Integrand size = 31, antiderivative size = 482 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=-\frac {\left (128 a^4-580 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{640 a^2 d}+\frac {b \left (36 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{64 a^2 d}+\frac {\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{80 a^2 d}+\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}-\frac {\left (128 a^4-2476 a^2 b^2-15 b^4\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{640 a^2 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (128 a^4+492 a^2 b^2-5 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{640 a d \sqrt {a+b \sin (c+d x)}}+\frac {3 b \left (80 a^4-40 a^2 b^2+b^4\right ) \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{128 a^2 d \sqrt {a+b \sin (c+d x)}} \]
1/64*b*(36*a^2-b^2)*cot(d*x+c)*csc(d*x+c)*(a+b*sin(d*x+c))^(3/2)/a^2/d+1/8 0*(32*a^2-b^2)*cot(d*x+c)*csc(d*x+c)^2*(a+b*sin(d*x+c))^(5/2)/a^2/d+3/40*b *cot(d*x+c)*csc(d*x+c)^3*(a+b*sin(d*x+c))^(7/2)/a^2/d-1/5*cot(d*x+c)*csc(d *x+c)^4*(a+b*sin(d*x+c))^(7/2)/a/d-1/640*(128*a^4-580*a^2*b^2+15*b^4)*cot( d*x+c)*(a+b*sin(d*x+c))^(1/2)/a^2/d+1/640*(128*a^4-2476*a^2*b^2-15*b^4)*(s in(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)/a^2/ d/((a+b*sin(d*x+c))/(a+b))^(1/2)-1/640*(128*a^4+492*a^2*b^2-5*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)*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)/a /d/(a+b*sin(d*x+c))^(1/2)-3/128*b*(80*a^4-40*a^2*b^2+b^4)*(sin(1/2*c+1/4*P i+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*sin(d*x+c))/(a+b))^(1/2)/a^2/d/( a+b*sin(d*x+c))^(1/2)
Result contains complex when optimal does not.
Time = 7.33 (sec) , antiderivative size = 545, normalized size of antiderivative = 1.13 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\frac {-4 \cot (c+d x) \left (128 a^4-1196 a^2 b^2-15 b^4+\left (-872 a^3 b+10 a b^3\right ) \csc (c+d x)-8 a^2 \left (32 a^2-31 b^2\right ) \csc ^2(c+d x)+336 a^3 b \csc ^3(c+d x)+128 a^4 \csc ^4(c+d x)\right ) \sqrt {a+b \sin (c+d x)}+b \left (-\frac {2 i \left (128 a^4-2476 a^2 b^2-15 b^4\right ) \cos (2 (c+d x)) \csc ^2(c+d x) \left (2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )-b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sec (c+d x) \sqrt {-\frac {b (-1+\sin (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sin (c+d x))}{a-b}}}{a b^2 \sqrt {-\frac {1}{a+b}} \left (-2+\csc ^2(c+d x)\right )}-\frac {8 a b \left (1484 a^2+5 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}-\frac {2 \left (2272 a^4+1276 a^2 b^2+45 b^4\right ) \operatorname {EllipticPi}\left (2,\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}\right )}{2560 a^2 d} \]
(-4*Cot[c + d*x]*(128*a^4 - 1196*a^2*b^2 - 15*b^4 + (-872*a^3*b + 10*a*b^3 )*Csc[c + d*x] - 8*a^2*(32*a^2 - 31*b^2)*Csc[c + d*x]^2 + 336*a^3*b*Csc[c + d*x]^3 + 128*a^4*Csc[c + d*x]^4)*Sqrt[a + b*Sin[c + d*x]] + b*(((-2*I)*( 128*a^4 - 2476*a^2*b^2 - 15*b^4)*Cos[2*(c + d*x)]*Csc[c + d*x]^2*(2*a*(a - b)*EllipticE[I*ArcSinh[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))])/(a*b^2*Sqrt[-(a + b)^(-1)]*(-2 + Csc[c + d*x]^2)) - (8*a*b*(1484*a ^2 + 5*b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Si n[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - (2*(2272*a^4 + 1276*a^2*b ^2 + 45*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]]))/(2560*a^2*d)
Time = 3.91 (sec) , antiderivative size = 486, normalized size of antiderivative = 1.01, number of steps used = 27, number of rules used = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.871, Rules used = {3042, 3372, 27, 3042, 3526, 27, 3042, 3526, 27, 3042, 3526, 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.
