Integrand size = 37, antiderivative size = 427 \[ \int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^n \, dx=-\frac {2 a^2 (A-B) \cos (e+f x) (c+d \sin (e+f x))^{1+n}}{d f (3+2 n) \sqrt {a+a \sin (e+f x)}}+\frac {2 a^2 B (3 c-d (11+4 n)) \cos (e+f x) (c+d \sin (e+f x))^{1+n}}{d^2 f (3+2 n) (5+2 n) \sqrt {a+a \sin (e+f x)}}-\frac {2 a B \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{1+n}}{d f (5+2 n)}+\frac {2 a^2 (A-B) (c-d (5+4 n)) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},\frac {d (1-\sin (e+f x))}{c+d}\right ) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n}}{d f (3+2 n) \sqrt {a+a \sin (e+f x)}}-\frac {2 a^2 B \left (3 c^2-2 c d (7+4 n)+d^2 \left (43+56 n+16 n^2\right )\right ) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},\frac {d (1-\sin (e+f x))}{c+d}\right ) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n}}{d^2 f (3+2 n) (5+2 n) \sqrt {a+a \sin (e+f x)}} \] Output:
-2*a^2*(A-B)*cos(f*x+e)*(c+d*sin(f*x+e))^(1+n)/d/f/(3+2*n)/(a+a*sin(f*x+e) )^(1/2)+2*a^2*B*(3*c-d*(11+4*n))*cos(f*x+e)*(c+d*sin(f*x+e))^(1+n)/d^2/f/( 3+2*n)/(5+2*n)/(a+a*sin(f*x+e))^(1/2)-2*a*B*cos(f*x+e)*(a+a*sin(f*x+e))^(1 /2)*(c+d*sin(f*x+e))^(1+n)/d/f/(5+2*n)+2*a^2*(A-B)*(c-d*(5+4*n))*cos(f*x+e )*hypergeom([1/2, -n],[3/2],d*(1-sin(f*x+e))/(c+d))*(c+d*sin(f*x+e))^n/d/f /(3+2*n)/(a+a*sin(f*x+e))^(1/2)/(((c+d*sin(f*x+e))/(c+d))^n)-2*a^2*B*(3*c^ 2-2*c*d*(7+4*n)+d^2*(16*n^2+56*n+43))*cos(f*x+e)*hypergeom([1/2, -n],[3/2] ,d*(1-sin(f*x+e))/(c+d))*(c+d*sin(f*x+e))^n/d^2/f/(3+2*n)/(5+2*n)/(a+a*sin (f*x+e))^(1/2)/(((c+d*sin(f*x+e))/(c+d))^n)
Time = 15.19 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.57 \[ \int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^n \, dx=-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n} \left (-30 (A+B) (c-d (5+4 n)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},-\frac {d (-1+\sin (e+f x))}{c+d}\right )+6 B d (3+2 n) \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-n,\frac {7}{2},-\frac {d (-1+\sin (e+f x))}{c+d}\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^4+20 B d (3+2 n) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-n,\frac {5}{2},-\frac {d (-1+\sin (e+f x))}{c+d}\right ) (-1+\sin (e+f x))+30 (A+B) (c+d) \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{1+n}\right )}{15 d f (3+2 n) \sqrt {a (1+\sin (e+f x))}} \] Input:
Integrate[(a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x])*(c + d*Sin[e + f *x])^n,x]
Output:
-1/15*(a^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^n*(-30*(A + B)*(c - d*(5 + 4* n))*Hypergeometric2F1[1/2, -n, 3/2, -((d*(-1 + Sin[e + f*x]))/(c + d))] + 6*B*d*(3 + 2*n)*Hypergeometric2F1[5/2, -n, 7/2, -((d*(-1 + Sin[e + f*x]))/ (c + d))]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4 + 20*B*d*(3 + 2*n)*Hyper geometric2F1[3/2, -n, 5/2, -((d*(-1 + Sin[e + f*x]))/(c + d))]*(-1 + Sin[e + f*x]) + 30*(A + B)*(c + d)*((c + d*Sin[e + f*x])/(c + d))^(1 + n)))/(d* f*(3 + 2*n)*Sqrt[a*(1 + Sin[e + f*x])]*((c + d*Sin[e + f*x])/(c + d))^n)
Time = 1.88 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.405, Rules used = {3042, 3466, 3042, 3242, 27, 2011, 3042, 3255, 80, 79, 3460, 3042, 3255, 80, 79}
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.
