Integrand size = 23, antiderivative size = 329 \[ \int \cos ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=-\frac {2 b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}-\frac {32 a \left (a^4-6 a^2 b^2-27 b^4\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{1155 b^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {8 \left (4 a^6-25 a^4 b^2+6 a^2 b^4+15 b^6\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{1155 b^4 d \sqrt {a+b \sin (c+d x)}}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (a^2+3 b^2+28 a b \sin (c+d x)\right )}{231 b d}-\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a^4-21 a^2 b^2-15 b^4-3 a b \left (a^2+31 b^2\right ) \sin (c+d x)\right )}{1155 b^3 d} \]
-2/11*b*cos(d*x+c)^5*(a+b*sin(d*x+c))^(1/2)/d+2/231*cos(d*x+c)^3*(a^2+3*b^ 2+28*a*b*sin(d*x+c))*(a+b*sin(d*x+c))^(1/2)/b/d-4/1155*cos(d*x+c)*(4*a^4-2 1*a^2*b^2-15*b^4-3*a*b*(a^2+31*b^2)*sin(d*x+c))*(a+b*sin(d*x+c))^(1/2)/b^3 /d+32/1155*a*(a^4-6*a^2*b^2-27*b^4)*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/si n(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^4/d/((a+b*sin(d*x+c))/(a+b))^(1/2)-8/1 155*(4*a^6-25*a^4*b^2+6*a^2*b^4+15*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)*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^4/d/(a+b*sin(d*x+c))^(1/2)
Time = 0.79 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.84 \[ \int \cos ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\frac {64 \left (b^2 \left (a^4-114 a^2 b^2-15 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right )+4 \left (a^5-6 a^3 b^2-27 a b^4\right ) \left ((a+b) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right )\right )\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}-b (a+b \sin (c+d x)) \left (2 \left (64 a^4-366 a^2 b^2+195 b^4\right ) \cos (c+d x)+5 b^2 \left (-4 a^2+93 b^2\right ) \cos (3 (c+d x))+105 b^4 \cos (5 (c+d x))-16 a b \left (3 a^2+128 b^2\right ) \sin (2 (c+d x))-280 a b^3 \sin (4 (c+d x))\right )}{9240 b^4 d \sqrt {a+b \sin (c+d x)}} \]
(64*(b^2*(a^4 - 114*a^2*b^2 - 15*b^4)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2* b)/(a + b)] + 4*(a^5 - 6*a^3*b^2 - 27*a*b^4)*((a + b)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] - a*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]))*Sqrt[(a + b*Sin[c + d*x])/(a + b)] - b*(a + b*Sin[c + d*x])*(2*(64 *a^4 - 366*a^2*b^2 + 195*b^4)*Cos[c + d*x] + 5*b^2*(-4*a^2 + 93*b^2)*Cos[3 *(c + d*x)] + 105*b^4*Cos[5*(c + d*x)] - 16*a*b*(3*a^2 + 128*b^2)*Sin[2*(c + d*x)] - 280*a*b^3*Sin[4*(c + d*x)]))/(9240*b^4*d*Sqrt[a + b*Sin[c + d*x ]])
Time = 1.76 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.03, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.783, Rules used = {3042, 3171, 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.
| \(\displaystyle \int \cos ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (c+d x)^4 (a+b \sin (c+d x))^{3/2}dx\) |
| \(\Big \downarrow \) 3171 |
| \(\displaystyle \frac {2}{11} \int \frac {\cos ^4(c+d x) \left (11 a^2+12 b \sin (c+d x) a+b^2\right )}{2 \sqrt {a+b \sin (c+d x)}}dx-\frac {2 b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \int \frac {\cos ^4(c+d x) \left (11 a^2+12 b \sin (c+d x) a+b^2\right )}{\sqrt {a+b \sin (c+d x)}}dx-\frac {2 b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \int \frac {\cos (c+d x)^4 \left (11 a^2+12 b \sin (c+d x) a+b^2\right )}{\sqrt {a+b \sin (c+d x)}}dx-\frac {2 b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\) |
| \(\Big \downarrow \) 3344 |
| \(\displaystyle \frac {1}{11} \left (\frac {4 \int \frac {3 \cos ^2(c+d x) \left (\left (29 a^2+3 b^2\right ) b^2+a \left (a^2+31 b^2\right ) \sin (c+d x) b\right )}{2 \sqrt {a+b \sin (c+d x)}}dx}{21 b^2}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (a^2+28 a b \sin (c+d x)+3 b^2\right )}{21 b d}\right )-\frac {2 b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {2 \int \frac {\cos ^2(c+d x) \left (\left (29 a^2+3 b^2\right ) b^2+a \left (a^2+31 b^2\right ) \sin (c+d x) b\right )}{\sqrt {a+b \sin (c+d x)}}dx}{7 b^2}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (a^2+28 a b \sin (c+d x)+3 b^2\right )}{21 b d}\right )-\frac {2 b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {2 \int \frac {\cos (c+d x)^2 \left (\left (29 a^2+3 b^2\right ) b^2+a \left (a^2+31 b^2\right ) \sin (c+d x) b\right )}{\sqrt {a+b \sin (c+d x)}}dx}{7 b^2}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (a^2+28 a b \sin (c+d x)+3 b^2\right )}{21 b d}\right )-\frac {2 b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\) |
| \(\Big \downarrow \) 3344 |
