Integrand size = 29, antiderivative size = 451 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=-\frac {2 \left (3 a^2+13 b^2\right ) \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{429 d}-\frac {2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{39 d}-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}+\frac {8 a \left (32 a^6-189 a^4 b^2+570 a^2 b^4+1635 b^6\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{45045 b^5 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {8 \left (32 a^8-197 a^6 b^2+615 a^4 b^4-255 a^2 b^6-195 b^8\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{45045 b^5 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^4-33 a^2 b^2-39 b^4-7 a b \left (a^2+63 b^2\right ) \sin (c+d x)\right )}{9009 b^2 d}+\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^6-165 a^4 b^2+450 a^2 b^4+195 b^6-24 a b \left (a^4-5 a^2 b^2-60 b^4\right ) \sin (c+d x)\right )}{45045 b^4 d} \]
-2/39*a*cos(d*x+c)^5*(a+b*sin(d*x+c))^(3/2)/d-2/15*cos(d*x+c)^5*(a+b*sin(d *x+c))^(5/2)/d-2/429*(3*a^2+13*b^2)*cos(d*x+c)^5*(a+b*sin(d*x+c))^(1/2)/d- 2/9009*cos(d*x+c)^3*(8*a^4-33*a^2*b^2-39*b^4-7*a*b*(a^2+63*b^2)*sin(d*x+c) )*(a+b*sin(d*x+c))^(1/2)/b^2/d+4/45045*cos(d*x+c)*(32*a^6-165*a^4*b^2+450* a^2*b^4+195*b^6-24*a*b*(a^4-5*a^2*b^2-60*b^4)*sin(d*x+c))*(a+b*sin(d*x+c)) ^(1/2)/b^4/d-8/45045*a*(32*a^6-189*a^4*b^2+570*a^2*b^4+1635*b^6)*(sin(1/2* c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*EllipticE(cos(1/2*c+1 /4*Pi+1/2*d*x),2^(1/2)*(b/(a+b))^(1/2))*(a+b*sin(d*x+c))^(1/2)/b^5/d/((a+b *sin(d*x+c))/(a+b))^(1/2)+8/45045*(32*a^8-197*a^6*b^2+615*a^4*b^4-255*a^2* b^6-195*b^8)*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x) *EllipticF(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2)*(b/(a+b))^(1/2))*((a+b*sin(d* x+c))/(a+b))^(1/2)/b^5/d/(a+b*sin(d*x+c))^(1/2)
Time = 16.06 (sec) , antiderivative size = 450, normalized size of antiderivative = 1.00 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\frac {-256 a \left (32 a^7+32 a^6 b-189 a^5 b^2-189 a^4 b^3+570 a^3 b^4+570 a^2 b^5+1635 a b^6+1635 b^7\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+256 \left (32 a^8-197 a^6 b^2+615 a^4 b^4-255 a^2 b^6-195 b^8\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+b \cos (c+d x) \left (4096 a^7-23936 a^5 b^2-36512 a^3 b^4+67584 a b^6+8 \left (32 a^5 b^2-18192 a^3 b^4-18741 a b^6\right ) \cos (2 (c+d x))-224 \left (161 a^3 b^4-54 a b^6\right ) \cos (4 (c+d x))+20328 a b^6 \cos (6 (c+d x))+1024 a^6 b \sin (c+d x)-5840 a^4 b^3 \sin (c+d x)+186768 a^2 b^5 \sin (c+d x)+8151 b^7 \sin (c+d x)-80 a^4 b^3 \sin (3 (c+d x))-101688 a^2 b^5 \sin (3 (c+d x))-22269 b^7 \sin (3 (c+d x))-46536 a^2 b^5 \sin (5 (c+d x))-2457 b^7 \sin (5 (c+d x))+3003 b^7 \sin (7 (c+d x))\right )}{1441440 b^5 d \sqrt {a+b \sin (c+d x)}} \]
(-256*a*(32*a^7 + 32*a^6*b - 189*a^5*b^2 - 189*a^4*b^3 + 570*a^3*b^4 + 570 *a^2*b^5 + 1635*a*b^6 + 1635*b^7)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/( a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + 256*(32*a^8 - 197*a^6*b^2 + 6 15*a^4*b^4 - 255*a^2*b^6 - 195*b^8)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b) /(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + b*Cos[c + d*x]*(4096*a^7 - 23936*a^5*b^2 - 36512*a^3*b^4 + 67584*a*b^6 + 8*(32*a^5*b^2 - 18192*a^3*b^ 4 - 18741*a*b^6)*Cos[2*(c + d*x)] - 224*(161*a^3*b^4 - 54*a*b^6)*Cos[4*(c + d*x)] + 20328*a*b^6*Cos[6*(c + d*x)] + 1024*a^6*b*Sin[c + d*x] - 5840*a^ 4*b^3*Sin[c + d*x] + 186768*a^2*b^5*Sin[c + d*x] + 8151*b^7*Sin[c + d*x] - 80*a^4*b^3*Sin[3*(c + d*x)] - 101688*a^2*b^5*Sin[3*(c + d*x)] - 22269*b^7 *Sin[3*(c + d*x)] - 46536*a^2*b^5*Sin[5*(c + d*x)] - 2457*b^7*Sin[5*(c + d *x)] + 3003*b^7*Sin[7*(c + d*x)]))/(1441440*b^5*d*Sqrt[a + b*Sin[c + d*x]] )
Time = 2.51 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.06, number of steps used = 24, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.828, Rules used = {3042, 3341, 27, 3042, 3341, 27, 3042, 3341, 27, 3042, 3344, 27, 3042, 3344, 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.
