\(\int \cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^3 \, dx\) [1191]

Optimal result

Integrand size = 29, antiderivative size = 623 \[ \int \cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {3 a \left (2 a^4 \left (6+5 n+n^2\right )+3 b^4 \left (35+12 n+n^2\right )-2 a^2 b^2 \left (58+16 n+n^2\right )\right ) \cos (c+d x) \sin ^{1+n}(c+d x)}{b^2 d (2+n) (4+n) (5+n) (6+n) (7+n)}+\frac {3 a \left (3 b^2 (1+n)+a^2 (6+n)\right ) \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2},\frac {3+n}{2},\sin ^2(c+d x)\right ) \sin ^{1+n}(c+d x)}{d (1+n) (2+n) (4+n) (6+n) \sqrt {\cos ^2(c+d x)}}-\frac {3 \left (2 a^4 \left (6+5 n+n^2\right )+b^4 \left (24+10 n+n^2\right )-2 a^2 b^2 \left (57+16 n+n^2\right )\right ) \cos (c+d x) \sin ^{2+n}(c+d x)}{b d (3+n) (4+n) (5+n) (6+n) (7+n)}+\frac {3 b \left (b^2 (2+n)+3 a^2 (7+n)\right ) \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+n}{2},\frac {4+n}{2},\sin ^2(c+d x)\right ) \sin ^{2+n}(c+d x)}{d (2+n) (3+n) (5+n) (7+n) \sqrt {\cos ^2(c+d x)}}-\frac {3 a \left (a^2 \left (6+5 n+n^2\right )-b^2 \left (53+15 n+n^2\right )\right ) \cos (c+d x) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^2}{b^2 d (4+n) (5+n) (6+n) (7+n)}-\frac {\left (a^2 (2+n) (3+n)-b^2 (6+n) (8+n)\right ) \cos (c+d x) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^3}{b^2 d (5+n) (6+n) (7+n)}+\frac {a (3+n) \cos (c+d x) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^4}{b^2 d (6+n) (7+n)}-\frac {\cos (c+d x) \sin ^{2+n}(c+d x) (a+b \sin (c+d x))^4}{b d (7+n)} \] Output:

-3*a*(2*a^4*(n^2+5*n+6)+3*b^4*(n^2+12*n+35)-2*a^2*b^2*(n^2+16*n+58))*cos(d 
*x+c)*sin(d*x+c)^(1+n)/b^2/d/(2+n)/(4+n)/(5+n)/(6+n)/(7+n)+3*a*(3*b^2*(1+n 
)+a^2*(6+n))*cos(d*x+c)*hypergeom([1/2, 1/2+1/2*n],[3/2+1/2*n],sin(d*x+c)^ 
2)*sin(d*x+c)^(1+n)/d/(1+n)/(2+n)/(4+n)/(6+n)/(cos(d*x+c)^2)^(1/2)-3*(2*a^ 
4*(n^2+5*n+6)+b^4*(n^2+10*n+24)-2*a^2*b^2*(n^2+16*n+57))*cos(d*x+c)*sin(d* 
x+c)^(2+n)/b/d/(3+n)/(4+n)/(5+n)/(6+n)/(7+n)+3*b*(b^2*(2+n)+3*a^2*(7+n))*c 
os(d*x+c)*hypergeom([1/2, 1+1/2*n],[2+1/2*n],sin(d*x+c)^2)*sin(d*x+c)^(2+n 
)/d/(2+n)/(3+n)/(5+n)/(7+n)/(cos(d*x+c)^2)^(1/2)-3*a*(a^2*(n^2+5*n+6)-b^2* 
(n^2+15*n+53))*cos(d*x+c)*sin(d*x+c)^(1+n)*(a+b*sin(d*x+c))^2/b^2/d/(4+n)/ 
(5+n)/(6+n)/(7+n)-(a^2*(2+n)*(3+n)-b^2*(6+n)*(8+n))*cos(d*x+c)*sin(d*x+c)^ 
(1+n)*(a+b*sin(d*x+c))^3/b^2/d/(5+n)/(6+n)/(7+n)+a*(3+n)*cos(d*x+c)*sin(d* 
x+c)^(1+n)*(a+b*sin(d*x+c))^4/b^2/d/(6+n)/(7+n)-cos(d*x+c)*sin(d*x+c)^(2+n 
)*(a+b*sin(d*x+c))^4/b/d/(7+n)
 