| \(\displaystyle \int (a \sin (e+f x)+a)^{3/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^n \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (e+f x)+a)^{3/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^ndx\) |
| \(\Big \downarrow \) 3466 |
| \(\displaystyle (A-B) \int (\sin (e+f x) a+a)^{3/2} (c+d \sin (e+f x))^ndx+\frac {B \int (\sin (e+f x) a+a)^{5/2} (c+d \sin (e+f x))^ndx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle (A-B) \int (\sin (e+f x) a+a)^{3/2} (c+d \sin (e+f x))^ndx+\frac {B \int (\sin (e+f x) a+a)^{5/2} (c+d \sin (e+f x))^ndx}{a}\) |
| \(\Big \downarrow \) 3242 |
| \(\displaystyle (A-B) \left (\frac {2 \int -\frac {(c+d \sin (e+f x))^n \left ((c-5 d-4 d n) a^2+(c-5 d-4 d n) \sin (e+f x) a^2\right )}{2 \sqrt {\sin (e+f x) a+a}}dx}{d (2 n+3)}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt {a \sin (e+f x)+a}}\right )+\frac {B \left (\frac {2 \int \frac {1}{2} \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^n \left (a^2 (c+d (4 n+7))-a^2 (3 c-11 d-4 d n) \sin (e+f x)\right )dx}{d (2 n+5)}-\frac {2 a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{n+1}}{d f (2 n+5)}\right )}{a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle (A-B) \left (-\frac {\int \frac {(c+d \sin (e+f x))^n \left ((c-5 d-4 d n) a^2+(c-5 d-4 d n) \sin (e+f x) a^2\right )}{\sqrt {\sin (e+f x) a+a}}dx}{d (2 n+3)}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt {a \sin (e+f x)+a}}\right )+\frac {B \left (\frac {\int \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^n \left (a^2 (c+d (4 n+7))-a^2 (3 c-11 d-4 d n) \sin (e+f x)\right )dx}{d (2 n+5)}-\frac {2 a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{n+1}}{d f (2 n+5)}\right )}{a}\) |
| \(\Big \downarrow \) 2011 |
| \(\displaystyle (A-B) \left (-\frac {a (c-4 d n-5 d) \int \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^ndx}{d (2 n+3)}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt {a \sin (e+f x)+a}}\right )+\frac {B \left (\frac {\int \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^n \left (a^2 (c+d (4 n+7))-a^2 (3 c-11 d-4 d n) \sin (e+f x)\right )dx}{d (2 n+5)}-\frac {2 a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{n+1}}{d f (2 n+5)}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle (A-B) \left (-\frac {a (c-4 d n-5 d) \int \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^ndx}{d (2 n+3)}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt {a \sin (e+f x)+a}}\right )+\frac {B \left (\frac {\int \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^n \left (a^2 (c+d (4 n+7))-a^2 (3 c-11 d-4 d n) \sin (e+f x)\right )dx}{d (2 n+5)}-\frac {2 a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{n+1}}{d f (2 n+5)}\right )}{a}\) |
| \(\Big \downarrow \) 3255 |
| \(\displaystyle \frac {B \left (\frac {\int \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^n \left (a^2 (c+d (4 n+7))-a^2 (3 c-11 d-4 d n) \sin (e+f x)\right )dx}{d (2 n+5)}-\frac {2 a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{n+1}}{d f (2 n+5)}\right )}{a}+(A-B) \left (-\frac {a^3 (c-4 d n-5 d) \cos (e+f x) \int \frac {(c+d \sin (e+f x))^n}{\sqrt {a-a \sin (e+f x)}}d\sin (e+f x)}{d f (2 n+3) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a}}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt {a \sin (e+f x)+a}}\right )\) |
| \(\Big \downarrow \) 80 |
| \(\displaystyle \frac {B \left (\frac {\int \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^n \left (a^2 (c+d (4 n+7))-a^2 (3 c-11 d-4 d n) \sin (e+f x)\right )dx}{d (2 n+5)}-\frac {2 a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{n+1}}{d f (2 n+5)}\right )}{a}+(A-B) \left (-\frac {a^3 (c-4 d n-5 d) \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n} \int \frac {\left (\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}\right )^n}{\sqrt {a-a \sin (e+f x)}}d\sin (e+f x)}{d f (2 n+3) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a}}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt {a \sin (e+f x)+a}}\right )\) |
| \(\Big \downarrow \) 79 |
| \(\displaystyle \frac {B \left (\frac {\int \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^n \left (a^2 (c+d (4 n+7))-a^2 (3 c-11 d-4 d n) \sin (e+f x)\right )dx}{d (2 n+5)}-\frac {2 a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{n+1}}{d f (2 n+5)}\right )}{a}+(A-B) \left (\frac {2 a^2 (c-4 d n-5 d) \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},\frac {d (1-\sin (e+f x))}{c+d}\right )}{d f (2 n+3) \sqrt {a \sin (e+f x)+a}}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt {a \sin (e+f x)+a}}\right )\) |
| \(\Big \downarrow \) 3460 |
