cos(d*x + c)/(b*cos(d*x + c) + a)^(5/2), x)
\[ \int \frac {\cos (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
integrate(cos(d*x+c)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(5/2 ),x, algorithm="giac")
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*cos(d*x + c)/(b*cos(d*x + c) + a)^(5/2), x)
Timed out. \[ \int \frac {\cos (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx=\int \frac {\cos \left (c+d\,x\right )\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \]