Integrand size = 31, antiderivative size = 484 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=-\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{640 a^3 d}+\frac {3 b \left (36 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{320 a^2 d}+\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{80 a^2 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}-\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{640 a^3 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (128 a^4+692 a^2 b^2+5 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{640 a^2 d \sqrt {a+b \sin (c+d x)}}+\frac {3 b \left (48 a^4+8 a^2 b^2-b^4\right ) \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{128 a^3 d \sqrt {a+b \sin (c+d x)}} \] Output:
-1/640*(128*a^4-116*a^2*b^2+15*b^4)*cot(d*x+c)*(a+b*sin(d*x+c))^(1/2)/a^3/ d+3/320*b*(36*a^2-5*b^2)*cot(d*x+c)*csc(d*x+c)*(a+b*sin(d*x+c))^(1/2)/a^2/ d+1/80*(32*a^2-5*b^2)*cot(d*x+c)*csc(d*x+c)^2*(a+b*sin(d*x+c))^(3/2)/a^2/d +1/8*b*cot(d*x+c)*csc(d*x+c)^3*(a+b*sin(d*x+c))^(5/2)/a^2/d-1/5*cot(d*x+c) *csc(d*x+c)^4*(a+b*sin(d*x+c))^(5/2)/a/d+1/640*(128*a^4-116*a^2*b^2+15*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)/a^3/d/((a+b*sin(d*x+c))/(a+b))^(1/2)+1/640*(128*a^4+692*a^2*b^ 2+5*b^4)*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)/a^2/d/(a+b*sin(d*x+c))^(1/2)-3/128*b*(48*a^4+8 *a^2*b^2-b^4)*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^3/d/(a+b*sin(d*x+c))^(1/2)
Result contains complex when optimal does not.
Time = 8.15 (sec) , antiderivative size = 700, normalized size of antiderivative = 1.45 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\frac {\left (\frac {\left (-128 a^4 \cos (c+d x)+116 a^2 b^2 \cos (c+d x)-15 b^4 \cos (c+d x)\right ) \csc (c+d x)}{640 a^3}+\frac {\left (236 a^2 b \cos (c+d x)+5 b^3 \cos (c+d x)\right ) \csc ^2(c+d x)}{320 a^2}+\frac {\left (32 a^2 \cos (c+d x)-b^2 \cos (c+d x)\right ) \csc ^3(c+d x)}{80 a}-\frac {11}{40} b \cot (c+d x) \csc ^3(c+d x)-\frac {1}{5} a \cot (c+d x) \csc ^4(c+d x)\right ) \sqrt {a+b \sin (c+d x)}}{d}+\frac {b \left (-\frac {2 \left (1616 a^3 b-20 a b^3\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (-c+\frac {\pi }{2}-d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}-\frac {2 \left (1312 a^4+356 a^2 b^2-45 b^4\right ) \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (-c+\frac {\pi }{2}-d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}-\frac {2 i \left (128 a^4-116 a^2 b^2+15 b^4\right ) \cos (c+d x) \cos (2 (c+d x)) \left (2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )-b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sqrt {\frac {b-b \sin (c+d x)}{a+b}} \sqrt {-\frac {b+b \sin (c+d x)}{a-b}}}{a \sqrt {-\frac {1}{a+b}} \sqrt {1-\sin ^2(c+d x)} \left (-2 a^2+b^2+4 a (a+b \sin (c+d x))-2 (a+b \sin (c+d x))^2\right ) \sqrt {-\frac {a^2-b^2-2 a (a+b \sin (c+d x))+(a+b \sin (c+d x))^2}{b^2}}}\right )}{2560 a^3 d} \] Input:
Integrate[Cot[c + d*x]^4*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(3/2),x]
Output:
((((-128*a^4*Cos[c + d*x] + 116*a^2*b^2*Cos[c + d*x] - 15*b^4*Cos[c + d*x] )*Csc[c + d*x])/(640*a^3) + ((236*a^2*b*Cos[c + d*x] + 5*b^3*Cos[c + d*x]) *Csc[c + d*x]^2)/(320*a^2) + ((32*a^2*Cos[c + d*x] - b^2*Cos[c + d*x])*Csc [c + d*x]^3)/(80*a) - (11*b*Cot[c + d*x]*Csc[c + d*x]^3)/40 - (a*Cot[c + d *x]*Csc[c + d*x]^4)/5)*Sqrt[a + b*Sin[c + d*x]])/d + (b*((-2*(1616*a^3*b - 20*a*b^3)*EllipticF[(-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - (2*(1312*a^4 + 356*a^2*b^2 - 45*b^4)*EllipticPi[2, (-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin [c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - ((2*I)*(128*a^4 - 116*a^2* b^2 + 15*b^4)*Cos[c + d*x]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSi nh[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)]))*Sqrt[(b - b*Sin[c + d*x])/(a + b)] *Sqrt[-((b + b*Sin[c + d*x])/(a - b))])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Si n[c + d*x]^2]*(-2*a^2 + b^2 + 4*a*(a + b*Sin[c + d*x]) - 2*(a + b*Sin[c + d*x])^2)*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Sin[c + d*x]) + (a + b*Sin[c + d*x ])^2)/b^2)])))/(2560*a^3*d)
Time = 4.12 (sec) , antiderivative size = 493, normalized size of antiderivative = 1.02, number of steps used = 27, number of rules used = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.871, Rules used = {3042, 3372, 27, 3042, 3526, 27, 3042, 3526, 27, 3042, 3534, 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.
| \(\displaystyle \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4 (a+b \sin (c+d x))^{3/2}}{\sin (c+d x)^6}dx\) |
| \(\Big \downarrow \) 3372 |
| \(\displaystyle -\frac {\int \frac {1}{4} \csc ^4(c+d x) (a+b \sin (c+d x))^{3/2} \left (-5 \left (16 a^2-b^2\right ) \sin ^2(c+d x)+6 a b \sin (c+d x)+3 \left (32 a^2-5 b^2\right )\right )dx}{20 a^2}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \csc ^4(c+d x) (a+b \sin (c+d x))^{3/2} \left (-5 \left (16 a^2-b^2\right ) \sin ^2(c+d x)+6 a b \sin (c+d x)+3 \left (32 a^2-5 b^2\right )\right )dx}{80 a^2}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {(a+b \sin (c+d x))^{3/2} \left (-5 \left (16 a^2-b^2\right ) \sin (c+d x)^2+6 a b \sin (c+d x)+3 \left (32 a^2-5 b^2\right )\right )}{\sin (c+d x)^4}dx}{80 a^2}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle -\frac {\frac {1}{3} \int \frac {3}{2} \csc ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (-b \left (128 a^2-5 b^2\right ) \sin ^2(c+d x)-2 a \left (16 a^2-b^2\right ) \sin (c+d x)+3 b \left (36 a^2-5 b^2\right )\right )dx-\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{80 a^2}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {1}{2} \int \csc ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (-b \left (128 a^2-5 b^2\right ) \sin ^2(c+d x)-2 a \left (16 a^2-b^2\right ) \sin (c+d x)+3 b \left (36 a^2-5 b^2\right )\right )dx-\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{80 a^2}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{2} \int \frac {\sqrt {a+b \sin (c+d x)} \left (-b \left (128 a^2-5 b^2\right ) \sin (c+d x)^2-2 a \left (16 a^2-b^2\right ) \sin (c+d x)+3 b \left (36 a^2-5 b^2\right )\right )}{\sin (c+d x)^3}dx-\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{80 a^2}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{2} \int -\frac {\csc ^2(c+d x) \left (128 a^4-116 b^2 a^2+2 b \left (212 a^2+b^2\right ) \sin (c+d x) a+15 b^4+b^2 \left (404 a^2-5 b^2\right ) \sin ^2(c+d x)\right )}{2 \sqrt {a+b \sin (c+d x)}}dx-\frac {3 b \left (36 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}\right )-\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{80 a^2}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {1}{2} \left (-\frac {1}{4} \int \frac {\csc ^2(c+d x) \left (128 a^4-116 b^2 a^2+2 b \left (212 a^2+b^2\right ) \sin (c+d x) a+15 b^4+b^2 \left (404 a^2-5 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx-\frac {3 b \left (36 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}\right )-\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{80 a^2}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{2} \left (-\frac {1}{4} \int \frac {128 a^4-116 b^2 a^2+2 b \left (212 a^2+b^2\right ) \sin (c+d x) a+15 b^4+b^2 \left (404 a^2-5 b^2\right ) \sin (c+d x)^2}{\sin (c+d x)^2 \sqrt {a+b \sin (c+d x)}}dx-\frac {3 b \left (36 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}\right )-\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{80 a^2}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {\int \frac {\csc (c+d x) \left (2 a \left (404 a^2-5 b^2\right ) \sin (c+d x) b^2-\left (128 a^4-116 b^2 a^2+15 b^4\right ) \sin ^2(c+d x) b+15 \left (48 a^4+8 b^2 a^2-b^4\right ) b\right )}{2 \sqrt {a+b \sin (c+d x)}}dx}{a}\right )-\frac {3 b \left (36 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}\right )-\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{80 a^2}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {\int \frac {\csc (c+d x) \left (2 a \left (404 a^2-5 b^2\right ) \sin (c+d x) b^2-\left (128 a^4-116 b^2 a^2+15 b^4\right ) \sin ^2(c+d x) b+15 \left (48 a^4+8 b^2 a^2-b^4\right ) b\right )}{\sqrt {a+b \sin (c+d x)}}dx}{2 a}\right )-\frac {3 b \left (36 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}\right )-\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{80 a^2}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {\int \frac {2 a \left (404 a^2-5 b^2\right ) \sin (c+d x) b^2-\left (128 a^4-116 b^2 a^2+15 b^4\right ) \sin (c+d x)^2 b+15 \left (48 a^4+8 b^2 a^2-b^4\right ) b}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{2 a}\right )-\frac {3 b \left (36 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}\right )-\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{80 a^2}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {-\left (\left (128 a^4-116 a^2 b^2+15 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx\right )-\frac {\int -\frac {\csc (c+d x) \left (15 \left (48 a^4+8 b^2 a^2-b^4\right ) b^2+a \left (128 a^4+692 b^2 a^2+5 b^4\right ) \sin (c+d x) b\right )}{\sqrt {a+b \sin (c+d x)}}dx}{b}}{2 a}\right )-\frac {3 b \left (36 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}\right )-\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{80 a^2}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {\frac {\int \frac {\csc (c+d x) \left (15 \left (48 a^4+8 b^2 a^2-b^4\right ) b^2+a \left (128 a^4+692 b^2 a^2+5 b^4\right ) \sin (c+d x) b\right )}{\sqrt {a+b \sin (c+d