Integrand size = 27, antiderivative size = 598 \[ \int (e \cos (c+d x))^{-4-m} (a+b \sin (c+d x))^m \, dx=-\frac {(e \cos (c+d x))^{-3-m} (a+b \sin (c+d x))^{1+m}}{(a-b) d e (3+m)}+\frac {2 b (e \cos (c+d x))^{-1-m} (a+b \sin (c+d x))^{1+m}}{(a-b)^2 d e^3 (1+m) (3+m)}+\frac {a (e \cos (c+d x))^{-3-m} (1+\sin (c+d x)) (a+b \sin (c+d x))^{1+m}}{\left (a^2-b^2\right ) d e (3+m)}+\frac {a (3 b+a (2+m)) (e \cos (c+d x))^{-3-m} (1-\sin (c+d x)) (1+\sin (c+d x)) (a+b \sin (c+d x))^{1+m}}{(a-b) (a+b)^2 d e (1+m) (3+m)}-\frac {2^{\frac {3}{2}-\frac {m}{2}} a b (e \cos (c+d x))^{-1-m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-1-m),\frac {1+m}{2},\frac {1-m}{2},\frac {(a-b) (1-\sin (c+d x))}{2 (a+b \sin (c+d x))}\right ) \left (\frac {(a+b) (1+\sin (c+d x))}{a+b \sin (c+d x)}\right )^{\frac {1+m}{2}} (a+b \sin (c+d x))^{1+m}}{(a-b)^2 (a+b) d e^3 (1+m) (3+m)}-\frac {2^{-\frac {1}{2}-\frac {m}{2}} a \left (2 a b-b^2+a^2 (2+m)\right ) (e \cos (c+d x))^{-3-m} \operatorname {Hypergeometric2F1}\left (\frac {1-m}{2},\frac {3+m}{2},\frac {3-m}{2},\frac {(a-b) (1-\sin (c+d x))}{2 (a+b \sin (c+d x))}\right ) (1-\sin (c+d x))^2 \left (\frac {(a+b) (1+\sin (c+d x))}{a+b \sin (c+d x)}\right )^{\frac {3+m}{2}} (a+b \sin (c+d x))^{1+m}}{(a-b) (a+b)^3 d e (1-m) (3+m)} \]
-(e*cos(d*x+c))^(-3-m)*(a+b*sin(d*x+c))^(1+m)/(a-b)/d/e/(3+m)+2*b*(e*cos(d *x+c))^(-1-m)*(a+b*sin(d*x+c))^(1+m)/(a-b)^2/d/e^3/(1+m)/(3+m)+a*(e*cos(d* x+c))^(-3-m)*(1+sin(d*x+c))*(a+b*sin(d*x+c))^(1+m)/(a^2-b^2)/d/e/(3+m)+a*( 3*b+a*(2+m))*(e*cos(d*x+c))^(-3-m)*(1-sin(d*x+c))*(1+sin(d*x+c))*(a+b*sin( d*x+c))^(1+m)/(a-b)/(a+b)^2/d/e/(1+m)/(3+m)-2^(3/2-1/2*m)*a*b*(e*cos(d*x+c ))^(-1-m)*hypergeom([-1/2-1/2*m, 1/2+1/2*m],[-1/2*m+1/2],1/2*(a-b)*(1-sin( d*x+c))/(a+b*sin(d*x+c)))*((a+b)*(1+sin(d*x+c))/(a+b*sin(d*x+c)))^(1/2+1/2 *m)*(a+b*sin(d*x+c))^(1+m)/(a-b)^2/(a+b)/d/e^3/(m^2+4*m+3)-2^(-1/2-1/2*m)* a*(2*a*b-b^2+a^2*(2+m))*(e*cos(d*x+c))^(-3-m)*hypergeom([3/2+1/2*m, -1/2*m +1/2],[3/2-1/2*m],1/2*(a-b)*(1-sin(d*x+c))/(a+b*sin(d*x+c)))*(1-sin(d*x+c) )^2*((a+b)*(1+sin(d*x+c))/(a+b*sin(d*x+c)))^(3/2+1/2*m)*(a+b*sin(d*x+c))^( 1+m)/(a-b)/(a+b)^3/d/e/(1-m)/(3+m)
Time = 6.08 (sec) , antiderivative size = 826, normalized size of antiderivative = 1.38 \[ \int (e \cos (c+d x))^{-4-m} (a+b \sin (c+d x))^m \, dx=\frac {\cos (c+d x) (e \cos (c+d x))^{-4-m} (a+b \sin (c+d x))^{1+m}}{(a-b) d (-3-m)}+\frac {2 b \cos ^{4+m}(c+d x) (e \cos (c+d x))^{-4-m} \left (\frac {\cos ^{-1-m}(c+d x) (a+b \sin (c+d x))^{1+m}}{(a-b) d (-1-m)}+\frac {2^{1+\frac {1}{2} (-1-m)} a \cos ^{-1-m}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-1-m),\frac {1+m}{2},1+\frac {1}{2} (-1-m),-\frac {(-a+b) (1-\sin (c+d x))}{2 (a+b \sin (c+d x))}\right ) (1-\sin (c+d x))^{\frac {1}{2} (-1-m)+\frac {1+m}{2}} (1+\sin (c+d x))^{\frac {1}{2} (-1-m)+\frac {1+m}{2}} \left (-\frac {(-a-b) (1+\sin (c+d x))}{a+b \sin (c+d x)}\right )^{\frac {1+m}{2}} (a+b \sin (c+d x))^{1+m}}{(-a-b) (a-b) d (-1-m)}\right )}{(a-b) (-3-m)}+\frac {a \cos (c+d x) (e \cos (c+d x))^{-4-m} (1-\sin (c+d x))^{\frac {3+m}{2}} (1+\sin (c+d x))^{\frac {3+m}{2}} \left (\frac {(1-\sin (c+d x))^{\frac {1}{2} (-3-m)} (1+\sin (c+d x))^{1+\frac {1}{2} (-3-m)} (a+b \sin (c+d x))^{1+m}}{(-a-b) (-3-m)}-\frac {-\frac {(3 b+a (2+m)) (1-\sin (c+d x))^{1+\frac {1}{2} (-3-m)} (1+\sin (c+d x))^{1+\frac {1}{2} (-3-m)} (a+b \sin (c+d x))^{1+m}}{2 (-a-b) \left (1+\frac {1}{2} (-3-m)\right )}-\frac {2^{-1+\frac {1}{2} (-3-m)} (1+m) \left (2 a b-b^2+a^2 (2+m)\right ) \operatorname {Hypergeometric2F1}\left (2+\frac {1}{2} (-3-m),\frac {3+m}{2},3+\frac {1}{2} (-3-m),-\frac {(-a+b) (1-\sin (c+d x))}{2 (a+b \sin (c+d x))}\right ) (1-\sin (c+d x))^{2+\frac {1}{2} (-3-m)} (1+\sin (c+d x))^{\frac {1}{2} (-3-m)} \left (-\frac {(-a-b) (1+\sin (c+d x))}{a+b \sin (c+d x)}\right )^{\frac {3+m}{2}} (a+b \sin (c+d x))^{1+m}}{(-a-b)^2 \left (1+\frac {1}{2} (-3-m)\right ) \left (2+\frac {1}{2} (-3-m)\right )}}{(-a-b) (-3-m)}\right )}{(a-b) d} \]
(Cos[c + d*x]*(e*Cos[c + d*x])^(-4 - m)*(a + b*Sin[c + d*x])^(1 + m))/((a - b)*d*(-3 - m)) + (2*b*Cos[c + d*x]^(4 + m)*(e*Cos[c + d*x])^(-4 - m)*((C os[c + d*x]^(-1 - m)*(a + b*Sin[c + d*x])^(1 + m))/((a - b)*d*(-1 - m)) + (2^(1 + (-1 - m)/2)*a*Cos[c + d*x]^(-1 - m)*Hypergeometric2F1[(-1 - m)/2, (1 + m)/2, 1 + (-1 - m)/2, -1/2*((-a + b)*(1 - Sin[c + d*x]))/(a + b*Sin[c + d*x])]*(1 - Sin[c + d*x])^((-1 - m)/2 + (1 + m)/2)*(1 + Sin[c + d*x])^( (-1 - m)/2 + (1 + m)/2)*(-(((-a - b)*(1 + Sin[c + d*x]))/(a + b*Sin[c + d* x])))^((1 + m)/2)*(a + b*Sin[c + d*x])^(1 + m))/((-a - b)*(a - b)*d*(-1 - m))))/((a - b)*(-3 - m)) + (a*Cos[c + d*x]*(e*Cos[c + d*x])^(-4 - m)*(1 - Sin[c + d*x])^((3 + m)/2)*(1 + Sin[c + d*x])^((3 + m)/2)*(((1 - Sin[c + d* x])^((-3 - m)/2)*(1 + Sin[c + d*x])^(1 + (-3 - m)/2)*(a + b*Sin[c + d*x])^ (1 + m))/((-a - b)*(-3 - m)) - (-1/2*((3*b + a*(2 + m))*(1 - Sin[c + d*x]) ^(1 + (-3 - m)/2)*(1 + Sin[c + d*x])^(1 + (-3 - m)/2)*(a + b*Sin[c + d*x]) ^(1 + m))/((-a - b)*(1 + (-3 - m)/2)) - (2^(-1 + (-3 - m)/2)*(1 + m)*(2*a* b - b^2 + a^2*(2 + m))*Hypergeometric2F1[2 + (-3 - m)/2, (3 + m)/2, 3 + (- 3 - m)/2, -1/2*((-a + b)*(1 - Sin[c + d*x]))/(a + b*Sin[c + d*x])]*(1 - Si n[c + d*x])^(2 + (-3 - m)/2)*(1 + Sin[c + d*x])^((-3 - m)/2)*(-(((-a - b)* (1 + Sin[c + d*x]))/(a + b*Sin[c + d*x])))^((3 + m)/2)*(a + b*Sin[c + d*x] )^(1 + m))/((-a - b)^2*(1 + (-3 - m)/2)*(2 + (-3 - m)/2)))/((-a - b)*(-3 - m))))/((a - b)*d)
Time = 1.45 (sec) , antiderivative size = 717, normalized size of antiderivative = 1.20, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {3042, 3179, 3042, 3178, 3042, 3399, 142, 144, 25, 172, 25, 27, 142}
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.
| \(\displaystyle \int (e \cos (c+d x))^{-m-4} (a+b \sin (c+d x))^m \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (e \cos (c+d x))^{-m-4} (a+b \sin (c+d x))^mdx\) |
| \(\Big \downarrow \) 3179 |
| \(\displaystyle -\frac {2 b \int (e \cos (c+d x))^{-m-2} (a+b \sin (c+d x))^mdx}{e^2 (m+3) (a-b)}+\frac {a \int \frac {(e \cos (c+d x))^{-m-2} (a+b \sin (c+d x))^m}{1-\sin (c+d x)}dx}{e^2 (a-b)}-\frac {(e \cos (c+d x))^{-m-3} (a+b \sin (c+d x))^{m+1}}{d e (m+3) (a-b)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 b \int (e \cos (c+d x))^{-m-2} (a+b \sin (c+d x))^mdx}{e^2 (m+3) (a-b)}+\frac {a \int \frac {(e \cos (c+d x))^{-m-2} (a+b \sin (c+d x))^m}{1-\sin (c+d x)}dx}{e^2 (a-b)}-\frac {(e \cos (c+d x))^{-m-3} (a+b \sin (c+d x))^{m+1}}{d e (m+3) (a-b)}\) |
| \(\Big \downarrow \) 3178 |
| \(\displaystyle \frac {a \int \frac {(e \cos (c+d x))^{-m-2} (a+b \sin (c+d x))^m}{1-\sin (c+d x)}dx}{e^2 (a-b)}-\frac {2 b \left (\frac {a \int \frac {(e \cos (c+d x))^{-m} (a+b \sin (c+d x))^m}{1-\sin (c+d x)}dx}{e^2 (a-b)}-\frac {(e \cos (c+d x))^{-m-1} (a+b \sin (c+d x))^{m+1}}{d e (m+1) (a-b)}\right )}{e^2 (m+3) (a-b)}-\frac {(e \cos (c+d x))^{-m-3} (a+b \sin (c+d x))^{m+1}}{d e (m+3) (a-b)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \int \frac {(e \cos (c+d x))^{-m-2} (a+b \sin (c+d x))^m}{1-\sin (c+d x)}dx}{e^2 (a-b)}-\frac {2 b \left (\frac {a \int \frac {(e \cos (c+d x))^{-m} (a+b \sin (c+d x))^m}{1-\sin (c+d x)}dx}{e^2 (a-b)}-\frac {(e \cos (c+d x))^{-m-1} (a+b \sin (c+d x))^{m+1}}{d e (m+1) (a-b)}\right )}{e^2 (m+3) (a-b)}-\frac {(e \cos (c+d x))^{-m-3} (a+b \sin (c+d x))^{m+1}}{d e (m+3) (a-b)}\) |
| \(\Big \downarrow \) 3399 |
| \(\displaystyle -\frac {2 b \left (\frac {a (1-\sin (c+d x))^{\frac {m+1}{2}} (\sin (c+d x)+1)^{\frac {m+1}{2}} (e \cos (c+d x))^{-m-1} \int (1-\sin (c+d x))^{\frac {1}{2} (-m-3)} (\sin (c+d x)+1)^{\frac {1}{2} (-m-1)} (a+b \sin (c+d x))^md\sin (c+d x)}{d e (a-b)}-\frac {(e \cos (c+d x))^{-m-1} (a+b \sin (c+d x))^{m+1}}{d e (m+1) (a-b)}\right )}{e^2 (m+3) (a-b)}+\frac {a (1-\sin (c+d x))^{\frac {m+3}{2}} (\sin (c+d x)+1)^{\frac {m+3}{2}} (e \cos (c+d x))^{-m-3} \int (1-\sin (c+d x))^{\frac {1}{2} (-m-5)} (\sin (c+d x)+1)^{\frac {1}{2} (-m-3)} (a+b \sin (c+d x))^md\sin (c+d x)}{d e (a-b)}-\frac {(e \cos (c+d x))^{-m-3} (a+b \sin (c+d x))^{m+1}}{d e (m+3) (a-b)}\) |
