Integrand size = 31, antiderivative size = 405 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=-\frac {8 \left (160 a^4-247 a^2 b^2+45 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3465 b^5 d}+\frac {8 a \left (120 a^2-179 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{3465 b^4 d}-\frac {2 \left (80 a^2-117 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{693 b^3 d}+\frac {20 a \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{99 b^2 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+b \sin (c+d x)}}{11 b d}-\frac {16 a \left (160 a^4-267 a^2 b^2+69 b^4\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{3465 b^6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {8 \left (320 a^6-614 a^4 b^2+249 a^2 b^4+45 b^6\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{3465 b^6 d \sqrt {a+b \sin (c+d x)}} \] Output:
-8/3465*(160*a^4-247*a^2*b^2+45*b^4)*cos(d*x+c)*(a+b*sin(d*x+c))^(1/2)/b^5 /d+8/3465*a*(120*a^2-179*b^2)*cos(d*x+c)*sin(d*x+c)*(a+b*sin(d*x+c))^(1/2) /b^4/d-2/693*(80*a^2-117*b^2)*cos(d*x+c)*sin(d*x+c)^2*(a+b*sin(d*x+c))^(1/ 2)/b^3/d+20/99*a*cos(d*x+c)*sin(d*x+c)^3*(a+b*sin(d*x+c))^(1/2)/b^2/d-2/11 *cos(d*x+c)*sin(d*x+c)^4*(a+b*sin(d*x+c))^(1/2)/b/d+16/3465*a*(160*a^4-267 *a^2*b^2+69*b^4)*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)+8/3465*(32 0*a^6-614*a^4*b^2+249*a^2*b^4+45*b^6)*InverseJacobiAM(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 = 5.28 (sec) , antiderivative size = 326, normalized size of antiderivative = 0.80 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\frac {128 a \left (160 a^5+160 a^4 b-267 a^3 b^2-267 a^2 b^3+69 a b^4+69 b^5\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}-64 \left (320 a^6-614 a^4 b^2+249 a^2 b^4+45 b^6\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+b \cos (c+d x) \left (-10240 a^5+16448 a^3 b^2-3718 a b^4-128 \left (5 a^3 b^2-6 a b^4\right ) \cos (2 (c+d x))+70 a b^4 \cos (4 (c+d x))-2560 a^4 b \sin (c+d x)+3752 a^2 b^3 \sin (c+d x)+990 b^5 \sin (c+d x)+200 a^2 b^3 \sin (3 (c+d x))-765 b^5 \sin (3 (c+d x))-315 b^5 \sin (5 (c+d x))\right )}{27720 b^6 d \sqrt {a+b \sin (c+d x)}} \] Input:
Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^2)/Sqrt[a + b*Sin[c + d*x]],x]
Output:
(128*a*(160*a^5 + 160*a^4*b - 267*a^3*b^2 - 267*a^2*b^3 + 69*a*b^4 + 69*b^ 5)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x ])/(a + b)] - 64*(320*a^6 - 614*a^4*b^2 + 249*a^2*b^4 + 45*b^6)*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]*(-10240*a^5 + 16448*a^3*b^2 - 3718*a*b^4 - 128*(5*a^3*b^2 - 6*a*b^4)*Cos[2*(c + d*x)] + 70*a*b^4*Cos[4*(c + d*x)] - 2560*a^4*b*Sin[c + d*x] + 3752*a^2*b^3*Sin[c + d*x] + 990*b^5*Sin[c + d*x] + 200*a^2*b^3*S in[3*(c + d*x)] - 765*b^5*Sin[3*(c + d*x)] - 315*b^5*Sin[5*(c + d*x)]))/(2 7720*b^6*d*Sqrt[a + b*Sin[c + d*x]])
Time = 2.45 (sec) , antiderivative size = 428, 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, 3374, 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.