\(\displaystyle \int \sin (c+d x) \cos ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (c+d x) \cos (c+d x)^4 (a+b \sin (c+d x))^{5/2}dx\) |
\(\Big \downarrow \) 3341 |
\(\displaystyle \frac {2}{15} \int \frac {5}{2} \cos ^4(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^{3/2}dx-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \int \cos ^4(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^{3/2}dx-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{3} \int \cos (c+d x)^4 (b+a \sin (c+d x)) (a+b \sin (c+d x))^{3/2}dx-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}\) |
\(\Big \downarrow \) 3341 |
\(\displaystyle \frac {1}{3} \left (\frac {2}{13} \int \frac {1}{2} \cos ^4(c+d x) \sqrt {a+b \sin (c+d x)} \left (16 a b+\left (3 a^2+13 b^2\right ) \sin (c+d x)\right )dx-\frac {2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\right )-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{13} \int \cos ^4(c+d x) \sqrt {a+b \sin (c+d x)} \left (16 a b+\left (3 a^2+13 b^2\right ) \sin (c+d x)\right )dx-\frac {2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\right )-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{13} \int \cos (c+d x)^4 \sqrt {a+b \sin (c+d x)} \left (16 a b+\left (3 a^2+13 b^2\right ) \sin (c+d x)\right )dx-\frac {2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\right )-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}\) |
\(\Big \downarrow \) 3341 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{13} \left (\frac {2}{11} \int \frac {\cos ^4(c+d x) \left (b \left (179 a^2+13 b^2\right )+3 a \left (a^2+63 b^2\right ) \sin (c+d x)\right )}{2 \sqrt {a+b \sin (c+d x)}}dx-\frac {2 \left (3 a^2+13 b^2\right ) \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\right )-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{13} \left (\frac {1}{11} \int \frac {\cos ^4(c+d x) \left (b \left (179 a^2+13 b^2\right )+3 a \left (a^2+63 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx-\frac {2 \left (3 a^2+13 b^2\right ) \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\right )-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{13} \left (\frac {1}{11} \int \frac {\cos (c+d x)^4 \left (b \left (179 a^2+13 b^2\right )+3 a \left (a^2+63 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx-\frac {2 \left (3 a^2+13 b^2\right ) \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\right )-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}\) |
\(\Big \downarrow \) 3344 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {4 \int -\frac {3 \cos ^2(c+d x) \left (b \left (a^4-474 b^2 a^2-39 b^4\right )+8 a \left (a^4-5 b^2 a^2-60 b^4\right ) \sin (c+d x)\right )}{2 \sqrt {a+b \sin (c+d x)}}dx}{21 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^4-7 a b \left (a^2+63 b^2\right ) \sin (c+d x)-33 a^2 b^2-39 b^4\right )}{21 b^2 d}\right )-\frac {2 \left (3 a^2+13 b^2\right ) \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\right )-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{13} \left (\frac {1}{11} \left (-\frac {2 \int \frac {\cos ^2(c+d x) \left (b \left (a^4-474 b^2 a^2-39 b^4\right )+8 a \left (a^4-5 b^2 a^2-60 b^4\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx}{7 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^4-7 a b \left (a^2+63 b^2\right ) \sin (c+d x)-33 a^2 b^2-39 b^4\right )}{21 b^2 d}\right )-\frac {2 \left (3 a^2+13 b^2\right ) \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\right )-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{13} \left (\frac {1}{11} \left (-\frac {2 \int \frac {\cos (c+d x)^2 \left (b \left (a^4-474 b^2 a^2-39 b^4\right )+8 a \left (a^4-5 b^2 a^2-60 b^4\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx}{7 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^4-7 a b \left (a^2+63 b^2\right ) \sin (c+d x)-33 a^2 b^2-39 b^4\right )}{21 b^2 d}\right )-\frac {2 \left (3 a^2+13 b^2\right ) \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\right )-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}\) |