| \(\displaystyle \frac {B \left (\frac {\frac {a^2 \left (3 c^2-2 c d (4 n+7)+d^2 \left (16 n^2+56 n+43\right )\right ) \int \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^ndx}{d (2 n+3)}+\frac {2 a^3 (3 c-d (4 n+11)) \cos (e+f x) (c+d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt {a \sin (e+f x)+a}}}{d (2 n+5)}-\frac {2 a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{n+1}}{d f (2 n+5)}\right )}{a}+(A-B) \left (\frac {2 a^2 (c-4 d n-5 d) \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},\frac {d (1-\sin (e+f x))}{c+d}\right )}{d f (2 n+3) \sqrt {a \sin (e+f x)+a}}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt {a \sin (e+f x)+a}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {B \left (\frac {\frac {a^2 \left (3 c^2-2 c d (4 n+7)+d^2 \left (16 n^2+56 n+43\right )\right ) \int \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^ndx}{d (2 n+3)}+\frac {2 a^3 (3 c-d (4 n+11)) \cos (e+f x) (c+d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt {a \sin (e+f x)+a}}}{d (2 n+5)}-\frac {2 a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{n+1}}{d f (2 n+5)}\right )}{a}+(A-B) \left (\frac {2 a^2 (c-4 d n-5 d) \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},\frac {d (1-\sin (e+f x))}{c+d}\right )}{d f (2 n+3) \sqrt {a \sin (e+f x)+a}}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt {a \sin (e+f x)+a}}\right )\) |
| \(\Big \downarrow \) 3255 |
| \(\displaystyle \frac {B \left (\frac {\frac {a^4 \left (3 c^2-2 c d (4 n+7)+d^2 \left (16 n^2+56 n+43\right )\right ) \cos (e+f x) \int \frac {(c+d \sin (e+f x))^n}{\sqrt {a-a \sin (e+f x)}}d\sin (e+f x)}{d f (2 n+3) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a}}+\frac {2 a^3 (3 c-d (4 n+11)) \cos (e+f x) (c+d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt {a \sin (e+f x)+a}}}{d (2 n+5)}-\frac {2 a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{n+1}}{d f (2 n+5)}\right )}{a}+(A-B) \left (\frac {2 a^2 (c-4 d n-5 d) \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},\frac {d (1-\sin (e+f x))}{c+d}\right )}{d f (2 n+3) \sqrt {a \sin (e+f x)+a}}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt {a \sin (e+f x)+a}}\right )\) |
| \(\Big \downarrow \) 80 |
| \(\displaystyle \frac {B \left (\frac {\frac {a^4 \left (3 c^2-2 c d (4 n+7)+d^2 \left (16 n^2+56 n+43\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n} \int \frac {\left (\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}\right )^n}{\sqrt {a-a \sin (e+f x)}}d\sin (e+f x)}{d f (2 n+3) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a}}+\frac {2 a^3 (3 c-d (4 n+11)) \cos (e+f x) (c+d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt {a \sin (e+f x)+a}}}{d (2 n+5)}-\frac {2 a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{n+1}}{d f (2 n+5)}\right )}{a}+(A-B) \left (\frac {2 a^2 (c-4 d n-5 d) \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},\frac {d (1-\sin (e+f x))}{c+d}\right )}{d f (2 n+3) \sqrt {a \sin (e+f x)+a}}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt {a \sin (e+f x)+a}}\right )\) |
| \(\Big \downarrow \) 79 |
| \(\displaystyle (A-B) \left (\frac {2 a^2 (c-4 d n-5 d) \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},\frac {d (1-\sin (e+f x))}{c+d}\right )}{d f (2 n+3) \sqrt {a \sin (e+f x)+a}}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt {a \sin (e+f x)+a}}\right )+\frac {B \left (\frac {\frac {2 a^3 (3 c-d (4 n+11)) \cos (e+f x) (c+d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt {a \sin (e+f x)+a}}-\frac {2 a^3 \left (3 c^2-2 c d (4 n+7)+d^2 \left (16 n^2+56 n+43\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},\frac {d (1-\sin (e+f x))}{c+d}\right )}{d f (2 n+3) \sqrt {a \sin (e+f x)+a}}}{d (2 n+5)}-\frac {2 a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{n+1}}{d f (2 n+5)}\right )}{a}\) |
Input:
Int[(a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^n ,x]
Output:
(A - B)*((-2*a^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(1 + n))/(d*f*(3 + 2*n) *Sqrt[a + a*Sin[e + f*x]]) + (2*a^2*(c - 5*d - 4*d*n)*Cos[e + f*x]*Hyperge ometric2F1[1/2, -n, 3/2, (d*(1 - Sin[e + f*x]))/(c + d)]*(c + d*Sin[e + f* x])^n)/(d*f*(3 + 2*n)*Sqrt[a + a*Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^n)) + (B*((-2*a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(1 + n))/(d*f*(5 + 2*n)) + ((2*a^3*(3*c - d*(11 + 4*n))*Cos[e + f*x ]*(c + d*Sin[e + f*x])^(1 + n))/(d*f*(3 + 2*n)*Sqrt[a + a*Sin[e + f*x]]) - (2*a^3*(3*c^2 - 2*c*d*(7 + 4*n) + d^2*(43 + 56*n + 16*n^2))*Cos[e + f*x]* Hypergeometric2F1[1/2, -n, 3/2, (d*(1 - Sin[e + f*x]))/(c + d)]*(c + d*Sin [e + f*x])^n)/(d*f*(3 + 2*n)*Sqrt[a + a*Sin[e + f*x]]*((c + d*Sin[e + f*x] )/(c + d))^n))/(d*(5 + 2*n))))/a