x)}}dx}{b}-\left (128 a^4-116 a^2 b^2+15 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx}{2 a}\right )-\frac {3 b \left (36 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}\right )-\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{80 a^2}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {\frac {\int \frac {15 \left (48 a^4+8 b^2 a^2-b^4\right ) b^2+a \left (128 a^4+692 b^2 a^2+5 b^4\right ) \sin (c+d x) b}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}-\left (128 a^4-116 a^2 b^2+15 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx}{2 a}\right )-\frac {3 b \left (36 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}\right )-\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{80 a^2}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {\frac {\int \frac {15 \left (48 a^4+8 b^2 a^2-b^4\right ) b^2+a \left (128 a^4+692 b^2 a^2+5 b^4\right ) \sin (c+d x) b}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}\right )-\frac {3 b \left (36 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}\right )-\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{80 a^2}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {\frac {\int \frac {15 \left (48 a^4+8 b^2 a^2-b^4\right ) b^2+a \left (128 a^4+692 b^2 a^2+5 b^4\right ) \sin (c+d x) b}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}\right )-\frac {3 b \left (36 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}\right )-\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{80 a^2}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {\frac {\int \frac {15 \left (48 a^4+8 b^2 a^2-b^4\right ) b^2+a \left (128 a^4+692 b^2 a^2+5 b^4\right ) \sin (c+d x) b}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {2 \left (128 a^4-116 a^2 b^2+15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}\right )-\frac {3 b \left (36 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}\right )-\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{80 a^2}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {\frac {a b \left (128 a^4+692 a^2 b^2+5 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx+15 b^2 \left (48 a^4+8 a^2 b^2-b^4\right ) \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {2 \left (128 a^4-116 a^2 b^2+15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}\right )-\frac {3 b \left (36 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}\right )-\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{80 a^2}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {\frac {15 b^2 \left (48 a^4+8 a^2 b^2-b^4\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx+a b \left (128 a^4+692 a^2 b^2+5 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {2 \left (128 a^4-116 a^2 b^2+15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}\right )-\frac {3 b \left (36 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}\right )-\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{80 a^2}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {\frac {15 b^2 \left (48 a^4+8 a^2 b^2-b^4\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx+\frac {a b \left (128 a^4+692 a^2 b^2+5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{b}-\frac {2 \left (128 a^4-116 a^2 b^2+15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}\right )-\frac {3 b \left (36 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}\right )-\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{80 a^2}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {\frac {15 b^2 \left (48 a^4+8 a^2 b^2-b^4\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx+\frac {a b \left (128 a^4+692 a^2 b^2+5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{b}-\frac {2 \left (128 a^4-116 a^2 b^2+15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}\right )-\frac {3 b \left (36 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}\right )-\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{80 a^2}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {\frac {15 b^2 \left (48 a^4+8 a^2 b^2-b^4\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx+\frac {2 a b \left (128 a^4+692 a^2 b^2+5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}-\frac {2 \left (128 a^4-116 a^2 b^2+15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}\right )-\frac {3 b \left (36 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}\right )-\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{80 a^2}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {\frac {\frac {15 b^2 \left (48 a^4+8 a^2 b^2-b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {\csc (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}+\frac {2 a b \left (128 a^4+692 a^2 b^2+5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}-\frac {2 \left (128 a^4-116 a^2 b^2+15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}\right )-\frac {3 b \left (36 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}\right )-\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{80 