\(\Big \downarrow \) 3344 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{13} \left (\frac {1}{11} \left (-\frac {2 \left (\frac {4 \int -\frac {b \left (8 a^6-45 b^2 a^4+1890 b^4 a^2+195 b^6\right )+a \left (32 a^6-189 b^2 a^4+570 b^4 a^2+1635 b^6\right ) \sin (c+d x)}{2 \sqrt {a+b \sin (c+d x)}}dx}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^6-165 a^4 b^2+450 a^2 b^4-24 a b \left (a^4-5 a^2 b^2-60 b^4\right ) \sin (c+d x)+195 b^6\right )}{15 b^2 d}\right )}{7 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^4-7 a b \left (a^2+63 b^2\right ) \sin (c+d x)-33 a^2 b^2-39 b^4\right )}{21 b^2 d}\right )-\frac {2 \left (3 a^2+13 b^2\right ) \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\right )-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{13} \left (\frac {1}{11} \left (-\frac {2 \left (-\frac {2 \int \frac {b \left (8 a^6-45 b^2 a^4+1890 b^4 a^2+195 b^6\right )+a \left (32 a^6-189 b^2 a^4+570 b^4 a^2+1635 b^6\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^6-165 a^4 b^2+450 a^2 b^4-24 a b \left (a^4-5 a^2 b^2-60 b^4\right ) \sin (c+d x)+195 b^6\right )}{15 b^2 d}\right )}{7 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^4-7 a b \left (a^2+63 b^2\right ) \sin (c+d x)-33 a^2 b^2-39 b^4\right )}{21 b^2 d}\right )-\frac {2 \left (3 a^2+13 b^2\right ) \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\right )-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{13} \left (\frac {1}{11} \left (-\frac {2 \left (-\frac {2 \int \frac {b \left (8 a^6-45 b^2 a^4+1890 b^4 a^2+195 b^6\right )+a \left (32 a^6-189 b^2 a^4+570 b^4 a^2+1635 b^6\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^6-165 a^4 b^2+450 a^2 b^4-24 a b \left (a^4-5 a^2 b^2-60 b^4\right ) \sin (c+d x)+195 b^6\right )}{15 b^2 d}\right )}{7 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^4-7 a b \left (a^2+63 b^2\right ) \sin (c+d x)-33 a^2 b^2-39 b^4\right )}{21 b^2 d}\right )-\frac {2 \left (3 a^2+13 b^2\right ) \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\right )-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}\) |
\(\Big \downarrow \) 3231 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{13} \left (\frac {1}{11} \left (-\frac {2 \left (-\frac {2 \left (\frac {a \left (32 a^6-189 a^4 b^2+570 a^2 b^4+1635 b^6\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}-\frac {\left (32 a^8-197 a^6 b^2+615 a^4 b^4-255 a^2 b^6-195 b^8\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^6-165 a^4 b^2+450 a^2 b^4-24 a b \left (a^4-5 a^2 b^2-60 b^4\right ) \sin (c+d x)+195 b^6\right )}{15 b^2 d}\right )}{7 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^4-7 a b \left (a^2+63 b^2\right ) \sin (c+d x)-33 a^2 b^2-39 b^4\right )}{21 b^2 d}\right )-\frac {2 \left (3 a^2+13 b^2\right ) \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\right )-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{13} \left (\frac {1}{11} \left (-\frac {2 \left (-\frac {2 \left (\frac {a \left (32 a^6-189 a^4 b^2+570 a^2 b^4+1635 b^6\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}-\frac {\left (32 a^8-197 a^6 b^2+615 a^4 b^4-255 a^2 b^6-195 b^8\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^6-165 a^4 b^2+450 a^2 b^4-24 a b \left (a^4-5 a^2 b^2-60 b^4\right ) \sin (c+d x)+195 b^6\right )}{15 b^2 d}\right )}{7 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^4-7 a b \left (a^2+63 b^2\right ) \sin (c+d x)-33 a^2 b^2-39 b^4\right )}{21 b^2 d}\right )-\frac {2 \left (3 a^2+13 b^2\right ) \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\right )-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}\) |
