Integrand size = 31, antiderivative size = 551 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=-\frac {b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{1536 a^3 d}-\frac {\left (16 a^4-56 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{256 a^2 d}+\frac {b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}+\frac {\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}-\frac {b \left (720 a^4-176 a^2 b^2+15 b^4\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{1536 a^3 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {b \left (816 a^4+1696 a^2 b^2+5 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{1536 a^2 d \sqrt {a+b \sin (c+d x)}}+\frac {\left (64 a^6+720 a^4 b^2+60 a^2 b^4-5 b^6\right ) \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{512 a^3 d \sqrt {a+b \sin (c+d x)}} \]
1/192*b*(52*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+1/96*(28*a^2-3*b^2)*cot(d*x+c)*csc(d*x+c)^3*(a+b*sin(d*x+c))^(5/2)/a^2/d +1/12*b*cot(d*x+c)*csc(d*x+c)^4*(a+b*sin(d*x+c))^(7/2)/a^2/d-1/6*cot(d*x+c )*csc(d*x+c)^5*(a+b*sin(d*x+c))^(7/2)/a/d-1/1536*b*(720*a^4-176*a^2*b^2+15 *b^4)*cot(d*x+c)*(a+b*sin(d*x+c))^(1/2)/a^3/d-1/256*(16*a^4-56*a^2*b^2+5*b ^4)*cot(d*x+c)*csc(d*x+c)*(a+b*sin(d*x+c))^(1/2)/a^2/d+1/1536*b*(720*a^4-1 76*a^2*b^2+15*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*s in(d*x+c))^(1/2)/a^3/d/((a+b*sin(d*x+c))/(a+b))^(1/2)-1/1536*b*(816*a^4+16 96*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*s in(d*x+c))/(a+b))^(1/2)/a^2/d/(a+b*sin(d*x+c))^(1/2)-1/512*(64*a^6+720*a^4 *b^2+60*a^2*b^4-5*b^6)*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*P i+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^3/d/(a+b*sin(d*x+c))^(1/2)
Result contains complex when optimal does not.
Time = 7.50 (sec) , antiderivative size = 771, normalized size of antiderivative = 1.40 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\frac {\left (\frac {\left (-720 a^4 b \cos (c+d x)+176 a^2 b^3 \cos (c+d x)-15 b^5 \cos (c+d x)\right ) \csc (c+d x)}{1536 a^3}+\frac {\left (-48 a^4 \cos (c+d x)+600 a^2 b^2 \cos (c+d x)+5 b^4 \cos (c+d x)\right ) \csc ^2(c+d x)}{768 a^2}+\frac {\left (164 a^2 b \cos (c+d x)-b^3 \cos (c+d x)\right ) \csc ^3(c+d x)}{192 a}+\frac {1}{96} \left (28 a^2 \cos (c+d x)-27 b^2 \cos (c+d x)\right ) \csc ^4(c+d x)-\frac {5}{12} a b \cot (c+d x) \csc ^4(c+d x)-\frac {1}{6} a^2 \cot (c+d x) \csc ^5(c+d x)\right ) \sqrt {a+b \sin (c+d x)}}{d}+\frac {-\frac {2 \left (192 a^5 b+3744 a^3 b^3-20 a b^5\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (-c+\frac {\pi }{2}-d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}-\frac {2 \left (384 a^6+3600 a^4 b^2+536 a^2 b^4-45 b^6\right ) \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (-c+\frac {\pi }{2}-d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}-\frac {2 i \left (720 a^4 b^2-176 a^2 b^4+15 b^6\right ) \cos (c+d x) \cos (2 (c+d x)) \left (2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )-b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sqrt {\frac {b-b \sin (c+d x)}{a+b}} \sqrt {-\frac {b+b \sin (c+d x)}{a-b}}}{a \sqrt {-\frac {1}{a+b}} \sqrt {1-\sin ^2(c+d x)} \left (-2 a^2+b^2+4 a (a+b \sin (c+d x))-2 (a+b \sin (c+d x))^2\right ) \sqrt {-\frac {a^2-b^2-2 a (a+b \sin (c+d x))+(a+b \sin (c+d x))^2}{b^2}}}}{6144 a^3 d} \]
((((-720*a^4*b*Cos[c + d*x] + 176*a^2*b^3*Cos[c + d*x] - 15*b^5*Cos[c + d* x])*Csc[c + d*x])/(1536*a^3) + ((-48*a^4*Cos[c + d*x] + 600*a^2*b^2*Cos[c + d*x] + 5*b^4*Cos[c + d*x])*Csc[c + d*x]^2)/(768*a^2) + ((164*a^2*b*Cos[c + d*x] - b^3*Cos[c + d*x])*Csc[c + d*x]^3)/(192*a) + ((28*a^2*Cos[c + d*x ] - 27*b^2*Cos[c + d*x])*Csc[c + d*x]^4)/96 - (5*a*b*Cot[c + d*x]*Csc[c + d*x]^4)/12 - (a^2*Cot[c + d*x]*Csc[c + d*x]^5)/6)*Sqrt[a + b*Sin[c + d*x]] )/d + ((-2*(192*a^5*b + 3744*a^3*b^3 - 20*a*b^5)*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*(384*a^6 + 3600*a^4*b^2 + 536*a^2*b^4 - 45*b^6)*EllipticPi[2, (-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/S qrt[a + b*Sin[c + d*x]] - ((2*I)*(720*a^4*b^2 - 176*a^2*b^4 + 15*b^6)*Cos[ c + d*x]*Cos[2*(c + d*x)]*(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*ArcS inh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] - b*El lipticPi[(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 - Sin[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)]))/(614 4*a^3*d)
Time = 4.37 (sec) , antiderivative size = 562, normalized size of antiderivative = 1.02, number of steps used = 30, number of rules used = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.968, Rules used = {3042, 3372, 27, 3042, 3526, 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.
\(\displaystyle \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (c+d x)^4 (a+b \sin (c+d x))^{5/2}}{\sin (c+d x)^7}dx\) |
\(\Big \downarrow \) 3372 |
\(\displaystyle -\frac {\int \frac {5}{4} \csc ^5(c+d x) (a+b \sin (c+d x))^{5/2} \left (28 a^2+2 b \sin (c+d x) a-3 b^2-\left (24 a^2-b^2\right ) \sin ^2(c+d x)\right )dx}{30 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int \csc ^5(c+d x) (a+b \sin (c+d x))^{5/2} \left (28 a^2+2 b \sin (c+d x) a-3 b^2-\left (24 a^2-b^2\right ) \sin ^2(c+d x)\right )dx}{24 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\int \frac {(a+b \sin (c+d x))^{5/2} \left (28 a^2+2 b \sin (c+d x) a-3 b^2-\left (24 a^2-b^2\right ) \sin (c+d x)^2\right )}{\sin (c+d x)^5}dx}{24 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}\) |
\(\Big \downarrow \) 3526 |
\(\displaystyle -\frac {\frac {1}{4} \int \frac {1}{2} \csc ^4(c+d x) (a+b \sin (c+d x))^{3/2} \left (-b \left (164 a^2-5 b^2\right ) \sin ^2(c+d x)-6 a \left (4 a^2-b^2\right ) \sin (c+d x)+3 b \left (52 a^2-5 b^2\right )\right )dx-\frac {\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 d}}{24 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\frac {1}{8} \int \csc ^4(c+d x) (a+b \sin (c+d x))^{3/2} \left (-b \left (164 a^2-5 b^2\right ) \sin ^2(c+d x)-6 a \left (4 a^2-b^2\right ) \sin (c+d x)+3 b \left (52 a^2-5 b^2\right )\right )dx-\frac {\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 d}}{24 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {1}{8} \int \frac {(a+b \sin (c+d x))^{3/2} \left (-b \left (164 a^2-5 b^2\right ) \sin (c+d x)^2-6 a \left (4 a^2-b^2\right ) \sin (c+d x)+3 b \left (52 a^2-5 b^2\right )\right )}{\sin (c+d x)^4}dx-\frac {\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 d}}{24 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}\) |
\(\Big \downarrow \) 3526 |
\(\displaystyle -\frac {\frac {1}{8} \left (\frac {1}{3} \int -\frac {3}{2} \csc ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (b^2 \left (276 a^2-5 b^2\right ) \sin ^2(c+d x)+2 a b \left (84 a^2-b^2\right ) \sin (c+d x)+3 \left (16 a^4-56 b^2 a^2+5 b^4\right )\right )dx-\frac {b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 d}}{24 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\frac {1}{8} \left (-\frac {1}{2} \int \csc ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (b^2 \left (276 a^2-5 b^2\right ) \sin ^2(c+d x)+2 a b \left (84 a^2-b^2\right ) \sin (c+d x)+3 \left (16 a^4-56 b^2 a^2+5 b^4\right )\right )dx-\frac {b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 d}}{24 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {1}{8} \left (-\frac {1}{2} \int \frac {\sqrt {a+b \sin (c+d x)} \left (b^2 \left (276 a^2-5 b^2\right ) \sin (c+d x)^2+2 a b \left (84 a^2-b^2\right ) \sin (c+d x)+3 \left (16 a^4-56 b^2 a^2+5 b^4\right )\right )}{\sin (c+d x)^3}dx-\frac {b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 d}}{24 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}\) |
\(\Big \downarrow \) 3526 |
\(\displaystyle -\frac {\frac {1}{8} \left (\frac {1}{2} \left (\frac {3 \left (16 a^4-56 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}-\frac {1}{2} \int \frac {\csc ^2(c+d x) \left (b \left (48 a^4+936 b^2 a^2-5 b^4\right ) \sin ^2(c+d x)+2 a \left (48 a^4+720 b^2 a^2+b^4\right ) \sin (c+d x)+b \left (720 a^4-176 b^2 a^2+15 b^4\right )\right )}{2 \sqrt {a+b \sin (c+d x)}}dx\right )-\frac {b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 d}}{24 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\frac {1}{8} \left (\frac {1}{2} \left (\frac {3 \left (16 a^4-56 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}-\frac {1}{4} \int \frac {\csc ^2(c+d x) \left (b \left (48 a^4+936 b^2 a^2-5 b^4\right ) \sin ^2(c+d x)+2 a \left (48 a^4+720 b^2 a^2+b^4\right ) \sin (c+d x)+b \left (720 a^4-176 b^2 a^2+15 b^4\right )\right )}{\sqrt {a+b \sin (c+d x)}}dx\right )-\frac {b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 d}}{24 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {1}{8} \left (\frac {1}{2} \left (\frac {3 \left (16 a^4-56 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}-\frac {1}{4} \int \frac {b \left (48 a^4+936 b^2 a^2-5 b^4\right ) \sin (c+d x)^2+2 a \left (48 a^4+720 b^2 a^2+b^4\right ) \sin (c+d x)+b \left (720 a^4-176 b^2 a^2+15 b^4\right )}{\sin (c+d x)^2 \sqrt {a+b \sin (c+d x)}}dx\right )-\frac {b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 d}}{24 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}\) |
