Integrand size = 35, antiderivative size = 458 \[ \int \cos ^2(e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{4/3} \, dx=-\frac {9 \left (64 a b c d-26 a^2 d^2-b^2 \left (18 c^2-13 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{2080 d^3 f}-\frac {9 b (3 b c-2 a d) \cos (e+f x) \sin (e+f x) (c+d \sin (e+f x))^{7/3}}{208 d^2 f}+\frac {3 \cos (e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{7/3}}{16 d f}-\frac {3 (c+d)^2 \left (208 a^2 c d^2-64 a b d \left (3 c^2-5 d^2\right )+b^2 c \left (54 c^2+d^2\right )\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {7}{3},\frac {3}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right ) \cos (e+f x) \sqrt [3]{c+d \sin (e+f x)}}{1040 \sqrt {2} d^4 f \sqrt {1+\sin (e+f x)} \sqrt [3]{\frac {c+d \sin (e+f x)}{c+d}}}-\frac {3 (c-d) (c+d)^2 \left (192 a b c d-208 a^2 d^2-b^2 \left (54 c^2+91 d^2\right )\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {4}{3},\frac {3}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right ) \cos (e+f x) \sqrt [3]{c+d \sin (e+f x)}}{1040 \sqrt {2} d^4 f \sqrt {1+\sin (e+f x)} \sqrt [3]{\frac {c+d \sin (e+f x)}{c+d}}} \]
-9/2080*(64*a*b*c*d-26*a^2*d^2-b^2*(18*c^2-13*d^2))*cos(f*x+e)*(c+d*sin(f* x+e))^(7/3)/d^3/f-9/208*b*(-2*a*d+3*b*c)*cos(f*x+e)*sin(f*x+e)*(c+d*sin(f* x+e))^(7/3)/d^2/f+3/16*cos(f*x+e)*(a+b*sin(f*x+e))^2*(c+d*sin(f*x+e))^(7/3 )/d/f-3/2080*(c+d)^2*(208*a^2*c*d^2-64*a*b*d*(3*c^2-5*d^2)+b^2*c*(54*c^2+d ^2))*AppellF1(1/2,-7/3,1/2,3/2,d*(1-sin(f*x+e))/(c+d),1/2-1/2*sin(f*x+e))* cos(f*x+e)*(c+d*sin(f*x+e))^(1/3)/d^4/f/((c+d*sin(f*x+e))/(c+d))^(1/3)*2^( 1/2)/(1+sin(f*x+e))^(1/2)-3/2080*(c-d)*(c+d)^2*(192*a*b*c*d-208*a^2*d^2-b^ 2*(54*c^2+91*d^2))*AppellF1(1/2,-4/3,1/2,3/2,d*(1-sin(f*x+e))/(c+d),1/2-1/ 2*sin(f*x+e))*cos(f*x+e)*(c+d*sin(f*x+e))^(1/3)/d^4/f/((c+d*sin(f*x+e))/(c +d))^(1/3)*2^(1/2)/(1+sin(f*x+e))^(1/2)
Time = 7.60 (sec) , antiderivative size = 573, normalized size of antiderivative = 1.25 \[ \int \cos ^2(e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{4/3} \, dx=-\frac {3 \sec (e+f x) \sqrt [3]{c+d \sin (e+f x)} \left (12 \left (c^2-d^2\right ) \left (-128 a b c d \left (6 c^2-17 d^2\right )+208 a^2 d^2 \left (4 c^2+7 d^2\right )+b^2 \left (216 c^4-248 c^2 d^2+637 d^4\right )\right ) \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},\frac {c+d \sin (e+f x)}{c-d},\frac {c+d \sin (e+f x)}{c+d}\right ) \sqrt {-\frac {d (-1+\sin (e+f x))}{c+d}} \sqrt {-\frac {d (1+\sin (e+f x))}{c-d}}-3 \left (208 a^2 c d^2 \left (4 c^2+51 d^2\right )+128 a b d \left (-6 c^4+21 c^2 d^2+40 d^4\right )+b^2 \left (216 c^5-392 c^3 d^2+3201 c d^4\right )\right ) \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {1}{2},\frac {7}{3},\frac {c+d \sin (e+f x)}{c-d},\frac {c+d \sin (e+f x)}{c+d}\right ) \sqrt {-\frac {d (-1+\sin (e+f x))}{c+d}} \sqrt {-\frac {d (1+\sin (e+f x))}{c-d}} (c+d \sin (e+f x))+4 d^2 \cos ^2(e+f x) \left (14 d^2 \left (448 a b c d+208 a^2 d^2+b^2 \left (4 c^2+91 d^2\right )\right ) \cos (2 (e+f x))-455 b^2 d^4 \cos (4 (e+f x))+2 \left (-108 b^2 c^4+384 a b c^3 d-416 a^2 c^2 d^2+152 b^2 c^2 d^2+2048 a b c d^3+728 a^2 d^4+546 b^2 d^4-d \left (4576 a^2 c d^2+32 a b d \left (8 c^2+45 d^2\right )+b^2 \left (-72 c^3+687 c d^2\right )\right ) \sin (e+f x)+35 b d^3 (17 b c+32 a d) \sin (3 (e+f x))\right )\right )\right )}{232960 d^5 f} \]
(-3*Sec[e + f*x]*(c + d*Sin[e + f*x])^(1/3)*(12*(c^2 - d^2)*(-128*a*b*c*d* (6*c^2 - 17*d^2) + 208*a^2*d^2*(4*c^2 + 7*d^2) + b^2*(216*c^4 - 248*c^2*d^ 2 + 637*d^4))*AppellF1[1/3, 1/2, 1/2, 4/3, (c + d*Sin[e + f*x])/(c - d), ( c + d*Sin[e + f*x])/(c + d)]*Sqrt[-((d*(-1 + Sin[e + f*x]))/(c + d))]*Sqrt [-((d*(1 + Sin[e + f*x]))/(c - d))] - 3*(208*a^2*c*d^2*(4*c^2 + 51*d^2) + 128*a*b*d*(-6*c^4 + 21*c^2*d^2 + 40*d^4) + b^2*(216*c^5 - 392*c^3*d^2 + 32 01*c*d^4))*AppellF1[4/3, 1/2, 1/2, 7/3, (c + d*Sin[e + f*x])/(c - d), (c + d*Sin[e + f*x])/(c + d)]*Sqrt[-((d*(-1 + Sin[e + f*x]))/(c + d))]*Sqrt[-( (d*(1 + Sin[e + f*x]))/(c - d))]*(c + d*Sin[e + f*x]) + 4*d^2*Cos[e + f*x] ^2*(14*d^2*(448*a*b*c*d + 208*a^2*d^2 + b^2*(4*c^2 + 91*d^2))*Cos[2*(e + f *x)] - 455*b^2*d^4*Cos[4*(e + f*x)] + 2*(-108*b^2*c^4 + 384*a*b*c^3*d - 41 6*a^2*c^2*d^2 + 152*b^2*c^2*d^2 + 2048*a*b*c*d^3 + 728*a^2*d^4 + 546*b^2*d ^4 - d*(4576*a^2*c*d^2 + 32*a*b*d*(8*c^2 + 45*d^2) + b^2*(-72*c^3 + 687*c* d^2))*Sin[e + f*x] + 35*b*d^3*(17*b*c + 32*a*d)*Sin[3*(e + f*x)]))))/(2329 60*d^5*f)
Time = 1.94 (sec) , antiderivative size = 480, normalized size of antiderivative = 1.05, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.486, Rules used = {3042, 3401, 3042, 3529, 25, 3042, 3512, 27, 3042, 3502, 27, 3042, 3235, 3042, 3144, 156, 155}
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.