| \(\displaystyle \frac {1}{11} \left (\frac {2 \left (\frac {4 \int -\frac {\left (a^4-114 b^2 a^2-15 b^4\right ) b^2+4 a \left (a^4-6 b^2 a^2-27 b^4\right ) \sin (c+d x) b}{2 \sqrt {a+b \sin (c+d x)}}dx}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a^4-3 a b \left (a^2+31 b^2\right ) \sin (c+d x)-21 a^2 b^2-15 b^4\right )}{15 b d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (a^2+28 a b \sin (c+d x)+3 b^2\right )}{21 b d}\right )-\frac {2 b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {2 \left (-\frac {2 \int \frac {\left (a^4-114 b^2 a^2-15 b^4\right ) b^2+4 a \left (a^4-6 b^2 a^2-27 b^4\right ) \sin (c+d x) b}{\sqrt {a+b \sin (c+d x)}}dx}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a^4-3 a b \left (a^2+31 b^2\right ) \sin (c+d x)-21 a^2 b^2-15 b^4\right )}{15 b d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (a^2+28 a b \sin (c+d x)+3 b^2\right )}{21 b d}\right )-\frac {2 b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {2 \left (-\frac {2 \int \frac {\left (a^4-114 b^2 a^2-15 b^4\right ) b^2+4 a \left (a^4-6 b^2 a^2-27 b^4\right ) \sin (c+d x) b}{\sqrt {a+b \sin (c+d x)}}dx}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a^4-3 a b \left (a^2+31 b^2\right ) \sin (c+d x)-21 a^2 b^2-15 b^4\right )}{15 b d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (a^2+28 a b \sin (c+d x)+3 b^2\right )}{21 b d}\right )-\frac {2 b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {1}{11} \left (\frac {2 \left (-\frac {2 \left (4 a \left (a^4-6 a^2 b^2-27 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx-\left (4 a^6-25 a^4 b^2+6 a^2 b^4+15 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a^4-3 a b \left (a^2+31 b^2\right ) \sin (c+d x)-21 a^2 b^2-15 b^4\right )}{15 b d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (a^2+28 a b \sin (c+d x)+3 b^2\right )}{21 b d}\right )-\frac {2 b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {2 \left (-\frac {2 \left (4 a \left (a^4-6 a^2 b^2-27 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx-\left (4 a^6-25 a^4 b^2+6 a^2 b^4+15 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a^4-3 a b \left (a^2+31 b^2\right ) \sin (c+d x)-21 a^2 b^2-15 b^4\right )}{15 b d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (a^2+28 a b \sin (c+d x)+3 b^2\right )}{21 b d}\right )-\frac {2 b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {1}{11} \left (\frac {2 \left (-\frac {2 \left (\frac {4 a \left (a^4-6 a^2 b^2-27 b^4\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\left (4 a^6-25 a^4 b^2+6 a^2 b^4+15 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a^4-3 a b \left (a^2+31 b^2\right ) \sin (c+d x)-21 a^2 b^2-15 b^4\right )}{15 b d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (a^2+28 a b \sin (c+d x)+3 b^2\right )}{21 b d}\right )-\frac {2 b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {2 \left (-\frac {2 \left (\frac {4 a \left (a^4-6 a^2 b^2-27 b^4\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\left (4 a^6-25 a^4 b^2+6 a^2 b^4+15 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a^4-3 a b \left (a^2+31 b^2\right ) \sin (c+d x)-21 a^2 b^2-15 b^4\right )}{15 b d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (a^2+28 a b \sin (c+d x)+3 b^2\right )}{21 b d}\right )-\frac {2 b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {1}{11} \left (\frac {2 \left (-\frac {2 \left (\frac {8 a \left (a^4-6 a^2 b^2-27 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\left (4 a^6-25 a^4 b^2+6 a^2 b^4+15 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a^4-3 a b \left (a^2+31 b^2\right ) \sin (c+d x)-21 a^2 b^2-15 b^4\right )}{15 b d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (a^2+28 a b \sin (c+d x)+3 b^2\right )}{21 b d}\right )-\frac {2 b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {1}{11} \left (\frac {2 \left (-\frac {2 \left (\frac {8 a \left (a^4-6 a^2 b^2-27 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (4 a^6-25 a^4 b^2+6 a^2 b^4+15 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a^4-3 a b \left (a^2+31 b^2\right ) \sin (c+d x)-21 a^2 b^2-15 b^4\right )}{15 b d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (a^2+28 a b \sin (c+d x)+3 b^2\right )}{21 b d}\right )-\frac {2 b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {2 \left (-\frac {2 \left (\frac {8 a \left (a^4-6 a^2 b^2-27 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (4 a^6-25 a^4 b^2+6 a^2 b^4+15 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a^4-3 a b \left (a^2+31 b^2\right ) \sin (c+d x)-21 a^2 b^2-15 b^4\right )}{15 b d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (a^2+28 a b \sin (c+d x)+3 b^2\right )}{21 b d}\right )-\frac {2 b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {1}{11} \left (\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (a^2+28 a b \sin (c+d x)+3 b^2\right )}{21 b d}+\frac {2 \left (-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a^4-3 a b \left (a^2+31 b^2\right ) \sin (c+d x)-21 a^2 b^2-15 b^4\right )}{15 b d}-\frac {2 \left (\frac {8 a \left (a^4-6 a^2 b^2-27 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 \left (4 a^6-25 a^4 b^2+6 a^2 b^4+15 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}\right )}{15 b^2}\right )}{7 b^2}\right )-\frac {2 b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\) |