\(\Big \downarrow \) 3534 |
\(\displaystyle -\frac {\frac {1}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {\int \frac {\csc (c+d x) \left (-b^2 \left (720 a^4-176 b^2 a^2+15 b^4\right ) \sin ^2(c+d x)+2 a b \left (48 a^4+936 b^2 a^2-5 b^4\right ) \sin (c+d x)+3 \left (64 a^6+720 b^2 a^4+60 b^4 a^2-5 b^6\right )\right )}{2 \sqrt {a+b \sin (c+d x)}}dx}{a}\right )+\frac {3 \left (16 a^4-56 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}\right )-\frac {b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 d}}{24 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\frac {1}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {\int \frac {\csc (c+d x) \left (-b^2 \left (720 a^4-176 b^2 a^2+15 b^4\right ) \sin ^2(c+d x)+2 a b \left (48 a^4+936 b^2 a^2-5 b^4\right ) \sin (c+d x)+3 \left (64 a^6+720 b^2 a^4+60 b^4 a^2-5 b^6\right )\right )}{\sqrt {a+b \sin (c+d x)}}dx}{2 a}\right )+\frac {3 \left (16 a^4-56 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}\right )-\frac {b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 d}}{24 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {1}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {\int \frac {-b^2 \left (720 a^4-176 b^2 a^2+15 b^4\right ) \sin (c+d x)^2+2 a b \left (48 a^4+936 b^2 a^2-5 b^4\right ) \sin (c+d x)+3 \left (64 a^6+720 b^2 a^4+60 b^4 a^2-5 b^6\right )}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{2 a}\right )+\frac {3 \left (16 a^4-56 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}\right )-\frac {b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 d}}{24 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}\) |
\(\Big \downarrow \) 3538 |
\(\displaystyle -\frac {\frac {1}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {-b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx-\frac {\int -\frac {\csc (c+d x) \left (a \left (816 a^4+1696 b^2 a^2+5 b^4\right ) \sin (c+d x) b^2+3 \left (64 a^6+720 b^2 a^4+60 b^4 a^2-5 b^6\right ) b\right )}{\sqrt {a+b \sin (c+d x)}}dx}{b}}{2 a}\right )+\frac {3 \left (16 a^4-56 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}\right )-\frac {b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 d}}{24 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\frac {1}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {\frac {\int \frac {\csc (c+d x) \left (a \left (816 a^4+1696 b^2 a^2+5 b^4\right ) \sin (c+d x) b^2+3 \left (64 a^6+720 b^2 a^4+60 b^4 a^2-5 b^6\right ) b\right )}{\sqrt {a+b \sin (c+d x)}}dx}{b}-b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx}{2 a}\right )+\frac {3 \left (16 a^4-56 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}\right )-\frac {b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 d}}{24 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {1}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {\frac {\int \frac {a \left (816 a^4+1696 b^2 a^2+5 b^4\right ) \sin (c+d x) b^2+3 \left (64 a^6+720 b^2 a^4+60 b^4 a^2-5 b^6\right ) b}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}-b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx}{2 a}\right )+\frac {3 \left (16 a^4-56 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}\right )-\frac {b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 d}}{24 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}\) |
\(\Big \downarrow \) 3134 |