Mathematica [A] (verified)

Time = 0.92 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.31 \[ \int \cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {\sqrt {\cos ^2(c+d x)} \sec (c+d x) \sin ^{1+n}(c+d x) \left (\frac {a^3 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1+n}{2},\frac {3+n}{2},\sin ^2(c+d x)\right )}{1+n}+b \sin (c+d x) \left (\frac {3 a^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {2+n}{2},\frac {4+n}{2},\sin ^2(c+d x)\right )}{2+n}+b \sin (c+d x) \left (\frac {3 a \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3+n}{2},\frac {5+n}{2},\sin ^2(c+d x)\right )}{3+n}+\frac {b \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {4+n}{2},\frac {6+n}{2},\sin ^2(c+d x)\right ) \sin (c+d x)}{4+n}\right )\right )\right )}{d} \] Input:

Integrate[Cos[c + d*x]^4*Sin[c + d*x]^n*(a + b*Sin[c + d*x])^3,x]
 

Output:

(Sqrt[Cos[c + d*x]^2]*Sec[c + d*x]*Sin[c + d*x]^(1 + n)*((a^3*Hypergeometr 
ic2F1[-3/2, (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2])/(1 + n) + b*Sin[c + d*x 
]*((3*a^2*Hypergeometric2F1[-3/2, (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2])/( 
2 + n) + b*Sin[c + d*x]*((3*a*Hypergeometric2F1[-3/2, (3 + n)/2, (5 + n)/2 
, Sin[c + d*x]^2])/(3 + n) + (b*Hypergeometric2F1[-3/2, (4 + n)/2, (6 + n) 
/2, Sin[c + d*x]^2]*Sin[c + d*x])/(4 + n)))))/d
 

Rubi [A] (verified)

Time = 3.02 (sec) , antiderivative size = 579, normalized size of antiderivative = 0.93, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.552, Rules used = {3042, 3374, 3042, 3528, 27, 3042, 3528, 3042, 3512, 3042, 3502, 25, 3042, 3227, 3042, 3122}

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.

\(\displaystyle \int \cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^3 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cos (c+d x)^4 \sin (c+d x)^n (a+b \sin (c+d x))^3dx\)

\(\Big \downarrow \) 3374

\(\displaystyle -\frac {\int \sin ^n(c+d x) (a+b \sin (c+d x))^3 \left ((n+1) (n+3) a^2+3 b \sin (c+d x) a-\left (a^2 (n+2) (n+3)-b^2 (n+6) (n+8)\right ) \sin ^2(c+d x)-b^2 (n+6) (n+7)\right )dx}{b^2 (n+6) (n+7)}+\frac {a (n+3) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^4}{b^2 d (n+6) (n+7)}-\frac {\cos (c+d x) \sin ^{n+2}(c+d x) (a+b \sin (c+d x))^4}{b d (n+7)}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int \sin (c+d x)^n (a+b \sin (c+d x))^3 \left ((n+1) (n+3) a^2+3 b \sin (c+d x) a-\left (a^2 (n+2) (n+3)-b^2 (n+6) (n+8)\right ) \sin (c+d x)^2-b^2 (n+6) (n+7)\right )dx}{b^2 (n+6) (n+7)}+\frac {a (n+3) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^4}{b^2 d (n+6) (n+7)}-\frac {\cos (c+d x) \sin ^{n+2}(c+d x) (a+b \sin (c+d x))^4}{b d (n+7)}\)

