Integrand size = 31, antiderivative size = 411 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=-\frac {2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}+\frac {2 \left (11 a^2-3 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {8 \left (32 a^2-11 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{21 b^5 d}+\frac {8 \left (24 a^2-7 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{21 a b^4 d}-\frac {2 \left (80 a^2-21 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{21 a^2 b^3 d}-\frac {16 a \left (32 a^2-15 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{21 b^6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {8 \left (64 a^4-46 a^2 b^2+3 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{21 b^6 d \sqrt {a+b \sin (c+d x)}} \]
-2/3*(a^2-b^2)*cos(d*x+c)*sin(d*x+c)^3/a/b^2/d/(a+b*sin(d*x+c))^(3/2)+2/3* (11*a^2-3*b^2)*cos(d*x+c)*sin(d*x+c)^3/a^2/b^2/d/(a+b*sin(d*x+c))^(1/2)-8/ 21*(32*a^2-11*b^2)*cos(d*x+c)*(a+b*sin(d*x+c))^(1/2)/b^5/d+8/21*(24*a^2-7* b^2)*cos(d*x+c)*sin(d*x+c)*(a+b*sin(d*x+c))^(1/2)/a/b^4/d-2/21*(80*a^2-21* b^2)*cos(d*x+c)*sin(d*x+c)^2*(a+b*sin(d*x+c))^(1/2)/a^2/b^3/d+16/21*a*(32* a^2-15*b^2)*(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^6/d/((a+b*sin(d*x+c))/(a+b))^(1/2)-8/21*(64*a^4-46*a^2*b^2+3*b ^4)*(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)*Elliptic F(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^6/d/(a+b*sin(d*x+c))^(1/2)
Time = 5.96 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.63 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\frac {32 a (a+b)^2 \left (32 a^2-15 b^2\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \left (\frac {a+b \sin (c+d x)}{a+b}\right )^{3/2}-16 (a+b) \left (64 a^4-46 a^2 b^2+3 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \left (\frac {a+b \sin (c+d x)}{a+b}\right )^{3/2}-\frac {1}{2} b \cos (c+d x) \left (1024 a^4-288 a^2 b^2-27 b^4-8 \left (8 a^2 b^2-3 b^4\right ) \cos (2 (c+d x))+3 b^4 \cos (4 (c+d x))+1280 a^3 b \sin (c+d x)-516 a b^3 \sin (c+d x)+12 a b^3 \sin (3 (c+d x))\right )}{42 b^6 d (a+b \sin (c+d x))^{3/2}} \]
(32*a*(a + b)^2*(32*a^2 - 15*b^2)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/( a + b)]*((a + b*Sin[c + d*x])/(a + b))^(3/2) - 16*(a + b)*(64*a^4 - 46*a^2 *b^2 + 3*b^4)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*((a + b*Sin[ c + d*x])/(a + b))^(3/2) - (b*Cos[c + d*x]*(1024*a^4 - 288*a^2*b^2 - 27*b^ 4 - 8*(8*a^2*b^2 - 3*b^4)*Cos[2*(c + d*x)] + 3*b^4*Cos[4*(c + d*x)] + 1280 *a^3*b*Sin[c + d*x] - 516*a*b^3*Sin[c + d*x] + 12*a*b^3*Sin[3*(c + d*x)])) /2)/(42*b^6*d*(a + b*Sin[c + d*x])^(3/2))
Time = 2.45 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.06, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.677, Rules used = {3042, 3370, 27, 3042, 3528, 27, 3042, 3528, 27, 3042, 3502, 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.
| \(\displaystyle \int \frac {\sin ^2(c+d x) \cos ^4(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^2 \cos (c+d x)^4}{(a+b \sin (c+d x))^{5/2}}dx\) |
| \(\Big \downarrow \) 3370 |
| \(\displaystyle -\frac {4 \int \frac {\sin ^2(c+d x) \left (-\left (\left (80 a^2-21 b^2\right ) \sin ^2(c+d x)\right )-2 a b \sin (c+d x)+15 \left (4 a^2-b^2\right )\right )}{4 \sqrt {a+b \sin (c+d x)}}dx}{3 a^2 b^2}+\frac {2 \left (11 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {\sin ^2(c+d x) \left (-\left (\left (80 a^2-21 b^2\right ) \sin ^2(c+d x)\right )-2 a b \sin (c+d x)+15 \left (4 a^2-b^2\right )\right )}{\sqrt {a+b \sin (c+d x)}}dx}{3 a^2 b^2}+\frac {2 \left (11 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {\sin (c+d x)^2 \left (-\left (\left (80 a^2-21 b^2\right ) \sin (c+d x)^2\right )-2 a b \sin (c+d x)+15 \left (4 a^2-b^2\right )\right )}{\sqrt {a+b \sin (c+d x)}}dx}{3 a^2 b^2}+\frac {2 \left (11 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle -\frac {\frac {2 \int -\frac {2 \sin (c+d x) \left (-5 b \sin (c+d x) a^2-5 \left (24 a^2-7 b^2\right ) \sin ^2(c+d x) a+\left (80 a^2-21 b^2\right ) a\right )}{\sqrt {a+b \sin (c+d x)}}dx}{7 b}+\frac {2 \left (80 a^2-21 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}}{3 a^2 b^2}+\frac {2 \left (11 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-21 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \int \frac {\sin (c+d x) \left (-5 b \sin (c+d x) a^2-5 \left (24 a^2-7 b^2\right ) \sin ^2(c+d x) a+\left (80 a^2-21 b^2\right ) a\right )}{\sqrt {a+b \sin (c+d x)}}dx}{7 b}}{3 a^2 b^2}+\frac {2 \left (11 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-21 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \int \frac {\sin (c+d x) \left (-5 b \sin (c+d x) a^2-5 \left (24 a^2-7 b^2\right ) \sin (c+d x)^2 a+\left (80 a^2-21 b^2\right ) a\right )}{\sqrt {a+b \sin (c+d x)}}dx}{7 b}}{3 a^2 b^2}+\frac {2 \left (11 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-21 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {2 \int -\frac {5 \left (-8 b \sin (c+d x) a^3-3 \left (32 a^2-11 b^2\right ) \sin ^2(c+d x) a^2+2 \left (24 a^2-7 b^2\right ) a^2\right )}{2 \sqrt {a+b \sin (c+d x)}}dx}{5 b}+\frac {2 a \left (24 a^2-7 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}\right )}{7 b}}{3 a^2 b^2}+\frac {2 \left (11 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-21 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {2 a \left (24 a^2-7 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}-\frac {\int \frac {-8 b \sin (c+d x) a^3-3 \left (32 a^2-11 b^2\right ) \sin ^2(c+d x) a^2+2 \left (24 a^2-7 b^2\right ) a^2}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{7 b}}{3 a^2 b^2}+\frac {2 \left (11 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-21 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {2 a \left (24 a^2-7 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}-\frac {\int \frac {-8 b \sin (c+d x) a^3-3 \left (32 a^2-11 b^2\right ) \sin (c+d x)^2 a^2+2 \left (24 a^2-7 b^2\right ) a^2}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{7 b}}{3 a^2 b^2}+\frac {2 \left (11 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-21 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {2 a \left (24 a^2-7 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}-\frac {\frac {2 \int \frac {3 \left (2 \left (32 a^2-15 b^2\right ) \sin (c+d x) a^3+b \left (16 a^2-3 b^2\right ) a^2\right )}{2 \sqrt {a+b \sin (c+d x)}}dx}{3 b}+\frac {2 a^2 \left (32 a^2-11 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{b}\right )}{7 b}}{3 a^2 b^2}+\frac {2 \left (11 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-21 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {2 a \left (24 a^2-7 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}-\frac {\frac {\int \frac {2 \left (32 a^2-15 b^2\right ) \sin (c+d x) a^3+b \left (16 a^2-3 b^2\right ) a^2}{\sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 a^2 \left (32 a^2-11 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{b}\right )}{7 b}}{3 a^2 b^2}+\frac {2 \left (11 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-21 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {2 a \left (24 a^2-7 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}-\frac {\frac {\int \frac {2 \left (32 a^2-15 b^2\right ) \sin (c+d x) a^3+b \left (16 a^2-3 b^2\right ) a^2}{\sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 a^2 \left (32 a^2-11 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{b}\right )}{7 b}}{3 a^2 b^2}+\frac {2 \left (11 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-21 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {2 a \left (24 a^2-7 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}-\frac {\frac {\frac {2 a^3 \left (32 a^2-15 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}-\frac {a^2 \left (64 a^4-46 a^2 b^2+3 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}}{b}+\frac {2 a^2 \left (32 a^2-11 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{b}\right )}{7 b}}{3 a^2 b^2}+\frac {2 \left (11 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-21 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {2 a \left (24 a^2-7 