| \(\Big \downarrow \) 142 |
| \(\displaystyle \frac {a (1-\sin (c+d x))^{\frac {m+3}{2}} (\sin (c+d x)+1)^{\frac {m+3}{2}} (e \cos (c+d x))^{-m-3} \int (1-\sin (c+d x))^{\frac {1}{2} (-m-5)} (\sin (c+d x)+1)^{\frac {1}{2} (-m-3)} (a+b \sin (c+d x))^md\sin (c+d x)}{d e (a-b)}-\frac {2 b \left (\frac {a 2^{\frac {1}{2}-\frac {m}{2}} (1-\sin (c+d x))^{\frac {1}{2} (-m-1)+\frac {m+1}{2}} (\sin (c+d x)+1)^{\frac {1}{2} (-m-1)+\frac {m+1}{2}} (e \cos (c+d x))^{-m-1} \left (\frac {(a+b) (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right )^{\frac {m+1}{2}} (a+b \sin (c+d x))^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-m-1),\frac {m+1}{2},\frac {1-m}{2},\frac {(a-b) (1-\sin (c+d x))}{2 (a+b \sin (c+d x))}\right )}{d e (m+1) (a-b) (a+b)}-\frac {(e \cos (c+d x))^{-m-1} (a+b \sin (c+d x))^{m+1}}{d e (m+1) (a-b)}\right )}{e^2 (m+3) (a-b)}-\frac {(e \cos (c+d x))^{-m-3} (a+b \sin (c+d x))^{m+1}}{d e (m+3) (a-b)}\) |
| \(\Big \downarrow \) 144 |
| \(\displaystyle \frac {a (1-\sin (c+d x))^{\frac {m+3}{2}} (\sin (c+d x)+1)^{\frac {m+3}{2}} (e \cos (c+d x))^{-m-3} \left (\frac {(1-\sin (c+d x))^{\frac {1}{2} (-m-3)} (\sin (c+d x)+1)^{\frac {1}{2} (-m-1)} (a+b \sin (c+d x))^{m+1}}{(m+3) (a+b)}-\frac {\int -(1-\sin (c+d x))^{\frac {1}{2} (-m-3)} (\sin (c+d x)+1)^{\frac {1}{2} (-m-3)} (a+b \sin (c+d x))^m (\sin (c+d x) b+2 b+a (m+2))d\sin (c+d x)}{(m+3) (a+b)}\right )}{d e (a-b)}-\frac {2 b \left (\frac {a 2^{\frac {1}{2}-\frac {m}{2}} (1-\sin (c+d x))^{\frac {1}{2} (-m-1)+\frac {m+1}{2}} (\sin (c+d x)+1)^{\frac {1}{2} (-m-1)+\frac {m+1}{2}} (e \cos (c+d x))^{-m-1} \left (\frac {(a+b) (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right )^{\frac {m+1}{2}} (a+b \sin (c+d x))^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-m-1),\frac {m+1}{2},\frac {1-m}{2},\frac {(a-b) (1-\sin (c+d x))}{2 (a+b \sin (c+d x))}\right )}{d e (m+1) (a-b) (a+b)}-\frac {(e \cos (c+d x))^{-m-1} (a+b \sin (c+d x))^{m+1}}{d e (m+1) (a-b)}\right )}{e^2 (m+3) (a-b)}-\frac {(e \cos (c+d x))^{-m-3} (a+b \sin (c+d x))^{m+1}}{d e (m+3) (a-b)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a (1-\sin (c+d x))^{\frac {m+3}{2}} (\sin (c+d x)+1)^{\frac {m+3}{2}} (e \cos (c+d x))^{-m-3} \left (\frac {\int (1-\sin (c+d x))^{\frac {1}{2} (-m-3)} (\sin (c+d x)+1)^{\frac {1}{2} (-m-3)} (a+b \sin (c+d x))^m (\sin (c+d x) b+2 b+a (m+2))d\sin (c+d x)}{(m+3) (a+b)}+\frac {(\sin (c+d x)+1)^{\frac {1}{2} (-m-1)} (1-\sin (c+d x))^{\frac {1}{2} (-m-3)} (a+b \sin (c+d x))^{m+1}}{(m+3) (a+b)}\right )}{d e (a-b)}-\frac {2 b \left (\frac {a 2^{\frac {1}{2}-\frac {m}{2}} (1-\sin (c+d x))^{\frac {1}{2} (-m-1)+\frac {m+1}{2}} (\sin (c+d x)+1)^{\frac {1}{2} (-m-1)+\frac {m+1}{2}} (e \cos (c+d x))^{-m-1} \left (\frac {(a+b) (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right )^{\frac {m+1}{2}} (a+b \sin (c+d x))^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-m-1),\frac {m+1}{2},\frac {1-m}{2},\frac {(a-b) (1-\sin (c+d x))}{2 (a+b \sin (c+d x))}\right )}{d e (m+1) (a-b) (a+b)}-\frac {(e \cos (c+d x))^{-m-1} (a+b \sin (c+d x))^{m+1}}{d e (m+1) (a-b)}\right )}{e^2 (m+3) (a-b)}-\frac {(e \cos (c+d x))^{-m-3} (a+b \sin (c+d x))^{m+1}}{d e (m+3) (a-b)}\) |
| \(\Big \downarrow \) 172 |
| \(\displaystyle \frac {a (1-\sin (c+d x))^{\frac {m+3}{2}} (\sin (c+d x)+1)^{\frac {m+3}{2}} (e \cos (c+d x))^{-m-3} \left (\frac {\frac {(a (m+2)+3 b) (1-\sin (c+d x))^{\frac {1}{2} (-m-1)} (\sin (c+d x)+1)^{\frac {1}{2} (-m-1)} (a+b \sin (c+d x))^{m+1}}{(m+1) (a+b)}-\frac {\int -\left ((m+1) \left ((m+2) a^2+2 b a-b^2\right ) (1-\sin (c+d x))^{\frac {1}{2} (-m-1)} (\sin (c+d x)+1)^{\frac {1}{2} (-m-3)} (a+b \sin (c+d x))^m\right )d\sin (c+d x)}{(m+1) (a+b)}}{(m+3) (a+b)}+\frac {(\sin (c+d x)+1)^{\frac {1}{2} (-m-1)} (1-\sin (c+d x))^{\frac {1}{2} (-m-3)} (a+b \sin (c+d x))^{m+1}}{(m+3) (a+b)}\right )}{d e (a-b)}-\frac {2 b \left (\frac {a 2^{\frac {1}{2}-\frac {m}{2}} (1-\sin (c+d x))^{\frac {1}{2} (-m-1)+\frac {m+1}{2}} (\sin (c+d x)+1)^{\frac {1}{2} (-m-1)+\frac {m+1}{2}} (e \cos (c+d x))^{-m-1} \left (\frac {(a+b) (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right )^{\frac {m+1}{2}} (a+b \sin (c+d x))^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-m-1),\frac {m+1}{2},\frac {1-m}{2},\frac {(a-b) (1-\sin (c+d x))}{2 (a+b \sin (c+d x))}\right )}{d e (m+1) (a-b) (a+b)}-\frac {(e \cos (c+d x))^{-m-1} (a+b \sin (c+d x))^{m+1}}{d e (m+1) (a-b)}\right )}{e^2 (m+3) (a-b)}-\frac {(e \cos (c+d x))^{-m-3} (a+b \sin (c+d x))^{m+1}}{d e (m+3) (a-b)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a (1-\sin (c+d x))^{\frac {m+3}{2}} (\sin (c+d x)+1)^{\frac {m+3}{2}} (e \cos (c+d x))^{-m-3} \left (\frac {\frac {\int (m+1) \left ((m+2) a^2+2 b a-b^2\right ) (1-\sin (c+d x))^{\frac {1}{2} (-m-1)} (\sin (c+d x)+1)^{\frac {1}{2} (-m-3)} (a+b \sin (c+d x))^md\sin (c+d x)}{(m+1) (a+b)}+\frac {(a (m+2)+3 b) (1-\sin (c+d x))^{\frac {1}{2} (-m-1)} (\sin (c+d x)+1)^{\frac {1}{2} (-m-1)} (a+b \sin (c+d x))^{m+1}}{(m+1) (a+b)}}{(m+3) (a+b)}+\frac {(\sin (c+d x)+1)^{\frac {1}{2} (-m-1)} (1-\sin (c+d x))^{\frac {1}{2} (-m-3)} (a+b \sin (c+d x))^{m+1}}{(m+3) (a+b)}\right )}{d e (a-b)}-\frac {2 b \left (\frac {a 2^{\frac {1}{2}-\frac {m}{2}} (1-\sin (c+d x))^{\frac {1}{2} (-m-1)+\frac {m+1}{2}} (\sin (c+d x)+1)^{\frac {1}{2} (-m-1)+\frac {m+1}{2}} (e \cos (c+d x))^{-m-1} \left (\frac {(a+b) (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right )^{\frac {m+1}{2}} (a+b \sin (c+d x))^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-m-1),\frac {m+1}{2},\frac {1-m}{2},\frac {(a-b) (1-\sin (c+d x))}{2 (a+b \sin (c+d x))}\right )}{d e (m+1) (a-b) (a+b)}-\frac {(e \cos (c+d x))^{-m-1} (a+b \sin (c+d x))^{m+1}}{d e (m+1) (a-b)}\right )}{e^2 (m+3) (a-b)}-\frac {(e \cos (c+d x))^{-m-3} (a+b \sin (c+d x))^{m+1}}{d e (m+3) (a-b)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a (1-\sin (c+d x))^{\frac {m+3}{2}} (\sin (c+d x)+1)^{\frac {m+3}{2}} (e \cos (c+d x))^{-m-3} \left (\frac {\frac {\left (a^2 (m+2)+2 a b-b^2\right ) \int (1-\sin (c+d x))^{\frac {1}{2} (-m-1)} (\sin (c+d x)+1)^{\frac {1}{2} (-m-3)} (a+b \sin (c+d x))^md\sin (c+d x)}{a+b}+\frac {(a (m+2)+3 b) (1-\sin (c+d x))^{\frac {1}{2} (-m-1)} (\sin (c+d x)+1)^{\frac {1}{2} (-m-1)} (a+b \sin (c+d x))^{m+1}}{(m+1) (a+b)}}{(m+3) (a+b)}+\frac {(\sin (c+d x)+1)^{\frac {1}{2} (-m-1)} (1-\sin (c+d x))^{\frac {1}{2} (-m-3)} (a+b \sin (c+d x))^{m+1}}{(m+3) (a+b)}\right )}{d e (a-b)}-\frac {2 b \left (\frac {a 2^{\frac {1}{2}-\frac {m}{2}} (1-\sin (c+d x))^{\frac {1}{2} (-m-1)+\frac {m+1}{2}} (\sin (c+d x)+1)^{\frac {1}{2} (-m-1)+\frac {m+1}{2}} (e \cos (c+d x))^{-m-1} \left (\frac {(a+b) (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right )^{\frac {m+1}{2}} (a+b \sin (c+d x))^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-m-1),\frac {m+1}{2},\frac {1-m}{2},\frac {(a-b) (1-\sin (c+d x))}{2 (a+b \sin (c+d x))}\right )}{d e (m+1) (a-b) (a+b)}-\frac {(e \cos (c+d x))^{-m-1} (a+b \sin (c+d x))^{m+1}}{d e (m+1) (a-b)}\right )}{e^2 (m+3) (a-b)}-\frac {(e \cos (c+d x))^{-m-3} (a+b \sin (c+d x))^{m+1}}{d e (m+3) (a-b)}\) |
| \(\Big \downarrow \) 142 |
| \(\displaystyle \frac {a (1-\sin (c+d x))^{\frac {m+3}{2}} (\sin (c+d x)+1)^{\frac {m+3}{2}} (e \cos (c+d x))^{-m-3} \left (\frac {\frac {(a (m+2)+3 b) (1-\sin (c+d x))^{\frac {1}{2} (-m-1)} (\sin (c+d x)+1)^{\frac {1}{2} (-m-1)} (a+b \sin (c+d x))^{m+1}}{(m+1) (a+b)}-\frac {2^{-\frac {m}{2}-\frac {1}{2}} \left (a^2 (m+2)+2 a b-b^2\right ) (1-\sin (c+d x))^{\frac {1-m}{2}} (\sin (c+d x)+1)^{\frac {1}{2} (-m-3)} \left (\frac {(a+b) (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right )^{\frac {m+3}{2}} (a+b \sin (c+d x))^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1-m}{2},\frac {m+3}{2},\frac {3-m}{2},\frac {(a-b) (1-\sin (c+d x))}{2 (a+b \sin (c+d x))}\right )}{(1-m) (a+b)^2}}{(m+3) (a+b)}+\frac {(\sin (c+d x)+1)^{\frac {1}{2} (-m-1)} (1-\sin (c+d x))^{\frac {1}{2} (-m-3)} (a+b \sin (c+d x))^{m+1}}{(m+3) (a+b)}\right )}{d e (a-b)}-\frac {2 b \left (\frac {a 2^{\frac {1}{2}-\frac {m}{2}} (1-\sin (c+d x))^{\frac {1}{2} (-m-1)+\frac {m+1}{2}} (\sin (c+d x)+1)^{\frac {1}{2} (-m-1)+\frac {m+1}{2}} (e \cos (c+d x))^{-m-1} \left (\frac {(a+b) (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right )^{\frac {m+1}{2}} (a+b \sin (c+d x))^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-m-1),\frac {m+1}{2},\frac {1-m}{2},\frac {(a-b) (1-\sin (c+d x))}{2 (a+b \sin (c+d x))}\right )}{d e (m+1) (a-b) (a+b)}-\frac {(e \cos (c+d x))^{-m-1} (a+b \sin (c+d x))^{m+1}}{d e (m+1) (a-b)}\right )}{e^2 (m+3) (a-b)}-\frac {(e \cos (c+d x))^{-m-3} (a+b \sin (c+d x))^{m+1}}{d e (m+3) (a-b)}\) |
-(((e*Cos[c + d*x])^(-3 - m)*(a + b*Sin[c + d*x])^(1 + m))/((a - b)*d*e*(3 + m))) - (2*b*(-(((e*Cos[c + d*x])^(-1 - m)*(a + b*Sin[c + d*x])^(1 + m)) /((a - b)*d*e*(1 + m))) + (2^(1/2 - m/2)*a*(e*Cos[c + d*x])^(-1 - m)*Hyper geometric2F1[(-1 - m)/2, (1 + m)/2, (1 - m)/2, ((a - b)*(1 - Sin[c + d*x]) )/(2*(a + b*Sin[c + d*x]))]*(1 - Sin[c + d*x])^((-1 - m)/2 + (1 + m)/2)*(1 + Sin[c + d*x])^((-1 - m)/2 + (1 + m)/2)*(((a + b)*(1 + Sin[c + d*x]))/(a + b*Sin[c + d*x]))^((1 + m)/2)*(a + b*Sin[c + d*x])^(1 + m))/((a - b)*(a + b)*d*e*(1 + m))))/((a - b)*e^2*(3 + m)) + (a*(e*Cos[c + d*x])^(-3 - m)*( 1 - Sin[c + d*x])^((3 + m)/2)*(1 + Sin[c + d*x])^((3 + m)/2)*(((1 - Sin[c + d*x])^((-3 - m)/2)*(1 + Sin[c + d*x])^((-1 - m)/2)*(a + b*Sin[c + d*x])^ (1 + m))/((a + b)*(3 + m)) + (((3*b + a*(2 + m))*(1 - Sin[c + d*x])^((-1 - m)/2)*(1 + Sin[c + d*x])^((-1 - m)/2)*(a + b*Sin[c + d*x])^(1 + m))/((a + b)*(1 + m)) - (2^(-1/2 - m/2)*(2*a*b - b^2 + a^2*(2 + m))*Hypergeometric2 F1[(1 - m)/2, (3 + m)/2, (3 - m)/2, ((a - b)*(1 - Sin[c + d*x]))/(2*(a + b *Sin[c + d*x]))]*(1 - Sin[c + d*x])^((1 - m)/2)*(1 + Sin[c + d*x])^((-3 - m)/2)*(((a + b)*(1 + Sin[c + d*x]))/(a + b*Sin[c + d*x]))^((3 + m)/2)*(a + b*Sin[c + d*x])^(1 + m))/((a + b)^2*(1 - m)))/((a + b)*(3 + m))))/((a - b )*d*e)
3.7.47.3.1 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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((b*e - a*f)*(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2, (-(d*e - c*f))*((a + b*x)/((b*c - a*d)*(e + f*x)))])/((b*e - a*f)*((c + d*x)/((b*c - a*d)*(e + f *x))))^n, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[m + n + p + 2, 0] && !IntegerQ[n]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> With[{mnp = Simplify[m + n + p]}, Simp[b*(a + b*x)^(m + 1)*( c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x )^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(mnp + 