a^2}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {\frac {\frac {15 b^2 \left (48 a^4+8 a^2 b^2-b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sin (c+d x) \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}+\frac {2 a b \left (128 a^4+692 a^2 b^2+5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}-\frac {2 \left (128 a^4-116 a^2 b^2+15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}\right )-\frac {3 b \left (36 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}\right )-\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{80 a^2}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {\frac {\frac {2 a b \left (128 a^4+692 a^2 b^2+5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}+\frac {30 b^2 \left (48 a^4+8 a^2 b^2-b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}-\frac {2 \left (128 a^4-116 a^2 b^2+15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}\right )-\frac {3 b \left (36 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}\right )-\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{80 a^2}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}\) |
Input:
Int[Cot[c + d*x]^4*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(3/2),x]
Output:
(b*Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^(5/2))/(8*a^2*d) - (Co t[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^(5/2))/(5*a*d) - (-(((32*a^ 2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(3/2))/d) + (( -3*b*(36*a^2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/ (2*d) + (((128*a^4 - 116*a^2*b^2 + 15*b^4)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(a*d) - ((-2*(128*a^4 - 116*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 + 692*a^2*b^2 + 5*b^4)*EllipticF[(c - P i/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*(48*a^4 + 8*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*Sqr t[a + b*Sin[c + d*x]]))/b)/(2*a))/4)/2)/(80*a^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[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_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int [(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A *b^2 - a*b*B + a^2*C)*(m + n + 3)*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] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ [n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) | | EqQ[a, 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. \(2074\) vs. \(2(454)=908\).
Time = 2.89 (sec) , antiderivative size = 2075, normalized size of antiderivative = 4.29
Input:
int(cot(d*x+c)^4*csc(d*x+c)^2*(a+b*sin(d*x+c))^(3/2),x,method=_RETURNVERBO SE)
Output:
-1/640*(15*((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)*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)*(-b*(1+sin(d*x+c))/(a-b))^(1/2)*Ell ipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^7*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)*(-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*sin(d*x+c)^5+720*((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)*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)^ 5-720*((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)*EllipticPi(((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+120*((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)*Elli pticPi(((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)^5-120*((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)*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)*(-b*(1+sin(d*x+c))/(a-b)) ^(1/2)*EllipticPi(((a+b*sin(d*x+c))/(a-b))^(1/2),(a-b)/a,((a-b)/(a+b))^...
Timed out. \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^2*(a+b*sin(d*x+c))^(3/2),x, algorithm="f ricas")
Output:
Timed out
Timed out. \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**4*csc(d*x+c)**2*(a+b*sin(d*x+c))**(3/2),x)
Output:
Timed out
\[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cot \left (d x + c\right )^{4} \csc \left (d x + c\right )^{2} \,d x } \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^2*(a+b*sin(d*x+c))^(3/2),x, algorithm="m axima")
Output:
integrate((b*sin(d*x + c) + a)^(3/2)*cot(d*x + c)^4*csc(d*x + c)^2, x)
Timed out. \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^2*(a+b*sin(d*x+c))^(3/2),x, algorithm="g iac")
Output:
Timed out
Timed out. \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\text {Hanged} \] Input:
int((cot(c + d*x)^4*(a + b*sin(c + d*x))^(3/2))/sin(c + d*x)^2,x)
Output:
\text{Hanged}
\[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\left (\int \sqrt {\sin \left (d x +c \right ) b +a}\, \cot \left (d x +c \right )^{4} \csc \left (d x +c \right )^{2} \sin \left (d x +c \right )d x \right ) b +\left (\int \sqrt {\sin \left (d x +c \right ) b +a}\, \cot \left (d x +c \right )^{4} \csc \left (d x +c \right )^{2}d x \right ) a \] Input:
int(cot(d*x+c)^4*csc(d*x+c)^2*(a+b*sin(d*x+c))^(3/2),x)
Output:
int(sqrt(sin(c + d*x)*b + a)*cot(c + d*x)**4*csc(c + d*x)**2*sin(c + d*x), x)*b + int(sqrt(sin(c + d*x)*b + a)*cot(c + d*x)**4*csc(c + d*x)**2,x)*a