| \(\displaystyle \int \cos ^2(e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{4/3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (e+f x)^2 (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{4/3}dx\) |
| \(\Big \downarrow \) 3401 |
| \(\displaystyle \int \left (1-\sin ^2(e+f x)\right ) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{4/3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (1-\sin (e+f x)^2\right ) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{4/3}dx\) |
| \(\Big \downarrow \) 3529 |
| \(\displaystyle \frac {3 \int -\left ((a+b \sin (e+f x)) (c+d \sin (e+f x))^{4/3} \left (-\left ((3 b c-2 a d) \sin ^2(e+f x)\right )-(a c+b d) \sin (e+f x)+2 b c-3 a d\right )\right )dx}{16 d}+\frac {3 \cos (e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{7/3}}{16 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3 \cos (e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{7/3}}{16 d f}-\frac {3 \int (a+b \sin (e+f x)) (c+d \sin (e+f x))^{4/3} \left (-\left ((3 b c-2 a d) \sin ^2(e+f x)\right )-(a c+b d) \sin (e+f x)+2 b c-3 a d\right )dx}{16 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \cos (e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{7/3}}{16 d f}-\frac {3 \int (a+b \sin (e+f x)) (c+d \sin (e+f x))^{4/3} \left (-\left ((3 b c-2 a d) \sin (e+f x)^2\right )-(a c+b d) \sin (e+f x)+2 b c-3 a d\right )dx}{16 d}\) |
| \(\Big \downarrow \) 3512 |
| \(\displaystyle \frac {3 \cos (e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{7/3}}{16 d f}-\frac {3 \left (\frac {3 \int -\frac {1}{3} (c+d \sin (e+f x))^{4/3} \left (9 b^2 c^2-32 a b d c+39 a^2 d^2+\left (-\left (\left (18 c^2-13 d^2\right ) b^2\right )+64 a c d b-26 a^2 d^2\right ) \sin ^2(e+f x)+d \left (13 c a^2+32 b d a+4 b^2 c\right ) \sin (e+f x)\right )dx}{13 d}+\frac {3 b (3 b c-2 a d) \sin (e+f x) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{13 d f}\right )}{16 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \cos (e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{7/3}}{16 d f}-\frac {3 \left (\frac {3 b (3 b c-2 a d) \sin (e+f x) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{13 d f}-\frac {\int (c+d \sin (e+f x))^{4/3} \left (9 b^2 c^2-32 a b d c+39 a^2 d^2+\left (-\left (\left (18 c^2-13 d^2\right ) b^2\right )+64 a c d b-26 a^2 d^2\right ) \sin ^2(e+f x)+d \left (13 c a^2+32 b d a+4 b^2 c\right ) \sin (e+f x)\right )dx}{13 d}\right )}{16 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \cos (e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{7/3}}{16 d f}-\frac {3 \left (\frac {3 b (3 b c-2 a d) \sin (e+f x) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{13 d f}-\frac {\int (c+d \sin (e+f x))^{4/3} \left (9 b^2 c^2-32 a b d c+39 a^2 d^2+\left (-\left (\left (18 c^2-13 d^2\right ) b^2\right )+64 a c d b-26 a^2 d^2\right ) \sin (e+f x)^2+d \left (13 c a^2+32 b d a+4 b^2 c\right ) \sin (e+f x)\right )dx}{13 d}\right )}{16 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {3 \cos (e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{7/3}}{16 d f}-\frac {3 \left (\frac {3 b (3 b c-2 a d) \sin (e+f x) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{13 d f}-\frac {\frac {3 \int \frac {1}{3} (c+d \sin (e+f x))^{4/3} \left (d \left (-\left (\left (36 c^2-91 d^2\right ) b^2\right )+128 a c d b+208 a^2 d^2\right )+\left (c \left (54 c^2+d^2\right ) b^2-64 a d \left (3 c^2-5 d^2\right ) b+208 a^2 c d^2\right ) \sin (e+f x)\right )dx}{10 d}-\frac {3 \left (-26 a^2 d^2+64 a b c d-\left (b^2 \left (18 c^2-13 