\(\Big \downarrow \) 3528

\(\displaystyle -\frac {\frac {\int 3 \sin ^n(c+d x) (a+b \sin (c+d x))^2 \left (-a \left (a^2 \left (n^2+5 n+6\right )-b^2 \left (n^2+15 n+53\right )\right ) \sin ^2(c+d x)+b \left (2 a^2-b^2 (n+6)\right ) \sin (c+d x)+a \left (a^2 \left (n^2+4 n+3\right )-b^2 \left (n^2+15 n+54\right )\right )\right )dx}{n+5}+\frac {\left (a^2 (n+2) (n+3)-b^2 (n+6) (n+8)\right ) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^3}{d (n+5)}}{b^2 (n+6) (n+7)}+\frac {a (n+3) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^4}{b^2 d (n+6) (n+7)}-\frac {\cos (c+d x) \sin ^{n+2}(c+d x) (a+b \sin (c+d x))^4}{b d (n+7)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {3 \int \sin ^n(c+d x) (a+b \sin (c+d x))^2 \left (-a \left (a^2 \left (n^2+5 n+6\right )-b^2 \left (n^2+15 n+53\right )\right ) \sin ^2(c+d x)+b \left (2 a^2-b^2 (n+6)\right ) \sin (c+d x)+a \left (a^2 \left (n^2+4 n+3\right )-b^2 \left (n^2+15 n+54\right )\right )\right )dx}{n+5}+\frac {\left (a^2 (n+2) (n+3)-b^2 (n+6) (n+8)\right ) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^3}{d (n+5)}}{b^2 (n+6) (n+7)}+\frac {a (n+3) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^4}{b^2 d (n+6) (n+7)}-\frac {\cos (c+d x) \sin ^{n+2}(c+d x) (a+b \sin (c+d x))^4}{b d (n+7)}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {3 \int \sin (c+d x)^n (a+b \sin (c+d x))^2 \left (-a \left (a^2 \left (n^2+5 n+6\right )-b^2 \left (n^2+15 n+53\right )\right ) \sin (c+d x)^2+b \left (2 a^2-b^2 (n+6)\right ) \sin (c+d x)+a \left (a^2 \left (n^2+4 n+3\right )-b^2 \left (n^2+15 n+54\right )\right )\right )dx}{n+5}+\frac {\left (a^2 (n+2) (n+3)-b^2 (n+6) (n+8)\right ) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^3}{d (n+5)}}{b^2 (n+6) (n+7)}+\frac {a (n+3) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^4}{b^2 d (n+6) (n+7)}-\frac {\cos (c+d x) \sin ^{n+2}(c+d x) (a+b \sin (c+d x))^4}{b d (n+7)}\)

\(\Big \downarrow \) 3528

\(\displaystyle -\frac {\frac {3 \left (\frac {\int \sin ^n(c+d x) (a+b \sin (c+d x)) \left (\left (2 a^2 \left (n^2+4 n+3\right )-b^2 \left (3 n^2+46 n+163\right )\right ) a^2+b \left (2 a^2-b^2 \left (2 n^2+26 n+81\right )\right ) \sin (c+d x) a-\left (2 \left (n^2+5 n+6\right ) a^4-2 b^2 \left (n^2+16 n+57\right ) a^2+b^4 \left (n^2+10 n+24\right )\right ) \sin ^2(c+d x)\right )dx}{n+4}+\frac {a \left (a^2 \left (n^2+5 n+6\right )-b^2 \left (n^2+15 n+53\right )\right ) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^2}{d (n+4)}\right )}{n+5}+\frac {\left (a^2 (n+2) (n+3)-b^2 (n+6) (n+8)\right ) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^3}{d (n+5)}}{b^2 (n+6) (n+7)}+\frac {a (n+3) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^4}{b^2 d (n+6) (n+7)}-\frac {\cos (c+d x) \sin ^{n+2}(c+d x) (a+b \sin (c+d x))^4}{b d (n+7)}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {3 \left (\frac {\int \sin (c+d x)^n (a+b \sin (c+d x)) \left (\left (2 a^2 \left (n^2+4 n+3\right )-b^2 \left (3 n^2+46 n+163\right )\right ) a^2+b \left (2 a^2-b^2 \left (2 n^2+26 n+81\right )\right ) \sin (c+d x) a-\left (2 \left (n^2+5 n+6\right ) a^4-2 b^2 \left (n^2+16 n+57\right ) a^2+b^4 \left (n^2+10 n+24\right )\right ) \sin (c+d x)^2\right )dx}{n+4}+\frac {a \left (a^2 \left (n^2+5 n+6\right )-b^2 \left (n^2+15 n+53\right )\right ) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^2}{d (n+4)}\right )}{n+5}+\frac {\left (a^2 (n+2) (n+3)-b^2 (n+6) (n+8)\right ) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^3}{d (n+5)}}{b^2 (n+6) (n+7)}+\frac {a (n+3) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^4}{b^2 d (n+6) (n+7)}-\frac {\cos (c+d x) \sin ^{n+2}(c+d x) (a+b \sin (c+d x))^4}{b d (n+7)}\)