(-2*b*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]]*(a^2 + 3*b^2 + 28*a*b*Sin[c + d*x]))/(21*b*d) + (2*((-2*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(4*a^4 - 21*a^2*b^2 - 15*b^4 - 3*a*b*(a^2 + 31*b^2)*Sin[c + d*x]))/(15*b*d) - (2*((8*a*(a^4 - 6*a^2*b^ 2 - 27*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (2*(4*a^6 - 25*a^4*b^2 + 6*a^2*b^4 + 15*b^6)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(d*Sqrt[a + b*Sin[c + d*x]])))/(15*b^2)))/(7*b ^2))/11
3.5.89.3.1 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[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(-b)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m - 1)/(f*g*(m + p))), x] + Simp[1/(m + p) Int[(g*Cos[e + f*x])^p* (a + b*Sin[e + f*x])^(m - 2)*(b^2*(m - 1) + a^2*(m + p) + a*b*(2*m + p - 1) *Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && NeQ[m + p, 0] && (IntegersQ[2*m, 2*p] || IntegerQ[m])
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[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. \(1354\) vs. \(2(371)=742\).
Time = 3.72 (sec) , antiderivative size = 1355, normalized size of antiderivative = 4.12
-2/1155*(60*b^7*sin(d*x+c)+60*((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^7-16*((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)*Ellipti cE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^7+16*((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^6 *b-12*((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^2-100*((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^4*b^3-360*((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)*Ellip ticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^3*b^4+24*((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^2*b^5+372*((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*b^6+112*((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*...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 583, normalized size of antiderivative = 1.77 \[ \int \cos ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\frac {2 \, {\left (2 \, \sqrt {2} {\left (8 \, a^{6} - 51 \, a^{4} b^{2} + 126 \, a^{2} b^{4} + 45 \, b^{6}\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 (8 \, a^{6} - 51 \, a^{4} b^{2} + 126 \, a^{2} b^{4} + 45 \, b^{6}\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 ) - 24 \, \sqrt {2} {\left (-i \, a^{5} b + 6 i \, a^{3} b^{3} + 27 i \, a b^{5}\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 ) - 24 \, \sqrt {2} {\left (i \, a^{5} b - 6 i \, a^{3} b^{3} - 27 i \, a b^{5}\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 (105 \, b^{6} \cos \left (d x + c\right )^{5} - 5 \, {\left (a^{2} b^{4} + 3 \, b^{6}\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (4 \, a^{4} b^{2} - 21 \, a^{2} b^{4} - 15 \, b^{6}\right )} \cos \left (d x + c\right ) - 2 \, {\left (70 \, a b^{5} \cos \left (d x + c\right )^{3} + 3 \, {\left (a^{3} b^{3} + 31 \, a b^{5}\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 )}}{3465 \, b^{5} d} \]
2/3465*(2*sqrt(2)*(8*a^6 - 51*a^4*b^2 + 126*a^2*b^4 + 45*b^6)*sqrt(I*b)*we ierstrassPInverse(-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)*(8*a ^6 - 51*a^4*b^2 + 126*a^2*b^4 + 45*b^6)*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) - 24*sqrt(2)*(-I*a^5*b + 6*I*a^3*b^3 + 27*I*a*b^5)*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)) - 24*sqrt(2)*(I*a^5*b - 6*I*a^3*b^3 - 27*I*a*b^5)*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*(105*b ^6*cos(d*x + c)^5 - 5*(a^2*b^4 + 3*b^6)*cos(d*x + c)^3 + 2*(4*a^4*b^2 - 21 *a^2*b^4 - 15*b^6)*cos(d*x + c) - 2*(70*a*b^5*cos(d*x + c)^3 + 3*(a^3*b^3 + 31*a*b^5)*cos(d*x + c))*sin(d*x + c))*sqrt(b*sin(d*x + c) + a))/(b^5*d)
\[ \int \cos ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\int \left (a + b \sin {\left (c + d x \right )}\right )^{\frac {3}{2}} \cos ^{4}{\left (c + d x \right )}\, dx \]
\[ \int \cos ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{4} \,d x } \]
\[ \int \cos ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{4} \,d x } \]
Timed out. \[ \int \cos ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\int {\cos \left (c+d\,x\right )}^4\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2} \,d x \]