\(\Big \downarrow \) 3134 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{13} \left (\frac {1}{11} \left (-\frac {2 \left (-\frac {2 \left (\frac {a \left (32 a^6-189 a^4 b^2+570 a^2 b^4+1635 b^6\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (32 a^8-197 a^6 b^2+615 a^4 b^4-255 a^2 b^6-195 b^8\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^6-165 a^4 b^2+450 a^2 b^4-24 a b \left (a^4-5 a^2 b^2-60 b^4\right ) \sin (c+d x)+195 b^6\right )}{15 b^2 d}\right )}{7 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^4-7 a b \left (a^2+63 b^2\right ) \sin (c+d x)-33 a^2 b^2-39 b^4\right )}{21 b^2 d}\right )-\frac {2 \left (3 a^2+13 b^2\right ) \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\right )-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{13} \left (\frac {1}{11} \left (-\frac {2 \left (-\frac {2 \left (\frac {a \left (32 a^6-189 a^4 b^2+570 a^2 b^4+1635 b^6\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (32 a^8-197 a^6 b^2+615 a^4 b^4-255 a^2 b^6-195 b^8\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^6-165 a^4 b^2+450 a^2 b^4-24 a b \left (a^4-5 a^2 b^2-60 b^4\right ) \sin (c+d x)+195 b^6\right )}{15 b^2 d}\right )}{7 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^4-7 a b \left (a^2+63 b^2\right ) \sin (c+d x)-33 a^2 b^2-39 b^4\right )}{21 b^2 d}\right )-\frac {2 \left (3 a^2+13 b^2\right ) \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\right )-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}\) |
\(\Big \downarrow \) 3132 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{13} \left (\frac {1}{11} \left (-\frac {2 \left (-\frac {2 \left (\frac {2 a \left (32 a^6-189 a^4 b^2+570 a^2 b^4+1635 b^6\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (32 a^8-197 a^6 b^2+615 a^4 b^4-255 a^2 b^6-195 b^8\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^6-165 a^4 b^2+450 a^2 b^4-24 a b \left (a^4-5 a^2 b^2-60 b^4\right ) \sin (c+d x)+195 b^6\right )}{15 b^2 d}\right )}{7 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^4-7 a b \left (a^2+63 b^2\right ) \sin (c+d x)-33 a^2 b^2-39 b^4\right )}{21 b^2 d}\right )-\frac {2 \left (3 a^2+13 b^2\right ) \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\right )-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}\) |
\(\Big \downarrow \) 3142 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{13} \left (\frac {1}{11} \left (-\frac {2 \left (-\frac {2 \left (\frac {2 a \left (32 a^6-189 a^4 b^2+570 a^2 b^4+1635 b^6\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (32 a^8-197 a^6 b^2+615 a^4 b^4-255 a^2 b^6-195 b^8\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{b \sqrt {a+b \sin (c+d x)}}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^6-165 a^4 b^2+450 a^2 b^4-24 a b \left (a^4-5 a^2 b^2-60 b^4\right ) \sin (c+d x)+195 b^6\right )}{15 b^2 d}\right )}{7 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^4-7 a b \left (a^2+63 b^2\right ) \sin (c+d x)-33 a^2 b^2-39 b^4\right )}{21 b^2 d}\right )-\frac {2 \left (3 a^2+13 b^2\right ) \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\right )-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{13} \left (\frac {1}{11} \left (-\frac {2 \left (-\frac {2 \left (\frac {2 a \left (32 a^6-189 a^4 b^2+570 a^2 