\(\Big \downarrow \) 3512

\(\displaystyle -\frac {\frac {3 \left (\frac {\frac {\int \sin ^n(c+d x) \left ((n+3) \left (2 a^2 \left (n^2+4 n+3\right )-b^2 \left (3 n^2+46 n+163\right )\right ) a^3-(n+3) \left (2 \left (n^2+5 n+6\right ) a^4-2 b^2 \left (n^2+16 n+58\right ) a^2+3 b^4 \left (n^2+12 n+35\right )\right ) \sin ^2(c+d x) a-b^3 \left (n^2+10 n+24\right ) \left (3 (n+7) a^2+b^2 (n+2)\right ) \sin (c+d x)\right )dx}{n+3}+\frac {b \left (2 a^4 \left (n^2+5 n+6\right )-2 a^2 b^2 \left (n^2+16 n+57\right )+b^4 \left (n^2+10 n+24\right )\right ) \cos (c+d x) \sin ^{n+2}(c+d x)}{d (n+3)}}{n+4}+\frac {a \left (a^2 \left (n^2+5 n+6\right )-b^2 \left (n^2+15 n+53\right )\right ) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^2}{d (n+4)}\right )}{n+5}+\frac {\left (a^2 (n+2) (n+3)-b^2 (n+6) (n+8)\right ) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^3}{d (n+5)}}{b^2 (n+6) (n+7)}+\frac {a (n+3) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^4}{b^2 d (n+6) (n+7)}-\frac {\cos (c+d x) \sin ^{n+2}(c+d x) (a+b \sin (c+d x))^4}{b d (n+7)}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {3 \left (\frac {\frac {\int \sin (c+d x)^n \left ((n+3) \left (2 a^2 \left (n^2+4 n+3\right )-b^2 \left (3 n^2+46 n+163\right )\right ) a^3-(n+3) \left (2 \left (n^2+5 n+6\right ) a^4-2 b^2 \left (n^2+16 n+58\right ) a^2+3 b^4 \left (n^2+12 n+35\right )\right ) \sin (c+d x)^2 a-b^3 \left (n^2+10 n+24\right ) \left (3 (n+7) a^2+b^2 (n+2)\right ) \sin (c+d x)\right )dx}{n+3}+\frac {b \left (2 a^4 \left (n^2+5 n+6\right )-2 a^2 b^2 \left (n^2+16 n+57\right )+b^4 \left (n^2+10 n+24\right )\right ) \cos (c+d x) \sin ^{n+2}(c+d x)}{d (n+3)}}{n+4}+\frac {a \left (a^2 \left (n^2+5 n+6\right )-b^2 \left (n^2+15 n+53\right )\right ) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^2}{d (n+4)}\right )}{n+5}+\frac {\left (a^2 (n+2) (n+3)-b^2 (n+6) (n+8)\right ) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^3}{d (n+5)}}{b^2 (n+6) (n+7)}+\frac {a (n+3) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^4}{b^2 d (n+6) (n+7)}-\frac {\cos (c+d x) \sin ^{n+2}(c+d x) (a+b \sin (c+d x))^4}{b d (n+7)}\)