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}-\frac {\frac {\frac {2 a^3 \left (32 a^2-15 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}-\frac {a^2 \left (64 a^4-46 a^2 b^2+3 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}}{b}+\frac {2 a^2 \left (32 a^2-11 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{b}\right )}{7 b}}{3 a^2 b^2}+\frac {2 \left (11 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-21 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {2 a \left (24 a^2-7 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}-\frac {\frac {\frac {2 a^3 \left (32 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {a^2 \left (64 a^4-46 a^2 b^2+3 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}}{b}+\frac {2 a^2 \left (32 a^2-11 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{b}\right )}{7 b}}{3 a^2 b^2}+\frac {2 \left (11 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-21 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {2 a \left (24 a^2-7 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}-\frac {\frac {\frac {2 a^3 \left (32 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {a^2 \left (64 a^4-46 a^2 b^2+3 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}}{b}+\frac {2 a^2 \left (32 a^2-11 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{b}\right )}{7 b}}{3 a^2 b^2}+\frac {2 \left (11 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-21 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {2 a \left (24 a^2-7 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}-\frac {\frac {\frac {4 a^3 \left (32 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {a^2 \left (64 a^4-46 a^2 b^2+3 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}}{b}+\frac {2 a^2 \left (32 a^2-11 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{b}\right )}{7 b}}{3 a^2 b^2}+\frac {2 \left (11 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-21 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {2 a \left (24 a^2-7 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}-\frac {\frac {\frac {4 a^3 \left (32 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {a^2 \left (64 a^4-46 a^2 b^2+3 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{b \sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 a^2 \left (32 a^2-11 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{b}\right )}{7 b}}{3 a^2 b^2}+\frac {2 \left (11 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-21 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {2 a \left (24 a^2-7 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}-\frac {\frac {\frac {4 a^3 \left (32 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {a^2 \left (64 a^4-46 a^2 b^2+3 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{b \sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 a^2 \left (32 a^2-11 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{b}\right )}{7 b}}{3 a^2 b^2}+\frac {2 \left (11 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {2 \left (11 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}-\frac {\frac {2 \left (80 a^2-21 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {2 a \left (24 a^2-7 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}-\frac {\frac {2 a^2 \left (32 a^2-11 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}+\frac {\frac {4 a^3 \left (32 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 a^2 \left (64 a^4-46 a^2 b^2+3 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \sin (c+d x)}}}{b}}{b}\right )}{7 b}}{3 a^2 b^2}\) |
(-2*(a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^3)/(3*a*b^2*d*(a + b*Sin[c + d*x ])^(3/2)) + (2*(11*a^2 - 3*b^2)*Cos[c + d*x]*Sin[c + d*x]^3)/(3*a^2*b^2*d* Sqrt[a + b*Sin[c + d*x]]) - ((2*(80*a^2 - 21*b^2)*Cos[c + d*x]*Sin[c + d*x ]^2*Sqrt[a + b*Sin[c + d*x]])/(7*b*d) - (4*((2*a*(24*a^2 - 7*b^2)*Cos[c + d*x]*Sin[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(b*d) - ((2*a^2*(32*a^2 - 11*b ^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(b*d) + ((4*a^3*(32*a^2 - 15*b^ 2)*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*a^2*(64*a^4 - 46*a^2*b^2 + 3 *b^4)*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]]))/b)/b))/(7*b))/(3*a^2*b^2)
3.12.84.3.1 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_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(a^2 - b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((d*Sin[e + f*x])^(n + 1)/(a*b^2*d*f*(m + 1))), x] + (Simp[(a^2*(n - m + 1) - b^2*(m + n + 2))*Cos[e + f*x]*(a + b*S in[e + f*x])^(m + 2)*((d*Sin[e + f*x])^(n + 1)/(a^2*b^2*d*f*(m + 1)*(m + 2) )), x] - Simp[1/(a^2*b^2*(m + 1)*(m + 2)) Int[(a + b*Sin[e + f*x])^(m + 2 )*(d*Sin[e + f*x])^n*Simp[a^2*(n + 1)*(n + 3) - b^2*(m + n + 2)*(m + n + 3) + a*b*(m + 2)*Sin[e + f*x] - (a^2*(n + 2)*(n + 3) - b^2*(m + n + 2)*(m + n + 4))*Sin[e + f*x]^2, x], x], x]) /; FreeQ[{a, b, d, e, f, n}, x] && NeQ[a ^2 - b^2, 0] && IntegersQ[2*m, 2*n] && LtQ[m, -1] && !LtQ[n, -1] && (LtQ[m , -2] || EqQ[m + n + 4, 0])
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_)])^(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])))
Leaf count of result is larger than twice the leaf count of optimal. \(1641\) vs. \(2(445)=890\).