\(\displaystyle -\frac {\frac {1}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {\frac {\int \frac {a \left (816 a^4+1696 b^2 a^2+5 b^4\right ) \sin (c+d x) b^2+3 \left (64 a^6+720 b^2 a^4+60 b^4 a^2-5 b^6\right ) b}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}\right )+\frac {3 \left (16 a^4-56 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}\right )-\frac {b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 d}}{24 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {1}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {\frac {\int \frac {a \left (816 a^4+1696 b^2 a^2+5 b^4\right ) \sin (c+d x) b^2+3 \left (64 a^6+720 b^2 a^4+60 b^4 a^2-5 b^6\right ) b}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}\right )+\frac {3 \left (16 a^4-56 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}\right )-\frac {b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 d}}{24 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}\) |
\(\Big \downarrow \) 3132 |
\(\displaystyle -\frac {\frac {1}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {\frac {\int \frac {a \left (816 a^4+1696 b^2 a^2+5 b^4\right ) \sin (c+d x) b^2+3 \left (64 a^6+720 b^2 a^4+60 b^4 a^2-5 b^6\right ) b}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {2 b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}\right )+\frac {3 \left (16 a^4-56 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}\right )-\frac {b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 d}}{24 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}\) |
\(\Big \downarrow \) 3481 |
\(\displaystyle -\frac {\frac {1}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {\frac {a b^2 \left (816 a^4+1696 a^2 b^2+5 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx+3 b \left (64 a^6+720 a^4 b^2+60 a^2 b^4-5 b^6\right ) \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {2 b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}\right )+\frac {3 \left (16 a^4-56 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}\right )-\frac {b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 d}}{24 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {1}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {\frac {a b^2 \left (816 a^4+1696 a^2 b^2+5 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx+3 b \left (64 a^6+720 a^4 b^2+60 a^2 b^4-5 b^6\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {2 b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}\right )+\frac {3 \left (16 a^4-56 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}\right )-\frac {b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 d}}{24 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}\) |
\(\Big \downarrow \) 3142 |
\(\displaystyle -\frac {\frac {1}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {\frac {\frac {a b^2 \left (816 a^4+1696 a^2 b^2+5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}+3 b \left (64 a^6+720 a^4 b^2+60 a^2 b^4-5 b^6\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {2 b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}\right )+\frac {3 \left (16 a^4-56 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}\right )-\frac {b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 d}}{24 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {1}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {\frac {\frac {a b^2 \left (816 a^4+1696 a^2 b^2+5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}+3 b \left (64 a^6+720 a^4 b^2+60 a^2 b^4-5 b^6\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {2 b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}\right )+\frac {3 \left (16 a^4-56 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}\right )-\frac {b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 d}}{24 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}\) |
\(\Big \downarrow \) 3140 |
\(\displaystyle -\frac {\frac {1}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {\frac {3 b \left (64 a^6+720 a^4 b^2+60 a^2 b^4-5 b^6\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx+\frac {2 a b^2 \left (816 a^4+1696 a^2 b^2+5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}-\frac {2 b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}\right )+\frac {3 \left (16 a^4-56 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}\right )-\frac {b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 d}}{24 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}\) |
\(\Big \downarrow \) 3286 |