| \(\displaystyle \int \frac {\sin ^2(c+d x) \cos ^4(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^2 \cos (c+d x)^4}{\sqrt {a+b \sin (c+d x)}}dx\) |
| \(\Big \downarrow \) 3374 |
| \(\displaystyle -\frac {4 \int \frac {\sin ^2(c+d x) \left (-\left (\left (80 a^2-117 b^2\right ) \sin ^2(c+d x)\right )-2 a b \sin (c+d x)+3 \left (20 a^2-33 b^2\right )\right )}{4 \sqrt {a+b \sin (c+d x)}}dx}{99 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{99 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {\sin ^2(c+d x) \left (-\left (\left (80 a^2-117 b^2\right ) \sin ^2(c+d x)\right )-2 a b \sin (c+d x)+3 \left (20 a^2-33 b^2\right )\right )}{\sqrt {a+b \sin (c+d x)}}dx}{99 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{99 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {\sin (c+d x)^2 \left (-\left (\left (80 a^2-117 b^2\right ) \sin (c+d x)^2\right )-2 a b \sin (c+d x)+3 \left (20 a^2-33 b^2\right )\right )}{\sqrt {a+b \sin (c+d x)}}dx}{99 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{99 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle -\frac {\frac {2 \int -\frac {2 \sin (c+d x) \left (-a \left (120 a^2-179 b^2\right ) \sin ^2(c+d x)-b \left (5 a^2-27 b^2\right ) \sin (c+d x)+a \left (80 a^2-117 b^2\right )\right )}{\sqrt {a+b \sin (c+d x)}}dx}{7 b}+\frac {2 \left (80 a^2-117 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}}{99 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{99 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-117 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \int \frac {\sin (c+d x) \left (-a \left (120 a^2-179 b^2\right ) \sin ^2(c+d x)-b \left (5 a^2-27 b^2\right ) \sin (c+d x)+a \left (80 a^2-117 b^2\right )\right )}{\sqrt {a+b \sin (c+d x)}}dx}{7 b}}{99 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{99 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-117 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \int \frac {\sin (c+d x) \left (-a \left (120 a^2-179 b^2\right ) \sin (c+d x)^2-b \left (5 a^2-27 b^2\right ) \sin (c+d x)+a \left (80 a^2-117 b^2\right )\right )}{\sqrt {a+b \sin (c+d x)}}dx}{7 b}}{99 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{99 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-117 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {2 \int -\frac {2 \left (120 a^2-179 b^2\right ) a^2-8 b \left (5 a^2-6 b^2\right ) \sin (c+d x) a-3 \left (160 a^4-247 b^2 a^2+45 b^4\right ) \sin ^2(c+d x)}{2 \sqrt {a+b \sin (c+d x)}}dx}{5 b}+\frac {2 a \left (120 a^2-179 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}\right )}{7 b}}{99 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{99 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-117 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {2 a \left (120 a^2-179 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\int \frac {2 \left (120 a^2-179 b^2\right ) a^2-8 b \left (5 a^2-6 b^2\right ) \sin (c+d x) a-3 \left (160 a^4-247 b^2 a^2+45 b^4\right ) \sin ^2(c+d x)}{\sqrt {a+b \sin (c+d x)}}dx}{5 b}\right )}{7 b}}{99 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{99 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-117 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {2 a \left (120 a^2-179 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\int \frac {2 \left (120 a^2-179 b^2\right ) a^2-8 b \left (5 a^2-6 b^2\right ) \sin (c+d x) a-3 \left (160 a^4-247 b^2 a^2+45 b^4\right ) \sin (c+d x)^2}{\sqrt {a+b \sin (c+d x)}}dx}{5 b}\right )}{7 b}}{99 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{99 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-117 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {2 a \left (120 a^2-179 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {2 \int \frac {3 \left (b \left (80 a^4-111 b^2 a^2-45 b^4\right )+2 a \left (160 a^4-267 b^2 a^2+69 b^4\right ) \sin (c+d x)\right )}{2 \sqrt {a+b \sin (c+d x)}}dx}{3 b}+\frac {2 \left (160 a^4-247 a^2 b^2+45 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}}{99 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{99 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-117 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {2 a \left (120 a^2-179 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {\int \frac {b \left (80 a^4-111 b^2 a^2-45 b^4\right )+2 a \left (160 a^4-267 b^2 a^2+69 b^4\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 \left (160 a^4-247 a^2 b^2+45 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}}{99 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{99 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-117 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {2 a \left (120 a^2-179 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {\int \frac {b \left (80 a^4-111 b^2 a^2-45 b^4\right )+2 a \left (160 a^4-267 b^2 a^2+69 b^4\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 \left (160 a^4-247 a^2 b^2+45 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}}{99 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{99 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b d}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-117 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {2 a \left (120 a^2-179 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {\frac {2 a \left (160 a^4-267 a^2 b^2+69 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}-\frac {\left (320 a^6-614 a^4 b^2+249 a^2 b^4+45 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}}{b}+\frac {2 \left (160 a^4-247 a^2 b^2+45 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}}{99 