b^4+1635 b^6\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (32 a^8-197 a^6 b^2+615 a^4 b^4-255 a^2 b^6-195 b^8\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{b \sqrt {a+b \sin (c+d x)}}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^6-165 a^4 b^2+450 a^2 b^4-24 a b \left (a^4-5 a^2 b^2-60 b^4\right ) \sin (c+d x)+195 b^6\right )}{15 b^2 d}\right )}{7 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^4-7 a b \left (a^2+63 b^2\right ) \sin (c+d x)-33 a^2 b^2-39 b^4\right )}{21 b^2 d}\right )-\frac {2 \left (3 a^2+13 b^2\right ) \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\right )-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}\) |
\(\Big \downarrow \) 3140 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{13} \left (\frac {1}{11} \left (-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^4-7 a b \left (a^2+63 b^2\right ) \sin (c+d x)-33 a^2 b^2-39 b^4\right )}{21 b^2 d}-\frac {2 \left (-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^6-165 a^4 b^2+450 a^2 b^4-24 a b \left (a^4-5 a^2 b^2-60 b^4\right ) \sin (c+d x)+195 b^6\right )}{15 b^2 d}-\frac {2 \left (\frac {2 a \left (32 a^6-189 a^4 b^2+570 a^2 b^4+1635 b^6\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 \left (32 a^8-197 a^6 b^2+615 a^4 b^4-255 a^2 b^6-195 b^8\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \sin (c+d x)}}\right )}{15 b^2}\right )}{7 b^2}\right )-\frac {2 \left (3 a^2+13 b^2\right ) \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\right )-\frac {2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}\right )-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}\) |
(-2*Cos[c + d*x]^5*(a + b*Sin[c + d*x])^(5/2))/(15*d) + ((-2*a*Cos[c + d*x ]^5*(a + b*Sin[c + d*x])^(3/2))/(13*d) + ((-2*(3*a^2 + 13*b^2)*Cos[c + d*x ]^5*Sqrt[a + b*Sin[c + d*x]])/(11*d) + ((-2*Cos[c + d*x]^3*Sqrt[a + b*Sin[ c + d*x]]*(8*a^4 - 33*a^2*b^2 - 39*b^4 - 7*a*b*(a^2 + 63*b^2)*Sin[c + d*x] ))/(21*b^2*d) - (2*((-2*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(32*a^6 - 16 5*a^4*b^2 + 450*a^2*b^4 + 195*b^6 - 24*a*b*(a^4 - 5*a^2*b^2 - 60*b^4)*Sin[ c + d*x]))/(15*b^2*d) - (2*((2*a*(32*a^6 - 189*a^4*b^2 + 570*a^2*b^4 + 163 5*b^6)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x ]])/(b*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (2*(32*a^8 - 197*a^6*b^2 + 615*a^4*b^4 - 255*a^2*b^6 - 195*b^8)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/( a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(b*d*Sqrt[a + b*Sin[c + d*x]]) ))/(15*b^2)))/(7*b^2))/11)/13)/3
3.12.60.3.1 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[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)* (g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(f*g*(m + p + 1))), x] + S imp[1/(m + p + 1) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1)*Sim p[a*c*(m + p + 1) + b*d*m + (a*d*m + b*c*(m + p + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && !LtQ[p, -1] && IntegerQ[2*m] && !(EqQ[m, 1] && NeQ[c^2 - d^2, 0] && S implerQ[c + d*x, a + b*x])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g*(g* Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c*(m + p + 1) - a*d* p + b*d*(m + p)*Sin[e + f*x])/(b^2*f*(m + p)*(m + p + 1))), x] + Simp[g^2*( (p - 1)/(b^2*(m + p)*(m + p + 1))) Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Si n[e + f*x])^m*Simp[b*(a*d*m + b*c*(m + p + 1)) + (a*b*c*(m + p + 1) - d*(a^ 2*p - b^2*(m + p)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && GtQ[p, 1] && NeQ[m + p, 0] && NeQ[m + p + 1 , 0] && IntegerQ[2*m]
Leaf count of result is larger than twice the leaf count of optimal. \(1800\) vs. \(2(485)=970\).