\(\Big \downarrow \) 3502

\(\displaystyle -\frac {\frac {3 \left (\frac {\frac {\frac {\int -\sin ^n(c+d x) \left ((n+2) \left (n^2+10 n+24\right ) \left (3 (n+7) a^2+b^2 (n+2)\right ) \sin (c+d x) b^3+a (n+3) \left (n^2+12 n+35\right ) \left ((n+6) a^2+3 b^2 (n+1)\right ) b^2\right )dx}{n+2}+\frac {a (n+3) \left (2 a^4 \left (n^2+5 n+6\right )-2 a^2 b^2 \left (n^2+16 n+58\right )+3 b^4 \left (n^2+12 n+35\right )\right ) \cos (c+d x) \sin ^{n+1}(c+d x)}{d (n+2)}}{n+3}+\frac {b \left (2 a^4 \left (n^2+5 n+6\right )-2 a^2 b^2 \left (n^2+16 n+57\right )+b^4 \left (n^2+10 n+24\right )\right ) \cos (c+d x) \sin ^{n+2}(c+d x)}{d (n+3)}}{n+4}+\frac {a \left (a^2 \left (n^2+5 n+6\right )-b^2 \left (n^2+15 n+53\right )\right ) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^2}{d (n+4)}\right )}{n+5}+\frac {\left (a^2 (n+2) (n+3)-b^2 (n+6) (n+8)\right ) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^3}{d (n+5)}}{b^2 (n+6) (n+7)}+\frac {a (n+3) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^4}{b^2 d (n+6) (n+7)}-\frac {\cos (c+d x) \sin ^{n+2}(c+d x) (a+b \sin (c+d x))^4}{b d (n+7)}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {3 \left (\frac {\frac {\frac {a (n+3) \left (2 a^4 \left (n^2+5 n+6\right )-2 a^2 b^2 \left (n^2+16 n+58\right )+3 b^4 \left (n^2+12 n+35\right )\right ) \cos (c+d x) \sin ^{n+1}(c+d x)}{d (n+2)}-\frac {\int \sin ^n(c+d x) \left ((n+2) \left (n^2+10 n+24\right ) \left (3 (n+7) a^2+b^2 (n+2)\right ) \sin (c+d x) b^3+a (n+3) \left (n^2+12 n+35\right ) \left ((n+6) a^2+3 b^2 (n+1)\right ) b^2\right )dx}{n+2}}{n+3}+\frac {b \left (2 a^4 \left (n^2+5 n+6\right )-2 a^2 b^2 \left (n^2+16 n+57\right )+b^4 \left (n^2+10 n+24\right )\right ) \cos (c+d x) \sin ^{n+2}(c+d x)}{d (n+3)}}{n+4}+\frac {a \left (a^2 \left (n^2+5 n+6\right )-b^2 \left (n^2+15 n+53\right )\right ) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^2}{d (n+4)}\right )}{n+5}+\frac {\left (a^2 (n+2) (n+3)-b^2 (n+6) (n+8)\right ) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^3}{d (n+5)}}{b^2 (n+6) (n+7)}+\frac {a (n+3) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^4}{b^2 d (n+6) (n+7)}-\frac {\cos (c+d x) \sin ^{n+2}(c+d x) (a+b \sin (c+d x))^4}{b d (n+7)}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {3 \left (\frac {\frac {\frac {a (n+3) \left (2 a^4 \left (n^2+5 n+6\right )-2 a^2 b^2 \left (n^2+16 n+58\right )+3 b^4 \left (n^2+12 n+35\right )\right ) \cos (c+d x) \sin ^{n+1}(c+d x)}{d (n+2)}-\frac {\int \sin (c+d x)^n \left ((n+2) \left (n^2+10 n+24\right ) \left (3 (n+7) a^2+b^2 (n+2)\right ) \sin (c+d x) b^3+a (n+3) \left (n^2+12 n+35\right ) \left ((n+6) a^2+3 b^2 (n+1)\right ) b^2\right )dx}{n+2}}{n+3}+\frac {b \left (2 a^4 \left (n^2+5 n+6\right )-2 a^2 b^2 \left (n^2+16 n+57\right )+b^4 \left (n^2+10 n+24\right )\right ) \cos (c+d x) \sin ^{n+2}(c+d x)}{d (n+3)}}{n+4}+\frac {a \left (a^2 \left (n^2+5 n+6\right )-b^2 \left (n^2+15 n+53\right )\right ) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^2}{d (n+4)}\right )}{n+5}+\frac {\left (a^2 (n+2) (n+3)-b^2 (n+6) (n+8)\right ) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^3}{d (n+5)}}{b^2 (n+6) (n+7)}+\frac {a (n+3) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^4}{b^2 d (n+6) (n+7)}-\frac {\cos (c+d x) \sin ^{n+2}(c+d x) (a+b \sin (c+d x))^4}{b d (n+7)}\)