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{99 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-117 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {2 a \left (120 a^2-179 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {\frac {2 a \left (160 a^4-267 a^2 b^2+69 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}-\frac {\left (320 a^6-614 a^4 b^2+249 a^2 b^4+45 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}}{b}+\frac {2 \left (160 a^4-247 a^2 b^2+45 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}}{99 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{99 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-117 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {2 a \left (120 a^2-179 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {\frac {2 a \left (160 a^4-267 a^2 b^2+69 b^4\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (320 a^6-614 a^4 b^2+249 a^2 b^4+45 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}}{b}+\frac {2 \left (160 a^4-247 a^2 b^2+45 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}}{99 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{99 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-117 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {2 a \left (120 a^2-179 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {\frac {2 a \left (160 a^4-267 a^2 b^2+69 b^4\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (320 a^6-614 a^4 b^2+249 a^2 b^4+45 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}}{b}+\frac {2 \left (160 a^4-247 a^2 b^2+45 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}}{99 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{99 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-117 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {2 a \left (120 a^2-179 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {\frac {4 a \left (160 a^4-267 a^2 b^2+69 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (320 a^6-614 a^4 b^2+249 a^2 b^4+45 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}}{b}+\frac {2 \left (160 a^4-247 a^2 b^2+45 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}}{99 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{99 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-117 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {2 a \left (120 a^2-179 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {\frac {4 a \left (160 a^4-267 a^2 b^2+69 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (320 a^6-614 a^4 b^2+249 a^2 b^4+45 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{b \sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 \left (160 a^4-247 a^2 b^2+45 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}}{99 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{99 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-117 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {2 a \left (120 a^2-179 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {\frac {4 a \left (160 a^4-267 a^2 b^2+69 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (320 a^6-614 a^4 b^2+249 a^2 b^4+45 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{b \sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 \left (160 a^4-247 a^2 b^2+45 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}}{99 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{99 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle -\frac {\frac {2 \left (80 a^2-117 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \left (\frac {2 a \left (120 a^2-179 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {2 \left (160 a^4-247 a^2 b^2+45 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}+\frac {\frac {4 a \left (160 a^4-267 a^2 b^2+69 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 \left (320 a^6-614 a^4 b^2+249 a^2 b^4+45 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \sin (c+d x)}}}{b}}{5 b}\right )}{7 b}}{99 b^2}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{99 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b d}\) |
Input:
Int[(Cos[c + d*x]^4*Sin[c + d*x]^2)/Sqrt[a + b*Sin[c + d*x]],x]
Output:
(20*a*Cos[c + d*x]*Sin[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]])/(99*b^2*d) - ( 2*Cos[c + d*x]*Sin[c + d*x]^4*Sqrt[a + b*Sin[c + d*x]])/(11*b*d) - ((2*(80 *a^2 - 117*b^2)*Cos[c + d*x]*Sin[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]])/(7*b *d) - (4*((2*a*(120*a^2 - 179*b^2)*Cos[c + d*x]*Sin[c + d*x]*Sqrt[a + b*Si n[c + d*x]])/(5*b*d) - ((2*(160*a^4 - 247*a^2*b^2 + 45*b^4)*Cos[c + d*x]*S qrt[a + b*Sin[c + d*x]])/(b*d) + ((4*a*(160*a^4 - 267*a^2*b^2 + 69*b^4)*El lipticE[(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*(320*a^6 - 614*a^4*b^2 + 249*a^2* b^4 + 45*b^6)*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)/(5*b)))/(7*b))/(99 *b^2)
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*(n + 3)*Cos[e + f* x]*(d*Sin[e + f*x])^(n + 1)*((a + b*Sin[e + f*x])^(m + 1)/(b^2*d*f*(m + n + 3)*(m + n + 4))), x] + (-Simp[Cos[e + f*x]*(d*Sin[e + f*x])^(n + 2)*((a + b*Sin[e + f*x])^(m + 1)/(b*d^2*f*(m + n + 4))), x] - Simp[1/(b^2*(m + n + 3 )*(m + n + 4)) Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^m*Simp[a^2*(n + 1)*(n + 3) - b^2*(m + n + 3)*(m + n + 4) + a*b*m*Sin[e + f*x] - (a^2*(n + 2)*(n + 3) - b^2*(m + n + 3)*(m + n + 5))*Sin[e + f*x]^2, x], x], x]) /; F reeQ[{a, b, d, e, f, m, n}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || Integ ersQ[2*m, 2*n]) && !m < -1 && !LtQ[n, -1] && NeQ[m + n + 3, 0] && NeQ[m + n + 4, 0]
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. \(1355\) vs. \(2(378)=756\).
Time = 1.48 (sec) , antiderivative size = 1356, normalized size of antiderivative = 3.35
Input:
int(cos(d*x+c)^4*sin(d*x+c)^2/(a+b*sin(d*x+c))^(1/2),x,method=_RETURNVERBO SE)
Output:
-2/3465*(180*a*b^6+3416*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/ (a+b))^(1/2)*(-b*(1+sin(d*x+c))/(a-b))^(1/2)*EllipticE(((a+b*sin(d*x+c))/( a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^5*b^2-2688*((a+b*sin(d*x+c))/(a-b))^(1/ 2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-b*(1+sin(d*x+c))/(a-b))^(1/2)*Ellipti cE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^3*b^4+552*((a+b*s in(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-b*(1+sin(d*x+c)) /(a-b))^(1/2)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2) )*a*b^6+1280*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2 )*(-b*(1+sin(d*x+c))/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2) ,((a-b)/(a+b))^(1/2))*a^6*b-960*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+ c)-1)*b/(a+b))^(1/2)*(-b*(1+sin(d*x+c))/(a-b))^(1/2)*EllipticF(((a+b*sin(d *x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^5*b^2-2456*((a+b*sin(d*x+c))/(a -b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-b*(1+sin(d*x+c))/(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+169 2*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-b*(1+si n(d*x+c))/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+ b))^(1/2))*a^3*b^4+996*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/( a+b))^(1/2)*(-b*(1+sin(d*x+c))/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a -b))^(1/2),((a-b)/(a+b))^(1/2))*a^2*b^5-732*((a+b*sin(d*x+c))/(a-b))^(1/2) *(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-b*(1+sin(d*x+c))/(a-b))^(1/2)*Ellipt...
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 573, normalized size of antiderivative = 1.41 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx =\text {Too large to display} \] Input:
integrate(cos(d*x+c)^4*sin(d*x+c)^2/(a+b*sin(d*x+c))^(1/2),x, algorithm="f ricas")
Output:
2/10395*(4*(640*a^6 - 1308*a^4*b^2 + 609*a^2*b^4 + 135*b^6)*sqrt(1/2*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) + 4*(640*a^6 - 1308*a^4*b^2 + 609*a^2*b^4 + 135*b^6)*sqrt(-1/2*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*(-160*I*a^5*b + 267*I*a^3*b^ 3 - 69*I*a*b^5)*sqrt(1/2*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*(160*I*a^5*b - 267*I*a^3*b^3 + 69*I*a*b^5)*sqrt(-1/ 2*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 *(315*b^6*cos(d*x + c)^5 - 5*(80*a^2*b^4 + 9*b^6)*cos(d*x + c)^3 + 2*(320* a^4*b^2 - 294*a^2*b^4 - 45*b^6)*cos(d*x + c) + 2*(175*a*b^5*cos(d*x + c)^3 - 3*(80*a^3*b^3 - 61*a*b^5)*cos(d*x + c))*sin(d*x + c))*sqrt(b*sin(d*x + c) + a))/(b^7*d)
\[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\int \frac {\sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{\sqrt {a + b \sin {\left (c + d x \right )}}}\, dx \] Input:
integrate(cos(d*x+c)**4*sin(d*x+c)**2/(a+b*sin(d*x+c))**(1/2),x)
Output:
Integral(sin(c + d*x)**2*cos(c + d*x)**4/sqrt(a + b*sin(c + d*x)), x)
\[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right )^{4} \sin \left (d x + c\right )^{2}}{\sqrt {b \sin \left (d x + c\right ) + a}} \,d x } \] Input:
integrate(cos(d*x+c)^4*sin(d*x+c)^2/(a+b*sin(d*x+c))^(1/2),x, algorithm="m axima")
Output:
integrate(cos(d*x + c)^4*sin(d*x + c)^2/sqrt(b*sin(d*x + c) + a), x)
Timed out. \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)^4*sin(d*x+c)^2/(a+b*sin(d*x+c))^(1/2),x, algorithm="g iac")
Output:
Timed out
Timed out. \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4\,{\sin \left (c+d\,x\right )}^2}{\sqrt {a+b\,\sin \left (c+d\,x\right )}} \,d x \] Input:
int((cos(c + d*x)^4*sin(c + d*x)^2)/(a + b*sin(c + d*x))^(1/2),x)
Output:
int((cos(c + d*x)^4*sin(c + d*x)^2)/(a + b*sin(c + d*x))^(1/2), x)
\[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\int \frac {\sqrt {\sin \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{4} \sin \left (d x +c \right )^{2}}{\sin \left (d x +c \right ) b +a}d x \] Input:
int(cos(d*x+c)^4*sin(d*x+c)^2/(a+b*sin(d*x+c))^(1/2),x)
Output:
int((sqrt(sin(c + d*x)*b + a)*cos(c + d*x)**4*sin(c + d*x)**2)/(sin(c + d* x)*b + a),x)