| \(\displaystyle \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4 (a+b \sin (c+d x))^{5/2}}{\sin (c+d x)^6}dx\) |
| \(\Big \downarrow \) 3372 |
| \(\displaystyle -\frac {\int \frac {1}{4} \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2} \left (-\left (\left (80 a^2+3 b^2\right ) \sin ^2(c+d x)\right )+10 a b \sin (c+d x)+3 \left (32 a^2-b^2\right )\right )dx}{20 a^2}+\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2} \left (-\left (\left (80 a^2+3 b^2\right ) \sin ^2(c+d x)\right )+10 a b \sin (c+d x)+3 \left (32 a^2-b^2\right )\right )dx}{80 a^2}+\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {(a+b \sin (c+d x))^{5/2} \left (-\left (\left (80 a^2+3 b^2\right ) \sin (c+d x)^2\right )+10 a b \sin (c+d x)+3 \left (32 a^2-b^2\right )\right )}{\sin (c+d x)^4}dx}{80 a^2}+\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle -\frac {\frac {1}{3} \int \frac {3}{2} \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2} \left (-b \left (192 a^2+5 b^2\right ) \sin ^2(c+d x)-2 a \left (16 a^2-5 b^2\right ) \sin (c+d x)+5 b \left (36 a^2-b^2\right )\right )dx-\frac {\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{80 a^2}+\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {1}{2} \int \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2} \left (-b \left (192 a^2+5 b^2\right ) \sin ^2(c+d x)-2 a \left (16 a^2-5 b^2\right ) \sin (c+d x)+5 b \left (36 a^2-b^2\right )\right )dx-\frac {\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{80 a^2}+\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{2} \int \frac {(a+b \sin (c+d x))^{3/2} \left (-b \left (192 a^2+5 b^2\right ) \sin (c+d x)^2-2 a \left (16 a^2-5 b^2\right ) \sin (c+d x)+5 b \left (36 a^2-b^2\right )\right )}{\sin (c+d x)^3}dx-\frac {\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{80 a^2}+\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{2} \int -\frac {1}{2} \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)} \left (128 a^4-580 b^2 a^2+2 b \left (268 a^2-5 b^2\right ) \sin (c+d x) a+15 b^4+3 b^2 \left (316 a^2+5 b^2\right ) \sin ^2(c+d x)\right )dx-\frac {5 b \left (36 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}\right )-\frac {\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{80 a^2}+\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {1}{2} \left (-\frac {1}{4} \int \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)} \left (128 a^4-580 b^2 a^2+2 b \left (268 a^2-5 b^2\right ) \sin (c+d x) a+15 b^4+3 b^2 \left (316 a^2+5 b^2\right ) \sin ^2(c+d x)\right )dx-\frac {5 b \left (36 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}\right )-\frac {\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{80 a^2}+\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{2} \left (-\frac {1}{4} \int \frac {\sqrt {a+b \sin (c+d x)} \left (128 a^4-580 b^2 a^2+2 b \left (268 a^2-5 b^2\right ) \sin (c+d x) a+15 b^4+3 b^2 \left (316 a^2+5 b^2\right ) \sin (c+d x)^2\right )}{\sin (c+d x)^2}dx-\frac {5 b \left (36 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}\right )-\frac {\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{80 a^2}+\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {\left (128 a^4-580 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{d}-\int \frac {\csc (c+d x) \left (2 a \left (1484 a^2+5 b^2\right ) \sin (c+d x) b^2-\left (128 a^4-2476 b^2 a^2-15 b^4\right ) \sin ^2(c+d x) b+15 \left (80 a^4-40 b^2 a^2+b^4\right ) b\right )}{2 \sqrt {a+b \sin (c+d x)}}dx\right )-\frac {5 b \left (36 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}\right )-\frac {\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{80 a^2}+\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {\left (128 a^4-580 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{d}-\frac {1}{2} \int \frac {\csc (c+d x) \left (2 a \left (1484 a^2+5 b^2\right ) \sin (c+d x) b^2-\left (128 a^4-2476 b^2 a^2-15 b^4\right ) \sin ^2(c+d x) b+15 \left (80 a^4-40 b^2 a^2+b^4\right ) b\right )}{\sqrt {a+b \sin (c+d x)}}dx\right )-\frac {5 b \left (36 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}\right )-\frac {\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{80 a^2}+\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {\left (128 a^4-580 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{d}-\frac {1}{2} \int \frac {2 a \left (1484 a^2+5 b^2\right ) \sin (c+d x) b^2-\left (128 a^4-2476 b^2 a^2-15 b^4\right ) \sin (c+d x)^2 b+15 \left (80 a^4-40 b^2 a^2+b^4\right ) b}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx\right )-\frac {5 b \left (36 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}\right )-\frac {\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{80 a^2}+\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{2} \left (\left (128 a^4-2476 a^2 b^2-15 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx+\frac {\int -\frac {\csc (c+d x) \left (15 \left (80 a^4-40 b^2 a^2+b^4\right ) b^2+a \left (128 a^4+492 b^2 a^2-5 b^4\right ) \sin (c+d x) b\right )}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )+\frac {\left (128 a^4-580 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {5 b \left (36 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}\right )-\frac {\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{80 a^2}+\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{2} \left (\left (128 a^4-2476 a^2 b^2-15 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx-\frac {\int \frac {\csc (c+d x) \left (15 \left (80 a^4-40 b^2 a^2+b^4\right ) b^2+a \left (128 a^4+492 b^2 a^2-5 b^4\right ) \sin (c+d x) b\right )}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )+\frac {\left (128 a^4-580 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {5 b \left (36 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}\right )-\frac {\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{80 a^2}+\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{2} \left (\left (128 a^4-2476 a^2 b^2-15 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx-\frac {\int \frac {15 \left (80 a^4-40 b^2 a^2+b^4\right ) b^2+a \left (128 a^4+492 b^2 a^2-5 b^4\right ) \sin (c+d x) b}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}\right )+\frac {\left (128 a^4-580 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {5 b \left (36 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}\right )-\frac {\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{80 a^2}+\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {\left (128 a^4-2476 a^2 b^2-15 b^4\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\int \frac {15 \left (80 a^4-40 b^2 a^2+b^4\right ) b^2+a \left (128 a^4+492 b^2 a^2-5 b^4\right ) \sin (c+d x) b}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}\right )+\frac {\left (128 a^4-580 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {5 b \left (36 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}\right )-\frac {\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{80 a^2}+\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {\left (128 a^4-2476 a^2 b^2-15 b^4\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\int \frac {15 \left (80 a^4-40 b^2 a^2+b^4\right ) b^2+a \left (128 a^4+492 b^2 a^2-5 b^4\right ) \sin (c+d x) b}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}\right )+\frac {\left (128 a^4-580 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {5 b \left (36 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}\right )-\frac {\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{80 a^2}+\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {2 \left (128 a^4-2476 a^2 b^2-15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\int \frac {15 \left (80 a^4-40 b^2 a^2+b^4\right ) b^2+a \left (128 a^4+492 b^2 a^2-5 b^4\right ) \sin (c+d x) b}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}\right )+\frac {\left (128 a^4-580 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {5 b \left (36 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}\right )-\frac {\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{80 a^2}+\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {2 \left (128 a^4-2476 a^2 b^2-15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {a b \left (128 a^4+492 a^2 b^2-5 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx+15 b^2 \left (80 a^4-40 a^2 b^2+b^4\right ) \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )+\frac {\left (128 a^4-580 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {5 b \left (36 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}\right )-\frac {\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{80 a^2}+\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {2 \left (128 a^4-2476 a^2 b^2-15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {15 b^2 \left (80 a^4-40 a^2 b^2+b^4\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx+a b \left (128 a^4+492 a^2 b^2-5 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )+\frac {\left (128 a^4-580 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {5 b \left (36 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}\right )-\frac {\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{80 a^2}+\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {2 \left (128 a^4-2476 a^2 b^2-15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {15 b^2 \left (80 a^4-40 a^2 b^2+b^4\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx+\frac {a b \left (128 a^4+492 a^2 b^2-5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{b}\right )+\frac {\left (128 a^4-580 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {5 b \left (36 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}\right )-\frac {\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{80 a^2}+\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {2 \left (128 a^4-2476 a^2 b^2-15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {15 b^2 \left (80 a^4-40 a^2 b^2+b^4\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx+\frac {a b \left (128 a^4+492 a^2 b^2-5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{b}\right )+\frac {\left (128 a^4-580 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {5 b \left (36 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}\right )-\frac {\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{80 a^2}+\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {2 \left (128 a^4-2476 a^2 b^2-15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {15 b^2 \left (80 a^4-40 a^2 b^2+b^4\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx+\frac {2 a b \left (128 a^4+492 a^2 b^2-5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}\right )+\frac {\left (128 a^4-580 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {5 b \left (36 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}\right )-\frac {\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{80 a^2}+\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {2 \left (128 a^4-2476 