\(\Big \downarrow \) 3227

\(\displaystyle -\frac {\frac {3 \left (\frac {\frac {\frac {a (n+3) \left (2 a^4 \left (n^2+5 n+6\right )-2 a^2 b^2 \left (n^2+16 n+58\right )+3 b^4 \left (n^2+12 n+35\right )\right ) \cos (c+d x) \sin ^{n+1}(c+d x)}{d (n+2)}-\frac {a b^2 (n+3) \left (n^2+12 n+35\right ) \left (a^2 (n+6)+3 b^2 (n+1)\right ) \int \sin ^n(c+d x)dx+b^3 (n+2) \left (n^2+10 n+24\right ) \left (3 a^2 (n+7)+b^2 (n+2)\right ) \int \sin ^{n+1}(c+d x)dx}{n+2}}{n+3}+\frac {b \left (2 a^4 \left (n^2+5 n+6\right )-2 a^2 b^2 \left (n^2+16 n+57\right )+b^4 \left (n^2+10 n+24\right )\right ) \cos (c+d x) \sin ^{n+2}(c+d x)}{d (n+3)}}{n+4}+\frac {a \left (a^2 \left (n^2+5 n+6\right )-b^2 \left (n^2+15 n+53\right )\right ) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^2}{d (n+4)}\right )}{n+5}+\frac {\left (a^2 (n+2) (n+3)-b^2 (n+6) (n+8)\right ) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^3}{d (n+5)}}{b^2 (n+6) (n+7)}+\frac {a (n+3) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^4}{b^2 d (n+6) (n+7)}-\frac {\cos (c+d x) \sin ^{n+2}(c+d x) (a+b \sin (c+d x))^4}{b d (n+7)}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {3 \left (\frac {\frac {\frac {a (n+3) \left (2 a^4 \left (n^2+5 n+6\right )-2 a^2 b^2 \left (n^2+16 n+58\right )+3 b^4 \left (n^2+12 n+35\right )\right ) \cos (c+d x) \sin ^{n+1}(c+d x)}{d (n+2)}-\frac {a b^2 (n+3) \left (n^2+12 n+35\right ) \left (a^2 (n+6)+3 b^2 (n+1)\right ) \int \sin (c+d x)^ndx+b^3 (n+2) \left (n^2+10 n+24\right ) \left (3 a^2 (n+7)+b^2 (n+2)\right ) \int \sin (c+d x)^{n+1}dx}{n+2}}{n+3}+\frac {b \left (2 a^4 \left (n^2+5 n+6\right )-2 a^2 b^2 \left (n^2+16 n+57\right )+b^4 \left (n^2+10 n+24\right )\right ) \cos (c+d x) \sin ^{n+2}(c+d x)}{d (n+3)}}{n+4}+\frac {a \left (a^2 \left (n^2+5 n+6\right )-b^2 \left (n^2+15 n+53\right )\right ) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^2}{d (n+4)}\right )}{n+5}+\frac {\left (a^2 (n+2) (n+3)-b^2 (n+6) (n+8)\right ) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^3}{d (n+5)}}{b^2 (n+6) (n+7)}+\frac {a (n+3) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^4}{b^2 d (n+6) (n+7)}-\frac {\cos (c+d x) \sin ^{n+2}(c+d x) (a+b \sin (c+d x))^4}{b d (n+7)}\)

\(\Big \downarrow \) 3122

\(\displaystyle -\frac {\frac {\left (a^2 (n+2) (n+3)-b^2 (n+6) (n+8)\right ) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^3}{d (n+5)}+\frac {3 \left (\frac {a \left (a^2 \left (n^2+5 n+6\right )-b^2 \left (n^2+15 n+53\right )\right ) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^2}{d (n+4)}+\frac {\frac {b \left (2 a^4 \left (n^2+5 n+6\right )-2 a^2 b^2 \left (n^2+16 n+57\right )+b^4 \left (n^2+10 n+24\right )\right ) \cos (c+d x) \sin ^{n+2}(c+d x)}{d (n+3)}+\frac {\frac {a (n+3) \left (2 a^4 \left (n^2+5 n+6\right )-2 a^2 b^2 \left (n^2+16 n+58\right )+3 b^4 \left (n^2+12 n+35\right )\right ) \cos (c+d x) \sin ^{n+1}(c+d x)}{d (n+2)}-\frac {\frac {a b^2 (n+3) \left (n^2+12 n+35\right ) \left (a^2 (n+6)+3 b^2 (n+1)\right ) \cos (c+d x) \sin ^{n+1}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\sin ^2(c+d x)\right )}{d (n+1) \sqrt {\cos ^2(c+d x)}}+\frac {b^3 \left (n^2+10 n+24\right ) \left (3 a^2 (n+7)+b^2 (n+2)\right ) \cos (c+d x) \sin ^{n+2}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+2}{2},\frac {n+4}{2},\sin ^2(c+d x)\right )}{d \sqrt {\cos ^2(c+d x)}}}{n+2}}{n+3}}{n+4}\right )}{n+5}}{b^2 (n+6) (n+7)}+\frac {a (n+3) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^4}{b^2 d (n+6) (n+7)}-\frac {\cos (c+d x) \sin ^{n+2}(c+d x) (a+b \sin (c+d x))^4}{b d (n+7)}\)