3)*x, x], x], x] /; ILtQ[mnp + 2, 0] && (SumSimplerQ[m, 1] | | ( !SumSimplerQ[n, 1] && !SumSimplerQ[p, 1]))] /; FreeQ[{a, b, c, d, e, f , m, n, p}, x] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> With[{mnp = Simplify[m + n + p]}, Simp[ (b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1) *(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f )*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(mnp + 3)*x, x], x], x] /; ILtQ[mnp + 2, 0] && (SumSimplerQ[m, 1] | | ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) && !(NeQ[p, -1] && SumSimplerQ[p, 1 ])))] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && NeQ[m, -1]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x]) ^(m + 1)/(f*g*(a - b)*(p + 1))), x] + Simp[a/(g^2*(a - b)) Int[(g*Cos[e + f*x])^(p + 2)*((a + b*Sin[e + f*x])^m/(1 - Sin[e + f*x])), x], x] /; FreeQ [{a, b, e, f, g, m, p}, x] && NeQ[a^2 - b^2, 0] && EqQ[m + p + 2, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x]) ^(m + 1)/(f*g*(a - b)*(p + 1))), x] + (-Simp[b*((m + p + 2)/(g^2*(a - b)*(p + 1))) Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^m, x], x] + Sim p[a/(g^2*(a - b)) Int[(g*Cos[e + f*x])^(p + 2)*((a + b*Sin[e + f*x])^m/(1 - Sin[e + f*x])), x], x]) /; FreeQ[{a, b, e, f, g, m, p}, x] && NeQ[a^2 - b^2, 0] && ILtQ[m + p + 2, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[a^ m*g*((g*Cos[e + f*x])^(p - 1)/(f*(1 + Sin[e + f*x])^((p - 1)/2)*(1 - Sin[e + f*x])^((p - 1)/2))) Subst[Int[(1 + (b/a)*x)^(m + (p - 1)/2)*(1 - (b/a)* x)^((p - 1)/2)*(c + d*x)^n, x], x, Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m]
\[\int \left (e \cos \left (d x +c \right )\right )^{-4-m} \left (a +b \sin \left (d x +c \right )\right )^{m}d x\]
\[ \int (e \cos (c+d x))^{-4-m} (a+b \sin (c+d x))^m \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{-m - 4} {\left (b \sin \left (d x + c\right ) + a\right )}^{m} \,d x } \]
\[ \int (e \cos (c+d x))^{-4-m} (a+b \sin (c+d x))^m \, dx=\int \left (e \cos {\left (c + d x \right )}\right )^{- m - 4} \left (a + b \sin {\left (c + d x \right )}\right )^{m}\, dx \]
\[ \int (e \cos (c+d x))^{-4-m} (a+b \sin (c+d x))^m \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{-m - 4} {\left (b \sin \left (d x + c\right ) + a\right )}^{m} \,d x } \]
\[ \int (e \cos (c+d x))^{-4-m} (a+b \sin (c+d x))^m \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{-m - 4} {\left (b \sin \left (d x + c\right ) + a\right )}^{m} \,d x } \]
Timed out. \[ \int (e \cos (c+d x))^{-4-m} (a+b \sin (c+d x))^m \, dx=\int \frac {{\left (a+b\,\sin \left (c+d\,x\right )\right )}^m}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{m+4}} \,d x \]