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{10 d f}}{13 d}\right )}{16 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \cos (e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{7/3}}{16 d f}-\frac {3 \left (\frac {3 b (3 b c-2 a d) \sin (e+f x) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{13 d f}-\frac {\frac {\int (c+d \sin (e+f x))^{4/3} \left (d \left (-\left (\left (36 c^2-91 d^2\right ) b^2\right )+128 a c d b+208 a^2 d^2\right )+\left (c \left (54 c^2+d^2\right ) b^2-64 a d \left (3 c^2-5 d^2\right ) b+208 a^2 c d^2\right ) \sin (e+f x)\right )dx}{10 d}-\frac {3 \left (-26 a^2 d^2+64 a b c d-\left (b^2 \left (18 c^2-13 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{10 d f}}{13 d}\right )}{16 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \cos (e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{7/3}}{16 d f}-\frac {3 \left (\frac {3 b (3 b c-2 a d) \sin (e+f x) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{13 d f}-\frac {\frac {\int (c+d \sin (e+f x))^{4/3} \left (d \left (-\left (\left (36 c^2-91 d^2\right ) b^2\right )+128 a c d b+208 a^2 d^2\right )+\left (c \left (54 c^2+d^2\right ) b^2-64 a d \left (3 c^2-5 d^2\right ) b+208 a^2 c d^2\right ) \sin (e+f x)\right )dx}{10 d}-\frac {3 \left (-26 a^2 d^2+64 a b c d-\left (b^2 \left (18 c^2-13 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{10 d f}}{13 d}\right )}{16 d}\) |
| \(\Big \downarrow \) 3235 |
| \(\displaystyle \frac {3 \cos (e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{7/3}}{16 d f}-\frac {3 \left (\frac {3 b (3 b c-2 a d) \sin (e+f x) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{13 d f}-\frac {\frac {\frac {\left (c^2-d^2\right ) \left (-208 a^2 d^2+192 a b c d-\left (b^2 \left (54 c^2+91 d^2\right )\right )\right ) \int (c+d \sin (e+f x))^{4/3}dx}{d}+\frac {\left (208 a^2 c d^2-64 a b d \left (3 c^2-5 d^2\right )+b^2 c \left (54 c^2+d^2\right )\right ) \int (c+d \sin (e+f x))^{7/3}dx}{d}}{10 d}-\frac {3 \left (-26 a^2 d^2+64 a b c d-\left (b^2 \left (18 c^2-13 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{10 d f}}{13 d}\right )}{16 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \cos (e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{7/3}}{16 d f}-\frac {3 \left (\frac {3 b (3 b c-2 a d) \sin (e+f x) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{13 d f}-\frac {\frac {\frac {\left (c^2-d^2\right ) \left (-208 a^2 d^2+192 a b c d-\left (b^2 \left (54 c^2+91 d^2\right )\right )\right ) \int (c+d \sin (e+f x))^{4/3}dx}{d}+\frac {\left (208 a^2 c d^2-64 a b d \left (3 c^2-5 d^2\right )+b^2 c \left (54 c^2+d^2\right )\right ) \int (c+d \sin (e+f x))^{7/3}dx}{d}}{10 d}-\frac {3 \left (-26 a^2 d^2+64 a b c d-\left (b^2 \left (18 c^2-13 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{10 d f}}{13 d}\right )}{16 d}\) |
| \(\Big \downarrow \) 3144 |
| \(\displaystyle \frac {3 \cos (e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{7/3}}{16 d f}-\frac {3 \left (\frac {3 b (3 b c-2 a d) \sin (e+f x) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{13 d f}-\frac {\frac {\frac {\left (c^2-d^2\right ) \left (-208 a^2 d^2+192 a b c d-\left (b^2 \left (54 c^2+91 d^2\right )\right )\right ) \cos (e+f x) \int \frac {(c+d \sin (e+f x))^{4/3}}{\sqrt {1-\sin (e+f x)} \sqrt {\sin (e+f x)+1}}d\sin (e+f x)}{d f \sqrt {1-\sin (e+f x)} \sqrt {\sin (e+f x)+1}}+\frac {\left (208 a^2 c d^2-64 a b d \left (3 c^2-5 d^2\right )+b^2 c \left (54 c^2+d^2\right )\right ) \cos (e+f x) \int \frac {(c+d \sin (e+f x))^{7/3}}{\sqrt {1-\sin (e+f x)} \sqrt {\sin (e+f x)+1}}d\sin (e+f x)}{d f \sqrt {1-\sin (e+f x)} \sqrt {\sin (e+f x)+1}}}{10 d}-\frac {3 \left (-26 a^2 d^2+64 a b c d-\left (b^2 \left (18 c^2-13 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{10 d f}}{13 d}\right )}{16 d}\) |