\(\displaystyle -\frac {\frac {1}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {\frac {\frac {3 b \left (64 a^6+720 a^4 b^2+60 a^2 b^4-5 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {\csc (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}+\frac {2 a b^2 \left (816 a^4+1696 a^2 b^2+5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}-\frac {2 b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}\right )+\frac {3 \left (16 a^4-56 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}\right )-\frac {b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 d}}{24 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {1}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {\frac {\frac {3 b \left (64 a^6+720 a^4 b^2+60 a^2 b^4-5 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sin (c+d x) \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}+\frac {2 a b^2 \left (816 a^4+1696 a^2 b^2+5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}-\frac {2 b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}\right )+\frac {3 \left (16 a^4-56 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}\right )-\frac {b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 d}}{24 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}\) |
\(\Big \downarrow \) 3284 |
\(\displaystyle \frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\frac {1}{8} \left (\frac {1}{2} \left (\frac {3 \left (16 a^4-56 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 d}+\frac {1}{4} \left (\frac {b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {\frac {\frac {2 a b^2 \left (816 a^4+1696 a^2 b^2+5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}+\frac {6 b \left (64 a^6+720 a^4 b^2+60 a^2 b^4-5 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}-\frac {2 b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}\right )\right )-\frac {b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 d}}{24 a^2}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}\) |
(b*Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^(7/2))/(12*a^2*d) - (C ot[c + d*x]*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^(7/2))/(6*a*d) - (-1/4*((2 8*a^2 - 3*b^2)*Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^(5/2))/d + (-((b*(52*a^2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^( 3/2))/d) + ((3*(16*a^4 - 56*a^2*b^2 + 5*b^4)*Cot[c + d*x]*Csc[c + d*x]*Sqr t[a + b*Sin[c + d*x]])/(2*d) + ((b*(720*a^4 - 176*a^2*b^2 + 15*b^4)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(a*d) - ((-2*b*(720*a^4 - 176*a^2*b^2 + 1 5*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^2*(816*a^4 + 1696*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]]) + (6*b*(64*a^6 + 720*a^4 *b^2 + 60*a^2*b^4 - 5*b^6)*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* a))/4)/2)/8)/(24*a^2)
3.12.67.3.1 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_)])^(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. \(2457\) vs. \(2(608)=1216\).
Time = 121.33 (sec) , antiderivative size = 2458, normalized size of antiderivative = 4.46
1/1536*(-192*((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^7*sin(d*x+c)^6-15*((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)*El lipticPi(((a+b*sin(d*x+c))/(a-b))^(1/2),(a-b)/a,((a-b)/(a+b))^(1/2))*b^7*s in(d*x+c)^6+96*((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^7*sin(d*x+c)^6+720*((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)*Elliptic E(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^7*sin(d*x+c)^6-256 *a^7-5*((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*sin(d*x+c)^6+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)*EllipticF(((a +b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a*b^6*sin(d*x+c)^6-896*(( 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*sin(d*x+c)^6+191*((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^3*b^4*sin(d*x+c)^6-15*((a+b*...
Timed out. \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]
Timed out. \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]
\[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cot \left (d x + c\right )^{4} \csc \left (d x + c\right )^{3} \,d x } \]
Timed out. \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]
Timed out. \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Hanged} \]