Input:

Int[Cos[c + d*x]^4*Sin[c + d*x]^n*(a + b*Sin[c + d*x])^3,x]
 

Output:

(a*(3 + n)*Cos[c + d*x]*Sin[c + d*x]^(1 + n)*(a + b*Sin[c + d*x])^4)/(b^2* 
d*(6 + n)*(7 + n)) - (Cos[c + d*x]*Sin[c + d*x]^(2 + n)*(a + b*Sin[c + d*x 
])^4)/(b*d*(7 + n)) - (((a^2*(2 + n)*(3 + n) - b^2*(6 + n)*(8 + n))*Cos[c 
+ d*x]*Sin[c + d*x]^(1 + n)*(a + b*Sin[c + d*x])^3)/(d*(5 + n)) + (3*((a*( 
a^2*(6 + 5*n + n^2) - b^2*(53 + 15*n + n^2))*Cos[c + d*x]*Sin[c + d*x]^(1 
+ n)*(a + b*Sin[c + d*x])^2)/(d*(4 + n)) + ((b*(2*a^4*(6 + 5*n + n^2) + b^ 
4*(24 + 10*n + n^2) - 2*a^2*b^2*(57 + 16*n + n^2))*Cos[c + d*x]*Sin[c + d* 
x]^(2 + n))/(d*(3 + n)) + ((a*(3 + n)*(2*a^4*(6 + 5*n + n^2) + 3*b^4*(35 + 
 12*n + n^2) - 2*a^2*b^2*(58 + 16*n + n^2))*Cos[c + d*x]*Sin[c + d*x]^(1 + 
 n))/(d*(2 + n)) - ((a*b^2*(3 + n)*(35 + 12*n + n^2)*(3*b^2*(1 + n) + a^2* 
(6 + n))*Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sin[c + 
 d*x]^2]*Sin[c + d*x]^(1 + n))/(d*(1 + n)*Sqrt[Cos[c + d*x]^2]) + (b^3*(24 
 + 10*n + n^2)*(b^2*(2 + n) + 3*a^2*(7 + n))*Cos[c + d*x]*Hypergeometric2F 
1[1/2, (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + n))/(d*Sqrt 
[Cos[c + d*x]^2]))/(2 + n))/(3 + n))/(4 + n)))/(5 + n))/(b^2*(6 + n)*(7 + 
n))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3122

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( 
b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]))*Hypergeometric2 
F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2], x] /; FreeQ[{b, c, d, n}, x] 
 &&  !IntegerQ[2*n]
 

rule 3227

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x 
_)]), x_Symbol] :> Simp[c   Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b   Int 
[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
 

rule 3374

Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + 
(b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[a*(n + 3)*Cos[e + f* 
x]*(d*Sin[e + f*x])^(n + 1)*((a + b*Sin[e + f*x])^(m + 1)/(b^2*d*f*(m + n + 
 3)*(m + n + 4))), x] + (-Simp[Cos[e + f*x]*(d*Sin[e + f*x])^(n + 2)*((a + 
b*Sin[e + f*x])^(m + 1)/(b*d^2*f*(m + n + 4))), x] - Simp[1/(b^2*(m + n + 3 
)*(m + n + 4))   Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^m*Simp[a^2*(n 
+ 1)*(n + 3) - b^2*(m + n + 3)*(m + n + 4) + a*b*m*Sin[e + f*x] - (a^2*(n + 
 2)*(n + 3) - b^2*(m + n + 3)*(m + n + 5))*Sin[e + f*x]^2, x], x], x]) /; F 
reeQ[{a, b, d, e, f, m, n}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || Integ 
ersQ[2*m, 2*n]) &&  !m < -1 &&  !LtQ[n, -1] && NeQ[m + n + 3, 0] && NeQ[m + 
 n + 4, 0]
 

rule 3502

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) 
+ (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co 
s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m 
+ 2))   Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m 
 + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] 
 &&  !LtQ[m, -1]
 

rule 3512

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[(-C)*d*Cos[e + f*x]*Sin[e + f*x]*((a + b*Si 
n[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Simp[1/(b*(m + 3))   Int[(a + b*Si 
n[e + f*x])^m*Simp[a*C*d + A*b*c*(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + 
A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e + f*x]^2 
, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 
0] && NeQ[a^2 - b^2, 0] &&  !LtQ[m, -1]
 

rule 3528

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[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x 
])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + 
n + 2))   Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* 
d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a 
*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + 
 n + 2))*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] && GtQ[ 
m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
 