Time = 38.12 (sec) , antiderivative size = 1801, normalized size of antiderivative = 3.99
2/45045*(-780*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/ 2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2 ),((a-b)/(a+b))^(1/2))*b^9-128*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c )-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticE(((a+b*sin(d* x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^9-370*a^5*b^4+64*a^7*b^2+128*((a +b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c ))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^( 1/2))*a^8*b-96*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1 /2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/ 2),((a-b)/(a+b))^(1/2))*a^7*b^2-788*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin( d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b*s in(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^6*b^3+576*((a+b*sin(d*x+c)) /(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1 /2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^5*b^4+ 2460*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+s in(d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/ (a+b))^(1/2))*a^4*b^5+5280*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1) *b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c) )/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^3*b^6-1020*((a+b*sin(d*x+c))/(a-b))^ (1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*E...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.19 (sec) , antiderivative size = 688, normalized size of antiderivative = 1.53 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=-\frac {2 \, {\left (2 \, \sqrt {2} {\left (64 \, a^{8} - 402 \, a^{6} b^{2} + 1275 \, a^{4} b^{4} - 2400 \, a^{2} b^{6} - 585 \, b^{8}\right )} \sqrt {i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right ) + 2 \, \sqrt {2} {\left (64 \, a^{8} - 402 \, a^{6} b^{2} + 1275 \, a^{4} b^{4} - 2400 \, a^{2} b^{6} - 585 \, b^{8}\right )} \sqrt {-i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right ) + 6 \, \sqrt {2} {\left (32 i \, a^{7} b - 189 i \, a^{5} b^{3} + 570 i \, a^{3} b^{5} + 1635 i \, a b^{7}\right )} \sqrt {i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right )\right ) + 6 \, \sqrt {2} {\left (-32 i \, a^{7} b + 189 i \, a^{5} b^{3} - 570 i \, a^{3} b^{5} - 1635 i \, a b^{7}\right )} \sqrt {-i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right )\right ) - 3 \, {\left (3003 \, b^{8} \cos \left (d x + c\right )^{7} - 21 \, {\left (213 \, a^{2} b^{6} + 208 \, b^{8}\right )} \cos \left (d x + c\right )^{5} - 5 \, {\left (8 \, a^{4} b^{4} - 33 \, a^{2} b^{6} - 39 \, b^{8}\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (32 \, a^{6} b^{2} - 165 \, a^{4} b^{4} + 450 \, a^{2} b^{6} + 195 \, b^{8}\right )} \cos \left (d x + c\right ) - {\left (7161 \, a b^{7} \cos \left (d x + c\right )^{5} - 35 \, {\left (a^{3} b^{5} + 63 \, a b^{7}\right )} \cos \left (d x + c\right )^{3} + 48 \, {\left (a^{5} b^{3} - 5 \, a^{3} b^{5} - 60 \, a b^{7}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}\right )}}{135135 \, b^{6} d} \]
-2/135135*(2*sqrt(2)*(64*a^8 - 402*a^6*b^2 + 1275*a^4*b^4 - 2400*a^2*b^6 - 585*b^8)*sqrt(I*b)*weierstrassPInverse(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8 *I*a^3 - 9*I*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) - 3*I*b*sin(d*x + c) - 2*I* a)/b) + 2*sqrt(2)*(64*a^8 - 402*a^6*b^2 + 1275*a^4*b^4 - 2400*a^2*b^6 - 58 5*b^8)*sqrt(-I*b)*weierstrassPInverse(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(-8* I*a^3 + 9*I*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*I*a )/b) + 6*sqrt(2)*(32*I*a^7*b - 189*I*a^5*b^3 + 570*I*a^3*b^5 + 1635*I*a*b^ 7)*sqrt(I*b)*weierstrassZeta(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*I*a^3 - 9* I*a*b^2)/b^3, weierstrassPInverse(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*I*a^3 - 9*I*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) - 3*I*b*sin(d*x + c) - 2*I*a)/b)) + 6*sqrt(2)*(-32*I*a^7*b + 189*I*a^5*b^3 - 570*I*a^3*b^5 - 1635*I*a*b^7)* sqrt(-I*b)*weierstrassZeta(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(-8*I*a^3 + 9*I *a*b^2)/b^3, weierstrassPInverse(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(-8*I*a^3 + 9*I*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*I*a)/b)) - 3*(3003*b^8*cos(d*x + c)^7 - 21*(213*a^2*b^6 + 208*b^8)*cos(d*x + c)^5 - 5*(8*a^4*b^4 - 33*a^2*b^6 - 39*b^8)*cos(d*x + c)^3 + 2*(32*a^6*b^2 - 165 *a^4*b^4 + 450*a^2*b^6 + 195*b^8)*cos(d*x + c) - (7161*a*b^7*cos(d*x + c)^ 5 - 35*(a^3*b^5 + 63*a*b^7)*cos(d*x + c)^3 + 48*(a^5*b^3 - 5*a^3*b^5 - 60* a*b^7)*cos(d*x + c))*sin(d*x + c))*sqrt(b*sin(d*x + c) + a))/(b^6*d)
Timed out. \[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]
\[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) \,d x } \]
\[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) \,d x } \]
Timed out. \[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\int {\cos \left (c+d\,x\right )}^4\,\sin \left (c+d\,x\right )\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{5/2} \,d x \]