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_), x_Symbol] :> Simp[(( a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 , m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] || !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) ^FracPart[n]) Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) ), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !Integ erQ[n] && (RationalQ[m] || !SimplerQ[n + 1, m + 1])
Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Simp[(b/d)^m Int[u*(c + d*v)^(m + n), x], x] /; FreeQ[{a, b, c, d, n}, x ] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c + d*x , a + b*x])
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Simp[1/(d*(m + n)) Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^n*Simp[a*b*c* (m - 2) + b^2*d*(n + 1) + a^2*d*(m + n) - b*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && !LtQ[ n, -1] && (IntegersQ[2*m, 2*n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[ c, 0]))
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[a^2*(Cos[e + f*x]/(f*Sqrt[a + b*Sin[e + f*x]]*Sqrt[a - b*Sin[e + f*x]])) Subst[Int[(c + d*x)^n/Sqrt[a - b*x], x] , x, Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !IntegerQ[2*n]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + ( f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp [-2*b*B*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(2*n + 3)*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(b *d*(2*n + 3)) Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ[n, -1]
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 - a*B)/b Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n, x], x ] + Simp[B/b Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ [a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && NeQ[A*b + a*B, 0]
\[\int \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \left (A +B \sin \left (f x +e \right )\right ) \left (c +d \sin \left (f x +e \right )\right )^{n}d x\]
Input:
int((a+a*sin(f*x+e))^(3/2)*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^n,x)
Output:
int((a+a*sin(f*x+e))^(3/2)*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^n,x)
\[ \int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^n \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^(3/2)*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^n,x, al gorithm="fricas")
Output:
integral(-(B*a*cos(f*x + e)^2 - (A + B)*a*sin(f*x + e) - (A + B)*a)*sqrt(a *sin(f*x + e) + a)*(d*sin(f*x + e) + c)^n, x)
Timed out. \[ \int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^n \, dx=\text {Timed out} \] Input:
integrate((a+a*sin(f*x+e))**(3/2)*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))**n,x)
Output:
Timed out
\[ \int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^n \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^(3/2)*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^n,x, al gorithm="maxima")
Output:
integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^(3/2)*(d*sin(f*x + e) + c)^n, x)
\[ \int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^n \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^(3/2)*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^n,x, al gorithm="giac")
Output:
integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^(3/2)*(d*sin(f*x + e) + c)^n, x)
Timed out. \[ \int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^n \, dx=\int \left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n \,d x \] Input:
int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))^n ,x)
Output:
int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))^n , x)
\[ \int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^n \, dx=\sqrt {a}\, a \left (\left (\int \left (\sin \left (f x +e \right ) d +c \right )^{n} \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}d x \right ) b +\left (\int \left (\sin \left (f x +e \right ) d +c \right )^{n} \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )d x \right ) a +\left (\int \left (\sin \left (f x +e \right ) d +c \right )^{n} \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )d x \right ) b +\left (\int \left (\sin \left (f x +e \right ) d +c \right )^{n} \sqrt {\sin \left (f x +e \right )+1}d x \right ) a \right ) \] Input:
int((a+a*sin(f*x+e))^(3/2)*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^n,x)
Output:
sqrt(a)*a*(int((sin(e + f*x)*d + c)**n*sqrt(sin(e + f*x) + 1)*sin(e + f*x) **2,x)*b + int((sin(e + f*x)*d + c)**n*sqrt(sin(e + f*x) + 1)*sin(e + f*x) ,x)*a + int((sin(e + f*x)*d + c)**n*sqrt(sin(e + f*x) + 1)*sin(e + f*x),x) *b + int((sin(e + f*x)*d + c)**n*sqrt(sin(e + f*x) + 1),x)*a)