Maple [F]

\[\int \cos \left (d x +c \right )^{4} \sin \left (d x +c \right )^{n} \left (a +b \sin \left (d x +c \right )\right )^{3}d x\]

Input:

int(cos(d*x+c)^4*sin(d*x+c)^n*(a+b*sin(d*x+c))^3,x)
 

Output:

int(cos(d*x+c)^4*sin(d*x+c)^n*(a+b*sin(d*x+c))^3,x)
 

Fricas [F]

\[ \int \cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^3 \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{3} \sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{4} \,d x } \] Input:

integrate(cos(d*x+c)^4*sin(d*x+c)^n*(a+b*sin(d*x+c))^3,x, algorithm="frica 
s")
 

Output:

integral(-(3*a*b^2*cos(d*x + c)^6 - (a^3 + 3*a*b^2)*cos(d*x + c)^4 + (b^3* 
cos(d*x + c)^6 - (3*a^2*b + b^3)*cos(d*x + c)^4)*sin(d*x + c))*sin(d*x + c 
)^n, x)
 

Sympy [F(-1)]

Timed out. \[ \int \cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^3 \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)**4*sin(d*x+c)**n*(a+b*sin(d*x+c))**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^3 \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{3} \sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{4} \,d x } \] Input:

integrate(cos(d*x+c)^4*sin(d*x+c)^n*(a+b*sin(d*x+c))^3,x, algorithm="maxim 
a")
                                                                                    
                                                                                    
 

Output:

integrate((b*sin(d*x + c) + a)^3*sin(d*x + c)^n*cos(d*x + c)^4, x)
 

Giac [F]

\[ \int \cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^3 \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{3} \sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{4} \,d x } \] Input:

integrate(cos(d*x+c)^4*sin(d*x+c)^n*(a+b*sin(d*x+c))^3,x, algorithm="giac" 
)
 

Output:

integrate((b*sin(d*x + c) + a)^3*sin(d*x + c)^n*cos(d*x + c)^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^3 \, dx=\int {\cos \left (c+d\,x\right )}^4\,{\sin \left (c+d\,x\right )}^n\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^3 \,d x \] Input:

int(cos(c + d*x)^4*sin(c + d*x)^n*(a + b*sin(c + d*x))^3,x)
 

Output:

int(cos(c + d*x)^4*sin(c + d*x)^n*(a + b*sin(c + d*x))^3, x)
 

Reduce [F]

\[ \int \cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^3 \, dx=\left (\int \sin \left (d x +c \right )^{n} \cos \left (d x +c \right )^{4} \sin \left (d x +c \right )^{3}d x \right ) b^{3}+3 \left (\int \sin \left (d x +c \right )^{n} \cos \left (d x +c \right )^{4} \sin \left (d x +c \right )^{2}d x \right ) a \,b^{2}+3 \left (\int \sin \left (d x +c \right )^{n} \cos \left (d x +c \right )^{4} \sin \left (d x +c \right )d x \right ) a^{2} b +\left (\int \sin \left (d x +c \right )^{n} \cos \left (d x +c \right )^{4}d x \right ) a^{3} \] Input:

int(cos(d*x+c)^4*sin(d*x+c)^n*(a+b*sin(d*x+c))^3,x)
 

Output:

int(sin(c + d*x)**n*cos(c + d*x)**4*sin(c + d*x)**3,x)*b**3 + 3*int(sin(c 
+ d*x)**n*cos(c + d*x)**4*sin(c + d*x)**2,x)*a*b**2 + 3*int(sin(c + d*x)** 
n*cos(c + d*x)**4*sin(c + d*x),x)*a**2*b + int(sin(c + d*x)**n*cos(c + d*x 
)**4,x)*a**3