| \(\Big \downarrow \) 156 |
| \(\displaystyle \frac {3 \cos (e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{7/3}}{16 d f}-\frac {3 \left (\frac {3 b (3 b c-2 a d) \sin (e+f x) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{13 d f}-\frac {\frac {\frac {(c+d)^2 \left (208 a^2 c d^2-64 a b d \left (3 c^2-5 d^2\right )+b^2 c \left (54 c^2+d^2\right )\right ) \cos (e+f x) \sqrt [3]{c+d \sin (e+f x)} \int \frac {\left (\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}\right )^{7/3}}{\sqrt {1-\sin (e+f x)} \sqrt {\sin (e+f x)+1}}d\sin (e+f x)}{d f \sqrt {1-\sin (e+f x)} \sqrt {\sin (e+f x)+1} \sqrt [3]{\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (c^2-d^2\right ) (c+d) \left (-208 a^2 d^2+192 a b c d-\left (b^2 \left (54 c^2+91 d^2\right )\right )\right ) \cos (e+f x) \sqrt [3]{c+d \sin (e+f x)} \int \frac {\left (\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}\right )^{4/3}}{\sqrt {1-\sin (e+f x)} \sqrt {\sin (e+f x)+1}}d\sin (e+f x)}{d f \sqrt {1-\sin (e+f x)} \sqrt {\sin (e+f x)+1} \sqrt [3]{\frac {c+d \sin (e+f x)}{c+d}}}}{10 d}-\frac {3 \left (-26 a^2 d^2+64 a b c d-\left (b^2 \left (18 c^2-13 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{10 d f}}{13 d}\right )}{16 d}\) |
| \(\Big \downarrow \) 155 |
| \(\displaystyle \frac {3 \cos (e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{7/3}}{16 d f}-\frac {3 \left (\frac {3 b (3 b c-2 a d) \sin (e+f x) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{13 d f}-\frac {\frac {-\frac {\sqrt {2} (c+d)^2 \left (208 a^2 c d^2-64 a b d \left (3 c^2-5 d^2\right )+b^2 c \left (54 c^2+d^2\right )\right ) \cos (e+f x) \sqrt [3]{c+d \sin (e+f x)} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {7}{3},\frac {3}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right )}{d f \sqrt {\sin (e+f x)+1} \sqrt [3]{\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\sqrt {2} \left (c^2-d^2\right ) (c+d) \left (-208 a^2 d^2+192 a b c d-\left (b^2 \left (54 c^2+91 d^2\right )\right )\right ) \cos (e+f x) \sqrt [3]{c+d \sin (e+f x)} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {4}{3},\frac {3}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right )}{d f \sqrt {\sin (e+f x)+1} \sqrt [3]{\frac {c+d \sin (e+f x)}{c+d}}}}{10 d}-\frac {3 \left (-26 a^2 d^2+64 a b c d-\left (b^2 \left (18 c^2-13 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{10 d f}}{13 d}\right )}{16 d}\) |
(3*Cos[e + f*x]*(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(7/3))/(16*d*f ) - (3*((3*b*(3*b*c - 2*a*d)*Cos[e + f*x]*Sin[e + f*x]*(c + d*Sin[e + f*x] )^(7/3))/(13*d*f) - ((-3*(64*a*b*c*d - 26*a^2*d^2 - b^2*(18*c^2 - 13*d^2)) *Cos[e + f*x]*(c + d*Sin[e + f*x])^(7/3))/(10*d*f) + (-((Sqrt[2]*(c + d)^2 *(208*a^2*c*d^2 - 64*a*b*d*(3*c^2 - 5*d^2) + b^2*c*(54*c^2 + d^2))*AppellF 1[1/2, 1/2, -7/3, 3/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d )]*Cos[e + f*x]*(c + d*Sin[e + f*x])^(1/3))/(d*f*Sqrt[1 + Sin[e + f*x]]*(( c + d*Sin[e + f*x])/(c + d))^(1/3))) - (Sqrt[2]*(c + d)*(c^2 - d^2)*(192*a *b*c*d - 208*a^2*d^2 - b^2*(54*c^2 + 91*d^2))*AppellF1[1/2, 1/2, -4/3, 3/2 , (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^(1/3))/(d*f*Sqrt[1 + Sin[e + f*x]]*((c + d*Sin[e + f*x])/( c + d))^(1/3)))/(10*d))/(13*d)))/(16*d)