Time = 1.94 (sec) , antiderivative size = 1642, normalized size of antiderivative = 4.00
-2/21*(3*b^6*cos(d*x+c)^6+6*a*b^5*cos(d*x+c)^4*sin(d*x+c)+(-16*a^2*b^4+3*b ^6)*cos(d*x+c)^4+(160*a^3*b^3-66*a*b^5)*cos(d*x+c)^2*sin(d*x+c)+(128*a^4*b ^2-28*a^2*b^4-6*b^6)*cos(d*x+c)^2+4*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*(b /(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*b*(64 *EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^4*b-4 8*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^3*b^ 2-46*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^2 *b^3+27*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))* a*b^4+3*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))* b^5-64*EllipticE((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a ^5+94*EllipticE((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^ 3*b^2-30*EllipticE((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2)) *a*b^4)*sin(d*x+c)+256*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*(-b/(a-b)*sin(d *x+c)-b/(a-b))^(1/2)*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/( a+b))^(1/2))*(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*a^5*b-192*(-b/(a+b)*sin(d* x+c)+b/(a+b))^(1/2)*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*EllipticF((b/(a-b) *sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*(b/(a-b)*sin(d*x+c)+a/(a-b ))^(1/2)*a^4*b^2-184*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*(-b/(a-b)*sin(d*x +c)-b/(a-b))^(1/2)*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+ b))^(1/2))*(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*a^3*b^3+108*(-b/(a+b)*sin...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.24 (sec) , antiderivative size = 873, normalized size of antiderivative = 2.12 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\text {Too large to display} \]
2/63*(2*(sqrt(2)*(128*a^4*b^2 - 108*a^2*b^4 + 9*b^6)*cos(d*x + c)^2 - 2*sq rt(2)*(128*a^5*b - 108*a^3*b^3 + 9*a*b^5)*sin(d*x + c) - sqrt(2)*(128*a^6 + 20*a^4*b^2 - 99*a^2*b^4 + 9*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) + 2*(sqrt(2)*(128*a^4*b^2 - 108*a^2*b^4 + 9*b^6)*cos(d*x + c)^2 - 2*sqrt(2)*(128*a^5*b - 108*a^3*b^3 + 9*a*b^5)*sin( d*x + c) - sqrt(2)*(128*a^6 + 20*a^4*b^2 - 99*a^2*b^4 + 9*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) + 12*(sqrt(2 )*(32*I*a^3*b^3 - 15*I*a*b^5)*cos(d*x + c)^2 + 2*sqrt(2)*(-32*I*a^4*b^2 + 15*I*a^2*b^4)*sin(d*x + c) + sqrt(2)*(-32*I*a^5*b - 17*I*a^3*b^3 + 15*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)) + 12*(sqrt(2)*(-32*I*a^3*b^3 + 15*I*a*b^5)*cos(d*x + c)^2 + 2*sqrt(2)* (32*I*a^4*b^2 - 15*I*a^2*b^4)*sin(d*x + c) + sqrt(2)*(32*I*a^5*b + 17*I*a^ 3*b^3 - 15*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*(3*b^6*cos(d*x + c)^5 - (16*a^2*b^4 - 3*b^6)*...
Timed out. \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\int { \frac {\cos \left (d x + c\right )^{4} \sin \left (d x + c\right )^{2}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4\,{\sin \left (c+d\,x\right )}^2}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \]