Integrand size = 29, antiderivative size = 447 \[ \int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=-\frac {2 \left (8 a^4-141 a^2 b^2+36 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{693 b^2 d}-\frac {2 a \left (8 a^2-131 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{693 b^2 d}-\frac {2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{693 b^2 d}+\frac {8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac {2 \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}+\frac {2 a \left (8 a^4-147 a^2 b^2+444 b^4\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{693 b^3 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 \left (8 a^6-149 a^4 b^2-516 a^2 b^4-36 b^6\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{693 b^3 d \sqrt {a+b \sin (c+d x)}}+\frac {2 a^3 \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{d \sqrt {a+b \sin (c+d x)}} \]
-2/693*a*(8*a^2-131*b^2)*cos(d*x+c)*(a+b*sin(d*x+c))^(3/2)/b^2/d-2/693*(8* a^2-117*b^2)*cos(d*x+c)*(a+b*sin(d*x+c))^(5/2)/b^2/d+8/99*a*cos(d*x+c)*(a+ b*sin(d*x+c))^(7/2)/b^2/d-2/11*cos(d*x+c)*sin(d*x+c)*(a+b*sin(d*x+c))^(7/2 )/b/d-2/693*(8*a^4-141*a^2*b^2+36*b^4)*cos(d*x+c)*(a+b*sin(d*x+c))^(1/2)/b ^2/d-2/693*a*(8*a^4-147*a^2*b^2+444*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)*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^3/d/((a+b*sin(d*x+c))/(a+b))^(1/2 )+2/693*(8*a^6-149*a^4*b^2-516*a^2*b^4-36*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^3/d/(a+b*sin(d*x+c) )^(1/2)-2*a^3*(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 )*EllipticPi(cos(1/2*c+1/4*Pi+1/2*d*x),2,2^(1/2)*(b/(a+b))^(1/2))*((a+b*si n(d*x+c))/(a+b))^(1/2)/d/(a+b*sin(d*x+c))^(1/2)
Result contains complex when optimal does not.
Time = 3.18 (sec) , antiderivative size = 521, normalized size of antiderivative = 1.17 \[ \int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\frac {-2 \left (\frac {2 i \left (8 a^4-147 a^2 b^2+444 b^4\right ) \left (-2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (-2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )+b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sec (c+d x) \sqrt {-\frac {b (-1+\sin (c+d x))}{a+b}} \sqrt {\frac {b (1+\sin (c+d x))}{-a+b}}}{b^2 \sqrt {-\frac {1}{a+b}}}+\frac {8 b \left (a^4+480 a^2 b^2+18 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}+\frac {2 a \left (8 a^4+1239 a^2 b^2+444 b^4\right ) \operatorname {EllipticPi}\left (2,\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}\right )+\cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^4+2660 a^2 b^2-9 b^4+4 \left (113 a^2 b^2-54 b^4\right ) \cos (2 (c+d x))-63 b^4 \cos (4 (c+d x))-24 a^3 b \sin (c+d x)+1954 a b^3 \sin (c+d x)+322 a b^3 \sin (3 (c+d x))\right )}{2772 b^2 d} \]
(-2*(((2*I)*(8*a^4 - 147*a^2*b^2 + 444*b^4)*(-2*a*(a - b)*EllipticE[I*ArcS inh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*(- 2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt [a + b*Sin[c + d*x]]], (a + b)/(a - b)]))*Sec[c + d*x]*Sqrt[-((b*(-1 + Sin [c + d*x]))/(a + b))]*Sqrt[(b*(1 + Sin[c + d*x]))/(-a + b)])/(b^2*Sqrt[-(a + b)^(-1)]) + (8*b*(a^4 + 480*a^2*b^2 + 18*b^4)*EllipticF[(-2*c + Pi - 2* d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[ c + d*x]] + (2*a*(8*a^4 + 1239*a^2*b^2 + 444*b^4)*EllipticPi[2, (-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b *Sin[c + d*x]]) + Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(32*a^4 + 2660*a^2 *b^2 - 9*b^4 + 4*(113*a^2*b^2 - 54*b^4)*Cos[2*(c + d*x)] - 63*b^4*Cos[4*(c + d*x)] - 24*a^3*b*Sin[c + d*x] + 1954*a*b^3*Sin[c + d*x] + 322*a*b^3*Sin [3*(c + d*x)]))/(2772*b^2*d)
Time = 3.55 (sec) , antiderivative size = 458, normalized size of antiderivative = 1.02, number of steps used = 27, number of rules used = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.931, Rules used = {3042, 3374, 27, 3042, 3528, 27, 3042, 3528, 27, 3042, 3528, 27, 3042, 3538, 25, 3042, 3134, 3042, 3132, 3481, 3042, 3142, 3042, 3140, 3286, 3042, 3284}
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.
| \(\displaystyle \int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4 (a+b \sin (c+d x))^{5/2}}{\sin (c+d x)}dx\) |
| \(\Big \downarrow \) 3374 |
| \(\displaystyle -\frac {4 \int -\frac {1}{4} \csc (c+d x) (a+b \sin (c+d x))^{5/2} \left (99 b^2-10 a \sin (c+d x) b+\left (8 a^2-117 b^2\right ) \sin ^2(c+d x)\right )dx}{99 b^2}+\frac {8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac {2 \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \csc (c+d x) (a+b \sin (c+d x))^{5/2} \left (99 b^2-10 a \sin (c+d x) b+\left (8 a^2-117 b^2\right ) \sin ^2(c+d x)\right )dx}{99 b^2}+\frac {8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac {2 \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(a+b \sin (c+d x))^{5/2} \left (99 b^2-10 a \sin (c+d x) b+\left (8 a^2-117 b^2\right ) \sin (c+d x)^2\right )}{\sin (c+d x)}dx}{99 b^2}+\frac {8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac {2 \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {2}{7} \int \frac {1}{2} \csc (c+d x) (a+b \sin (c+d x))^{3/2} \left (693 a b^2-6 \left (5 a^2-18 b^2\right ) \sin (c+d x) b+5 a \left (8 a^2-131 b^2\right ) \sin ^2(c+d x)\right )dx-\frac {2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{99 b^2}+\frac {8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac {2 \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \int \csc (c+d x) (a+b \sin (c+d x))^{3/2} \left (693 a b^2-6 \left (5 a^2-18 b^2\right ) \sin (c+d x) b+5 a \left (8 a^2-131 b^2\right ) \sin ^2(c+d x)\right )dx-\frac {2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{99 b^2}+\frac {8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac {2 \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \int \frac {(a+b \sin (c+d x))^{3/2} \left (693 a b^2-6 \left (5 a^2-18 b^2\right ) \sin (c+d x) b+5 a \left (8 a^2-131 b^2\right ) \sin (c+d x)^2\right )}{\sin (c+d x)}dx-\frac {2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{99 b^2}+\frac {8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac {2 \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {2}{5} \int \frac {15}{2} \csc (c+d x) \sqrt {a+b \sin (c+d x)} \left (231 a^2 b^2-2 a \left (a^2-68 b^2\right ) \sin (c+d x) b+\left (8 a^4-141 b^2 a^2+36 b^4\right ) \sin ^2(c+d x)\right )dx-\frac {2 a \left (8 a^2-131 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{99 b^2}+\frac {8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac {2 \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \left (3 \int \csc (c+d x) \sqrt {a+b \sin (c+d x)} \left (231 a^2 b^2-2 a \left (a^2-68 b^2\right ) \sin (c+d x) b+\left (8 a^4-141 b^2 a^2+36 b^4\right ) \sin ^2(c+d x)\right )dx-\frac {2 a \left (8 a^2-131 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{99 b^2}+\frac {8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac {2 \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (3 \int \frac {\sqrt {a+b \sin (c+d x)} \left (231 a^2 b^2-2 a \left (a^2-68 b^2\right ) \sin (c+d x) b+\left (8 a^4-141 b^2 a^2+36 b^4\right ) \sin (c+d x)^2\right )}{\sin (c+d x)}dx-\frac {2 a \left (8 a^2-131 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{99 b^2}+\frac {8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac {2 \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {1}{7} \left (3 \left (\frac {2}{3} \int \frac {\csc (c+d x) \left (693 b^2 a^3+\left (8 a^4-147 b^2 a^2+444 b^4\right ) \sin ^2(c+d x) a+2 b \left (a^4+480 b^2 a^2+18 b^4\right ) \sin (c+d x)\right )}{2 \sqrt {a+b \sin (c+d x)}}dx-\frac {2 \left (8 a^4-141 a^2 b^2+36 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )-\frac {2 a \left (8 a^2-131 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{99 b^2}+\frac {8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac {2 \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \left (3 \left (\frac {1}{3} \int \frac {\csc (c+d x) \left (693 b^2 a^3+\left (8 a^4-147 b^2 a^2+444 b^4\right ) \sin ^2(c+d x) a+2 b \left (a^4+480 b^2 a^2+18 b^4\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx-\frac {2 \left (8 a^4-141 a^2 b^2+36 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )-\frac {2 a \left (8 a^2-131 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{99 b^2}+\frac {8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac {2 \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (3 \left (\frac {1}{3} \int \frac {693 b^2 a^3+\left (8 a^4-147 b^2 a^2+444 b^4\right ) \sin (c+d x)^2 a+2 b \left (a^4+480 b^2 a^2+18 b^4\right ) \sin (c+d x)}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx-\frac {2 \left (8 a^4-141 a^2 b^2+36 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )-\frac {2 a \left (8 a^2-131 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{99 b^2}+\frac {8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac {2 \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle \frac {\frac {1}{7} \left (3 \left (\frac {1}{3} \left (\frac {a \left (8 a^4-147 a^2 b^2+444 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}-\frac {\int -\frac {\csc (c+d x) \left (693 a^3 b^3-\left (8 a^6-149 b^2 a^4-516 b^4 a^2-36 b^6\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )-\frac {2 \left (8 a^4-141 a^2 b^2+36 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )-\frac {2 a \left (8 a^2-131 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{99 b^2}+\frac {8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac {2 \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{7} \left (3 \left (\frac {1}{3} \left (\frac {a \left (8 a^4-147 a^2 b^2+444 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}+\frac {\int \frac {\csc (c+d x) \left (693 a^3 b^3-\left (8 a^6-149 b^2 a^4-516 b^4 a^2-36 b^6\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )-\frac {2 \left (8 a^4-141 a^2 b^2+36 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )-\frac {2 a \left (8 a^2-131 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{99 b^2}+\frac {8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac {2 \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (3 \left (\frac {1}{3} \left (\frac {a \left (8 a^4-147 a^2 b^2+444 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}+\frac {\int \frac {693 a^3 b^3-\left (8 a^6-149 b^2 a^4-516 b^4 a^2-36 b^6\right ) \sin (c+d x)}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}\right )-\frac {2 \left (8 a^4-141 a^2 b^2+36 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )-\frac {2 a \left (8 a^2-131 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{99 b^2}+\frac {8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac {2 \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {1}{7} \left (3 \left (\frac {1}{3} \left (\frac {a \left (8 a^4-147 a^2 b^2+444 b^4\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\int \frac {693 a^3 b^3-\left (8 a^6-149 b^2 a^4-516 b^4 a^2-36 b^6\right ) \sin (c+d x)}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}\right )-\frac {2 \left (8 a^4-141 a^2 b^2+36 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )-\frac {2 a \left (8 a^2-131 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{99 b^2}+\frac {8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac {2 \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (3 \left (\frac {1}{3} \left (\frac {a \left (8 a^4-147 a^2 b^2+444 b^4\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\int \frac {693 a^3 b^3-\left (8 a^6-149 b^2 a^4-516 b^4 a^2-36 b^6\right ) \sin (c+d x)}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}\right )-\frac {2 \left (8 a^4-141 a^2 b^2+36 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )-\frac {2 a \left (8 a^2-131 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{99 b^2}+\frac {8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac {2 \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {1}{7} \left (3 \left (\frac {1}{3} \left (\frac {\int \frac {693 a^3 b^3-\left (8 a^6-149 b^2 a^4-516 b^4 a^2-36 b^6\right ) \sin (c+d x)}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 a \left (8 a^4-147 a^2 b^2+444 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {2 \left (8 a^4-141 a^2 b^2+36 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )-\frac {2 a \left (8 a^2-131 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{99 b^2}+\frac {8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac {2 \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle \frac {\frac {1}{7} \left (3 \left (\frac {1}{3} \left (\frac {693 a^3 b^3 \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx-\left (8 a^6-149 a^4 b^2-516 a^2 b^4-36 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 a \left (8 a^4-147 a^2 b^2+444 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {2 \left (8 a^4-141 a^2 b^2+36 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )-\frac {2 a \left (8 a^2-131 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{99 b^2}+\frac {8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac {2 \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (3 \left (\frac {1}{3} \left (\frac {693 a^3 b^3 \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx-\left (8 a^6-149 a^4 b^2-516 a^2 b^4-36 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 a \left (8 a^4-147 a^2 b^2+444 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {2 \left (8 a^4-141 a^2 b^2+36 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )-\frac {2 a \left (8 a^2-131 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{99 b^2}+\frac {8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac {2 \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {1}{7} \left (3 \left (\frac {1}{3} \left (\frac {693 a^3 b^3 \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx-\frac {\left (8 a^6-149 a^4 b^2-516 a^2 b^4-36 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 a \left (8 a^4-147 a^2 b^2+444 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {2 \left (8 a^4-141 a^2 b^2+36 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )-\frac {2 a \left (8 a^2-131 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{99 b^2}+\frac {8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac {2 \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (3 \left (\frac {1}{3} \left (\frac {693 a^3 b^3 \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx-\frac {\left (8 a^6-149 a^4 b^2-516 a^2 b^4-36 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 a \left (8 a^4-147 a^2 b^2+444 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {2 \left (8 a^4-141 a^2 b^2+36 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )-\frac {2 a \left (8 a^2-131 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{99 b^2}+\frac {8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac {2 \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {1}{7} \left (3 \left (\frac {1}{3} \left (\frac {693 a^3 b^3 \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx-\frac {2 \left (8 a^6-149 a^4 b^2-516 a^2 b^4-36 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 a \left (8 a^4-147 a^2 b^2+444 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {2 \left (8 a^4-141 a^2 b^2+36 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )-\frac {2 a \left (8 a^2-131 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{99 b^2}+\frac {8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac {2 \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {\frac {1}{7} \left (3 \left (\frac {1}{3} \left (\frac {\frac {693 a^3 b^3 \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {\csc (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}-\frac {2 \left (8 a^6-149 a^4 b^2-516 a^2 b^4-36 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 a \left (8 a^4-147 a^2 b^2+444 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {2 \left (8 a^4-141 a^2 b^2+36 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )-\frac {2 a \left (8 a^2-131 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{99 b^2}+\frac {8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac {2 \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (3 \left (\frac {1}{3} \left (\frac {\frac {693 a^3 b^3 \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sin (c+d x) \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}-\frac {2 \left (8 a^6-149 a^4 b^2-516 a^2 b^4-36 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 a \left (8 a^4-147 a^2 b^2+444 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {2 \left (8 a^4-141 a^2 b^2+36 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )-\frac {2 a \left (8 a^2-131 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{99 b^2}+\frac {8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac {2 \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {\frac {1}{7} \left (3 \left (\frac {1}{3} \left (\frac {2 a \left (8 a^4-147 a^2 b^2+444 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\frac {1386 a^3 b^3 \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (8 a^6-149 a^4 b^2-516 a^2 b^4-36 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}\right )-\frac {2 \left (8 a^4-141 a^2 b^2+36 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )-\frac {2 a \left (8 a^2-131 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{99 b^2}+\frac {8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac {2 \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}\) |
(8*a*Cos[c + d*x]*(a + b*Sin[c + d*x])^(7/2))/(99*b^2*d) - (2*Cos[c + d*x] *Sin[c + d*x]*(a + b*Sin[c + d*x])^(7/2))/(11*b*d) + ((-2*(8*a^2 - 117*b^2 )*Cos[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(7*d) + ((-2*a*(8*a^2 - 131*b^2 )*Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/d + 3*((-2*(8*a^4 - 141*a^2*b^2 + 36*b^4)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(3*d) + ((2*a*(8*a^4 - 1 47*a^2*b^2 + 444*b^4)*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*(8*a^6 - 149*a^4*b^2 - 516*a^2*b^4 - 36*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]]) + (1386*a^3*b^3*EllipticPi[2, (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]]))/b)/3))/7)/(99*b^2 )
3.12.61.3.1 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[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[ 2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a, b, c , d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt [c + d*Sin[e + f*x]] Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/(c + d))*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] && NeQ[c^2 - d^2, 0] && !GtQ[c + d, 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_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[ B/d Int[(a + b*Sin[e + f*x])^m, x], x] - Simp[(B*c - A*d)/d Int[(a + b* Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
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])))
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[C/(b*d) Int[Sqrt[a + b*Sin[e + f*x]], x] , x] - Simp[1/(b*d) Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[ e + f*x], x]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; Fre eQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0 ] && NeQ[c^2 - d^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1572\) vs. \(2(510)=1020\).
Time = 89.30 (sec) , antiderivative size = 1573, normalized size of antiderivative = 3.52
-2/693*(-8*((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+6*((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+149*((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)*Elli pticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^4*b^3-1107*((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^4+516*((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+408*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(s in(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^6-155*((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^5*b^2 +591*((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)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/ (a+b))^(1/2))*a^3*b^4-444*((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+...
\[ \int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{3} \cot \left (d x + c\right ) \,d x } \]
integral((2*a*b*cos(d*x + c)^3*cot(d*x + c)*sin(d*x + c) - (b^2*cos(d*x + c)^5 - (a^2 + b^2)*cos(d*x + c)^3)*cot(d*x + c))*sqrt(b*sin(d*x + c) + a), x)
Timed out. \[ \int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]
\[ \int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{3} \cot \left (d x + c\right ) \,d x } \]
Timed out. \[ \int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]
Timed out. \[ \int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\int {\cos \left (c+d\,x\right )}^3\,\mathrm {cot}\left (c+d\,x\right )\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{5/2} \,d x \]