3.16.16.3.1 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)/(b*(m + 1)*Simplify[b/(b*c - a*d)]^n* Simplify[b/(b*e - a*f)]^p))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/ (b*c - a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[m] && !IntegerQ[n] && !IntegerQ[p] && GtQ[Sim plify[b/(b*c - a*d)], 0] && GtQ[Simplify[b/(b*e - a*f)], 0] && !(GtQ[Simpl ify[d/(d*a - c*b)], 0] && GtQ[Simplify[d/(d*e - c*f)], 0] && SimplerQ[c + d *x, a + b*x]) && !(GtQ[Simplify[f/(f*a - e*b)], 0] && GtQ[Simplify[f/(f*c - e*d)], 0] && SimplerQ[e + f*x, a + b*x])
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) ^(p_), x_] :> Simp[(e + f*x)^FracPart[p]/(Simplify[b/(b*e - a*f)]^IntPart[p ]*(b*((e + f*x)/(b*e - a*f)))^FracPart[p]) Int[(a + b*x)^m*(c + d*x)^n*Si mp[b*(e/(b*e - a*f)) + b*f*(x/(b*e - a*f)), x]^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[m] && !IntegerQ[n] && !IntegerQ[p] & & GtQ[Simplify[b/(b*c - a*d)], 0] && !GtQ[Simplify[b/(b*e - a*f)], 0]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]/(d*Sqrt[1 + Sin[c + d*x]]*Sqrt[1 - Sin[c + d*x]]) Subst[Int[(a + b*x )^n/(Sqrt[1 + x]*Sqrt[1 - x]), x], x, Sin[c + d*x]], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[a^2 - b^2, 0] && !IntegerQ[2*n]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)/b Int[(a + b*Sin[e + f*x])^m, x], x] + Simp[d/b Int[(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
Int[cos[(e_.) + (f_.)*(x_)]^2*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)* ((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*(1 - Sin[e + f*x]^2), x] /; FreeQ[{a, b, c , d, e, f, m, n}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n])
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]
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]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (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)) + (A*b*d*(m + n + 2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + C* (a*d*m - b*c*(m + 1))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f , A, 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])))
\[\int \left (\cos ^{2}\left (f x +e \right )\right ) \left (a +b \sin \left (f x +e \right )\right )^{2} \left (c +d \sin \left (f x +e \right )\right )^{\frac {4}{3}}d x\]
\[ \int \cos ^2(e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{4/3} \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{2} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {4}{3}} \cos \left (f x + e\right )^{2} \,d x } \]
integral(-((b^2*c + 2*a*b*d)*cos(f*x + e)^4 - (2*a*b*d + (a^2 + b^2)*c)*co s(f*x + e)^2 + (b^2*d*cos(f*x + e)^4 - (2*a*b*c + (a^2 + b^2)*d)*cos(f*x + e)^2)*sin(f*x + e))*(d*sin(f*x + e) + c)^(1/3), x)
Timed out. \[ \int \cos ^2(e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{4/3} \, dx=\text {Timed out} \]
\[ \int \cos ^2(e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{4/3} \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{2} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {4}{3}} \cos \left (f x + e\right )^{2} \,d x } \]
\[ \int \cos ^2(e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{4/3} \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{2} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {4}{3}} \cos \left (f x + e\right )^{2} \,d x } \]
Timed out. \[ \int \cos ^2(e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{4/3} \, dx=\int {\cos \left (e+f\,x\right )}^2\,{\left (a+b\,\sin \left (e+f\,x\right )\right )}^2\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{4/3} \,d x \]