a^2 b^2-15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\frac {15 b^2 \left (80 a^4-40 a^2 b^2+b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {\csc (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}+\frac {2 a b \left (128 a^4+492 a^2 b^2-5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}\right )+\frac {\left (128 a^4-580 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {5 b \left (36 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}\right )-\frac {\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{80 a^2}+\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {2 \left (128 a^4-2476 a^2 b^2-15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\frac {15 b^2 \left (80 a^4-40 a^2 b^2+b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sin (c+d x) \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}+\frac {2 a b \left (128 a^4+492 a^2 b^2-5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}\right )+\frac {\left (128 a^4-580 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {5 b \left (36 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}\right )-\frac {\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{80 a^2}+\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {\left (128 a^4-580 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{d}+\frac {1}{2} \left (\frac {2 \left (128 a^4-2476 a^2 b^2-15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\frac {2 a b \left (128 a^4+492 a^2 b^2-5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}+\frac {30 b^2 \left (80 a^4-40 a^2 b^2+b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}\right )\right )-\frac {5 b \left (36 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}\right )-\frac {\left (32 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{80 a^2}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}\) |
(3*b*Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^(7/2))/(40*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^(7/2))/(5*a*d) - (-(((32 *a^2 - b^2)*Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(5/2))/d) + ( (-5*b*(36*a^2 - b^2)*Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^(3/2)) /(2*d) + (((128*a^4 - 580*a^2*b^2 + 15*b^4)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/d + ((2*(128*a^4 - 2476*a^2*b^2 - 15*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*a*b*(128*a^4 + 492*a^2*b^2 - 5*b^4)*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]]) + (30*b^2*(80*a^4 - 40*a^2*b^2 + b^4)*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)/2)/4)/2)/(80*a^2)
3.12.66.3.1 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[Cos[e + f*x]*(a + b* Sin[e + f*x])^(m + 1)*((d*Sin[e + f*x])^(n + 1)/(a*d*f*(n + 1))), x] + (-Si mp[b*(m + n + 2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((d*Sin[e + f*x] )^(n + 2)/(a^2*d^2*f*(n + 1)*(n + 2))), x] - Simp[1/(a^2*d^2*(n + 1)*(n + 2 )) Int[(a + b*Sin[e + f*x])^m*(d*Sin[e + f*x])^(n + 2)*Simp[a^2*n*(n + 2) - b^2*(m + n + 2)*(m + n + 3) + a*b*m*Sin[e + f*x] - (a^2*(n + 1)*(n + 2) - b^2*(m + n + 2)*(m + n + 4))*Sin[e + f*x]^2, x], x], x]) /; FreeQ[{a, b, d, e, f, m}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n]) && !m < -1 && LtQ[n, -1] && (LtQ[n, -2] || EqQ[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^2*C - B*c*d + A*d^2))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x ] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f *x]^2, x], x], x] /; FreeQ[{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] && GtQ[m, 0] && LtQ[n, -1]
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. \(2074\) vs. \(2(543)=1086\).
Time = 50.17 (sec) , antiderivative size = 2075, normalized size of antiderivative = 4.30
1/640*(15*((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)*EllipticPi(((a+b*sin(d*x+c))/(a-b))^(1/2),( a-b)/a,((a-b)/(a+b))^(1/2))*b^7*sin(d*x+c)^5+128*((a+b*sin(d*x+c))/(a-b))^ (1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*Elli pticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^7*sin(d*x+c)^5 -1200*((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)*EllipticPi(((a+b*sin(d*x+c))/(a-b))^(1/2),(a-b) /a,((a-b)/(a+b))^(1/2))*a^5*b^2*sin(d*x+c)^5+1200*((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)*Ell ipticPi(((a+b*sin(d*x+c))/(a-b))^(1/2),(a-b)/a,((a-b)/(a+b))^(1/2))*a^4*b^ 3*sin(d*x+c)^5+600*((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)*EllipticPi(((a+b*sin(d*x+c))/(a-b) )^(1/2),(a-b)/a,((a-b)/(a+b))^(1/2))*a^3*b^4*sin(d*x+c)^5-600*((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)*EllipticPi(((a+b*sin(d*x+c))/(a-b))^(1/2),(a-b)/a,((a-b)/(a+b))^( 1/2))*a^2*b^5*sin(d*x+c)^5-15*((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)*EllipticPi(((a+b*sin(d* x+c))/(a-b))^(1/2),(a-b)/a,((a-b)/(a+b))^(1/2))*a*b^6*sin(d*x+c)^5-2604*(( 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...
\[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cot \left (d x + c\right )^{4} \csc \left (d x + c\right )^{2} \,d x } \]
integral((2*a*b*cot(d*x + c)^4*csc(d*x + c)^2*sin(d*x + c) - (b^2*cos(d*x + c)^2 - a^2 - b^2)*cot(d*x + c)^4*csc(d*x + c)^2)*sqrt(b*sin(d*x + c) + a ), x)
Timed out. \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]
\[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cot \left (d x + c\right )^{4} \csc \left (d x + c\right )^{2} \,d x } \]
Timed out. \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]
Timed out. \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Hanged} \]